Application of multi-objective optimization based on Sobol sensitivity analysis in solar single-double-effect LiBr−H₂O absorption refrigeration

Abstract

To improve the adaptability of solar refrigeration systems to different heat sources, a single-double-effect LiBr−H₂O absorption refrigeration system (ARS) driven by solar energy was designed and analyzed. The system was optimized using a multi-objective optimization method based on Sobol sensitivity analysis to enhance solar energy efficiency and reduce costs. The model of the solar single-double-effect LiBr−H₂O ARS was developed, and the continuous operation characteristics of the system in different configurations were simulated and compared. The results show that the average cooling time of the system without auxiliary heat source is approximately 8.5 h per day, and the double-effect mode (DEM) generates about 11 kW of cooling capacity during continuous operation for one week under the designated conditions, and the system with adding auxiliary heat source meet the requirements of daily cooling time, the solar fraction (SF) of the system reaches 59.29%. The collector area has a greater effect on SF, while the flowrate of the hot water circulating pump and the volume of storage tank have little effect on SF. The optimized SF increases by 3.22% and the levelized cost decreases by 10.18%. Moreover, compared with the solar single-effect LiBr−H₂O ARS, the SF of the system is increased by 15.51% and 17.42% respectively after optimization.

Graphical abstract

Keywords

solar energy / absorption refrigeration / single-double-effect / solar fraction (SF) / multi-objective optimization

1 Introduction

1.1 Background

With the rapid economic growth, the demand for energy in various fields is on the rise, and the energy supply cannot meet the demand for production. According to the report released by the International Renewable Energy Agency, CO₂ emissions related to energy are slightly projected to decrease by 2050. To reduce emission of CO₂ and the consumption of the primary energy, over 50 countries worldwide have set the goal of achieving net-zero emissions by transforming the supply, conversion, and utilization of energy. This shift toward clean energy, including wind [1], ocean [2], and solar power, is increasingly being adopted across various fields [3,4]. Among those clean energy sources, solar energy is convenient to access and abundant in resources. It has a great significance for protecting the environment due to its large energy capacity. At present, the solar thermal utilization technology is used in many fields, such as refrigeration, heating, domestic hot water preparation, and cooking [5] in the low-temperature field. However, the demand and consumption of primary energy for refrigeration are huge, and will increase by 2.5 times by 2050 [6].

Therefore, the development and utilization of solar absorption refrigeration is one of crucial technological approaches to achieve net-zero emissions in the refrigeration technology field. A recent series of studies on the LiBr−H₂O absorption refrigeration technology demonstrates that solar energy could be effectively harnessed in an absorption refrigeration system (ARS) [7,8]. In addition, the advantage of solar refrigeration is the good matching of refrigeration requirements with the seasons. Generally, the hotter the summer, the higher intensity of solar irradiation, and the greater cooling capacity of solar ARS. However, solar energy is inherently intermittent and unstable [9]. To fully leverage solar energy and minimize auxiliary energy usage, it is essential to match heat sources of varying temperatures with corresponding operational modes of the solar refrigeration system. Additionally, optimizing the solar refrigeration system or integrating energy storage devices could maximize the utilization of solar fraction (SF) [10,11]. Currently, enhancing the commercial applicability of the solar refrigeration system is the most pressing objective [12,13].

With the continuous deepening of ARS research, many scholars have improved the system by expanding the application range to enhance the applicability. To further improve the performance of the system, they have proposed various new circulation methods, such as coupling the traditional single-effect with the double-effect system. For instance, Lubis et al. [14] conducted experimental studies on a solar-driven single-double-effect ARS suitable for tropical Asian climates. Compared with conventional vapor compression refrigeration systems, the use of single-double-effect ARS could reduce energy consumption by 11% to 48%. Xu & Wang [15,16] proposed a solar-driven variable-effect ARS, which could operate in single-effect, double-effect, and 1.n-effect absorption refrigeration modes to adapt to changes in the heat source temperature. In the 1.n-effect mode, the proportion of steam generated by the high-pressure generator into the high-pressure absorber was controlled based on different working conditions to reduce concentration of the solution when it entered the low-pressure generator. This resulted in achieving 1.n-effect refrigeration, with a cycle working temperature range of (85–150 °C). Ma et al. [17] proposed a solar single-double-effect LiBr−H₂O ARS and analyzed its operation characteristics and economics. The system has the potential to make full use of solar energy, but its stability requires further study.

In addition to adjusting the existing loop structure, the simulation optimization method can also be used to improve the system performance. In general, system optimization can be divided into single objective optimization and multi-objective optimization according to the different number of objectives. In the single objective optimization problem, the most direct objective is usually used for optimization. In a refrigeration system, the coefficient of performance (COP) [18–20] is generally regarded as the optimization goal. However, the multi-objective optimization method provides an efficient way to simultaneously solve the conflicting objectives in the optimization problem. For instance, when optimizing solar ARSs, objective such as cost and SF often conflict. Typically, achieving a higher SF requires a larger heat collection area, which in turn leads to higher system costs [21]. Lee et al. [22] examined and optimized the generator of a multi-source lithium bromide/water absorption chiller. Unlike previous optimization approaches that focused on individual components, they employed metamodels constructed using the kriging method for multi-objective optimization. By considering the five design variables of the generator, the conflicting optimization objectives were resolved, resulting in the attainment of both the minimum total volume and the maximum total generation rate objectives. In the pursuit of enhanced cooling performance of the single-effect ARS, Sharifi et al. [23] adopted a multi-objective optimization method to optimize the system by considering the generator and evaporator temperatures as variables. The optimization results demonstrated an approximate improvement of 9.1% in COP and 3.0% in exergy efficiency, respectively.

Multi-objective optimization not only theoretically aids in enhancing system performance, but is also widely applied in practical scenarios to optimize various parameters, exhibiting a wide range of applicable fields. Ghersi et al. [24] conducted optimization on performance indicators, including primary energy consumption, annual total cost, and CO₂ emissions of a combined cooling and thermal power generation system. The optimal system capacity was determined by employing a multi-objective optimization method. Hai et al. [25] improved a geothermal cycle, analyzed and compared the energy, exergy, and exergy economy of the new cycle, extracted the optimal parameters of the system through artificial neural network and multi-objective optimization, and then obtained the ideal result and the best selection point. Hou et al. [26], choosing an apartment building in Guangzhou chosen as an example, introduced a strategy called the point of best operation condition point (BOCP) and applied it to solar-assisted cogeneration systems. They adopted the particle swarm optimization (PSO) algorithm to optimize the system, aiming to improve the performance and economic benefits. Nateghi et al. [27] used the EnergyPlus software to simulate the energy demand of a hotel building, adopted the non-dominated sorting genetic algorithm (NSGA) to optimize the building parameters, and conducted sensitivity analysis to analyze and calculate the three building related parameters. The results showed that the long-term cost of hotel optimization could be invested in the installation of solar energy system.

To improve the utilization range of heat source temperature, increase the adaptability of solar refrigeration system to heat source with different temperatures, and give full play to the advantages of solar refrigeration system, a solar ARS scheme consisting of two independent chillers and solar heat collection system is proposed. When the solar energy is used as the heat source driving system, the single-effect system runs for a long time, but its performance is low. The double-effect system has a higher performance, but the running time is long. Based on these two points, the solar single-double-effect switching ARS can switch between single-effect models (SEMs) and DEMs according to the change of solar irradiance, which further improves the performance of the system. However, the high cost of the system limits its wide application. To promote the development and progress of this field, the Sobol sensitivity analysis method is used to assess the sensitivity of each design and operating parameter to SF in this work. Based on the results of sensitivity analysis, the multi-objective optimization of the solar ARS is conducted to maximize the solar energy utilization fraction and minimize the levelized cost.

2 System description

The system diagram of the solar single-double-effect LiBr−H₂O ARS is shown in Fig.1. The main components of the system consist of a solar collector, a storage tank, a double-effect absorption chiller (DACH), a single-effect absorption chiller (SACH), pumps, and electric valves (EVs), etc.

The solar energy is collected by the collectors to heat water and stores it in the storage tank. Once the temperature of the hot water stored reaches a certain threshold, it is directed from the storage tank to either the single-effect or the double-effect chiller. After undergoing the cooling process, the water returns to the storage tank. In situations where the solar irradiance is adequate but not high enough to reach the startup temperature of the double-effect chiller, the system operates in the SEM. This is achieved by opening the electric valve EV2 and closing EV3, allowing the heat source water to flow into the single-effect chiller. However, when the solar irradiance is abundant and the temperature of the hot water in the storage tank reaches the startup temperature of the double-effect chiller, the system switches from SEM to DEM by opening EV3 and closing EV2. In cases where the solar irradiance is insufficient to heat the hot water to the startup temperature of the chiller, the auxiliary heater (AUX) is activated. The AUX, powered by natural gas, effectively raises the temperature of the hot water to the required level, ensuring the smooth operation of the chiller under the rated condition of SEM. The controller (CT) in Fig.1 controls the start and stop of the pumps, EVs, AUX, and chillers. The control lines of the EVs are not indicated in Fig.1 to ensure the beauty of the picture. The control flowchart of the system is shown in Fig.2.

According to Shirazi et al. [21], the startup temperature of SEM was set at 75 °C to ensure the stable operation of the system in the SEM. Under the design conditions, the switching temperatures of SEM and DEM were established at 130 °C, to fulfill the minimum colling requirements of users. Setting the switching temperature too low would result in a small solution concentration difference, leading to a reduced cooling capacity and rendering the system unable to start in the DEM. On the other hand, setting the switching temperature too high would increase the risk of crystallization in the lithium bromide solution in SEM.

3 Models and methods

This section presents the development of a model for the solar single-double-effect LiBr−H₂O ARS, the mathematical model of the whole system including solar collector, energy storage tank and single-double-effect LiBr−H₂O ARS and introduces the methods employed for system analysis.

3.1 System modeling

The system was modeled in TRNSYS. The solar collector, storage tank, AUX, water pumps, CTs and other components were built in modules in the software, and were interconnected according to the actual operation process and control mode of the system [28]. However, there was no module of single- and double-effect refrigeration system in TRNSYS. Therefore, this refrigerator model was established in EES, and the formula was provided for calculating the physical parameters of LiBr solution. Since TRNSYS cannot interact with EES, the Monte Carlo method in ISIGHT was used to generate sample data, which was then imported into MATLAB, and trained into a single- and double-effect refrigeration system model through artificial neural network algorithm to facilitate the call in TRNSYS. The TRNSYS modules used in the building process and the main input parameters are listed in Tab.1 while the model construction process is depicted in Fig.3.

There are many kinds of solar collectors, including flat plate collector, vacuum tube collectors, etc. In this study, the solar collectors utilized were compound parabolic collector (CPC) vacuum tube collectors, which were selected due to their cost-effectiveness. Additionally, the collector can withstand a maximum pressure of approximately 4 MPa (with a saturation temperature of 250 °C), while the storage tank can withstand a maximum pressure of approximately 5.5 MPa (with a saturation temperature of 270 °C), meeting the operational requirements [29], which meets the use requirements. The heat collected from the solar collector is calculated as [30]

(1)q_{\text{c}}=A_{\text{c}}\cdot F_{\text{R}}[(\alpha \tau )\cdot I-U_{\text{c}}(T_{\text{ci}}-T_{\text{a}}) ],

where q_c is the heat gain of the solar collector in unit time, W; A_c is solar collector area, m²; F_R is the heat transfer factor of the solar collector; ατ is the absorption conversion factor; U_{\text{c}} is the heat loss coefficient of the solar collector, W/(m²·°C); T_{\text{ci}} is the inlet temperature of the solar collector, °C; T_{\text{a}} is ambient temperature, °C; and I is the total radiation intensity on the inclined plane, W/m².

The energy conservation equation of the storage tank is

(2)C_{\text{p,st}}\frac{\text{d}{T}_{\text{st}}}{\text{d}t}=Q_{\text{u}}-Q_{\text{effect,st}}- Q_{\text{loss,st}},

where C_{\text{p,st}} is the heat capacity of the storage tank, J/(kg·°C); Q_{\text{u}} is the effective heat output per unit time of the collector, W; Q_{\text{effect,st}} is the heat transfer inside the storage tank, including the inlet flow heat transfer and the node mixed heat transfer, W; and Q_{\text{loss,st}} is the heat loss between storage tank and environment, W; T_st is the temperature of hot water in the storage tank, °C.

The solar single-double-effect system has a rated cooling capacity of 5 kW in SEM and 10 kW in DEM. To facilitate the development and solution of the model, a steady-state model without considering thermal inertia and time delay was adopted, and the mathematical model of the absorption chiller was formulated under the assumptions [31] that the pressure drop of the pipe and heat exchange components is ignored; the refrigerant at the outlet of the condenser and evaporator is in a saturated state; the throttling devices are adiabatic; the power of the solution pump is negligible; and the reference environment temperature is 25 °C.

The system model was developed based on the thermophysical properties of the working fluid, as well as the mass and energy conservation equations, the heat transfer equations, and other relevant equations [32,33]. The equations used for each component are [34]

Mass balance equation:

(3){\sum }_{i=1}^n({\dot{m}}_i{)}_{\text{in}}={\sum }_{i=1}^n( {\dot{m}}_i{)}_{\text{out}},

Mass balance equation of LiBr−H₂O:

(4){\sum }_{i=1}^n({\dot{m}}_i{x}_i {)}_{\text{in}}={\sum }_{i=1}^n( {\dot{m}}_i{x}_i {)}_{\text{out}},

Energy balance equation:

(5){\sum }_{i=1}^n({\dot{m}}_i{h}_i {)}_{\text{in}}+{\dot{Q}}_i={\sum }_{i=1}^n({\dot{m}}_i{h}_i{)}_{\text{out}},

where \dot{m} represents the mass flowrate of working fluid, kg/s; h is expressed as the specific enthalpy of the working fluid, kJ/kg; x represents the concentration of LiBr−H₂O solution; and \dot{Q} is expressed as the heat transfer rate of each component, kW.

Heat transfer equation:

(6){\dot{Q}}_i=C_{\text{p},i}{\dot{m}}_i(T_{\text{in}}-T_{\text{out}} )=KA\mathrm{\Delta }{T}_{\text{m}} ,

where K is the total heat transfer coefficient of components, kW/°C, and

\mathrm{\Delta }{T}_{\mathrm{m}}={\displaystyle \frac{\mathrm{\Delta }{T}_{max}-\mathrm{\Delta }{T}_{min}}{\ln {\displaystyle \frac{\mathrm{\Delta }{T}_{max}}{\mathrm{\Delta }{T}_{max}}}}}.

The design parameters and operating parameters of the solar single-double-effect LiBr−H₂O ARS are tabulated in Tab.2.

3.2 Performance evaluation

SF is defined as the ratio of the energy supplied by solar energy to the total energy demand of the system. Analyzing the SF of the ARS is crucial for minimizing the reliance on primary energy sources and reducing the operational cost of the system. The calculation expression for SF is given by [33]

(7)\text{SF =} \frac{{\displaystyle \sum Q_{\text{solar}}}}{{\displaystyle \sum Q_{\text{solar}}+\sum Q_{\text{aux}}}},

where SF is the fraction of solar energy utilized over a specific period of time, %; {\displaystyle \sum Q_{\text{solar}}} is the accumulated heat derived from solar energy, kJ; and {\displaystyle \sum Q_{\text{aux}}} is the accumulated heat from auxiliary heat sources, kJ.

In this work, to facilitate the comparison of cooling capacity per unit collector area before and after optimization, a cooling power is introduced as an evaluation index, which is defined as the ratio of cumulative cooling capacity to collector area and expressed as

(8)P_{\text{c}} \text{=}\frac{{\displaystyle \sum Q_{\text{d}}}}{A_{\text{c}}},

where {\displaystyle \sum Q_{\text{d}}} is the cumulative cooling capacity of the solar refrigeration system in a day, kWh; and P_c is the cumulative daily cooling capacity per unit collector area, kWh/m².

3.3 Economic analysis

Taking the solar ARS as an example, it has the typical characteristics of high initial investment cost but low operating cost. Unlike the compression refrigeration system, the solar refrigeration consumes the minimal electricity or gas to operate the pump or serve as an auxiliary heat source. Consequently, the operational expenses of the system are lower. However, the initial investment cost is mainly attributed to the cost of solar LiBr−H₂O ARS devices and installation, with the former being significantly higher than that of compression refrigeration systems. Hence, conducting an economic analysis of the solar refrigeration system is of utmost importance. In this study, the annual cost method was used to assess the economic feasibility of the system. The approach involves converting all cash flows of the system into equal annuity payments [35]. The investment of system componrnts and parameter values for the economic analysis are summarized in Tab.3 based on actual prices and literatures [37,38].

The levelized annual cost C_tot,L includes the levelized annual capital investment CI_L and the levelized annual fuel cost FC_L:

(9)C_{\mathrm{t}\mathrm{o}\mathrm{t},\mathrm{L}}=\mathrm{C}{\mathrm{I}}_{\mathrm{L}} +\mathrm{F}{\mathrm{C}}_{\text{L}} .

CI_L consists of the equipment investment cost {\displaystyle \sum Z_{\text{k}}} and the equipment installation cost C_INSTL, $ (viz. US dollar). CI_L can be mathematically expressed as

(10){\mathrm{C}\mathrm{I}}_{\mathrm{L}} =\left(\sum Z_{\mathrm{k}}+C_{\mathrm{I}\mathrm{N}\mathrm{S}\mathrm{T}\mathrm{L}}\right)\mathrm{C}\mathrm{R}\mathrm{F},

where CRF is the capital recovery factor, which is a function of the system lifetime n and the interest rate i and can be expressed as [36]

(11)\text{CRF}=\frac{i{(1+i)}^n}{{(1+i)}^n-1}.

The FC_L calculation is expressed as

(12)\text{F}{\text{C}}_{\text{L}}=c_{\text{NG}}{V}_{\text{NG}},

where c_NG is the unit price of natural gas, S/m³; and V_NG is the natural gas consumption, m³.

The price of each part of the initial investment in equipment is shown in Fig.4.

3.4 CO₂ emissions

The issue of global warming resulting from CO₂ emissions has gained significant attention and required urgent solutions. In this study, CO₂ emissions are introduced as an evaluation metric to assess the performance of the solar single-double-effect refrigeration system before and after optimization. The calculated expression of carbon emission is

(13){\text{CDE}}_{\text{NG}} ={\text{E}}_{\text{NG}}\cdot {\text{EF}}_{\text{NG}},

where E_NG is natural gas consumption, kWh; EF_NG is the CO₂ emission factor of natural gas, kg/kWh; and \text{CD}{\text{E}}_{\text{NG}} is the CO₂ emissions from the auxiliary heat source of the solar single-double-effect LiBr−H₂O ARS, kg. The parameter values used for the analysis of CO₂ emissions and emission cost are shown in Tab.4.

4 System multi-objective optimization

4.1 Sensitivity analysis

Parameter sensitivity analysis is an important method to rank the influence of model input parameters on system response. The sensitivity analysis can select the parameters that have a great influence on the system response, eliminate the insensitive parameters, reduce the uncertainty of parameters, and reduce the calculation amount in the process of model solving. Sensitivity analysis can be divided into local sensitivity analysis and global sensitivity analysis. Local sensitivity analysis has a small amount of computation, but its parameter space searching ability is poor, and it cannot analyze the interaction between parameters. Although the global sensitivity analysis method has a large amount of computation, it can analyze the influence of the interaction between model parameters on the system-to-system response [41]. In the global sensitivity analysis method, the Sobol sensitivity analysis method [42] stands out among many sensitivity analysis methods with its advantages of high efficiency and high precision, and is widely used. At present, it is mainly used in economic forecasting, meteorological forecasting, and other fields, but it is relatively rarely used in engineering fields such as refrigeration and heating. This method mainly analyzes the impact of changes of design and operating parameters on the output parameters by calculating the influence of sampling variance of model design and operating parameters on the total variance of model output parameters, and helps identify the sensitivity of the system by determining the impact of each input parameter on the SF [43,44]. Conducting sensitivity analysis on the solar single-double-effect ARS allows for adjustments to the actual operation and optimization of the system. In this study, the Sobol method was used to analyze the effect of solar ARS parameters on the model response. The basic derivation process is outlined as follows [44]:

First, the response of the SF of the solar ARS to the change of design and operating parameters can be expressed as Y_{\text{SF}}=f(X ) which represents the k-dimensional design and operating variables such as collector area, tank volume, external fluid flow and temperature of the refrigeration system. Secondly, Y_SF is decomposed into a combination of single parameters and multi-parameters, as

(14)\begin{array}{rl}{Y}_{\mathrm{S}\mathrm{F}}=& f_0+{\sum }_{i=1}^k{f}_i(X_i)+{\sum }_{1\le i\le j}^k{f}_{ij}(x_i,x_j)+\cdots \\ & +f_{1,2,\dots ,k}(x_1,\dots ,x_k).\end{array}

Therefore, the response variance D(Y_{\text{SF}}) of Y_SF can be decomposed as

(15)

The calculation expression of the sensitivity analysis index is

(16)

where S_i represents the 1st-order Sobol sensitivity data of parameter x_i; S_ij represents the 2nd-order Sobol sensitivity data; S_ij···k represents the high-order Sobol sensitivity data; and S_Ti represents the total Sobol sensitivity data of parameter x_i.

In the sensitivity analysis of the solar LiBr−H₂O ARS model, the following parameters were selected: collector area A_c, tank volume V_st, flowrate of hot water circulating pump {\dot{m}}_{\text{p1}}, flowrate of hot water in SEM {\dot{m}}_{\text{shwi}}, inlet temperature of cooling water (CW) in SEM T_scwi, flowrate of CW in SEM {\dot{m}}_{\text{scwi}}, outlet temperature of chilled water in SEM T_scho, flowrate of chilled water in SEM {\dot{m}}_{\text{scho}}, flowrate of hot water in DEM {\dot{m}}_{\text{dhwi}}, inlet temperature of CW in DEM T_dcwi, flowrate of CW in DEM {\dot{m}}_{\text{dcwi}}, outlet temperature of chilled water in DEM T_dcho, flowrate of chilled water in DEM {\dot{m}}_{\text{dc}\text{ho}}. Based on the working principle of the solar LiBr−H₂O ARS and considering the subsequent experimental research and design conditions, the value range of these parameters was determined to ensure the normal operation of the system. The specific value ranges of each parameter are presented in Tab.5.

4.2 System optimization

When it comes to system optimization, single-objective optimization becomes relatively simple in finding the optimal solution, and only one objective can be optimized at a time. However, this approach often neglects the performance of other parameters, potentially resulting in suboptimal outcomes [4]. To address this issue, multi-objective optimization methods are used when dealing with conflicting objectives that require simultaneous consideration. To ensure the rationality of the operation of the solar refrigeration system and avoid the increase of the solving time of the optimization process due to the excessive optimization range, the value range of each optimization variable is constrained. In addition to constraints on optimization variables, in order to judge whether the optimization results are feasible and ensure the optimization quality, it is also necessary to restrict the intermediate parameters or operation results of the system. Additionally, taking into account the practical operating conditions, a constraint optimization model for the system was developed:

(17)\text{max}{f}_1 (x)=\text{SF},

(18)\text{min}{f}_2 (x)=C_{\text{tot,L}},

(19)\{\begin{array}{l}2\text{kW}⩽Q_{\text{eva}},\\ x_{\text{g}}⩽0.65\text{.}\end{array}

To ensure that the system aligns with the actual operating conditions and meet the user’s requirements, certain constraints were imposed. The constraint on cooling capacity guarantees that the refrigeration system can still deliver a minimum of 50% cooling capacity even during the periods of low solar irradiation intensity throughout the optimization process. Additionally, the solution mass fraction was controlled to prevent crystallization, which could adversely impact system performance. These constraints were put in place to ensure the economic viability and safe operation of the system.

The optimization process was completed by using the ISIGHT software. The multi-objective optimization algorithms provided in the software can be roughly divided into two categories: normalized algorithm and non-normalized algorithm. The normalization algorithm has the weighted coefficient method, which mainly converts multiple targets into a single target through the weighted way, and then uses the single objective optimization algorithm to solve. However, its disadvantage is that the optimization results are subject to subjective influence. The non-normalization algorithm deals with multiple targets directly through Pareto mechanism, which overcomes the shortcomings of the normalization method. The non-normalized algorithms mainly include neighborhood cultivation genetic algorithm (NCGA), NSGA-II, and archive-based micro genetic algorithm (AMGA), in which, NCGA is suitable for multi-objective optimization problems when the objective function has multiple peaks, while AMGA is suitable for real variables and highly constrained multi-objective optimization problems.

In this study, the concept of Pareto dominance was utilized to handle the mutual constraints among the optimization objectives, aiming to achieve solutions that are close to the optimal values [43,45]. Through the analysis of optimization problems, the NSGA-II algorithm with a good exploration performance and a strong Pareto forward ability was selected to complete the optimization process. The NSGA-II algorithm is a multi-objective optimization algorithm based on genetic algorithm. Genetic algorithm is an evolutionary algorithm based on biological evolution and biological genetics mechanism, which seeks the optimal solution by simulating the evolution process of natural organisms. The algorithm uses genetic operations such as selection, crossover, and variation to solve the optimal solution, in which, population size, iteration times, genetic factors and other factors constitute the core content of the algorithm. On the basis of genetic algorithm selection, crossover, mutation and other operations, NSGA-II carries out fast non-dominated ranking of population individuals, and introduces elite strategy, crowding degree, and crowding comparison operator, which can solve multi-objective optimization problems and optimize multiple variables at the same time, and rapidly improve the fitness level of population individuals while ensuring population diversity [46]. The optimization efficiency of the algorithm is improved. The optimization process of the system, along with the working principle of the NSGA-II algorithm [47], is demonstrated in Fig.5.

5 Results and discussion

5.1 Model verification

To prove the accuracy of the model, the single-effect and double-effect refrigeration systems were respectively verified. The SEM adopted the design and operation data reported by Wen et al. [48] to ensure the consistency of important parameters such as inlet flowrate of hot water and the circulation flowrate of solution, while the DEM adopted the design and operation parameters proposed by Wang [49]. In model verification, the data used in the SEM and the DEM are shown in Tab.6 and Tab.8 respectively, and the comparison results between the two models and the references are shown in Tab.7 [48] and 9. The analysis shows that the maximum deviation of SEM is about 2.62%, and that of the DEM is about 8.14%, which indicates that the model can effectively simulate the operating characteristics of the refrigeration system.

Based on the meteorological data of Qingdao, the system analysis was conducted using the weather data for a one-week period from June 8 to June 14. During this period, the variation of solar radiation intensity was obvious, which reflected the characteristics of summer climate. Fig.6 illustrates the solar irradiance and ambient temperature data during this period. Throughout the week, the solar irradiance intensity reached a maximum of 900 W/m², providing sufficient heat to raise the temperature of the heat source water to the temperature required for system operation during daylight hours under the design conditions. This ensured continuous operation of the chiller. In addition, it can be seen from the that there exists a lag in the variation of daily ambient temperature compared to the variations in solar irradiance.

Under the design conditions without the auxiliary heat source, the mode switching of the system and variations in cooling capacity during one week of operation are depicted in Fig.7. At the start of the simulation, on the first day, it took the collector a long time to heat the hot water to the startup temperature of the chiller. After 10:00 am, the hot water flowed to the single-effect generator, heating the LiBr−H₂O solution to produce refrigerant steam, and the system began operating in SEM continuously. The effective cooling time on that day was 7 h. On the second, third, and fourth days of the simulation, the weather was clear and sunny, with a high solar irradiance. The tank storage temperature reached the driving temperature (130 °C, referred to as the switching temperature) of the DEM at noon. As a result, the hot water flowed to the high-pressure generator, and the system switched from SEM to DEM. When the irradiation intensity weakened and the hot water temperature decreased, the system switched back from DEM to SEM. Compared to the first day of simulation, the cumulative cooling capacity of the system significantly increased on the second day. On the fifth day, the irradiation intensity and ambient temperature were relatively low, leading to a decrease in heat in the storage tank. Consequently, the chiller could only operate in SEM for 6.75 h, resulting in a cumulative cooling capacity reduction of 33 kWh relative to the second day. The dynamic behavior of the system operating for one week is presented in Fig.8. During system operation, the outlet temperature of the storage tank reached up to 150 °C, which enabled the system to produce approximately 11 kW of cooling capacity in the DEM mode. During this period, the daily solution pump power calculation accounted for about 0.2%–0.4% of the daily cumulative cooling capacity, which had a relatively small impact on cooling capacity and can be ignored.

The above analysis suggests that the average daily cooling time in Qingdao from June 8 to June 14, without the use of an auxiliary heat source, is less than 10 h. The duration of cooling and cumulative cooling capacity is significantly influenced by the weather condition. In practical applications, the addition of an auxiliary heat source is necessary to ensure continuous chiller operation over extended periods. Therefore, the operational characteristics of the system will be further analyzed conducting an engineering case study.

To investigate the operational characteristics of the single-double-effect LiBr−H₂O ARS in real-world scenarios and enhance its practicality, it was assumed that the system provides the cooling capacity for a laboratory with an area of about 30 m², a maximum hourly cooling load of about 5 kW, and a daily cooling time from 7:00 to 19:00. To meet the required cooling duration, an AUX was incorporated into the original system, the inclusion of which enabled the system to meet the desired cooling time. Fig.9 illustrates the mode switching and cooling capacity variation throughout the week after integrating the AUX. The aforementioned analysis evidently indicates that in weather conditions characterized by high solar irradiance intensity, the temperature of the hot water rises, enabling the system to switch from the SEM to the DEM. Conversely, when the hot water temperature is insufficient to drive the system, the auxiliary heat source is activated. In this case, the hot water circulates solely between the auxiliary heat source and the single-effect chiller, while the driving system operates in the SEM mode. By incorporating the AUX, the system operated for one week under the specified design conditions, resulting in an SF of 59.29% and a levelized annual cost of $732.12. To emphasize the advantages of the single-double-effect LiBr−H₂O ARS, a comparison was made with the solar single-effect LiBr−H₂O ARS. The solar single-effect LiBr−H₂O ARS operated for one week and achieved an SF of only 43.78%. This comparison highlights the superior performance of the solar single-double-effect LiBr−H₂O ARS, as it can effectively harness solar energy and reduce the reliance on primary energy sources. However, it is worth noting that the levelized cost of the solar single-double-effect LiBr−H₂O ARS is relatively high, and there is still room for improvement in achieving a higher SF.

In the solar refrigeration system with an auxiliary heat source, enhancing the utilization of solar energy and reducing the reliance on auxiliary energy is crucial for lowering the operational costs and enhancing the overall system performance. However, the SF is influenced by both uncontrollable factors like weather conditions and system-related factors such as A_c, V_st, and the temperature and flowrate of the external fluid. The variations of SF with A_c and V_st are depicted in Fig.10, respectively. Clearly, increasing the collector area results in a higher heat transfer from the collector to the chiller, thereby reducing the need for auxiliary energy and improving the SF. Furthermore, a larger collector area leads to an elevated heat storage temperature in the storage tank, allowing the system to operate for a longer duration in the DEM while shortening the operation time in the SEM. Additionally, since the COP of the DEM is higher than that of the SEM, the thermal advantage of the system is further enhanced. However, it is worth noting that a larger A_c corresponds to a higher cost. Similarly, within a certain range, increasing the V_st enhances the heat storage capacity of the storage tank, allowing for more heat to be provided to the chiller and reducing its reliance on the auxiliary heat source. Consequently, the SF of the system increases with an increase in V_st. However, if the V_st becomes too large, it may lead to higher thermal losses from the storage tank, which is not conducive to the effective utilization of solar energy by the system. The maximum SF achieved was 61.83% when A_c was 70 m² and V_st was 1.5 m³.

5.2 Sensitivity analysis results

In the Sobol method, the accuracy of sensitivity analysis was influenced by the number of samples used. A larger sample size results in a more stable estimation of sensitivity. In this study, a total of 15000 samples were utilized.

The 1st-order and total sensitivities of each parameter to the SF are presented in Fig.11. The 1st-order sensitivity represents the influence of an independent change of a parameter on the SF. The total sensitivity parameter not only represents the influence of parameter change on the output result, but also reflects the interaction effect between the parameter and other parameters. If there is a great difference between the first-order sensitivity and the total sensitivity of the parameter, it indicates that the interaction between the parameter and other parameters is apparent.

The sensitivity analysis reveals that the collector area had a significant impact on the SF. The reason for this is that the collector area directly influences the amount of heat collected by the system, which in turn determines the cooling capacity to a certain extent. On the other hand, V_st and {\dot{m}}_{\text{p1}} have a minimal effect on the SF. Furthermore, as the system operates in the DEM for a shorter duration compared to the SEM, the hot water flowrate, the CW inlet temperature and flowrate, the condensate outlet temperature and flowrate have a greater influence on the SF in the SEM than in the DEM.

The parameter of A_c directly affects the amount of solar heat collected to change the fraction of SF. The parameters of {\dot{m}}_{\text{shwi}} and {\dot{m}}_{\text{dhwi}} directly influence the heat load of the generator, thereby impacting the SF. On the other hand, the parameters of T_scwi, {\dot{m}}_{\text{scwi}}, T_scho, {\dot{m}}_{\text{s}\text{c}\text{ho}}, T_dcwi, {\dot{m}}_{\text{dcwi}}, T_dcho, and {\dot{m}}_{\text{dcho}} indirectly affect the heat load of the generator by modifying the evaporation pressure and condensation pressure, leading to variations in the SF, either increasing or decreasing it.

According to Fig.4 in Section 3.3, the equipment investment cost of solar collectors and absorption chillers has the greatest impact on the investment of the overall system. Therefore, the solar collector area is an important factor affecting the cost. The cost of the system is not only related to the initial investment of the equipment, but also related to the energy consumption during operation and price of energy. Moreover, the energy consumption has a high correlation with the performance of the refrigeration system. The performance of the refrigeration system is related to the operating parameters of the system. It can be seen that among the design parameters and operation parameters of the system, A_c, {\dot{m}}_{\text{shwi}}, {\dot{m}}_{\text{dhwi}}, T_scwi, {\dot{m}}_{\text{scwi}}, T_scho, {\dot{m}}_{\text{s}\text{c}\text{ho}}, T_dcwi, {\dot{m}}_{\text{dcwi}}, T_dcho, and {\dot{m}}_{\text{dcho}} have a significant impact on system performance.

Based on the sensitivity analysis, the following parameters were identified as optimization variables, i.e., A_c, {\dot{m}}_{\text{shwi}}, {\dot{m}}_{\text{dhwi}}, T_scwi, {\dot{m}}_{\text{scwi}}, T_scho, {\dot{m}}_{\text{s}\text{c}\text{ho}}, T_dcwi, {\dot{m}}_{\text{dcwi}}, T_dcho, and {\dot{m}}_{\text{dcho}}. However, during the optimization process, the volume of the storage tank and the flowrate of the hot water circulating pump were held constant at predetermined values.

5.3 Optimization results

A notable advantage of the ISIGHT software is its capability for multi-objective optimization, which allows the assignment of predefined weights to the objectives before the optimization to begin. Following the optimization process, the software automatically presents an optimal solution based on the assigned weights, eliminating the need for manual Pareto solution set sorting. In this study, equal weights were assigned to both objectives. The resulting optimal solution, as illustrated in Tab.10, showcases a SF of 61.2% and a levelized annual cost of $657.56. Comparatively, these values represented a 3.22% increase in SF and a significant 10.18% decrease in levelized annual cost, highlighting the improved performance of optimization. The comparison of SF among three configurations is presented in Fig.12. The solar single-effect LiBr−H₂O ARS was denoted as C-1, while the former optimized solar single-double-effect LiBr−H₂O ARS was denoted as C-2. Lastly, the optimized solar single-double-effect LiBr−H₂O ARS was denoted as C-3. Notably, the optimized SF exhibited a remarkable 17.42% increase compared to the solar single-effect LiBr−H₂O ARS. The corresponding optimization variables at this point are detailed in Tab.11. The collector area was reduced from 65 to 48.6 m², thus serving as the primary factor behind the reduced levelized cost of this system. Moreover, the flowrate, the temperature of the external fluid, and other parameters were optimized within the specified range. The magnitude and percentage of cost differentiation for each component after optimization are illustrated in Fig.13. The initial conditions showed high levelized costs for the absorption chiller and solar collector, while the operating costs were relatively low. After optimization, the levelized cost of the collector accounted for 28.9%, experiencing a reduction of approximately 25.23%. Although the operating cost of the system increased by 0.3% compared to the overall proportion, the energy consumption of auxiliary heat source was reduced by 23.53 m³, and the operating cost was reduced by about 8.27% compared with that before optimization. These optimization-driven improvements significantly enhanced the overall economy of the system. The initial investment cost of solar refrigeration systems remained relatively high, mainly due to the elevated prices of collectors and absorption chillers.

Fig.14 showed the cumulative cooling capacity, auxiliary heat source consumption and average cumulative cooling capacity per unit collector area in the week after optimization, and the comparison with that without auxiliary heat source and before optimization. Due to the addition of the auxiliary heat source, the cooling capacity of the solar single-double-effect LiBr−H₂O ARS was divided into two parts, including the cooling capacity generated by solar energy as a heat source, and the cooling capacity generated by the auxiliary heat source when the solar energy was insufficient. The results indicated a substantial improvement in the cooling capacity per unit collector area following optimization, surpassing the values obtained without the AUX. Under the design conditions, the average cooling capacity per unit collector area for a week was 0.8 kWh/m² without the AUX and increased to 1.05 kWh/m² when the AUX was incorporated. However, after the multi-objective optimization process, the value significantly rose to 1.16 kWh/m², representing a 10.5% improvement compared to the pre-optimization state and a remarkable 45% enhancement compared to the system without the AUX. While the auxiliary heat source consumption and solar collector area decreased, the average cumulative cooling capacity per unit area increased, these findings highlighted the positive impact of optimization on system performance, rendering it more efficient and practical.

To further validate the reliability of the optimization outcomes, the calculation and comparison of CO₂ emissions were performed after system optimization. The CO₂ emission of the auxiliary heat sources of the system was 45.17 kg, and after optimization, the CO₂ emissions was reduced to 41.95 kg, reflecting an impressive decrease of 8.29%. In the production and processing of each square meter of solar collector will produce 92.4 CO_2eq kg/m² [50], after optimization the area of the solar collector has changed, and after optimization, this will reduce 1515.36 CO_2eq kg/m². These results demonstrated that the multi-objective optimization of the system not only enhances its economic benefits but also contributes to notable environmental benefits by reducing CO₂ emissions.

6 Conclusions

This study aimed to enhance the adaptability of solar refrigeration systems across various heat sources, increase solar energy utilization, and improve system economics while ensuring stable operation. To achieve these objectives, the solar single-double-effect LiBr−H₂O ARS undergoes analysis and optimization. Through Sobol sensitivity analysis, parameter sensitivities are acquired, guiding the selection of optimization variables. Utilizing the NSGA-II algorithm, a multi-objective optimization was performed, prioritizing maximizing SF and minimizing the levelized cost. Key study findings and conclusions include:

1) The system adeptly transitions between SEMs and DEMs in a high solar radiation climate without employing AUXs for a week. It achieves an average daily cooling duration of around 8.5 h.

2) Sobol sensitivity analysis method is applied to assess the sensitivity of the solar single-double-effect LiBr−H₂O ARS. It identifies the sensitivity of each design and operating parameter affecting solar energy utilization. The parameters A_c, {\dot{m}}_{\text{shwi}}, {\dot{m}}_{\text{dhwi}}, T_scwi, {\dot{m}}_{\text{scwi}}, T_scho, {\dot{m}}_{\text{s}\text{c}\text{ho}}, T_dcwi, {\dot{m}}_{\text{dcwi}}, T_dcho, and {\dot{m}}_{\text{dcho}} are chosen as the optimization variables.

3) The NSGA-II algorithm is employed to optimize the solar LiBr−H₂O ARS, aiming to maximize solar energy utilization fraction and minimize the levelized cost. This optimization results in a 3.22% increase in SF and a 10.18% reduction in levelized cost. Compared to the single-effect LiBr−H₂O ARS, the single-double-effect LiBr−H₂O ARS showcases a significant enhancement in SF, boasting a 17.42% increase after optimization.

4) Through optimization, the daily cumulative cooling capacity per unit collector area increase significantly, rising from 1.05 to 1.16 kWh/m². Furthermore, CO₂ emissions are reduced by 8.29%. These optimization endeavors have significantly boosted the overall performance of system, resulting in enhanced efficiency and reduced environmental impact.

In summary, compared with the traditional single-effect or double-effect refrigeration system, the solar single-double-effect LiBr−H₂O ARS has a stronger adaptability to heat sources. In addition, the use of multi-objective optimization improves the solar energy utilization of the system while reducing the cost. This is an attempt to apply multi-objective optimization based on sensitivity analysis in a new system, in order to improve the universality of the system for practical application and the efficiency of the optimization method, for the system by ① building a solar single-double-effect ARS LiBr−H₂O ARS unit, combining the optimization with the experiment, and adjusting the experimental conditions with the best performance, and by ② training the model for the whole system to improve the efficiency of optimization.

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