Performance evaluation and optimization of a novel compressed CO₂ energy storage system based on gas–liquid phase change and cold-electricity cogeneration

Abstract

Compressed CO₂ energy storage (CCES) system has received widespread attention due to its superior performance. This paper proposes a novel CCES concept based on gas–liquid phase change and cold-electricity cogeneration. Thermodynamic and exergoeconomic analyses are performed under simulation conditions, followed by an investigation of the impacts of various decision parameters on the proposed system. Next, a multi-objective optimization is conducted with the total energy efficiency and total product unit cost as the objective functions. Finally, brief comparisons are made between the proposed system and existing systems. The results indicate that the total energy efficiency of the proposed system reaches 79.21% under the given simulation conditions, outperforming the electrical efficiency of 61.27%. Additionally, the total product unit cost of the system is 25.61 $/GJ. A key component, T1, plays an important role due to its large exergy destruction rate (1.0591 MW) and total investment cost rate (154.85 $/h). Despite this, the exergoeconomic factors of T1 is only 41.08%, indicating that investing in T1 to improve the efficiency is practicable. The analysis shows that a lower CO₂ condensation temperature benefits the proposed system performance. While improving the isentropic efficiencies of the compressors and turbines enhances total energy efficiency, excessive isentropic efficiencies can lead to a significant increase in total product unit cost. Through multi-objective optimization, an optimal favorable operating condition is identified, yielding a compromise result with a total energy efficiency of 111.91% and a total product unit cost of 28.35 $/GJ. The proposed CCES system efficiently delivers both power and cooling energy, demonstrating clear superiorities over previous systems.

Graphical abstract

Keywords

compressed CO₂ energy storage system (CCES) / gas–liquid phase change / cold-electricity cogeneration / thermodynamic and exergoeconomic analyses / multi-objective optimization

1 Introduction

With the development of the global economy, the rapid increase in fossil fuel consumption has significantly contributed to anthropogenic CO₂ emissions, leading to widespread global warming [1]. Human-induced greenhouse gas emissions have increased the global temperature by approximately 1 °C, bringing the world closer to the critical 2 °C threshold outlined in the Paris Agreement [2]. Therefore, the adoption of green and low-carbon development models has received increasing attention from nations around the world [3]. Renewable energy sources, such as solar, wind, geothermal, and bioenergy, offer clean and sustainable alternatives to fossil fuels. The widespread deployment of these renewable energy reduces dependence on fossil fuels and plays a critical role in mitigating greenhouse gas emissions. Consequently, integrating renewable energy into hybrid or standalone systems has become a key strategy in energy transition in many regions [4,5]. However, renewable energy generation is highly sensitive to environmental factors and weather conditions, leading to intermittency and volatility in supply. Relying on renewable energy as a sole power source can introduce significant uncertainties and mismatches in supply and demand [6]. Energy storage systems (ESSs) can help address these challenges by enabling peak load shifting and mitigating the impacts of large-scale integration of renewable energy into the grid. As a result, the ESS technology is increasingly considered as a critical enabler of renewable energy adoption [7,8].

Currently, various energy storage technologies including electrical, electrochemical, thermal, chemical and mechanical, have been developed [9]. Among these, electrochemical energy storage technologies are particularly suited for small-scale and medium-scale applications. However, their overall performance is usually limited to 50%–55%, primarily due to challenges such as heat generation and self-discharge [10]. In contrast, mechanical energy storage systems (MESSs) are generally used for long-term energy storage. As an evolution from the gas turbine technology, compressed air energy storage (CAES) is valued for its large capacity, long lifespan, and other advantages [11]. To date, two CAES power plants based on the traditional diabatic CAES (D-CAES) have been built and are commercially operating around the world [12]. In addition, in order to get rid of the dependence of traditional CAES on fossil fuel, researchers have proposed several advanced CAES schemes, such as advanced adiabatic compressed air energy storage (AA-CAES) [13], liquid air energy storage (LAES) [14], and isothermal compressed air energy storage (I-CAES) [15]. However, traditional systems like D-CAES, AA-CAES, and I-CAES face significant limitations due to their dependence on specific geographical conditions such as large underground caverns or abandoned mines for storage sites [16,17]. Although LAES offers a potential solution by eliminating the need for such storage sites, it remains difficult to implement due to the extremely low liquefaction temperatures required for air [18].

Compared to air, carbon dioxide (CO₂) has a much higher critical point (7.4 MPa and 31.4 °C), making it more easily brought to a supercritical or liquid state. Additionally, the density of supercritical CO₂ is higher than that of air at the same pressure, which allows for a significant reduction in the required volume of storage tanks for high-pressure fluids [19]. These advantages led to the proposal of compressed CO₂ energy storage (CCES) as a promising alternative.

Currently, research on CCES mainly focuses on the development of new system configurations to improve performance. One promising approach is the introduction of external heat sources to enhance the thermodynamic performance of the system. He et al. [20] conducted an exergy analysis of the super-critical CCES (SC-CCES) system which utilized external heat, achieving an exergy efficiency of 57.02% which outperformed the traditional CAES (50.86%). Fu et al. [21] coupled the CCES system with solar heat storage and studied its performance under both trans-critical and super-critical conditions. Their results showed significant improvements in cycle efficiency, reaching 77.75% and 67.72% respectively, under different conditions. Xu et al. [22] proposed the use of underwater energy bags in the CCES and SC-CCES systems, maintaining a constant gas pressure. Tang et al. [23] applied external heat to a liquid CO₂ energy storage (LCES) system, analyzing both basic and modified configurations followed by optimization. Wang et al. [24] developed a novel isothermal CCES system integrated with solar thermal storage, achieving an impressive round-trip efficiency of 107.14%. Additionally, some researchers have explored coupling thermal power plants with CCES systems to utilize the heat from steam cycles. These includes indirect methods, such as introducing a thermal storage system [25,26], as well as directly heating CO₂ with steam [27].

Combined cooling, heating, and power (CCHP) systems have received significant attention due to their higher energy efficiency and lower greenhouse gas emissions, in recent years [28]. When driven by renewable energy, CCHP systems can reduce energy consumption in residential buildings and industries settings, while also supporting hydrogen production and desalination processes to increase productivity [29]. Besides, CCHP systems can efficiently recover waste heat, further improve the performance of energy plants [30]. Therefore, poly-generation energy storage systems (ESSs) have also an important area of research. However, most current studies on CCHP systems based on ESS have focused on CAES systems [31,32], with relatively few investigations into CCES systems. Liu et al. [33] proposed a CCHP system based on trans-critical compressed CO₂ energy storage (TC-CCES). This system used part of the heat stored during the charging process for heating, which reduced the exhaust temperature of the turbine and allowed the system to provide cooling as well. The results showed that the system could achieve an energy efficiency of 100%, though it is more suitable for heating applications than cooling. Sun et al. [34] integrated a liquid CO₂ ESS with electricity and cooling reported, by exploiting the latent heat of CO₂ evaporation for refrigeration, resulting in a substantial cooling capacity. Deng et al. [35] designed a novel CCHP system based on LCES, which could flexibly meet the demands for refrigeration, heating, and power generation by adjusting the shunt ration of splitter. The advantages of this system were proved through their study.

Although existing research has greatly promoted the development of CCES, several challenges remain: While introducing the external heat can improve the performance of the system, there are still certain limitations to its application. The dependence of the system on the availability and stability of the heat source can pose issues, and the utilization of certain heat sources may lead to increased costs. Most current studies on CCES focus on TC-CCES and SC-CCES systems, which operate at very high-pressure levels, some reaching up to 27 MPa [34]. Excessive pressure not only complicates equipment design and manufacturing, but also poses challenges to CO₂ storage. Existing studies on CCES systems typically involve storing expanded CO₂ in liquid, often using low-pressure storage to extend the compression/expansion lines. This requires storing CO₂ at low-temperatures, which can be economically inefficient and challenging.

To address the aforementioned issues and facilitate the practical applications of CCES systems, a novel approach based on gas–liquid phase change (PC-CCES) was proposed [36]. This system uses a flexible gas holder to store CO₂ at ambient pressure, albeit with a larger volume requirement. The flexible gas holder can adapt to various application scenarios, overcoming the limitations imposed by topographic conditions. Additionally, the volume of the gas holder can adjust dynamically in response to the gas storage capacity during operation, ensuring that the pressure of CO₂ remains constant. Meanwhile, compressed CO₂ with 8.0 MPa is stored in a liquid storage tank. This configuration offers several advantages: it is not dependent on specific heat sources and it avoids the need for low-temperature storage. Furthermore, by simply storing liquid CO₂ at a low pressure, the system can extend the compression/expansion lines, which also greatly reduces the challenges associated with high-pressure CO₂ storage and simplifies equipment design. Despite these improvements, the system still has room for further optimization. As shown in Fig.1, the specific heat capacity of CO₂ changes significantly with the increase of temperature near the critical pressure (7.0 and 8.0 MPa). This rapid variation in specific heat leads to a smaller temperature difference between the hot and cold fluids in the heat exchanger during compression heat recovery. To maintain effective heat transfer, the heat exchanger requires a large temperature gradient, which in turn increases heat exchanger losses and negatively impacts the performance of the system. On the other hand, when the pressure is farther from the critical point (e.g., 4.0 or 5.0 MPa), the change in specific heat capacity of CO₂ is more gradual. Therefore, if the compression heat can be recovered at a lower pressure, it may significantly reduce the loss of heat exchanger and improve system efficiency.

In addition, the challenge of further pressurizing CO₂ after heat exchange remains a key issue. The method of CO₂ liquefaction and pumping, commonly used in CCS systems, has proven effective for CO₂ pressurization [37]. Consequently, a novel PC-CCES system is proposed in this paper. This system introduces a refrigeration cycle during the charging process. After CO₂ undergoes heat exchange, it is condensed using the refrigeration cycle and then pumped to the desired storage pressure. This paper is innovative because the compression heat carried by CO₂ is recovered at a lower pressure, effectively avoiding the pinch point problem in the heat exchanger. This also allows the end temperature difference of the heat exchanger to remain lower, reducing thermal losses and improving the performance of the system. Additionally, the proposed system adopts a liquefaction-pumping method in the charging process, in contrast to previous systems which directly compresses gaseous CO₂. This approach significantly reduces the power required for CO₂ pressurization, thereby enhancing system efficiency. Moreover, under certain conditions, CO₂ after pumping and expansion reaches a lower temperature, which can be used for refrigeration. This enables the proposed system to achieve cold-electricity cogeneration, enhancing its overall utility.

2 System description and assumptions

The simplified schematic diagram and the layout of the proposed PC-CCES system are presented in Fig.2. An energy storage subsystem and a refrigeration cycle constitute the whole system. Besides, the energy storage subsystem can be further divided into two processes: charge and discharge, whose T–s diagram is shown in Fig.3.

The charging process is conducted during the off-peak period. Gaseous CO₂ under ambient condition, stored in the flexible gas storage (FGS), is pressurized by two CO₂ compressors (C1 and C2) to reach the pre-set condensation pressure. Additionally, intercoolers installed after the two compressors (IC1 and IC2) are used to cool the compressed CO₂ (1–5). Subsequently, CO₂ is liquefied in the condenser (Cond1), after which the liquid CO₂ is pumped by pump (P1) (5–7). Finally, the high-pressure liquid CO₂ is heated in heater (H1) and then stored in the liquid storage tank (LST) (7–8). Meanwhile, the heat generated by CO₂ in IC1 and IC2 is absorbed by the low-temperature water stored in the cold-water storage tank (CST), which is finally contained in the hot-water storage tank (HST) (16–21). Besides, the cooling capacity of liquid CO₂ is carried away by ambient air in H1 (32–33), which is used to meet user demand.

When the peak period arrives, the discharge process of the system comes into operation. The liquid CO₂ leaving from the LST is first evaporated by the environment in the evaporator (Evap) (9–10). Next, gaseous CO₂ enters preheater1 (PRH1) to absorb heat and then generates power in turbine1 (T1) (10–12). Then, the CO₂ exhausted by T1 is heated in PRH2 and further expands in T2 for power generation (12–14). After that, the CO₂ with ambient pressure absorbs heat in H2 and returns to FGS, preparing for the next cycle (14–15). Finally, high-temperature water from HST enters CST for storage after releasing heat in the two preheaters (22–27). Similarly, H2 can also serve as the heat exchanger for producing low-temperature air (34–35).

In the charging process, the cooling capacity required by CO₂ in Cond1 is provided by the refrigeration cycle. There are plenty of choices for the working fluid, and ammonia is selected in this work since both its ozone depletion potential and global warming potential are 0 [37]. In the refrigeration cycle, the ammonia leaving from refrigerant compressor (C3) is first liquefied by ambience in condenser (Cond2) (30–31). The liquid refrigerant decreases pressure through an expansion valve and vaporizes in Cond1 for the liquefaction of CO₂ (31–29). Eventually, the gaseous ammonia is compressed in C3 for the next cycle (29–30).

To simplify the calculation without significantly affecting the results, it is assumed that the system operates under stable conditions, and the changes of the kinetic and potential energies as well as losses occurring in heat exchangers and pipelines are neglected [36]. The fluids enter and exit the storage unit in the same state [23]. The mass flow rate of CO₂ during both the charge and discharge processes is identical, and the operating time of each process is also the same [38]. CO₂ at the outlet of Cond1 (Point 6) is saturated liquid, and that exiting Evap (Point 10) is saturated gas. Similarly, the refrigerant leaves Cond1 (Point 29) as saturated vapor and Cond2 (Point 31) as saturated liquid [39]. The minimum temperature differences for the heat exchangers (IC1, IC2, PRH1, PRH2, H1 and H2) are set to 5 °C. Meanwhile, the pinch point temperature differences in the condensers (Cond1 and Cond2) and the evaporator (Evap) are set to 3 °C to ensure normal operation [39,40]. To prevent losses due to the mixing of fluids at different temperatures, the temperature of CO₂ leaving compressors can be controlled to remain uniform by adjusting the compression ratio of C1, based on the specific system design parameters. Similarly, the temperature of the cooling water leaving the preheaters can be kept equal by adjusting the expansion ratio of T1 [41].

3 Mathematical modeling

In this paper, the REFPROP9.1 provided by National Institute of Standards and Technology (NIST) is used to calculate the thermodynamic properties of CO₂ and water at each state point. Using known thermodynamic properties, the performance of the individual components and the entire system can be determined based on the conservations of mass, energy, exergy, and cost balances. All of these calculations are implemented through a MATLAB simulation program.

3.1 Definition of the decision parameters

Three kinds of variables are selected as the decision parameters for the PC-CCES system: CO₂ condensation temperature, and the isentropic efficiencies of compressors, pump, and turbines.

(1) The liquefaction temperature of CO₂, denoted as T₆, is directly related to the pressure ratio of compressors. This variable not only influences the power consumed by the compressor, but also affects the heat storage temperature. In addition, changes in T₆ leads to an inverse relationship between the power required for CO₂ pressurization and the power consumed by refrigeration cycle. This contradictory relationship underscores its importance in system optimization.

(2) The isentropic efficiencies of power units, denoted as {\eta }_{\text{C1}}, {\eta }_{\text{C2}}, {\eta }_{\text{C3}}, {\eta }_{\text{P1}}, {\eta }_{\text{T1}}, and {\eta }_{\text{T2}}, are crucial for the thermodynamic and economic performances of the proposed system [23]. Additionally, the efficiencies of C1 and C2 are also related to the heat storage temperature, while the isentropic efficiency of C3 affects the coefficient of performance of the refrigeration cycle.

The isentropic efficiencies of the components reflect their ability of energy conversion, which can be expressed as

(1){\eta }_{\mathrm{C}1}=\left(h_{2,\mathrm{s}}-h_1\right)/\left(h_2-h_1\right),

(2){\eta }_{\mathrm{C}2}=\left(h_{4,\mathrm{s}}-h_3\right)/\left(h_4-h_3\right),

(3){\eta }_{\mathrm{C}3}=\left(h_{30,\mathrm{s}}-h_{29}\right)/\left(h_{30}-h_{29}\right),

(4){\eta }_{\mathrm{P}1}=\left(h_{7,\mathrm{s}}-h_6\right)/\left(h_7-h_6\right),

(5){\eta }_{\mathrm{T}1}=\left(h_{11}-h_{12}\right)/\left(h_{11}-h_{12,\mathrm{s}}\right),

(6){\eta }_{\mathrm{T}2}=\left(h_{13}-h_{14}\right)/\left(h_{13}-h_{14,\mathrm{s}}\right),

where the subscript s means the isentropic process.

3.2 Thermodynamic analysis

As the foundation of the thermodynamic analysis, the balance equations of mass and energy can be written as [34]

(7)\sum {\dot{m}}_{\text{in}}=\sum {\dot{m}}_{\text{out}},

(8)\sum {(\dot{m}\cdot h)}_{\text{in}}-\sum {(\dot{m}\cdot h)}_{\text{out}}+\sum \dot{Q} +\sum \dot{W}=0.

Based on the above equations, the balance relation of components in the discussed system are summarized in Tab.1.

Except for balance equations, exergy is also a significant parameter in thermodynamic analysis. For the PC-CCES system designed in this paper, it is only necessary to take the physical exergy ({\dot{E}}_{\text{ph}}) of working fluids into account, since no chemical changes occur during the cyclic process. The {\dot{E}}_{\text{ph}} of fluid at point j can be calculated as [42]

(9){\dot{E}}_{\text{ph},j}={\dot{m}}_j\cdot [(h_j-h_0)-T_0\cdot (s_j-s_0)] ,

where the subscript 0 means the ambient state.

3.3 Exergoeconomic analysis

Exergoeconomic analysis can determine the per-unit exergy cost of the streams and identify potential cost-saving opportunities by combining exergy analysis and economic analysis. The SPECO (specific exergy cost) method, which is utilized for this purpose, facilitates the development of the exergoeconomic analysis [43].

The exergy of each fluid stream has been defined in Section 3.2, based on which the fuel exergy ({\dot{E}}_{\text{F}}) and the product exergy ({\dot{E}}_{\text{P}}) of each component can be calculated. The relationship between {\dot{E}}_{\text{F}} and {\dot{E}}_{\text{P}} can be expressed as [44]

(10){\dot{E}}_{\text{F}} ={\dot{E}}_{\text{P}}+{\dot{E}}_{\text{D}},

where {\dot{E}}_{\text{D}} represents the destruction exergy.

The definitions of fuel-product exergy for components in the PC-CCES system are given in Tab.2.

In addition, the exergy efficiency of component k can be defined as the ratio of {\dot{E}}_{\text{P}} to {\dot{E}}_{\text{F}} [45]

(11){\epsilon }_k=\frac{{\dot{E}}_{\text{P},k}}{{\dot{E}}_{\text{F},k}}.

After that, the cost balance equations together with auxiliary equations can be applied. The cost balance equation of component k can be expressed as [46]

(12)\sum {\dot{C}}_{\text{in},k}+{\dot{C}}_{\text{q},k}+{\dot{Z}}_k=\sum {\dot{C}}_{\text{out},k}+{\dot{C}}_{\text{w},k},

where {\dot{C}}_{\text{in,}k} is the cost rate associated with the exergy streams entering component k, and {\dot{C}}_{\text{out,}k} is the cost rate associated with the exergy streams exiting the component; {\dot{C}}_{\text{q},k} represents the cost rate of the energy input, and {\dot{C}}_{\text{w,}k} means the cost rate associated with the kth component; {\dot{Z}}_k denotes the capital investment cost rate of component k.

In addition, the above cost rate items can be respectively written as [26]

(13){\dot{C}}_j=c_j\cdot {\dot{E}}_j,

(14){\dot{C}}_{\text{q}} =c_{\text{q}}\cdot \dot{Q},

(15){\dot{C}}_{\text{w}} =c_{\text{w}}\cdot \dot{W},

(16)\dot{Z}=\phi \cdot \text{CRF}\cdot Z_k/N,

where c denotes the cost per unit exergy; N is running time of the system which is 2920 h per year for the energy storage system, and \phi represents the maintenance factor with a value of 1.06 [26]; Z_k refers to the capital investment for the kth equipment, with the capital models for each component summarized in Tab.3. Moreover, CRF stands for the capital recovery factor, which is given by [26]

(17)\text{CRF =}\frac{i\cdot (1+i{)}^Y}{(1+i{)}^Y-1},

where i denotes the interest rate with a value of 0.12, and Y is the lifespan of the PC-CCES system which is 20 years [26].

Since the capital investment models for the components were proposed in different years, the calculated cost need to be adjusted using the Chemical Engineering Plant Cost Index (CEPCI), which is defined as [50]

(18)Z_{\text{ref},k}=Z_{\text{ori},k} \frac{\text{CEPC}{\text{I}}_{\text{ref}}}{\text{CEPC}{\text{I}}_{\text{ori}}},

where the subscript ref means the reference year which is set to be 2023 with a CEPCI of 797.9.

In this paper, the logarithmic mean temperature difference (LMTD) and total heat transfer coefficient (U) of the heat exchanger have a bearing on the heat transfer area (A) [42]

(19)A_k=\frac{{\dot{Q}}_k}{U_k\cdot \text{LMT}{\text{D}}_k}.

In addition, the approximate U of intercoolers and preheaters are taken as 3000 W/(m²·K) [51], while that of heaters and evaporators are assumed to be 1600 W/(m²·K) [52]. A value of 2000 W/(m²·K) is used to calculate the A of condensers [51].

The cost balance equations and corresponding accessorial functions of each component in the PC-CCES system are illustrated in Tab.4. A closed linear system of equations is formed, and solving this allows the determination of c for each stream. Since the air introduced in the heat exchangers is at ambient temperature, it is reasonable to regard it as a free source with negligible cost. In addition, the power consumed by compressors and pumps is not considered as part of the output power of the turbines, owing to the time-divided operation of the energy storage systems. Therefore, the unit cost of input power is set at 0.05 $/kWh [48].

More relevant indicators, which are exhibits in Tab.5, for evaluating the exergoeconomic performance of the PC-CCES system can be derived.

3.4 Performance evaluation indicators

Three parameters are used to assess the proposed system from both thermodynamic and exergoeconomic perspectives: electrical efficiency (EE), total energy efficiency (TEE), and total product unit cost (c_{\text{P,tot}}).

EE is defined as the ratio of the total power generated during discharge process to the total power consumed during the charge process, which can be expressed as [33]

(20)\text{EE}=\frac{{\dot{W}}_{\text{dischar}}\cdot t_{\text{dischar}}}{{\dot{W}}_{\text{char}}\cdot t_{\text{char}}}=\frac{({\dot{W}}_{\text{T1}}+{\dot{W}}_{\text{T2}})\cdot t_{\text{dischar}}}{({\dot{W}}_{\text{C1}}+{\dot{W}}_{\text{C2}}+{\dot{W}}_{\text{P1}}+{\dot{W}}_{\text{C3}})\cdot t_{\text{char}}}.

Additionally, since the temperature of CO₂ exhaust from P1 and T2 is relatively low under certain conditions, low temperature air can be obtained through H1 and H2, which can then be used for refrigeration. Therefore, TEE is defined as the ratio of total output energy to input energy, and can be expressed as [33]

(21)\text{TEE}=\frac{({\dot{W}}_{\text{T1}}+{\dot{W}}_{\text{T2}}+{\dot{Q}}_{\text{H2}})\cdot t_{\text{dischar}}+{\dot{Q}}_{\text{H1}}\cdot t_{\text{char}}}{({\dot{W}}_{\text{C1}}+{\dot{W}}_{\text{C2}}+{\dot{W}}_{\text{P1}}+{\dot{W}}_{\text{C3}})\cdot t_{\text{char}}}.

It is important to note that the calculation of TEE is based on the first law of thermodynamics, which considers only the quantity of energy. In this paper, the cooling energy generated by the proposes system is considered as equally important as the electrical energy, and the difference in energy quality between the two is ignored. Hence, TEE may exceed 100% under certain conditions [33].

c_{\text{P,tot}} is used to characterize the system product unit cost, which can be calculated by the following equation [34]

(22)\begin{array}{rl}{c}_{\text{P,tot}}& =\frac{{\displaystyle \sum {\dot{Z}}_k+{\dot{C}}_{\text{W,C1}}+{\dot{C}}_{\text{W,C2}}+{\dot{C}}_{\text{W,C3}}+{\dot{C}}_{\text{W,P1}}}}{{\dot{W}}_{\text{T1}}+{\dot{W}}_{\text{T2}}+{\dot{Q}}_{\text{Cooling}}}\\ & =\frac{{\displaystyle \sum {\dot{Z}}_k+{\dot{C}}_{\text{W,C1}}+{\dot{C}}_{\text{W,C2}}+{\dot{C}}_{\text{W,C3}}+{\dot{C}}_{\text{W,P1}}}}{{\dot{W}}_{\text{T1}}+{\dot{W}}_{\text{T2}}+{\dot{Q}}_{\text{H1}}+{\dot{Q}}_{\text{H2}}}.\end{array}

The coefficient of performance (COP) is adopted to measure the cooling capacity of the proposed system, which can be written as [34]

(23)\text{COP}=\frac{{\dot{Q}}_{\text{H1}}\cdot t_{\text{char}}+{\dot{Q}}_{\text{H2}}\cdot t_{\text{dischar}}}{({\dot{W}}_{\text{C1}}+{\dot{W}}_{\text{C2}}+{\dot{W}}_{\text{P1}}+{\dot{W}}_{\text{C3}})\cdot t_{\text{char}}-(W_{\text{T1}}+W_{\text{T2}})\cdot t_{\text{dischar}}}.

It should be noted that when the exhaust temperature of P1 or T2 (T₇ or T₁₄) is relatively high, the PC-CCES system will not have cooling capacity. Considering the temperature difference of heat exchanger, this paper assumes that the corresponding heat exchangers (H1 and H2) will not be used for refrigeration when T₇ or T₁₄ exceeds 0 °C.

Based on the above assumption, an interesting phenomenon arises. Taking T₁₄ as an example, when it exceeds 0 °C, H2 is not used for refrigeration. Although heat exchange occurs in H2, {\dot{Q}}_{\text{H2}} in Eqs. (21) and (22) is 0. However, when T₁₄ decreases below 0 °C, the value of {\dot{Q}}_{\text{H2}} will jumps from 0 to the heat load of H2. As a result, TEE and c_{\text{P,tot}} will also change rapidly. This behavior will be evident in the parametric analysis in Section 5.3, specifically seen in the separation of the curves.

3.5 Multi-objective optimization

TEE and c_{\text{P,tot}} are selected as the performance indicators from both thermodynamic and exergoeconomic perspectives. However, thermodynamic and economic performances are usually in conflict, as achieving a higher TEE generally requires an increase in c_{\text{P,tot}}. Accordingly, the optimization of the proposed system should aim either to maximize TEE or minimize c_{\text{P,tot}}. The feasibility of the non-dominated sorting genetic algorithm (NSGA-II) has been demonstrated in several studies on thermal systems, including supercritical CO₂ cycle [53], cold-end system of thermal power plants [54], and waste heat recovery system of gas turbine [55]. In this paper, NSGA-II is applied to the multi-objective optimization of the system.

The mathematical description of the optimization problem for the PC-CCES system can be expressed as

(24)\{\begin{array}{l}\text{Max}\text{. TEE}(\mathbf{X}),\\ \text{Min}.c_{\text{P,tot}}(\mathbf{X}),\end{array}

where X indicates the matrix composed of decision parameters, which can be represented as

(25)\mathbf{X}=[T_6\text{,} {\eta }_{\text{C1}},{\eta }_{\text{C2}},{\eta }_{\text{C3}} ,{\eta }_{\text{P1}},{\eta }_{\text{T1}},{\eta }_{\text{T2}}{]}^{\text{T}}.

4 Model validation

Since the PC-CCES system discussed in this paper has not been previously reported in the existing literature, a typical CCES system, which can also be used for refrigeration is selected to validate the mathematical model. The typical CCES system stores both low-pressure and high-pressure CO₂ in liquid form and consists of components such as a compressor, turbine, heat exchanger, evaporator, and condenser. This system is structurally similar to the proposed PC-CCES system, making it suitable for model validation. The specific verification process can be summarized as follows: the mathematical model developed above is used to analyze the performance of the typical CCES system, using the input data from existing literature. The calculated results are then compared with those reported in the relevant publications, and the scientific validity of the mathematical model is conformed if the results show minimal discrepancies. This validation method has also been employed in previous studies [23,47]. The simulation data reported by Sun et al. [34] is used to validate the model for the main components and performance parameters, as summarized in Tab.6. The verification results are presented in Tab.7. As shown in Tab.7, the errors remain within a small range, indicating that the models proposed in this paper are accurate and reliable.

5 Results and discussion

5.1 Thermodynamic analysis results

The input parameters assumed for the baseline conditions are summarized in Tab.8, and the calculation flowchart of the PC-CCES system is presented in Fig.4.

Under the above conditions, thermodynamic parameters at each state point (such as temperature, and pressure) are listed in Tab.9.

Based on the data in Tab.9, the results of the exergy analysis are represented in Tab.10. Notably, the {\dot{E}}_{\text{F}} for the compressors and pump are 7.0363, 6.3289, 2.7758, and 0.1807 MW, respectively. Meanwhile, the {\dot{E}}_{\text{P}} for T1 and T2 are 5.6942 and 4.3058 MW. Therefore, EE of the proposed system is 61.27% under the baseline simulation conditions. Additionally, H1 can be used for refrigeration under the given parameters, with a cooling capacity of 2.9276 MW. Consequently, the TEE of the system reaches 79.21%, which is significantly higher than the EE. The COP of the PC-CCES system under these conditions is 46.31%. Therefore, the PC-CCES system is particularly suited for applications that require both electricity and cooling, such as commercial parks and data centers. In such scenarios, the proposed system not only facilitates peak load shifting but also reduces cooling energy consumption, leading to energy savings. Furthermore, users can benefit economically from the price difference between peak and off-peak electricity.

To clearly compare the exergy performance of each component, Fig.5 displays the {\dot{E}}_{\text{D}} and ε of components in descending order. According to Fig.5, T1 exhibits the highest {\dot{E}}_{\text{D}}, reaching 1.0591 MW. Additionally, C1, T2, and C2 also show significantly high exergy destruction, each exceeding 0.8 MW. Despite this, the exergy efficiencies of these four modules remain relatively high. On the other hand, due to the large temperature difference, module with the lowest exergy efficiency is PRH2, which is only 59.54%.

5.2 Exergoeconomic analysis results

Tab.11 reveals the results of c and \dot{C} for each stream in the proposed system. Based on Tab.11, the exergoeconomic performance of components is depicted in Tab.12. Synthesizing the results of thermodynamic and exergoeconomic analyses, the unit cost of per product of the PC-CCES system (c_{\text{P,tot}}) is 25.61 $/GJ.

Fig.6 categorizes the system components into four groups: charge, discharge, refrigeration cycle, and container, showing the ratio of \dot{Z} for each category and the proportion of each component in the corresponding group. Since T1 and T2 are expensive, the \dot{Z} of the discharge process is the highest, reaching 155.15 $/h and accounting for 41.30% of the total cost. Following this, the total cost rate of the containers is 119.12 $/h, which is attributed to the large volume of FGS, requiring significant investment. Additionally, the capital investment cost rate for the compressors (C1 and C2) accounts for more than 50% of the total cost rate of the charge process.

Accordingly, the \dot{Z}+{\dot{C}}_{\text{D}} and f are the two major parameters for exergoeconomic performance evaluation. A large \dot{Z}+{\dot{C}}_{\text{D}} indicates that the component is economically important, while the major resource of cost can be identified through f. A high f suggests that the cost rate of the component is mainly caused by capital investment and operation costs. Conversely, if exergy destruction is low, the cost rate caused by {\dot{E}}_{\text{D}} is dominant in total cost. The total \dot{Z}+{\dot{C}}_{\text{D}} of the proposed system reaches 822.05 $/h, with f accounting for only 45.70%. Thus, this indicates considerable exergy destruction in the proposed system, highlighting the need to optimize the critical components through additional investment. From Tab.12, the maximum \dot{Z}+{\dot{C}}_{\text{D}} is found for T1 (154.85 $/h), which can be attributed to its high-cost rate and exergy destruction. Similarly, the \dot{Z}+{\dot{C}}_{\text{D}} of T2 follows closely behind with a cost rate of 129.69 $/h, driven by the same factor as T1. However, the f of T1 and T2 is relatively low (41.80% and 40.03%), suggesting that the cost rate related to exergy destruction is the major contributor. Therefore, in future designs of the proposed system, it may be economically beneficial to increase the capital investment for T1 and T2 to improve their efficiencies. Similarly, C1 and C2 also have considerable \dot{Z}+{\dot{C}}_{\text{D}} (77.52 and 65.40 $/h), but low f (38.89% and 35.17%). Thus, these components should be considered in the same way as T1 and T2. Additionally, the \dot{Z}+{\dot{C}}_{\text{D}} for PRH2 is 30.93 $/h, which is moderate in comparison to other components. However, the f of PRH2 is only 9.69%, indicating that a significant portion of its cost is driven by exergy destruction. This finding aligns with the low exergy efficiency of PRH2, as shown in Fig.5.

5.3 Parametric analysis results

This section discusses the impacts of changes in decision variables on TEE and c_{\text{P,tot}}. The initial values of decision variables are listed in Tab.8, and their variation ranges are given in Tab.13. In addition, the relationship between some key parameters and the main thermodynamic variables is exhibited to clarify the results, including the temperature and mass flow rate of hot water (T_{21} and {\dot{m}}_{21}), inlet and outlet temperature of T2 (T_{13} and T_{14}), mass flow rate of CO₂ ({\dot{m}}_{\text{C}{\text{O}}_{\text{2}}}) along with expansion ratio of T2 (P_{13} /P_{14}).

The impacts of T_6 on the PC-CCES system are shown in Fig.7. From Fig.7(a), it is seen that although the growth in T_6 leads to a decrease in {\dot{W}}_{\text{char}}, the reduction in {\dot{Q}}_{\text{Cooling}} is more pronounced. Therefore, a higher T_6 is detrimental to the system, resulting in a lower TEE and a larger c_{\text{P,tot}}.

From Fig.7(b) and Fig.7(c), it is observed that since the compression ratios of compressors and T_6 are positively correlated, the increase in T_6 not only increases the specific compression power of C1 and C2, but also improves T₂₁. A higher T₂₁ increases the inlet temperature and specific output power of turbines. Since the power generation remains constant at 10 MW, {\dot{m}}_{\text{C}{\text{O}}_{\text{2}}} decreases. Under the combined influences of specific power consumption and {\dot{m}}_{\text{C}{\text{O}}_{\text{2}}}, {\dot{W}}_{\text{char}} gradually decreases. Additionally, the effects of T_6 on {\dot{Q}}_{\text{Cooling}} are reflected in two aspects. First, there is a direct correlation between T_6 and CO₂ inlet temperature of H1 (T₇). Consequently, an increase in T_6 reduces the refrigeration output of H1 ({\dot{Q}}_{\text{H1}}). Next, as previously mentioned, the growth of T_6 raises the inlet temperature of the turbines. For T1, its expansion ratio increases to maintain the exhaust temperature. Therefore, the elevated inlet temperature and reduced expansion ratio (see Fig.7(c)) cause the outlet temperature of T2 to rise. As a result, the cooling capacity of H2 ({\dot{Q}}_{\text{H2}}) decreases until it can no longer be used for refrigeration, which explains the separation of curves in Fig.7(a).

Fig.8 illustrates the influence of {\eta }_{\text{C1}} on the PC-CCES system. As shown in Fig.8 (a), the curves begin to diverge when {\eta }_{\text{C1}} changes from 82% to 83%. On both sides of the dividing line, {\eta }_{\text{C1}} is negatively correlated with {\dot{W}}_{\text{char}} and positively correlated with {\dot{Q}}_{\text{Cooling}}. As a result, TEE consistently increases with the increment of {\eta }_{\text{C1}}. However, the behavior of c_{\text{P,tot}} follows a different trend. When {\eta }_{\text{C1}} is less than 82%, changes in {\eta }_{\text{C1}} have a minimal impact on {\dot{Q}}_{\text{Cooling}}. The improvement in TEE mainly results from the reduction in {\dot{W}}_{\text{char}}, which is caused by the upgrading of C1. Better performance comes at the cost of higher investments, leading to an increase in c_{\text{P,tot}}. Once {\eta }_{\text{C1}} exceeds 83%, the increase in TEE is influenced by both {\dot{W}}_{\text{char}} and {\dot{Q}}_{\text{Cooling}}. Initially, {\dot{Q}}_{\text{Cooling}} has a more significant impact on c_{\text{P,tot}}, but as {\eta }_{\text{C1}} continues to increase, the investment cost for C1 sharply increases, and the effect of {\dot{Q}}_{\text{Cooling}} gradually weakens. Consequently, c_{\text{P,tot}} slightly decreases first and then rapidly increases.

The impact of {\eta }_{\text{C1}} on the major thermodynamic parameters is depicted in Fig.8(b) and Fig.8(c) to explain the changes in {\dot{W}}_{\text{char}} and {\dot{Q}}_{\text{Cooling}}. As C1 improves, both the specific input power and exhaust temperature of C1 decrease. The reduction in exhaust temperature ultimately leads to a decrease in the inlet temperature of turbines, which lowers the specific power generation of turbines and increases {\dot{m}}_{\text{C}{\text{O}}_{\text{2}}} accordingly. Consequently, {\dot{W}}_{\text{char}} gradually decreases because of the dominant effect of specific power consumption. Moreover, {\dot{Q}}_{\text{H1}} is proportional to {\dot{m}}_{\text{C}{\text{O}}_{\text{2}}} because T₆ remains unchanged. The decrease in inlet temperature causes the expansion ratio of T1 to reduce in order to maintain a constant outlet temperature. This not only reduces the intake temperature of T2 but also increases its expansion ratio (see Fig.8(c)). Consequently, T₁₄ significantly decreases, while {\dot{Q}}_{\text{H2}} increases. Thus, {\dot{Q}}_{\text{Cooling}} rises along with {\eta }_{\text{C1}}.

Fig.9 illustrates the relationship between {\eta }_{\text{C2}} and the proposed system. As shown in the three charts in Fig.9, the impact of {\eta }_{\text{C2}} on the system is essentially the same as {\eta }_{\text{C1}}, so further details are not repeated here.

Since the change in {\eta }_{\text{P1}} only affects the thermodynamic parameters at point 7, Fig.10 presents the variation trend of the performance parameters of the PC-CCES system as {\eta }_{\text{P1}} increases. As shown in Fig.10, the influence of {\eta }_{\text{P1}} is not significant. As {\eta }_{\text{P1}} increases, {\dot{W}}_{\text{char}} slightly decreases, while {\dot{Q}}_{\text{Cooling}} rises. Consequently, TEE and c_{\text{P,tot}} show opposite trends: TEE increases, while c_{\text{P,tot}} decreases. For P1, the increase in isentropic efficiency indicates an improvement in performance, which leads to a reduction in outlet temperature and power consumption. Therefore, {\dot{W}}_{\text{char}} and {\dot{Q}}_{\text{Cooling}} exhibit the pattern as shown in Fig.10.

Similarly, Fig.11 illustrates the impact of {\eta }_{\text{C3}} on the proposed system, as changes in {\eta }_{\text{C3}} only affects the refrigeration cycle. As {\eta }_{\text{C3}} increases, the power consumed by C3 decreases. Since the energy storage subsystem remains unchanged, {\dot{W}}_{\text{char}} decreases while {\dot{Q}}_{\text{Cooling}} stays constant. In other words, the increase in {\eta }_{\text{C3}} improves the COP of the refrigeration cycle, leading to an upward trend of TEE. However, the capital investment cost of C3 also increases as thermodynamic performance improves, causing c_{\text{P,tot}} to initially decrease and then increase.

Fig.12 demonstrates the effects of {\eta }_{\text{T1}} on the PC-CCES system. As shown in Fig.12(a), with the increase in {\eta }_{\text{T1}}, {\dot{W}}_{\text{char}} consistently decreases, while {\dot{Q}}_{\text{Cooling}} follows two different trends. When {\eta }_{\text{T1}} is below 87%, the growth of {\eta }_{\text{T1}} is unfavorable to {\dot{Q}}_{\text{Cooling}} and the fast decline of {\dot{W}}_{\text{char}} causes TEE to gradually increase. However, when {\eta }_{\text{T1}} exceeds 88%, {\dot{Q}}_{\text{Cooling}} and {\eta }_{\text{T1}} become positively correlated, and TEE begins to show an upward trend. In addition, under the combine effect of reduced power consumption and increasing capital investment cost of T1, c_{\text{P,tot}} initially decreases and then slightly increases when {\eta }_{\text{T1}} is below 87%. In contrast, c_{\text{P,tot}} experiences a rapid increase when {\eta }_{\text{T1}} exceeds 88%, which is caused by the dominant influence of high investment cost of T1.

Fig.12(b) and 12(c) illustrate the influence of {\eta }_{\text{T1}} on the thermodynamic parameters. It is important to note that the change in {\eta }_{\text{T1}} only affects the charge process through the mass flow rate, with no other impact. Therefore, the specific input power of the compressors and T₂₁ remain unchanged. The increase in {\eta }_{\text{T1}} enhances the specific output power of T1, which leads to a reduction in {\dot{m}}_{\text{C}{\text{O}}_2} and {\dot{W}}_{\text{char}}. Additionally, {\dot{m}}_{\text{21}} decreases along with {\dot{m}}_{\text{C}{\text{O}}_2}, resulting in insufficient hot water supply in PRH2 and a subsequent decrease in T₁₃. Meanwhile, the expansion ratio of T2 increases, as the exhaust temperature of T1 is assumed to remain constant. Consequently, T₁₄ exhibits a downward trend. T₁₄ remains higher than 0 °C when {\eta }_{\text{T1}} is less than 87%, and {\dot{Q}}_{\text{Cooling}}, which is equal to {\dot{Q}}_{\text{H2}}, is proportional to {\dot{m}}_{\text{C}{\text{O}}_2}. When {\eta }_{\text{T1}} exceeds 88%, H1 can be used for refrigeration, and its cooling capacity increases. The dominant effect of {\dot{Q}}_{\text{H1}} causes a rise in {\dot{Q}}_{\text{Cooling}}.

The relationships between the performance parameters of the PC-CCES system and {\eta }_{\text{T2}} are shown in Fig.13(a). It can be observed that the effect of {\eta }_{\text{T2}} on the proposed system is quite similar to that of {\eta }_{\text{T1}}. Further, referring to Fig.13 (b) and 13(c), the reasons for the observed results are also basically the same. The only difference is that {\eta }_{\text{T2}} has no effect on the operation conditions of T1. Therefore, the reduction in the exhaust temperature of T2 is solely due to the decrease in T₁₃, which is caused by drop in {\dot{m}}_{\text{21}}.

Fig.14 represents the first-order and total-order sensitivity indices for TEE and c_{\text{P,tot}}. In Fig.14, S₁ represents the first-order sensitivity index and S_T is the total-order sensitivity index. It can be observed that the performance of the proposed system is more sensitive to the CO₂ liquefaction temperature, as well as the isentropic efficiencies of CO₂ compressors (C1 and C2) and turbines (T1 and T2).

5.4 Multi-objective optimization results

Through the optimization of the PC-CCES system with TEE and c_{\text{P,tot}} as objectives, the Pareto frontier point clusters are obtained and shown in Fig.15.

After optimization, the non-dominated solution set for TEE ranges from 99.17% to 113.22%, while that for c_{\text{P,tot}} falls between 21.47 and 35.78 $/GJ. Compared to the results under the simulation conditions (79.21% and 25.61 $/GJ), the optimized TEE can be improved by 34.01%, while the optimized c_{\text{P,tot}} can be reduced by 4.14 $/GJ. It is important to note that under the simulation conditions, the exhaust temperature of T2 does not satisfy the requirements, which prevents H2 from being used for refrigeration. This is why the optimized TEE is much higher than that of the assumed simulation conditions. Additionally, the TEE of the ideal point is 113.22%, with c_{\text{P,tot}} of 21.47 $/GJ, as shown in Fig.15. However, due to the constraint relationship between TEE and c_{\text{P,tot}}, this ideal point is not realizable.

Five typical optimization results, labeled A-E, are selected based on interval selection, with the corresponding objective values and decision parameters listed in Tab.14. Both TEE and c_{\text{P,tot}} increase from point A to E, indicating that achieving a higher TEE requires a greater cost. Among these points, B, C, and D, strike a balance between TEE and c_{\text{P,tot}}, providing compromises for solving the multi-objective problem. Notably, Point C (111.91% and 28.35 $/GJ) appears to represent the optimal solution in this multi-objective optimization scenario.

In addition, the distributions of decision variables in the final generation of the population are given in Fig.16. As seen in Fig.16(a), the distribution of T₆ is concentrated at the lower boundary of its variation range. Moreover, as shown in Fig.16(b) and 16(c), the optimized efficiencies of the compressors and pump remain at high levels. In contrast, the optimized efficiencies of T1 and T2 show a more dispersed distribution, fluctuating between 85% and 92%. In general, the efficiencies of compressors, pump, and turbines are all above 85%. Based on the previous analysis, it is clear that C3 and P have relatively small impacts on the proposed system. This indicates that the key to optimizing the PC-CCES system lies in improving the performance of the CO₂ compressors and turbines (C1, C2, T1, and T2). Variations in their efficiencies will inevitably lead to deviations from optimum performance.

5.5 System comparison

In this section, the PC-CCES system is compared with three other energy storage systems based on reported data from the literature. The systems used for comparison include two CCHP systems presented in Liu et al. [33], one based on TCES and the other on air-based technology, as well as the liquid CO₂ energy storage system with integrated electricity and cooling (LCES-EC) reported by Sun et al. [34]. In addition, data of point C in Tab.14 of the proposed system is used for comparison. The main comparison results are summarized in Tab.15. Compared to the other systems, the proposed system demonstrates superior electrical efficiency. However, the TCES-based CCHP system in Liu et al. [33], which simultaneously generates cooling and heating at the cost of some power generation, has a higher total energy efficiency than the PC-CCES system. Nevertheless, the proposed system outperforms the other two systems in terms of total energy efficiency. Moreover, the total product unit cost of PC-CCES system is significantly lower than that of LCES-EC system from Sun et al. [34] due to its superior total energy efficiency. In summary, the PC-CCES system designed in this paper effectively achieves the combined supply of power and cooling energy, offering advantages over previously reported systems.

6 Conclusions

A novel CCES system based on gas–liquid phase change with cold-electricity cogeneration is proposed. First, the mathematical models of the system are developed, followed by thermodynamic and exergoeconomic analyses. The effects of seven key parameters are then examined through parametric analysis. Next, multi-objective optimization is conducted with TEE and c_{\text{P,tot}} as the objective functions. Finally, the proposed system is compared with three other reported systems. It is concluded that in terms of thernodynamic analysis, the maximum exergy destruction in the proposed system occurs in T1, reaching 1.0591 MW. C1, C2, and T2 also exhibit relatively high exergy destruction, each exceeding 0.8 MW. In addition, the lowest exergy efficiency is found in PRH2, due to the significant temperature difference during heat exchange.

The exergoeconomic analysis indicate that T1 has the highest \dot{Z}+{\dot{C}}_{\text{D}} (154.85 $/h) and is the most significant component in terms of the economic performance of the system. However, the f of T1 is relatively low (41.80%), indicating that it is imperative to consider appropriately increasing the capital investment cost to improve the performance. Similarly, T2, C1, and C2 show similar results, suggesting that enhancing the investment in these components is also necessary for overall system improvement.

The results of the parametric analysis reveal that the performance of the proposed system is most sensitive to the CO₂ liquefaction temperature and the isentropic efficiencies of CO₂ compressors (C1 and C2) and turbines (T1 and T2). A lower CO₂ condensation temperature benefits both the thermal and economic performance of the system. Moreover, increasing the isentropic efficiencies of the CO₂ compressors and turbines can significantly improve the total energy efficiency. However, excessively high efficiencies lead to a rapid rise in costs, requiring a a careful balance between thermal performance and economic efficiency.

In the muti-objective optimization, the Pareto front curve indicates that a total energy efficiency of 111.91% and a total product unit cost of 28.35 $/GJ could represent favorable operating conditions for the PC-CCES system.

Through comparative analysis, the proposed PC-CCES system efficiently provides both supply of power and cooling energy, offering superior compared to previously reported systems.

In conclusion, the PC-CCES system analyzed in this paper is proven to be feasible and high-performing, providing a promising new direction for the CCES systems. Future research will focus on three key areas: First, the design and optimization of system components, including compressors, turbines, and heat exchangers. Next, studies on system operation, such as off-design analysis and control strategies. Finally, the experimental verification to be developed to gather engineering data, to enable further optimization of the design and operation strategies of the proposed system.

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