Comprehensive assessment and optimization of a hybrid cogeneration system based on compressed air energy storage with high-temperature thermal energy storage

Abstract

Compressed air energy storage (CAES) is an effective technology for mitigating the fluctuations associated with renewable energy sources. In this work, a hybrid cogeneration energy system that integrates CAES with high-temperature thermal energy storage and a supercritical CO₂ Brayton cycle is proposed for enhancing the overall system performance. This proposal emphasizes system cost-effectiveness, eco-friendliness, and adaptability. Comprehensive analyses, including thermodynamic, exergoeconomic, economic, and sensitivity evaluations, are conducted to assess the viability of the system. The findings indicate that, under design conditions, the system achieves an energy storage density, a round-trip efficiency, an exergy efficiency, a unit product cost, and a dynamic payback period of 5.49 kWh/m³, 58.39%, 61.85%, 0.1421 $/kWh, and 4.81 years, respectively. The high-temperature thermal energy storage unit, intercoolers, and aftercooler show potential for optimization due to their suboptimal exergoeconomic performance. Sensitivity evaluation indicates that the operational effectiveness of the system is highly sensitive to the maximum and minimum air storage pressures, the outlet temperature of the high-temperature thermal energy storage unit, and the isentropic efficiencies of both compressors and turbines. Ultimately, the system is optimized for maximum exergy efficiency and minimum dynamic payback period. These findings demonstrate the significant potential of this system and provide valuable insights for its design and optimization.

Graphical abstract

Keywords

compressed air energy storage (CAES) / high-temperature thermal energy storage / supercritical CO₂ Brayton cycle / performance assessment / multi-objective optimization

1 Introduction

The global energy supply is currently heavily dependent on fossil fuels, leading to energy depletion and ecological damage [1]. To promote sustainable development, countries and energy agencies are swiftly transitioning from fossil fuels to renewable sources [2]. In 2022, 83% of new electricity capacity was derived from renewables, predominantly solar and wind, which together accounted for 90% of the latest renewable energy installations worldwide, with growth rates of 22% and 9%, respectively [3]. However, electricity generation from renewable sources such as wind and solar often faces challenges related to intermittency and instability, which can affect the peak regulation of the electricity grid [4,5]. Electrical energy storage (EES) technologies are crucial for orderly energy regulation. Integrating renewables with EES can help stabilize power output on the power grid [6]. Several EES options are available, with pumped hydro storage (PHS) and compressed air energy storage (CAES) being suitable for large-scale storage [7]. While PHS has geographical limitations and high costs [8], CAES offers advantages such as lower costs, flexibility, and a longer lifespan [9]. Therefore, CAES is particularly promising for researchers worldwide.

Traditional CAES (T-CAES) systems operate in two primary phases: energy accumulation and energy release. During the energy accumulation phase, surplus electrical energy is used to compresses air for storage. In the energy release phase, the pressurized air is unleashed, heated, and used to generate power [10,11]. Currently, there are two commercial CAES plants: Huntorf in Germany, which operates at a 42% efficiency, and McIntosh in the United States, with a 54% efficiency. Both plants rely on fossil fuel-fueled combustors, resulting in significant pollutant emissions [12].

In the pursuit of green development, replacing the combustor in T-CAES systems with alternative heating facilities is a promising strategy for reducing emissions and fuel costs. Researchers have explored various auxiliary heating options for CAES systems. For instance, Alirahmi et al. [13] developed an innovative CAES setup that combines solar thermal storage with seawater desalination to increase the turbine inlet temperature. Wu et al. [14] explored a system that integrates thermochemical storage with CAES, using solar energy to reduce Co₃O₄ to CoO for energy storage, while the reaction between CoO and O₂ releases heat to boost the turbine inlet temperature during energy release. Zhang et al. [15] presented an innovative energy storage system that integrates CAES with a solid oxide fuel cell, organic Rankine cycle (ORC), and gas turbine. By replacing the combustor with a solid oxide fuel cell-gas turbine system and coupling the ORC for waste heat recovery, this system achieved impressive efficiencies: a round-trip efficiency (RTE) of 76.07%, an exergy RTE of 61.78%, and a discharging electrical efficiency of 72.39%. Nabat et al. [2] introduced a cutting-edge liquid air energy storage (LAES) system, integrating high-temperature thermal energy storage (HTTES) with ORC. This combination raises the turbine inlet temperature via HTTES and efficiently recycles waste heat and compression heat by the ORC. The results indicated that this system outperforms traditional LAES by 6.59% in RTE (61.13%) and 3.28% in exergy RTE (52.84%). Mousavi et al. [16] explored a CAES system integrated with ORC, utilizing geothermal and solar energies as heat sources, achieving an exergy efficiency (EXE) of 31.17%, under design conditions.

In T-CAES systems, the underutilization of compression heat from compressed air and waste heat from the last-stage turbine reduces overall efficiency. Therefore, repurposing these heat sources presents a promising approach to improve system performance. Many researchers advocate for integrating a bottoming cycle for heat recovery. For example, Bagherzadeh et al. [17] introduced a synergistic system that combines CAES with an ORC, showcasing optimal RTE and energy efficiency using R123 as the organic working medium. Zhao et al. [18] examined a synergistic power system that merges CAES with a Kalina cycle, achieving a RTE of up to 76.07%, an exergetic RTE of 61.78%, and a discharging electrical efficiency of 72.39%. Sadeghi and Ahmadi [19] proposed an advanced energy system integrating a CO₂ ejector refrigeration cycle as the bottoming cycle within a CAES setup for waste heat recovery. He et al. [20] integrated a gas-steam combined cycle into a CAES system to enhance turbine inlet temperature and utilized compression heat, resulting in an EXE of 52.89%. Liu et al. [21] explored a CAES-based trigeneration scheme that encompasses an ORC, coupled with an absorption refrigeration, employing both as bottoming cycles to recuperate residual heat from turbines. Their study showed that this scheme achieves an RTE of 68.38% and a total investment cost of 0.1984 $/kWh.

Achieving a balance between economic and thermodynamic performance is essential for enhancing CAES systems and advancing their commercial viability. The literature extensively discusses a variety of tailored optimization strategies and objectives for CAES systems. For instance, Yao et al. [22] performed an exergoeconomic evaluation and optimization of a CAES system that incorporates services for heating, cooling, and electrical power. They optimized the system using the differential evolution algorithm, targeting cost per unit of output and EXE. Alirahmi et al. [23] conducted economic and exergoeconomic analyses, along with optimization, for a green hydrogen-CAES scheme. They employed four optimization methods (MOPSO, NSGA-II, PESA-II, and SPEA2), aiming to maximize exergy RTE and minimize per-unit production costs. Han et al. [24] performed an economic assessment and optimization for an advanced adiabatic CAES system that integrates power, cooling, and heating. In this study, the MOGWO algorithm was employed to maximize annual profit margins and energy storage density (ESD).

The literature review suggests that the underutilization of compression heat from compressed air and waste, along with the reliance on combustion equipment, results in low energy conversion efficiency and high fuel costs in CAES systems. Consequently, coupling CAES systems with various energy generation techniques has become a common research strategy to enhance their energy conversion capabilities. While these integrated systems offer performance benefits, their complex configurations and numerous components pose considerable challenges that affect investment, operation, maintenance, and commercial viability. Therefore, there is an urgent need for future research to develop and refine coupling strategies, carefully considering factors such as thermodynamic performance, environmental sustainability, cost-effectiveness, and adaptability.

Recently, the supercritical CO₂ Brayton cycle (SCO₂-BC) has gained recognition as a promising waste-heat recovery solution. Its advantages, including compact size, rapid response, improved efficiency, economic benefits, and environmental sustainability, have attracted growing interest from researchers worldwide. An SCO₂-BC system primarily consists of essential components such as compressors, turbines, heaters, coolers, and CO₂ serving as the working fluid [25,26]. However, it is worth noting that the SCO₂-BC may encounter challenges when operated independently, such as higher compressor power consumption, the need for heat sources, and increased system complexity. In addition, the rapid development of industrial technologies has led to the adoption of HTTES systems, which utilizes cost-effective materials such as refractory concrete, cement, or stone for solid storage. This technology has been employed by numerous researchers to integrate various renewable energy systems and optimize overall system performance [27,28].

Therefore, in this paper, a hybrid cogeneration energy system based on compressed air energy storage integrated with high-temperature thermal energy storage and a supercritical CO₂ Brayton cycle (HTTES-CAES-SCBC) is proposed, after an analysis of current research and existing technologies. The proposed system enhances specific power production by increasing inlet air temperature via the HTTES unit. The SCO₂-BC recaptures and repurposes waste heat from the last-stage turbine, collectively boosting the ESD and RTE of the system. In addition, by replacing the combustor with an HTTES unit, the system significantly reduces pollutant emissions, thereby enhancing the environmental sustainability of T-CAES systems. Moreover, the use of HTTES with cost-effective solid storage materials, combined with the economically favorable SCO₂-BC, reduces the total investment, operational, and maintenance costs of the integrated system. Furthermore, the system enhances the utilization efficiency of renewable energy by enabling the storage of stable, high-quality electricity through air compressors, as well as fluctuating, low-quality electricity via the HTTES unit using resistance wire, depending on the quality of electricity available.

This paper is contributive primarily because a novel HTTES-CAES-SCBC system is developed, considering thermodynamic performance, eco-friendliness, cost-effectiveness, and strong adaptability. The proposed integrated system is comprehensively assessed through energetic, exergetic, exergoeconomic, economic, and sensitivity analyses. Utilizing the NSGA-II, a multi-objective optimization algorithm, key design parameters are effectively optimized, with EXE and dynamic payback period (DPP) as the objective functions.

2 System description

The configuration layout of the HTTES-CAES-SCBC system is depicted in Fig.1. This design integrates a HTTES unit with a CAES unit, resulting in an elevated air turbine (AT) inlet temperature. Additionally, a SCO₂-BC unit is utilized to recover excess thermal energy discharged by AT₃ in the CAES unit, generating additional electricity amid peak times. Consequently, the proposed system consists of three main unit: the CAES unit, HTTES unit, and the SCO₂-BC unit. This configuration effectively supplies both heating and electricity within a practical framework for energy applications. During periods of low demand, by operating air compressors (ACs) and the HTTES unit, surplus electricity is converted to pressurized gas and solid sensible heat for storage. Conversely, during periods of high demand, the stored pressurized gas is discharged and heated in the HTTES unit to drive the ATs and the SCO₂-BC unit for electricity generation. Moreover, cold water streams capture the excess heat produced by various components of the system, transforming it into useful hot water for daily needs. The basic working principle of this integrated system is divided into charge and discharge processes, each described as follows.

During the charging process, only the CAES and HTTES units are operational. Stable, high-quality electricity is utilized to drive compressor set featuring intercoolers (ICs) to compress ambient air (state 1). Thermal energy from the pressurized airflow is extracted by a water stream (state 23) from the cool vessel (CV) in each intercooler before the air enters the next compression stage, significantly reducing the work required by the ACs. Subsequently, the compressed air (state 10) undergoes cooling in the aftercooler (AF) before being routed to the air storage reservoir (ASR) for storage. The hot water (state 35) generated during the compression phase is collected in a hot vessel (HV) for domestic hot water use. Simultaneously, fluctuating, low-quality electricity from renewable energy sources, which is unsuitable for ACs, is stored in the HTTES unit through thermal transformation via Joule heating. A conceptual design scheme for the HTTES unit is presented in Fig.2. Key components of the HTTES unit include inexpensive refractory concrete materials with high thermal performance and specific heat, airflow channels, alumina pockets surrounding nickel-chromium (NiCr) resistive wires, and ceramic layers around the concrete for preventing heat losses. Further details on its specific internal structure and working principles can be found in Refs. [27,29]. It is important to note that this combined system has a more powerful energy storage capability as the CAES and HTTES units can operate during different temporal segments.

During the discharging process, compressed gas from the ASR is released through a throttle valve (TV) to maintain consistent air pressure at the AT₁ inlet. The compressed air (state 13) is then sequentially preheated through heat exchanger 3 (HX₃) and HX₁, and heated further in the HTTES unit to reach the rated temperature before entering AT₁. The high-temperature and pressure air (state 16) expands within ATs to perform work. However, the expelled gases from AT₃ (state 19) still possesses substantial energy that can be recovered. Therefore, the SCO₂-BC unit is used to recuperate a portion of thermal energy in the expelled gases for achieving cascade utilization of energy. The remaining heat is recovered by HX₁ to warm the gas entering the HTTES unit (state 15) and by HX₅ to provide domestic hot water (state 40). In the SCO₂-BC unit, the heat released by the exhaust gas in the recuperator (REC) is absorbed by the supercritical CO₂ working fluid (state 41), which then expands within the supercritical CO₂ turbine (CT) to produce work. The exhaust gas exiting the CT (state 42) still retains considerable energy. Thus, this heat is captured successively by HX₂ and HX₃ to preheat the supercritical CO₂ working fluid (state 47) supplied to the REC, and the high-pressure air flowing into HX₁ (state 14), respectively. Afterwards, the exhaust gas (state 44) is compressed in the supercritical CO₂ compressor (CC) and pre-heated in HX₂ after being cooled in HX₄. Eventually, it (state 47) returns to the REC to absorb heat and complete the cycle.

3 Mathematical models

In this section, the energetic, exergetic, exergoeconomic, economic, and multi-objective optimization models for the compound scheme are established. The simulation is conducted on the MATLAB platform, utilizing REFPROP 9.1 software to obtain the necessary properties of the working fluid. To facilitate calculations and analysis, several essential and reasonable assumptions regarding the system are made, outlined as follows:

1) Air and CO₂ are treated as real gases.

2) All system units operate in a steady state, except for the ASR, with variations in potential and kinetic energies, as well as exergies, being neglected.

3) Heat losses in all units and connecting tubes are considered negligible, except in the HTTES unit.

4) Pressure losses are considered in the intercoolers, aftercooler, heat exchangers, and HTTES unit, but are deemed negligible in the connecting tubes.

5) Power consumption of the water pump is neglected.

3.1 Energetic model

The energetic model, based on the principle of energy conservation, quantifies the energy transfer and transformation among system components, with specific formulas detailed in Tab.1.

3.2 Exergetic model

The exergetic model is designed to guide the optimization and improvement of both the overall system and its individual components. By incorporating the principles of fuels and products, exergetic analysis evaluates the energy transfer or conversion capabilities of each component, allowing the exergy rate balance for each component to be expressed as [35]

(1)\dot{E}{x}_{\mathrm{F},k}= \dot{E}{x}_{\mathrm{P},k}+ \dot{E}{x}_{\mathrm{D},k},

where \dot{E}{x}_{\mathrm{F},k}, \dot{E}{x}_{\mathrm{P},k}, and \dot{E}{x}_{\mathrm{D},k} denote the exergy rates of supplied fuel, generated product, and the exergy destruction for the kth component, respectively.

The specific physical exergy of working fluids (ex_ph), defined relative to the ambient condition as the reference state, can be written as [35]

(2)e{x}_{\mathrm{p}\mathrm{h}}=h-h_0-T_0(s-s_0),

where h is the specific enthalpy of working fluids, s is the specific entropy of working fluids, and subscript 0 denotes the reference state.

The exergy efficiency of the kth component ({\eta }_{\mathrm{e}\mathrm{x},k}), defined as the ratio of the exergy rate of generated product to that of supplied fuel, can be expressed as [36]

(3)n_{\mathrm{e}\mathrm{x},k}=\frac{\dot{E}{x}_{\mathrm{P},k}}{\dot{E}{x}_{\mathrm{F},k}}.

Utilizing the aforementioned equations, the specific exergy equations for each system component are calculated and detailed in Tab.2.

3.3 Exergoeconomic model

Exergoeconomic analysis combines exergetic and economic evaluations to assess the performance of a component. By considering the conservation of cash flow related to fuels, products, and investment costs, the cost balance equation for the kth component is formulated as [35]

(4){\dot{C}}_{\mathrm{F},k}+{\dot{Z}}_k={\dot{C}}_{\mathrm{P},k},

in which,

(5){\dot{C}}_{\mathrm{F},k}=c_{\mathrm{F},k}\dot{E}{x}_{\mathrm{F},k},

(6){\dot{C}}_{\mathrm{P},k}=c_{\mathrm{P},k}\dot{E}{x}_{\mathrm{P},k},

where {\dot{C}}_{\mathrm{F},k} and {\dot{C}}_{\mathrm{P},k} are the cost rates of fuel and product, respectively; c_{\mathrm{F},k} and c_{\mathrm{P},k} are the cost of fuel and product per exergy unit, respectively; {\dot{Z}}_k represents the overall cost rates for the kth component, which includes initial investment, operation, and maintenance costs, which can be calculated using [36]

(7){\dot{Z}}_k=\frac{Z_k\cdot CRF\cdot \left(1+\phi \right)}{\tau },

where \tau is the annual operating time of the system, \phi is the operation and maintenance factor, and CRF is the capital recovery factor, which can be presented as [35]

(8)CRF=\frac{i_{\mathrm{r}}{\left(1+i_{\mathrm{r}}\right)}^{N_{\mathrm{y}}}}{{\left(1+i_{\mathrm{r}}\right)}^{N_{\mathrm{y}}}-1},

where i_{\mathrm{r}} is the discount rate, N_{\mathrm{y}} is the lifetime of system, and Z_k is the procurement costs of the kth component, as estimated by the cost equations in Tab.3.

The reference year cost values (Z_{\text{ry}}) for these functions are adjusted to the present year (Z_{\mathrm{p}\mathrm{y}}) by application of chemical engineering plant cost index (CEPCI). This adjustment is calculated using [36]

(9)Z_{\mathrm{p}\mathrm{y}}= Z_{\mathrm{r}\mathrm{y}}\times \frac{CEPC{I}_{\mathrm{p}\mathrm{y}}}{CEPC{I}_{\mathrm{r}\mathrm{y}}}.

Tab.4 presents the cost balance equation along with auxiliary equations for each component, derived from the previously discussed cost balance equation. Additionally, the levelized price of electricity for the charging period (c_{\mathrm{e},\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r},1}) can be calculated using [40]

(10)c_{\mathrm{e},\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r},1}= c_{\mathrm{e},\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}CEL F,

(11)CELF= \frac{1+r_i}{i_r-r_i}\left[1-{\left(\frac{1+r_i}{1+i_r}\right)}^{N_y}\right] CRF,

where c_{\mathrm{e},\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}} is the price of electricity for the charging period, CELF is the constant-escalation levelization factor, and r_i is the nominal escalation ratio.

The exergoeconomic factor (f_k) for the kth component is defined as the ratio of the investment cost rate to the sum of the investment cost rate and the exergy destruction cost rate. It can be mathematically expressed as [36]

(12)f_k=\frac{{\dot{Z}}_k}{{\dot{Z}}_k+{\dot{C}}_{\mathrm{D},k}},

in which

(13){\dot{C}}_{\mathrm{D},k}=c_{\mathrm{F},k}\dot{E}{x}_{\mathrm{D},k},

where {\dot{C}}_{\mathrm{D},k} is the cost rate of exergy destruction of the kth component.

The relative cost difference (r_k) for the kth component is defined as the ratio of the change in cost per unit of exergy from the fuel to the product, compared to the base cost per unit of exergy of the fuel, as can be mathematically represented as [36]

(14)r_k=\frac{c_{\mathrm{P},k}-c_{\mathrm{F},k}}{c_{\mathrm{F},k}}.

3.4 Economic model

To comprehensively assess the economies of the proposed system, conducting a life cycle cost analysis following the exergoeconomic study is essential. This step ensures that both indirect costs and partial direct investment costs are considered, providing a complete economic evaluation over the lifetime of the system.

The total fixed investment cost (C_TFI) of the compound scheme can be described as [28]

(15)C_{\mathrm{T}\mathrm{F}\mathrm{I}}=C_{\mathrm{T}\mathrm{D}\mathrm{I}}+ C_{\mathrm{T}\mathrm{I}\mathrm{I}},

where C_{\mathrm{T}\mathrm{D}\mathrm{I}} and C_{\mathrm{T}\mathrm{I}\mathrm{I}} are the total direct and indirect investment costs of the system, respectively. Tab.5 summarizes the assumptions for calculating the direct and indirect investment costs, which contribute to the total investment cost of the compound scheme.

The annual total cost (ATC) of the compound scheme includes the annual electricity purchase costs (A{C}_{\mathrm{e},\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}) and the annual operation and maintenance costs (AC_OM). These can be expressed as [41]

(16)ATC=A C_{\mathrm{e},\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}+A{C}_{\mathrm{O}\mathrm{M}},

(17)A{C}_{\mathrm{e},\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}= c_{\mathrm{e},\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}\cdot \left({\dot{W}}_{\mathrm{A}\mathrm{C}}+{\dot{W}}_{\mathrm{H}\mathrm{T}\mathrm{T}\mathrm{E}\mathrm{S}}\right)\cdot t_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}\cdot N_{\mathrm{d}}\cdot CELF,

(18)A{C}_{\mathrm{O}\mathrm{M}}=\phi \cdot C_{\mathrm{T}\mathrm{F}\mathrm{I}}\cdot CRF,

where N_{\mathrm{d}} is the annual operational days of the system.

The annual total revenue (ATR) of the compound scheme comprises the annual revenue from electricity sales (A{C}_{\mathrm{e},\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}) and hot water sales (A{C}_{\mathrm{h}\mathrm{w}}), expressed as [41]

(19)ATR=A C_{\mathrm{e},\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}+A{C}_{\mathrm{h}\mathrm{w}},

(20)A{C}_{\mathrm{e},\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}=c_{\mathrm{e},\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}\cdot \left({\dot{W}}_{\mathrm{A}\mathrm{T}}+{\dot{W}}_{\mathrm{C}\mathrm{T}}-{\dot{W}}_{\mathrm{C}\mathrm{C}}\right)\cdot t_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}\cdot N_{\mathrm{d}}\cdot CELF,

(21)A{C}_{\mathrm{h}\mathrm{w}}=c_{\mathrm{h}\mathrm{w}}\cdot {\dot{W}}_{\mathrm{h}\mathrm{w}}\cdot t_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}\cdot N_{\mathrm{d}}\cdot CELF,

where c_{\mathrm{e},\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}} and c_{\mathrm{h}\mathrm{w}} represent the electricity price for discharging periods and the price of hot water, respectively.

The annual total profit (ATP) of the compound scheme is calculated as [41]

(22)ATP=A TR-ATC.

The net present value (NPV) represents the present value of the cumulative system earnings over its lifetime. It is calculated by discounting the annual total system profit to its present value using a specified discount rate. The formula for NPV is [42]

(23)NPV=\sum _{i=1}^{N_{\mathrm{y}}}\left[\frac{ATP}{{\left(1+i_{\mathrm{r}}\right)}^t}\right]-C_{\mathrm{T}\mathrm{F}\mathrm{I}}.

The internal rate of return (IRR) of the compound system, defined as the discount rate that results in a NPV of zero with the recovery of C_TFI realized in the final year of its lifetime. This metric indicates the profitability of the system. The IRR is calculated as [41]

(24)NPV=\sum _{i=1}^{N_{\mathrm{y}}}\left(\frac{ATP}{(1+IRR{)}^t}\right)-c_{\mathrm{T}\mathrm{F}\mathrm{I}}= 0.

3.5 Performance indicators

The energy storage density (ESD), round-trip efficiency (RTE), exergy efficiency (EXE), sum unit cost of products (SUCP), and dynamic payback period (DPP) serve as performance indicators for the HTTES-CAES-SCBC system. These are defined as [1,28,32,38,42]

(25)ESD=\frac{\left({\dot{W}}_{\mathrm{A}\mathrm{T}}+{\dot{W}}_{\mathrm{G}\mathrm{T}}-{\dot{W}}_{\mathrm{C}\mathrm{C}}\right)t_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}}{V_{\mathrm{A}\mathrm{S}\mathrm{R}}},

(26)RTE=\frac{\left({\dot{W}}_{\mathrm{A}\mathrm{T}}+{\dot{W}}_{\mathrm{C}\mathrm{T}}-{\dot{W}}_{\mathrm{C}\mathrm{C}}\right)t_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}}{\left({\dot{W}}_{\mathrm{H}\mathrm{T}\mathrm{T}\mathrm{E}\mathrm{S}}+{\dot{W}}_{\mathrm{A}\mathrm{C}}\right)t_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}},

(27)EXE=\frac{\left({\dot{W}}_{\mathrm{A}\mathrm{T}}+{\dot{W}}_{\mathrm{C}\mathrm{T}}-{\dot{W}}_{\mathrm{C}\mathrm{C}}+\dot{E}{x}_{\mathrm{h}\mathrm{w}}\right)t_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}}{\left({\dot{W}}_{\mathrm{A}\mathrm{C}}+{\dot{W}}_{\mathrm{H}\mathrm{T}\mathrm{T}\mathrm{E}\mathrm{S}}\right)t_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}}},

(28)\begin{array}{rl}& SUCP=\\ & \frac{c_{{\mathrm{A}\mathrm{T}}_1}{\dot{W}}_{{\mathrm{A}\mathrm{T}}_1}+c_{{\mathrm{A}\mathrm{T}}_2}{\dot{W}}_{{\mathrm{A}\mathrm{T}}_2}+c_{{\mathrm{A}\mathrm{T}}_3}{\dot{W}}_{{\mathrm{A}\mathrm{T}}_3}+c_{\mathrm{C}\mathrm{T}}{\dot{W}}_{\mathrm{C}\mathrm{T}}+c_{\mathrm{h}\mathrm{w}}\dot{E}{x}_{\mathrm{h}\mathrm{w}}-c_{\mathrm{C}\mathrm{C}}{\dot{W}}_{\mathrm{C}\mathrm{C}}}{{\dot{W}}_{{\mathrm{A}\mathrm{T}}_1}+{\dot{W}}_{{\mathrm{A}\mathrm{T}}_2}+{\dot{W}}_{{\mathrm{A}\mathrm{T}}_3}+{\dot{W}}_{\mathrm{C}\mathrm{T}}+\dot{E}{x}_{\mathrm{h}\mathrm{w}}-{\dot{W}}_{\mathrm{C}\mathrm{C}}},\end{array}

(29)NPV=\sum _{i=1}^{DPP}\left[\frac{ATP}{{\left(1+i_{\mathrm{r}}\right)}^t}\right]-C_{\mathrm{T}\mathrm{F}\mathrm{I}}= 0.

3.6 Multi-objective optimization model

In engineering problem-solving, achieving optimal values across multiple conflicting indicators simultaneously is a complex challenge. The NSGA-II algorithm offers effective compromise solutions with advantages in both computing time and convergence rate. The application of this algorithm includes five key steps: generating an initial population; selecting, intersecting, and mutating primary progeny population; merging the parental and progeny populations; identifying a new parental population; and producing the final offspring population that meets the criteria [31,43].

In engineering, various optimization methods, such as NSGA-II, often yield a Pareto front, necessitating a systematic approach for final solution selection. TOPSIS excels in such cases, providing a comprehensive framework for distinguishing evaluation object disparities. In this study, the TOPSIS is utilized to assist in finding the final solution for the system. The fundamental principle of TOPSIS unfolds in three key steps: identifying the ideal and non-ideal solutions using the cosine method, computing the evaluative distance of each object from these solutions, and assessing solution quality based on relative proximity to the ideal scenario [23,40].

In the concept of relative nearness, a lower value indicates greater proximity to the ideal solution. The evaluation distances from the ith objective to both the ideal and non-ideal solutions are defined as [31]

(30)E{D}_{i+}=\sqrt{\sum _{j=1}^n{\left(f_{ij}-f_j^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}}\right)}^2},

(31)E{D}_{i-}= \sqrt{\sum _{j=1}^n{\left(f_{ij}-f_j^{\text{non-ideal}}\right)}^2} ,

where f_j^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}} and f_j^{\text{non-ideal}} refer to the ideal and non-ideal solutions, respectively. The relative nearness for each evaluated solution is computed using [31]

(32)\{\begin{array}{l}{Y}_i={\displaystyle \frac{E{D}_{i+}}{E{D}_{i-}+E{D}_{i+}}},\\ i_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}=i\in min\left\{Y_i\right\}.\end{array}

4 Results and discussion

4.1 Model validation

To assess the validity of proposed models, each unit in the system is independently verified by comparing it with previously published data in the literature, considering the novelty of the system structure. The design parameters for the verified models are aligned with those mentioned in the literature. The simulation results and the previously reported results are presented in Tab.6 for comparison. As shown, none of the errors exceed 2.43%, demonstrating the validity of the models developed in this study and confirming that the simulation results are accurate and reliable.

4.2 Base case results

The parameter specifications for the proposed system under the base case are provided in Tab.7.

Tab.8 presents the performance parameters of the hybrid system under the base case. During the charge phase, the power consumption of ACs in the CAES unit reaches 478.67 kW, accounting for approximately 33.50% of the overall power consumption in the system. In contrast, the power consumption of HTTES unit reaches 950.04 kW, making up roughly 66.50% of the overall power consumption.

During the discharge phase, the total electrical power output from the entire system reaches 834.23 kW, which corresponds to around 61.31% of the overall power yield. The hot water supply contributes 526.51 kW, making up approximately 38.69% of the overall power yield. The electrical power yield from the ATs within the CAES unit is 814.02 kW, representing about 97.58% of the overall electrical power output. Meanwhile, the net electrical power generated by the SCO₂-BC unit, which is the electrical power generation of the CT (33.89 kW) minus the electrical power consumption of the CC (13.68 kW), is 20.21 kW, which corresponds to approximately 2.42% of the overall electrical power output.

After completing a full load cycle, the RTE, EXE, and ESD of the hybrid system are 58.39%, 61.85%, and 5.49 kWh/m³, accordingly.

In the exergoeconomic assessment, when the price of electricity during the charging period is set at 0.061 $/kWh, the costs per unit power generated by AT₁, AT₂, AT₃, and CT are 0.1347, 0.1369, 0.14, and 0.2362 $/kWh, respectively. The cost per unit exergy for the hot water provided in the HV is 0.1918 $/kWh, while the SUCP of the whole system is 0.1421 $/kWh.

Additionally, upon lifetime evaluation, the cumulative NPV of the system over the entire lifecycle can reach $0.688 million, with total investment costs amounting to $0.468 million. The IRR and the DPP of the system are 23.51% and 4.81 years, respectively.

To better understand the exergy transfers and destructions of each component of the entire system, and to address issues of exergetic dissipation, a detailed Sankey diagram illustrating exergy flow rates is provided in Fig.3. The results of the exergetic analysis associated with each component in the designed system are listed in Tab.9. Fig.4 shows the distribution of the exergy loss rates across the system components, facilitating further analysis.

The results indicate that the HTTES unit exhibits the highest exergy loss rate at 358.63 kW, constituting 65.84% of the overall exergy loss of the entire system. Following this, the ATs has a loss rate of 43.19 kW (7.93%), the ACs 42.78 kW (7.85%), the ICs and AF 40.95 kW (7.52%), the ASR and TV 34.87 kW (6.4%), and others components account for 24.27 kW (4.46%).

Furthermore, the HX₃, HX₄, HTTES unit, and ICs and AF exhibit the lowest exergy efficiencies, ranging from 25.49% to 62.25%. In contrast, HX₂ and HX₅ have efficiencies of 74.64% and 77.62%, respectively, while HX₁ and REC exhibit efficiencies of 89.41% and 89.90%, respectively. Other components have exergy efficiencies of at least 90.17% (see Tab.9).

The analysis reveals that the HTTES unit significantly contributes to the overall exergy losses while demonstrating relatively low exergy efficiency. This inefficiency is largely due to substantial irreversible losses during the conversion of energy from electrical to thermal in the charging phase. The considerable exergy losses and low exergy efficiencies observed in the ICs and AF can be attributed to the large temperature difference between the cold and hot fluids. Meanwhile, the exergy losses in the TV, ACs, and ATs primarily results from pressure differences at the component entry and exit.

Therefore, to improve system performance, it is crucial to consider operating conditions, such as working pressure and temperature approaches of the heat exchangers. In particular, increasing the working temperature of the HTTES unit is deemed as the most effective method to mitigate its significant contribution to exergy destruction within the system, thereby enhancing whole system performance.

Tab.10 presents the exergoeconomic results for each component of the designed system. To aid data analysis, Fig.5 and Fig.6 display the distributions of exergy destruction cost rates and overall cost rates for the components, respectively.

Among all components, the HTTES unit exhibits the highest {\dot{C}}_{\mathrm{D},k}+{\dot{Z}}_k (31.448 $/h), followed by the ASR and TV (3.382 $/h), the AT (average 2.542 $/h), HX₂ (1.827 $/h), and other components (all below 1 $/h) (see Tab.10). Due to the inclusion of Joule heating, the exergy destruction cost of the HTTES unit notably contributes significantly to the entire system at 58.75%, whereas the contributions from the ATs, ICs and AF, ACs, ASR and TV, HXs, and other components are 11.29%, 7.54%, 7%, 6.41%, 6.24%, and 2.77%, respectively (see Fig.5).

Furthermore, the HTTES unit, driven by its elevated working temperature and substantial thermal storage capabilities, accounts for 32.53% of the total cost rate. In descending order, next largest contributions come from the ATs and CT, HXs, ACs and CC, ICs and AF, and other components, at 27.84%, 14.66%, 10.54%, 8.36%, and 6.07%, respectively (see Fig.6).

In Tab.10, the HTTES unit, ICs, and AF are characterized by a low exergoeconomic factor alongside a high relative cost difference. This indicates that the cost rates of exergy losses are the dominant contributors to the overall cost rates for these components. Consequently, optimizing their operating parameters emerges as the primary strategy for enhancing the exergoeconomic effectiveness.

In contrast, HX₄ and HX₅ exhibit higher exergoeconomic factors and relative cost differences, suggesting that investment, maintenance, and operating costs significantly contribute to the overall cost rates of these components. Selecting less expensive components or adjusting design parameters of the components to reduce these costs is the essential for improving exergoeconomic effectiveness.

Meanwhile, the HV demonstrates a high exergoeconomic factor and a low relative cost difference, signifying that its thermodynamic and economic performance is a relatively optimal. Similar conclusions can be drawn from comparing the exergoeconomic factors and {\dot{C}}_{\mathrm{D},k}+{\dot{Z}}_k. Therefore, it is crucial to optimize these components by considering both exergy efficiency and economic costs.

The setting parameters for four different economic scenarios are shown in Tab.11. The fluctuations in DPP and cumulative NPV of the designed system across these scenarios is depicted in Fig.7.

In Scenario A, the DPP and NPV are 4.81 years and $0.688 million, respectively. In Scenario B, the DPP increases to 5.18 years while the NPV increases to $0.622 million. Scenario C shows a significant increase in DPP to 15.83 years, with a corresponding drop in NPV to $0.077 million. For Scenario D, the DPP exceeds the life cycle of the system (25 years), indicating that the proposed system is unlikely to achieve profitability over its entire life cycle.

Using Scenario A as a baseline, the DPP and NPV in Scenario B are extended by 0.37 years and decreased by $0.066 million, respectively, with a decrease of about 24% in c_hw. The DPP and NPV in Scenario C are extended by 11.02 years and reduced by $0.611 million, respectively, with an increase of about 24% in c_e,char. The DPP in Scenario D exceeds 25 years, coinciding with a decrease of about 24% in c_e,dischar.

This comparison indicates that c_e,dischar has the most significant impact on the system profitability, followed by c_e,char. In contrast, c_hw has the least impact.

4.3 Sensitivity analysis

A sensitivity investigation is conducted to examine the impacts of various design parameters on system behavior. These parameters include ambient temperature (T₀), AC isentropic efficiency ({\eta }_{\mathrm{A}\mathrm{C},\mathrm{s}}), AT isentropic efficiency ({\eta }_{\mathrm{A}\mathrm{T},\mathrm{s}}), CC isentropic efficiency ({\eta }_{\mathrm{C}\mathrm{C},\mathrm{s}}), CT isentropic efficiency ({\eta }_{\mathrm{C}\mathrm{T},\mathrm{s}}), effectiveness of the IC, AF, HX₅ ({\epsilon }_{\mathrm{I}\mathrm{C},\mathrm{A}\mathrm{F},\mathrm{H}{\mathrm{X}}_5}), maximum air storage pressure (p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{a}\mathrm{x}}), minimum air storage pressure (p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{i}\mathrm{n}}), outlet temperature of the HTTES unit (T_{\mathrm{H}\mathrm{T}\mathrm{T}\mathrm{E}\mathrm{S},\mathrm{o}\mathrm{u}\mathrm{t}}), minimum temperature approach of the REC (ΔT_REC), minimum temperature approach of the HX₁, HX₂, and HX₃ (\mathrm{\Delta }{T}_{\mathrm{H}{\mathrm{X}}_1,\mathrm{H}{\mathrm{X}}_2,\mathrm{H}{\mathrm{X}}_3}), and maximum working pressure of the SCO₂-BC unit (p_{\mathrm{S}\mathrm{C}{\mathrm{O}}_2-\mathrm{B}\mathrm{C},\mathrm{m}\mathrm{a}\mathrm{x}}).

4.3.1 Effects of design parameters on ESD, RTE and EXE

Fig.8(a) depicts the sensitivity trend graph of the ESD with respect to variations in various parameters. It is observed that the ESD is the most sensitive to p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{a}\mathrm{x}}, exhibiting a significant increase as p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{a}\mathrm{x}} improves. This effect can be attributed to the fact that the higher p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{a}\mathrm{x}} enhances the total air mass within the ASR, extending the system discharge time and thereby increasing the ESD.

The second most sensitive parameter is p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{i}\mathrm{n}}, with the ESD decreasing as p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{i}\mathrm{n}} improves. This decrease occurs despite the higher power generation from the air turbine achieved with increased air pressure at the turbine inlet. An increase in p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{i}\mathrm{n}} reduces the total air mass within the ASR, ultimately diminishing the ESD.

The next critical parameters influencing the ESD are T_{\mathrm{H}\mathrm{T}\mathrm{T}\mathrm{E}\mathrm{S},\mathrm{o}\mathrm{u}\mathrm{t}} and {\eta }_{\mathrm{A}\mathrm{T},\mathrm{s}}. Both parameters exhibit a similar influence on the ESD, specifically that the ESD increases with higher T_{\mathrm{H}\mathrm{T}\mathrm{T}\mathrm{E}\mathrm{S},\mathrm{o}\mathrm{u}\mathrm{t}} and {\eta }_{\mathrm{A}\mathrm{T},\mathrm{s}}. This is because an increase in T_{\mathrm{H}\mathrm{T}\mathrm{T}\mathrm{E}\mathrm{S},\mathrm{o}\mathrm{u}\mathrm{t}} enhances the power yield from air turbines by boosting the air temperature at turbine inlet, while an increase in {\eta }_{\mathrm{A}\mathrm{T},\mathrm{s}} improves the air turbine efficiency.

In contrast, other parameters (T₀, {\eta }_{\mathrm{A}\mathrm{C},\mathrm{s}}, {\eta }_{\mathrm{C}\mathrm{C},\mathrm{s}}, {\eta }_{\mathrm{C}\mathrm{T},\mathrm{s}}, {\epsilon }_{\mathrm{I}\mathrm{C},\mathrm{A}\mathrm{F},\mathrm{H}{\mathrm{X}}_5}, ΔT_REC, \mathrm{\Delta }{T}_{\mathrm{H}{\mathrm{X}}_1,\mathrm{H}{\mathrm{X}}_2,\mathrm{H}{\mathrm{X}}_3}, and p_{\mathrm{S}\mathrm{C}{\mathrm{O}}_2-\mathrm{B}\mathrm{C},\mathrm{m}\mathrm{a}\mathrm{x}}) have a minimal impact on the ESD. As a result, these parameters are considered as secondary factors when optimizing the ESD.

Fig.8(b) displays the sensitivity trend graph of the RTE with respect to the variation of various parameters. It is observed that the RTE is the most sensitive to T_HTTES,out, with a significant increase as T_HTTES,out is raised. This sensitivity can be explained by two main factors. First, an increase in T_HTTES,out enhances the exergy efficiency of the HTTES unit during the conversion of electricity into heat. Second, an improvement in T_HTTES,out causes an elevated air temperature at turbine inlet, which boosts the power yield from air turbines and consequently increases the RTE.

The {\eta }_{\mathrm{A}\mathrm{C},\mathrm{s}} and {\eta }_{\mathrm{A}\mathrm{T},\mathrm{s}} emerge as the second most sensitive regarding the RTE. Both exhibit a similar trend on the RTE: the RTE increases with higher values of {\eta }_{\mathrm{A}\mathrm{C},\mathrm{s}} and {\eta }_{\mathrm{A}\mathrm{T},\mathrm{s}}. Specifically, an increase in {\eta }_{\mathrm{A}\mathrm{C},\mathrm{s}} reduces the power consumption of the air compressors, while an increase in {\eta }_{\mathrm{A}\mathrm{T},\mathrm{s}} boosts the power generation of air turbines.

Subsequently, vital parameters are the {\epsilon }_{\mathrm{I}\mathrm{C},\mathrm{A}\mathrm{F},\mathrm{H}{\mathrm{X}}_5} and p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{a}\mathrm{x}}, which display opposite trends in their influence on the RTE. Specifically, the RTE rises with an increase in {\epsilon }_{\mathrm{I}\mathrm{C},\mathrm{A}\mathrm{F},\mathrm{H}{\mathrm{X}}_5} but decreases with an increase in p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{a}\mathrm{x}}. The former occurs because raising {\epsilon }_{\mathrm{I}\mathrm{C},\mathrm{A}\mathrm{F},\mathrm{H}{\mathrm{X}}_5} lowers the cooled air temperature entering AC₂, AC₃, AC₄, and AC₅, thereby reducing the power consumption of air compressors. Conversely, the latter is attributed to the increased work consumption of air compressors resulting from a higher p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{a}\mathrm{x}}.

In contrast, other parameters (T₀, {\eta }_{\mathrm{C}\mathrm{C},\mathrm{s}}, {\eta }_{\mathrm{C}\mathrm{T},\mathrm{s}}, p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{i}\mathrm{n}}, ΔT_REC, \mathrm{\Delta } T_{\mathrm{H}{\mathrm{X}}_1,\mathrm{H}{\mathrm{X}}_2,\mathrm{H}{\mathrm{X}}_3}, and p_{\mathrm{S}\mathrm{C}{\mathrm{O}}_2-\mathrm{B}\mathrm{C},\mathrm{m}\mathrm{a}\mathrm{x}}) have a minor impact on the RTE. Additionally, the sensitivity trend graph for the EXE, upon varying these parameters, exhibits almost identical results to those of the RTE, as shown in Fig.8(c).

4.3.2 Effects of design parameters on SUCP and DPP

Fig.9(a) illustrates the sensitivity trend graph of the SUCP with respect to the variation of various parameters. It can be observed that the SUCP is most sensitive to T_{\mathrm{H}\mathrm{T}\mathrm{T}\mathrm{E}\mathrm{S},\mathrm{o}\mathrm{u}\mathrm{t}}, {\eta }_{\mathrm{A}\mathrm{C},\mathrm{s}} and {\eta }_{\mathrm{A}\mathrm{T},\mathrm{s}}, followed by parameters such as p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{i}\mathrm{n}}, p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{i}\mathrm{n}}, {\epsilon }_{\mathrm{I}\mathrm{C},\mathrm{A}\mathrm{F},\mathrm{H}{\mathrm{X}}_5}, and T₀. This indicates that optimizing these key parameters can lead to substantial improvements in the SUCP.

In contrast, other parameters, including {\eta }_{\mathrm{C}\mathrm{C},\mathrm{s}}, {\eta }_{\mathrm{C}\mathrm{T},\mathrm{s}}, ΔT_REC, \mathrm{\Delta }{T}_{\mathrm{H}{\mathrm{X}}_1,\mathrm{H}{\mathrm{X}}_2,\mathrm{H}{\mathrm{X}}_3}, and p_{\mathrm{S}\mathrm{C}{\mathrm{O}}_2-\mathrm{B}\mathrm{C},\mathrm{m}\mathrm{a}\mathrm{x}} exert a weaker effect on the SUCP. Therefore, they are considered secondary factors in the economic optimization process.

Fig.9(b) portrays the sensitivity trend graph of the DPP with respect to the variation of various parameters. Observing the graph, the DPP is the most sensitive to T_{\mathrm{H}\mathrm{T}\mathrm{T}\mathrm{E}\mathrm{S},\mathrm{o}\mathrm{u}\mathrm{t}} and p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{a}\mathrm{x}}, followed by p_{\mathrm{A}\mathrm{S}\mathrm{R},\mathrm{m}\mathrm{i}\mathrm{n}}, {\eta }_{\mathrm{A}\mathrm{T},\mathrm{s}}, and {\eta }_{\mathrm{A}\mathrm{C},\mathrm{s}}. Similar to the SUCP analysis, other parameters, including T₀, {\eta }_{\mathrm{C}\mathrm{C},\mathrm{s}}, {\eta }_{\mathrm{C}\mathrm{T},\mathrm{s}}, {\epsilon }_{\mathrm{I}\mathrm{C},\mathrm{A}\mathrm{F},\mathrm{H}{\mathrm{X}}_5}, \mathrm{\Delta }{T}_{\mathrm{R}\mathrm{E}\mathrm{C}}, \mathrm{\Delta } T_{\mathrm{H}{\mathrm{X}}_1,\mathrm{H}{\mathrm{X}}_2,\mathrm{H}{\mathrm{X}}_3}, and p_{\mathrm{S}\mathrm{C}{\mathrm{O}}_2-\mathrm{B}\mathrm{C},\mathrm{m}\mathrm{a}\mathrm{x}}, have a minor influence on the DPP and are also deemed as secondary factors.

Overall, these findings underscore the importance of focusing on a specific set of parameters that significantly affect both the SUCP and DPP. By targeting these critical areas, improvements can be achieved more effectively, ultimately leading to enhanced economic performance of the system.

4.4 Multi-objective optimization

4.4.1 Optimization settings and determination of final solution

The overall system performance, as evaluated through thermodynamic, exergoeconomic, economic, and sensitivity analyses, is strongly influenced by design parameters. Optimizing these parameters is crucial for successful commercial application. Tab.12 outlines the optimization range of decision variables. Both the population dimensions and the evolutionary cycles are adjusted to 100. In this study, EXE and DPP are selected as the objective functions, defined as

(33)F=[max(EXE),min (DPP ){]}^{\mathrm{T}}.

Fig.10 illustrates the Pareto optimal distribution for the EXE and DPP of the designed system, utilizing the NSGA-II algorithm. At point A, the DPP reaches an optimal value of 2.91 years, while point B represents the optimal EXE (64.36%). As the DPP rises with an increasing EXE, achieving optimal values for both simultaneously is not feasible. To identify the best compromise solution, the TOPSIS method is employed. As a result, the final solution at point P, with an EXE of 63.84% and a DPP of 4.70 years, is determined, as shown in Fig.10.

4.4.2 Analysis of optimization results

Tab.13 presents the optimization results corresponding to point P (in Fig.10) for the designed system. Through optimization, the charging/discharging time reaches 11.25 h. The system achieves an ESD, RTE, and EXE of 9.97 kWh/m³, 60.35%, and 63.84%, respectively.

The power consumption of the ACs, HTTES unit, and CC amounts to 477.56, 991.36, and 13.69 kW, respectively. Meanwhile, the power yield from ATs and CT, as well as the power of supplied as hot water, stands at 863.75, 36.49, and 545.88 kW, respectively.

Additionally, the costs per unit power generated by AT₁, AT₂, AT₃, and CT are determined to be 0.1335, 0.1353, 0.1378, and 0.2227 $/kWh, respectively. The cost per unit exergy of hot water supply is 0.1630 $/kWh and the SUCP is 0.1388 $/kWh. The C_TFI, NPV, IRR and DPP are estimated at $0.886 million, $1.347 million, 23.91%, and 4.70 years, respectively.

4.5 Comparison with the literature

Lastly, Tab.14 provides a comparative assessment of performance metrics for the designed system versus other air storage-based systems as detailed in recent literature. The results show the superior performance of the current HTTES-CAES-SCBC system. This improvement is attributed to two main factors: first, the combination of CAES and HTTES, which increases the inlet air temperature for the air turbines; and second, the integration of CAES with SCO₂-BC, allowing for the recovery of thermal energy from the exhaust gas of AT₃ and its conversion into electrical energy.

5 Conclusions

A hybrid cogeneration energy system integrating CAES, HTTES, and SCO₂-BC technologies is introduced in this paper, whose performance is assessed through energetic, exergetic, thermoeconomic, economic, and sensitivity analyses, with an emphasis on maximizing efficiency by optimizing key design parameters. Below is a concise summary of the key findings:

(1) Based on the thermodynamic analysis, the ESD, RTE, and EXE in the base state are calculated to be 5.49 kWh/m³, 58.39%, and 61.85%, respectively. Notably, the HTTES unit exhibits the highest exergy loss rates. Regarding exergy efficiency, the worst performers are HX₃, HX₄, the HTTES unit, and ICs and AF. Therefore, when optimizing the proposed system, it is crucial to apply careful consideration and in-depth analysis to these components.

(2) The exergoeconomic analysis reveals that, in the base state, the costs per unit power generated by AT₁, AT₂, AT₃, and CT are 0.1347, 0.1369, 0.14, and 0.2362 $/kWh, respectively. The cost per unit exergy of hot water from the HV amounts to 0.1918 $/kWh, and the SUCP for the entire system stands at 0.1421 $/kWh. The costs related to exergy losses significantly impact the overall costs of the HTTES unit, ICs, and AF, while investment, maintenance, and operational costs have a significant impact on the total costs of HX₄ and HX₅. Additionally, the economic analysis reveals that in the base case, the C_TFI, NPV, IRR and DPP are $0.468 million, $0.688 million, 23.51% and 4.81 years, respectively.

(3) The parametric sensitivity study yields significant insights. The ESD is the most sensitive to variations in both the maximum and minimum air storage pressures. In contrast, the RTE and EXE, and SCUP show the highest sensitivity to changes in the outlet temperature of the HTTES unit, as well as to the isentropic efficiency of both AC and AT. Moreover, the DPP displays considerable sensitivity to a range of factors, including the outlet temperature of the HTTES unit, the isentropic efficiency of both AC and AT, and both the maximum and minimum air storage pressures.

(4) After optimization utilizing the NSGA-II, the ESD, RTE, EXE, NPV, and IRR increase by 4.48 kWh/m³, 1.96%, 1.99%, $ 0.659 million, and 0.4%, respectively, compared with their values before optimization. Meanwhile, the SUCP and DPP decrease by 0.0033 $/kWh and 0.11 years, respectively. Additionally, the superior performance of the hybrid system is demonstrated through a comparative assessment with other air storage-based systems.

The findings from this research can significantly assist decision-makers in selecting appropriate components for optimal arrangement in CAES systems and other energy systems. Moreover, the results offer substantial guidance for energy system optimization. Further studies will also consider the environmental impact of operating the system to further enhance the comprehensive evaluation of the system.

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