Thermo-economic optimization of supercritical CO 2 Brayton cycle on the design point for application in solar power tower system

. The supercritical CO 2 Brayton cycle integrated with a solar power tower system has the advantages of high efficiency, compact cycle structure, strong scalability, and great power generation potential, which can positively deal with the energy crisis and global warming. The selection and optimization of design points are very important for actual operating situations. In this paper, the thermodynamic and economic models of the 10 MWe supercritical CO 2 Brayton cycle for application in solar power tower system are established. Multi-objective optimizations of the simple recuperative cycle, reheating cycle, and recompression cycle at different compressor inlet temperature are completed. The thermal efficiency and the levelized energy cost are selected as the fitness functions. The ranges of the optimal compressor inlet pressure and reheating pressure on the Pareto frontier are analyzed. Finally, multi-objective optimizations and analysis of the supercritical CO 2 Brayton cycle at different ambient temperature are carried out. This paper investigates the influence of the compressor inlet temperature and ambient temperature on the thermal efficiency and economic performance of the supercritical CO 2 Brayton cycle.


Introduction
The application of renewable energy is an important method to solve the energy crisis, deal with climate warming, and achieve sustainable development. Concentrated solar power (CSP) technology has got more and more attention. Among the common CSP technologies, the solar power tower (SPT) technology can offer high efficiency and promising scaling [1]. However, compared with the traditional coal-fired power generation plants, CSP still has no economic competitiveness [2]. Therefore, increasing the efficiency and reducing the cost for CSP plants are the future development directions [3].
Supercritical carbon dioxide (S-CO2) Brayton cycles are applied in the fields of nuclear power, coal-fired power, waste-heat recovery, CSP and so on. It is a promising option that S-CO2 Brayton cycles are integrated into the SPT system because of superior thermodynamic performance and compact layout [4]. The advantages of S-CO2 Brayton cycles in the CSP system are as follows [5]: less compression work is required; compact layout structure due to small turbomachinery and heat exchangers; droughty areas can be selected as application sites by dry cooling; less corrosiveness, fewer requirements for materials.
The thermodynamic performance of the S-CO2 Brayton cycle has received more and more attention. Turchi et al. [6] compared the thermodynamic performance of different forms of the S-CO2 Brayton cycle. Wang et al. [1] studied the influence of key parameters on thermal efficiency, specific work, and incorporation ability. He also compared different S-CO2 Brayton cycle layouts integrated into the SPT system based on multi-objective optimization, in which overall efficiency and specific work are selected performance criteria [7]. For the recompression S-CO2 Brayton cycle, Wang et al. [2] investigated the effects of key parameters on the overall SPT system and cycle exergy efficiency. Yang et al. [8,9] analysed and compared part-load and off-design thermodynamic performance of typical S-CO2 Brayton cycles. Many scholars have also studied the economic performance of the S-CO2 Brayton cycle. Neises et al. [10] investigated the minimize levelized cost of energy of different configurations of S-CO2 Brayton cycles. Ma et al. [11] utilized an exergoeconomic approach to minimize the total unit exergy cost for optimal integration of main compression intercooling S-CO2 Brayton cycles with or without reheat in the SPT system and established sensitivity analysis model. Therefore, it is essential to investigate the thermo-economic performance of the S-CO2 Brayton cycle in the future.
In the S-CO2 Brayton cycle integrated into the SPT system, the study of design points is still significant because the selection and optimization of design conditions are the guidance and basis for off-design or dynamic operation. In many studies, the compressor inlet temperature is set to a certain value [1,2,[7][8][9]. The influence of the ambient temperature of design points is also not negligible. However, few studies on the effects of compressor inlet temperature or ambient temperature on the multi-objective optimization comprehensively consider thermodynamic and economic performance. This paper establishes thermodynamic and economic models for the 10 MWe S-CO2 Brayton cycle applied to the solar power tower system. The thermodynamic and economic performance are selected as fitness functions. The parameters analysis of compressor inlet temperature and ambient temperature and multi-object optimization for the simple recuperative cycle, reheating cycle, and recompression cycle are investigated. The influence of variable compressor inlet temperature and ambient temperature on the Pareto frontiers are studied.

Configurations
This paper selects three layouts of the S-CO2 Brayton cycle. Among the common layouts, the simple recuperative cycle is the basic form. The reheating cycle can increase the power generation of the turbine. The recompression cycle is beneficial to heat recovery. The T-s diagrams are shown in Fig.1, Fig.2 and Fig.3.
The S-CO2 Brayton cycle components generally include compressors, turbines, recuperators, heaters, and air-coolers. In the spilt-shaft configuration, compressors are driven by a motor, and turbines connect a generator. In the simple recuperative cycle, from state 1 to 2, the S-CO2 is compressed to a high-pressure state. Then it is heated in the recuperator and the heater. From state 4 to 5, the S-CO2 expands in the turbine. In the recuperator, the hot stream transfers heat to the cold stream. Finally, the S-CO2 is cooled in the air-cooler to state 1.
The reheating cycle and recompression cycle process are similar to the simple recuperative cycle. Comparing to the simple recuperative cycle, the reheating cycle has an extra turbine and an additional reheating process in the re-heater. Comparing to the reheating cycle, the recompression cycle has an extra re-compressor.

Calculation models 3.1 Thermodynamic models
The power output of the S-CO2 Brayton cycle is 10 MWe in this work. The turbomachinery inlet and outlet enthalpy (h) values are determined based on the known temperature and pressure in the calculation process. The specific work consumed by the compressor and generated by the turbine can be indicated as follows: The specific work of the S-CO2 Brayton cycle is calculated as follows: The energy absorbed in the heater or re-heater can be calculated as follows: The thermal efficiency of the S-CO2 Brayton cycle is defined as follows:

Economic models
The S-CO2 Brayton cycle components cost models are proposed [12]. The compressor and turbine cost ($) models related to consuming or generating work (kW) are as follows: The levelized energy cost (LEC) represents the economic performance of the cycle in this paper. Annal operation and maintenance cost (COM) is 1.65 % of the total cost, and interest (r) is 5 % [13]. The operation time (top) is 7500 h/year and the lifetime (L) is 15 years.

Performance analysis
In this part, the influence of compressor inlet temperature and ambient temperature on the thermal efficiency and economic performance are analysed. Fig.4 shows thermal efficiency and economic performance change with the compressor inlet temperature of the simple recuperative cycle. As the compressor inlet temperature increases, the thermal efficiency decreases, and the LEC increases. Fig.5 shows the LEC of three cycle configurations changes with the ambient temperature. As the ambient temperature of the design condition increases, the air-cooler conductance and LEC increase, and the thermal efficiency does not change.

Multi-objective optimization
Genetic algorithm (GA) is an optimization method that mimics the natural selection of biological evolution [14]. NSGA-II is used in this paper for multi-objective optimization. The thermal efficiency and the LEC are selected as fitness functions. The main input parameters are shown in Table 1 Fig.6 shows the simple recuperative cycle, reheating cycle, and recompression cycle Pareto frontiers in different compressor inlet temperature. As we can see, the recompression cycle has the highest thermal efficiency and LEC, followed by the reheating cycle and the simple recuperative cycle. The lower the compressor inlet temperature, the higher the thermal efficiency and the lower LEC. On the Pareto frontiers, the compressor inlet temperature drops by 5 ℃, the thermal efficiency increases by 2 %, and the LEC decrease by 0.00176 $/kW . h on average when another fitness function remains the same.  Compressor isentropic efficiency (%) 89 [1] Turbine isentropic efficiency (%) 93 [1] Motor efficiency (%) 90 Generator efficiency (%) 97  Table 2 shows the highest thermal efficiency that can be achieved in a certain compressor inlet temperature and corresponding LEC. When the compressor inlet temperature increases by 5 ℃, the maximum achievable cycle thermal efficiency reduces by 1 %, and the LEC increases by an average of 0.001 $/kW . h in the same cycle layout. Fig.7 shows the minimum cycle pressure and reheating pressure values on Pareto frontiers points in different compressor inlet temperature. When the design compressor inlet temperature increases, the optimal minimum cycle pressure also increases (Table 3), and the optimal reheating pressure increases in the 14 MPa -18 MPa, and then it increases rapidly.

Conclusions
This paper studies the multi-objective optimization of the simple recuperative cycle, reheating cycle, and recompression cycle in different compressor inlet temperature and ambient temperature. The thermal efficiency and economic performance are the fitness functions. The main conclusions are listed as follows: (1) The lower the compressor inlet temperature, the higher the thermal efficiency, and the lower the LEC. The lower the ambient temperature of design conditions, the lower the LEC. The ambient temperature does not affect the cycle thermal efficiency. (2) On the Pareto frontiers of thermal efficiency and LEC, the compressor inlet temperature drops by 5 ℃, the thermal efficiency increases by 2 %, and the LEC decreases by 0.00176 $/kW . h on average when another fitness function remains the same. As the design compressor inlet temperature increases, the minimum cycle pressure of optimal points increases, and the reheating pressure increases in a certain range. (3) On the Pareto frontiers of thermal efficiency and economic performance with the different ambient temperature, the S-CO2 Brayton cycle LEC increases as the ambient temperature increases.
This work was supported by Beijing Natural Science Foundation (Grant No. 3202014).