Cutting-Edge Research: Artificial Intelligence Applications and Control Optimization in Advanced CO₂ Cycles

Abstract

In recent years, advanced CO₂ cycles, including supercritical CO₂ power cycles, transcritical CO₂ power cycles and refrigeration cycles, have demonstrated significant potential for application across a broad spectrum of energy conversion processes, owing to their high efficiency and compact components that are environmentally benign and non-polluting. This study presents a comprehensive review of the dynamic performance and control strategies of these advanced CO₂ cycles. It details the selection of system configurations and various control strategies, detailing the principles behind different control strategies, their applicable scopes, and their respective advantages. Furthermore, this study conducts a comparison between the joint control strategy and single control strategies for CO₂ cycles, demonstrating the superiority of the joint control strategy in CO₂ cycles. It then delves into the potential of novel control technologies for CO₂ cycles, using model-based control technology powered by artificial intelligence as a case study. This study also offers an extensive overview of control theory, methodology, scope of application, and the pros and cons of various control strategies, with examples including extreme value-seeking control, model predictive control (MPC) based on an artificial neural network model, and MPC based on particle swarm optimization. Finally, it explores the application of AI-controlled CO₂ cycles in new energy vehicles, solar power generation, aerospace, and other fields. It also provides an outlook on the development direction of CO₂ cycle control strategies in light of the evolving trends in the energy sector and advancements in AI methodologies.

1. Introduction

The widespread use of air-conditioning and heat pump technologies has greatly enhanced quality of life, yet it has also led to energy—consumption and environmental issues, such as high electricity use and the greenhouse effect and ozone—layer depletion from refrigerant leakage. To address these challenges, researchers are committed to developing energy-efficient and environmentally friendly green refrigeration technologies and new refrigerants [1]. Carbon dioxide (CO₂), as a natural refrigerant, exhibits the advantages of being non-toxic, non-flammable, low critical temperature, and widely available with an ozone depletion potential (ODP) of 0 and a global warming potential (GWP) of only 1. Therefore, CO₂-based recycling systems have been widely developed and applied, which is of great significance in promoting energy conservation and emission reduction and realizing sustainable development of society [2].

The supercritical carbon dioxide (sCO₂) Brayton cycle has potential applications in a wide range of energy conversion applications. It is widely recognized that the sCO₂ Brayton cycle has the potential to be compact, flexible, and efficient in design, and the working fluid of CO₂ is clean and non-polluting. In the supercritical cycle (Figure 1a), the working fluid operates entirely above the critical pressure, while in the transcritical cycle (Figure 1b), the working fluid passes through both subcritical and supercritical conditions [3]. Compared to the organic Rankine cycle (ORC) and the steam Rankine cycle (SRC), CO₂-based power cycles offer several significant advantages:

(a)

Compatibility with a wider temperature range (up to 1000 °C);

(b)

Reduced irreversible losses in heat recovery exchangers due to improved temperature matching between waste heat and the working fluid [4];

(c)

Enhanced system performance even at temperatures below 400 °C [5];

(d)

Non-toxic, non-flammable, and non-corrosive characteristics;

(e)

Low global warming potential (GWP);

(f)

More compact system compared to SRC [6].

Based on the excellent heat transfer characteristics of CO₂ in the supercritical and subcritical regions, Professor Gustav Lorentzen [7] first proposed the transcritical CO₂ cycle theory in 1993. This cycle model has received widespread attention for its environmental advantages and energy-efficient features. The transcritical CO₂ cycle system operates at a large pressure difference and can recover the energy loss in the expansion process through the expansion work recovery device, which improves the system efficiency.

However, due to the highly non-linear variation of the sCO₂ fluid near the critical zone with various forms of external perturbations, there are multiple challenges in the operation of CO₂ cycles such as the sCO₂ Brayton cycle and the CO₂ transcritical cycle. Among them, the supercritical CO₂ cycle, due to its continuous operation in the supercritical region near the critical point, is highly sensitive to the nonlinear variations in the thermophysical properties of the working fluid. It is particularly susceptible to dynamic fluctuations in key processes such as compression and heat transfer [8]. In contrast, the transcritical cycle mitigates the impact of property abrupt changes through the phase—change process and is more robust as some regions are far from the critical point. Therefore, compared to the transcritical CO₂ cycle, the design and control of the supercritical CO₂ cycle require more precise avoidance of the unstable regions near the critical point to ensure efficient and stable operation [9]. As a result, most researchers focus mainly on the advanced system design of the supercritical CO₂ cycle [10]. Many researchers have investigated the off-design performance of the sCO2 Brayton cycle. The cycle is operated under off-design conditions due to changes in turbine inlet parameters caused by variations in heat source temperature and mass flow rate, and changes in compressor inlet parameters caused by variations in ambient conditions [11,12]. It has been found that changes in ambient heat sink temperature have a greater effect on compressor inlet temperature, and hence cycle performance, than changes in cycle performance caused by changes in heat source temperature and mass flow rate. These studies have helped to understand the effect of changes in factors such as heat source and heat sink conditions on the off-design performance of the CO₂ cycle [13]. To address this issue, researchers have carried out system optimization and dynamic analysis of the sCO₂ Brayton cycle and transcritical refrigeration CO₂ cycle, etc., to optimize the dynamic performance and control characteristics of these systems. This paper focuses on different system selection methods, control strategies, and control technologies for CO₂ cycles and provides a comparative analysis to provide a comprehensive view of the design, optimization, and control of advanced CO₂ cycles, as well as the potential and challenges of these systems in modern energy conversion applications.

2. Advanced System Selection and Dynamic Performance Studies

The layout of a CO₂ power cycle has an extremely important impact on its performance, and the advanced system selection is crucial for the CO₂ power cycle, as it can significantly enhance the efficiency, compactness, and flexibility of the system [14]. In the case of the sCO₂ Brayton cycle, there are three main ways to improve the performance of the system: heat recovery, reduction of the compression work, and increase of the expansion work. The corresponding typical and most efficient layouts are the recompression cycle, the intercooling cycle, and the reheating cycle, respectively [15]. Based on their common characteristics (e.g., intercooling, recompression, and reheating) with the performance of each layout, Crespi [16] et al. found that the recompression cycle achieves the best results in terms of balancing the thermal efficiency and the system complexity. In the study of Bian [14], scholars concluded that the most typical and promising layouts are the recompression, intercooling, and reheating cycles after studying and analyzing sCO2 Brayton cycle with different layouts, and based on this, the dynamic load performance of cycles with different layouts was compared for partial loads, and by building a dynamic model that can reflect the performance of the actual partial loads, and based on the performance The optimal layout for different operating scenarios is determined based on this performance.

2.1. Simple Recuperative Cycle

The supercritical simple recuperative cycle, an optimized Brayton cycle, is designed to overcome limitations of the conventional Brayton cycle, such as excessive compression work and oversized heat transfer areas caused by high specific volume. By exploiting the thermophysical properties of carbon dioxide in the supercritical region, the cycle significantly reduces the compression work and results in a more compact system with less sensitivity to pressure drop. The structure of the simple recuperative cycle and the corresponding T-s diagram are shown in Figure 2. The sCO₂ is first compressed to a high-pressure state by a compressor, and then heated to a high temperature and a high-pressure state by an accumulator and a heater, respectively. The sCO₂ expands at a high temperature and pressure, performs work within the turbine, and then releases the heat back to the initial state through an accumulator and a cooler. A transcritical refrigeration CO₂ cycle was simultaneously derived from the supercritical simple regeneration cycle, which presents the same conceptual structure as the supercritical simple regeneration, as repeatedly presented in [17,18], and which is suitable for low-temperature waste heat recovery applications.

2.2. Recompression Cycle

The structure of the recompression cycle and the corresponding T-s diagram are shown in Figure 3. Firstly, the sCO₂ heated by the heater expands and does work in the turbine, and then flows through the high-temperature return heaters (HTR) and low-temperature return heaters (LTR) sequentially to dissipate the heat, after which the sCO₂ is divided into two branches: one flows through the cooler, and then it is compressed by the main compressor; then the sCO₂ enters the LTR to absorb the heat. The second branch of sCO₂ goes directly through the compressor and is compressed to high pressure. The two branches then converge and enter the HTR and heater, in turn absorbing heat. The cycle is thus completed. This layout reduces the problem of pinch points at the compressor, which can result in no heat transfer between the hot and cold streams, secondly, the heat load on the cooler is also reduced in this layout, and the size of the unit is therefore reduced. Due to the above advantages of the recompression cycle, the supercritical recompression layout is the most widely studied cycle in the literature [19].

Parameter optimization and dynamic control strategies for simple recuperator cycles, as well as recompression cycles, have been well studied to achieve higher thermal efficiencies under different operating conditions. To make the cycles achieve higher thermal efficiency, several new cycle layouts and applications under design conditions have been studied, these methods include increasing the turbine power by reheating, increasing the heat recovery of the return heaters by pre-compression, and reducing the compressor power consumption by intermediate or partial cooling [20], the new layouts, while increasing the complexity of the system, have also led to a further improvement.

2.3. Recompression Intercooling Cycle

The structure of the recompression intercooling cycle and the corresponding T-s diagram are shown in Figure 4. The recompression intercooling cycle is developed based on the recompression cycle. Unlike the recompression cycle, in the recompression intercooling cycle, the sCO₂ pressurized by the low-pressure main compressor enters the intercooler, which is cooled again and then compressed by the high-pressure main compressor and then enters the LTR. Under the design conditions, the efficiency of the recirculation cycle with intercooling is greatly improved compared with the cycle without the intercooler. In a study by [13], scholars optimized the parameters of four sCO₂ Brayton cycles and compared the thermal efficiencies at different part loads. The results of the study found that the intercooling cycle showed the greatest efficiency improvement over the recompression cycle under the design conditions, however, the intercooling cycle showed the worst response to the requirement for wide-range load regulation when the system load was reduced from 100 percent to 30 percent, with a reduction in the relative efficiency of up to 28.7 percent. The results indicate that the use of intercooling has a large negative impact on the off-design condition performance of the system.

2.4. Recompression Reheat Cycle

The recompression reheat cycle is a modification of the recompression cycle and its structure and corresponding T-s diagram are shown in Figure 5. Compared to the recompression cycle, the sCO₂ heated by the HTR and the heater in this cycle first expands to do work in the high pressure (HP) turbine and is then heated by the reheater. The sCO₂ then flows into the low-pressure (LP) turbine to do work. Compared to the recompression cycle, the introduction of recompression reheating results in a double improvement of the cycle performance: the expansion work is increased and the thermal stresses caused by the high pressure and temperature at the turbine inlet are greatly reduced [19].

Turchi [21] conducted a similar thermodynamic analysis of the central receiver and proposed the adoption of a reheating configuration to significantly enhance the system’s thermal efficiency. In his study, various cycle configurations were simulated, including the simple Brayton cycle, recompression cycle, partial cooling Brayton cycle, and recompression with intercooling in the main compression cycle. The results indicated that the reheating configuration has a significant advantage in improving thermal efficiency, with the cycle efficiency reaching 43.80%. In contrast, the efficiency of the recompression cycle is comparable to that of other configurations (such as the partial cooling Brayton cycle and the recompression with main compression intercooling), ranging from 36% to 40%. The recompression and reheating cycle hold considerable potential for enhancing thermal efficiency. Based on the thermal efficiency analysis of the recompression and reheating cycle, some scholars have conducted further exergy analysis. Since exergy analysis provides a detailed thermodynamic assessment of each cycle component, it is of great significance in the system optimization process. This method can deeply reveal the thermodynamic characteristics of each key component, identify the root causes and specific locations of thermodynamic losses, and provide strong support for determining the optimal operating conditions, thereby maximizing the system’s output power [22]. Through this process, exergy analysis not only promotes a precise understanding of system performance but also provides a scientific basis for optimizing design and operational strategies. In the study referenced in the literature [19], scholars conducted energy and exergy analyses of the sCO₂ recompression Brayton cycle, calculating the first law and exergy efficiencies with and without reheating. The results showed that incorporating a recompression and reheating cycle in the sCO₂ recompression Brayton cycle significantly improved both the thermal and exergy efficiencies, with the exergy efficiency increasing in a parabolic manner and reaching its highest value of 35.1% at a reheating temperature of 600 °C.

After analyzing various system configurations, the following conclusions were drawn. In terms of thermal efficiency, the simple recuperative cycle, although structurally simple, exhibits relatively low thermal efficiency, which can be somewhat improved through optimization. The recompression cycle, by mitigating pinch point issues, significantly enhances thermal efficiency and has become a widely chosen option in research. The recompression intercooling cycle shows a notable increase in efficiency under design conditions, but its efficiency declines significantly under off-design conditions. The recompression reheat cycle, by further increasing the work of expansion and thermal efficiency through reheat, achieves the highest level, demonstrating its significant advantage in thermal efficiency. Regarding adaptability to off-design conditions, the simple recuperative cycle exhibits strong adaptability under off-design conditions and can effectively cope with changes in operating conditions. The recompression cycle experiences a significant decrease in efficiency under partial load conditions, resulting in poor adaptability. The recompression intercooling cycle has the poorest adaptability under off-design conditions, with the most pronounced decrease in efficiency. In contrast, as an advanced CO₂ cycle, the recompression reheat cycle demonstrates good adaptability under off-design conditions and can effectively handle changes in operating conditions.

3. Control Strategies Applied to the CO₂ Cycle and Comparative Analyses

Control strategies are essential for CO₂ power cycles, as they ensure stable and safe system operation across various conditions. This involves precisely regulating key parameters like pressure and temperature to prevent equipment damage from issues such as overpressure, overheating, liquid hammer, and cavitation [23]. The operation and control of sCO₂ Brayton cycles are challenged by the highly nonlinear behavior of supercritical CO₂ fluids, various types and degrees of disturbances, and complex structure and operational variables. Moreover, these strategies can optimize the cycle process by adjusting the operating conditions during stages such as condensation and expansion, thereby reducing thermal and frictional losses and significantly enhancing energy conversion efficiency. Additionally, control strategies must adapt to load fluctuations and complex operating conditions to ensure stable operation under extreme weather, equipment failures, and other challenging circumstances [24]. More importantly, by integrating advanced control algorithms, sensor technology, and remote monitoring and fault diagnosis systems, these strategies can promote the development of CO₂ power cycles towards intelligent and automated directions. This not only improves operational efficiency but also reduces the risk of manual intervention, decreases downtime and maintenance costs, and thus comprehensively enhances the overall performance, reliability, and economic viability of the system.

3.1. Basic State Parameter Control Method

The CO₂ cycles are often used under variable heat source conditions, employing control strategies to ensure the stable and safe operation of the cycle. The main control parameters include the inlet temperature and pressure of the main compressor, as well as the inlet temperature and pressure of the turbine. They play a crucial role in preventing hot leg alarms or the cold leg from entering the subcritical region [25]. For each parameter, there are different control methods with varying manipulated variables, such as different valves, turbine mechanical speeds, and heat sources [26]. The detailed analysis of different control parameters is as follows.

Compressor inlet state parameters:

The state of CO₂ at the inlet of the main compressor is typically near the critical point (31.3 °C, 7.38 MPa), where the physical properties of CO₂ undergo drastic changes. Therefore, maintaining the stability of the compressor inlet state and preventing the cold leg from entering the subcritical region is of significant operational importance [27], to prevent drastic changes in the properties of carbon dioxide from affecting system performance. To this end, it is usually necessary to rely on adjusting the working state of the cold source or certain control valves to control the inlet temperature and pressure of the compressor to ensure the stable operation of the system [28]. The main compressor inlet temperature (MCIT) is typically controlled by adjusting the position of the cooling valve or the speed of the cooling pump, based on the difference between the actual inlet temperature of the compressor and the set temperature, thereby regulating the mass flow rate of the cooling source [29].

Turbine inlet state parameters:

Turbine inlet parameter control includes TIP control and TIT control. As shown in the system’s non-design configuration, TIP [30] control is achieved by adjusting the pressure-reducing valve located in front of the turbine to meet the output power requirements, which, like bypass control, can achieve a rapid response effect. Turbine inlet pressure control (TIP) stabilizes the output power by regulating the pressure of the working fluid entering the turbine. With the pressure-reducing valve designed at the turbine inlet, the compressor outlet pressure remains constant, and the output power demand is met by adjusting the pressure-reducing valve. This process is isenthalpic. Turbine inlet pressure control can also achieve a rapid response. Turbine inlet temperature control (TIT) is another control strategy for regulating output power, often achieved by changing the heat supply of the heat source. Turbine inlet temperature control requires a slower response speed to avoid the impact of thermal shock on the heater [31].

Compressor surge control:

Compressor surge is a phenomenon that causes abnormal vibration of the compressor, severely affecting system performance and even damaging turbine machinery [32]. Compressor surge protection is typically achieved through two main methods: adjusting the rotor speed and valve control. Adjusting the rotor speed is a control strategy that changes the working point of the compressor by altering its speed, thereby avoiding the surge region [31]. This method can be realized by real-time monitoring of the compressor’s working state and dynamically adjusting the speed [33].

3.2. Control Strategy

Control strategies are typically implemented using PID controllers, with common methods including bypass control, inventory control, and heat source control. The first two control methods achieve their objectives by altering the flow rate or pressure of CO₂, while heat source control adjusts cycle parameters by changing the heat input from outside the cycle. In literature [34], scholars have summarized the control strategies that have been researched in recent years, as shown in

Table 1. Summary of the investigated control strategies [34].

Author	Control Strategies
Anton Moisseytsev et al. (Moisseytsev et al., 2009) (Moisseytsev and Sienicki, 2011a) (Moisseytsev and Sienicki, 2011b) (Moisseytsev and Sienicki, 2012) (Moisseytsev and Sienicki, 2018)	turbine throttle valve, turbine bypass valve, inventory control, compressor throttle valve, cooler bypass valve
Minh Tri Luu et al. (Luu et al., 2017a)	inventory control, cooler bypass valve, compressor throttle valve, compressor throttle valve
Felipe G. Battisti et al. (Battisti et al., 2018)	Heat source control
Eric M. Clementoni et al. (Clementoni et al., 2016)	Turbine bypass valve, turbine throttle valve, recuperator bypass valve

.

Inventory control:

Inventory control, also known as pressure control, can be achieved by regulating the CO₂ flow rate of the system to match the required electrical load, thereby maintaining pressure control. Typically, inventory control is used to adjust the inlet temperature and pressure of the main compressor, as well as the inlet temperature of the turbine in the recompression cycle. Specifically, a CO₂ storage tank is installed between the precooler and the main compressor. Due to pressure differences, CO₂ is stored and released during load reduction and increase, respectively, to regulate the flow rate of CO₂ in the cycle [35]. CO₂ is released or filled into the storage tank to stabilize TIT. The CO₂ in the storage tank is modeled as having constant parameters, with pressure and temperature equal to the design values of the MCIT. The storage tank volume should be equal to the sum of the volumes of the cycle heat exchangers. By regulating the cooling water flow of the precooler, the MCIT is stabilized at the design value. The control strategy flowchart is shown in Figure 6 [36].

Carsten [37] controlled the TIT by adjusting the power of the heat source (nuclear reactor), while also employing low-temperature control, split-valve control, and throttle valve control to balance the outlet pressures of two compressors. Ultimately, inventory control was used to safely regulate the load from 100% to 10%. Additionally, it was noted that this control strategy is suitable for controlling slower transients. Therefore, inventory control applies to situations where the control target requires high thermal efficiency but the control speed is relatively slow, and it is not suitable for system conditions with sudden load changes, such as cycle startup control. This is also reflected in the research of other scholars, where Zhang et al. [38] analyzed the dynamic performance of the recompression sCO₂ Brayton cycle applied to CSP systems under partial load conditions from 100% to 50%. The results indicated that the inventory control strategy can maintain high efficiency during load reduction, but its response speed is relatively slow.

In summary, employing inventory control strategies can effectively regulate the CO₂ flow in the cycle, quickly stabilize parameters such as TIT and MCIT, and improve cycle efficiency, especially when the system load is reduced. However, inventory control also has limitations, such as a limited adjustment range constrained by the storage tank volume, and it may need to be combined with other control strategies when dealing with large-scale load changes in the system. Overall, inventory control is a simple and efficient control method that can significantly enhance the dynamic response and thermal efficiency of sCO₂ cycles, but its complexity and adaptability in practical applications should also be considered.

Bypass control:

Bypass control is achieved by regulating the mass flow rate through the turbine to match the required power load. The lower the load, the greater the bypass flow, which results in a reduced flow through the turbine. The significant advantage of this control strategy lies in its rapid response to parameter fluctuations and sudden load changes, such as during the startup and shutdown processes of the sCO₂ Brayton cycle. The startup and shutdown processes are important transitional phases that significantly impact the stability and safety of system operation. Hefetz et al. [39] proposed the concept of a dynamic sCO₂ recompression cycle controller, which can effectively respond to the demand for reduced generator power, and noted that the turbine bypass valve controller responds faster than the inventory controller. Bypass control is widely used in scenarios that require rapid load response, such as cycle engine startup control. Ding et al. [34] established a dynamic model of the sCO₂ recompression Brayton cycle and explored the system’s dynamic performance in maintaining the main compressor inlet pressure using throttle valve, bypass valve, and coupled control strategies after a change in thermal load. Simulation results indicate that using turbine and heater bypass valves can achieve the highest net efficiency. In Li’s [40] study, scholars analyzed the dynamic control, flow transition, and thermal inertia characteristics of the supercritical CO₂ recompression cycle power system. The difficulty of generator startup increases with the initial pressure, hence a low-pressure startup strategy based on bypass control was proposed and verified. The study found that compared to existing strategies, this new method can quickly and stably start the sCO₂ power system, significantly reduce working fluid consumption, and does not require external auxiliary heating or a complex cooling water regulation system. Additionally, in experimental research, Conboy et al. [41] tested a complete startup and shutdown process of the recompression sCO₂ cycle. Scholars transitioned the recompression cycle from cold startup conditions to power generation by two turbines, then to the required test conditions, and finally to the safe shutdown through control operations. In this process, factors such as the thrust state of the turbine compressor, the thermodynamic state of CO₂ at the compressor inlet, compressor surge and stall, turbine urea ratio, and many other factors must be considered. The bypass system plays an important role, allowing the hot leg to preheat fully before utilizing the turbine during startup, keeping the turbine in a normal startup state without causing system backflow.

In the selection of control valves for bypass control, different control valves such as turbine bypass valves and HTR bypass valves can be chosen according to different control needs, each with its own control characteristics. In Bian’s [42] study, scholars established a dynamic model of the recompression sCO₂ Brayton Cycle in the Matlab/Simulink environment and compared the open-loop dynamic performance of the entire system with five different control valves. Simulation results show that the turbine bypass valve and HTR bypass valve have better load regulation capabilities than other valves. When the valve opening is reduced to 50%, the load rates can reach 40.87% and 52.33%, respectively. Based on the control mechanisms of each control valve, a multi-valve control strategy was established to achieve the system’s long-term, wide-range load tracking capability under different operating conditions.

In Wang’s [43] study, scholars introduced six control valves, V1 to V6, into the recompression sCO₂ cycle, as shown in Figure 7. They also implemented supercritical state control, surge protection control using MC and RC bypass valves, and heater mass flow rate constraints. The study conducted a fundamental exploration of the impact of system startup and shutdown operation variables under different constraint conditions, as well as the actual constraints and corresponding countermeasures. It was found that to achieve successful startup and shutdown operations, it is necessary to simultaneously change the compressor speed and continuously control the rate of temperature change. Compared to the startup process, overload is more likely to occur during the shutdown process. After the system load decreases, promptly reducing the rotor speed can effectively prevent overload. There are various practical constraints, and there is a strong coupling relationship between the countermeasures and the constraints. This study illustrates the high importance of selecting different control valve strategies based on different operating states, requiring reasonable choices according to the actual operational constraint conditions of the system.

In summary, bypass control provides rapid response and extensive load regulation capabilities in advanced CO₂ cycle. Bypass control can quickly respond to load changes by adjusting the opening of the bypass valve, which can rapidly change the flow rate of the working fluid, thereby achieving rapid regulation of system load. It also allows the system to operate over a wide range of loads. By adjusting the bypass valve, load regulation from 100% to 0% can be achieved, effectively enhancing system safety and reducing pressure fluctuations. However, this is accompanied by efficiency losses, potential risks of motor overload, and increased system complexity, as part of the working fluid is bypassed and does not go through the complete cycle, thereby reducing the system’s output power [44]. Therefore, in practical applications, it is necessary to balance the advantages of rapid regulation with efficiency losses to achieve optimal system operation.

Turbine Speed Control:

Adjusting the compressor speed can alter the mass flow rate and pressure ratio of CO₂, thereby changing the system’s power. This control method is more widely applied in non-coaxial systems of expanders and compressors [29,45]. In Gao’s study [46], researchers investigated the off-design behavior of a combined sCO₂ cycle and organic Rankine cycle (sCO₂-ORC), with particular attention to optimizing system performance through speed control. To achieve operational objectives, a variable speed control strategy was employed to regulate the maximum pressure and mass flow rate, that is, by adjusting the speed of the compressor and turbine to control the pressure ratio of the sCO₂ cycle and the evaporation pressure of the ORC cycle. In the sCO₂ cycle, the low-pressure side pressure is maintained at the design value, while the high-pressure side pressure can be adjusted by changing the compressor speed. This control scheme is implemented by altering the speed of the rotating machinery, which is operationally simple. The study found that the speed control strategy effectively optimized the performance of the combined sCO₂-ORC cycle, enabling it to maintain high efficiency and good economic performance under various operating conditions.

3.3. Performance Comparison of Different Control Methods

Different control methods exhibit varying performance characteristics in different control scenarios. Ding et al. [34] compared four single control strategies for the sCO₂-RC: bypass control, throttle control, inventory control, and shaft speed control. They found that the shaft speed control of the main compressor has the highest efficiency, while the bypass control of the pre-cooler achieves the maximum turbine shaft power. Li et al. [47] compared five control strategies in the application of the sCO₂ cycle to the power grid: inventory control, two types of bypass control, and two types of turbine inlet parameter control. They discovered that inventory control has the highest thermal efficiency. However, the clean production coefficient of each control strategy decreases with the increase of grid load, necessitating the selection of different control strategies under various operating conditions. For the off-design operation of the sCO₂ cycle with an intercooled main compressor, Ma et al. [45] developed and compared four different control modes for the main compressor outlet pressure. The off-design results indicate that compared with constant pressure mode, the optimized cycle efficiency is significantly improved under variable pressure mode, but the net output power is reduced.

As stated in [42], compared with valve and speed control, heat source control and inventory control have slower response times. Moreover, inventory control is limited by the finite capacity of the storage tank, and interference with the cooler can lead to oscillations. Therefore, they are not suitable for rapid load regulation. The aforementioned studies also indicate that unless the loop has power regulation hardware to support a variable speed generator, the speed of the power turbine is usually fixed. The main reason is that the coaxial arrangement of the compressor and turbine poses significant challenges to control strategies. On the one hand, adjusting the speed at low loads makes the compressor more susceptible to surge; on the other hand, the turbine usually operates at a stable frequency synchronized with the power grid, and changing the speed poses a greater challenge to the grid. Therefore, in the sCO₂ Brayton cycle, bypass control has been widely applied due to its fast and flexible characteristics, but valve bypass control also has room for further development due to its relatively low improvement in thermal efficiency.

Different operating loads and objectives determine the choice of control strategies, and control strategies should also be matched with the system layout to achieve higher thermal efficiency or system response speed. In [14], scholars established dynamic models for four sCO₂ cycle layouts: simple recuperation cycle, recompression cycle, reheat cycle, and intercooled cycle, and studied the partial load performance of the four layouts under different control strategies. The results indicate that the thermal efficiency of the sCO₂ Brayton cycle is influenced by both system layout and control strategy. The control strategy mainly determines whether the compressor surges, whether the system is over-pressurized, and whether the drive motor is overloaded, among other safety performances. The thermal efficiency of the system under different thermal loads is mainly determined by the system layout. When using turbine bypass valves and following the load with bypass valves, the reheat cycle has the highest thermal efficiency at high loads, followed by intercooled, recompression, and simple recuperation cycles. However, when using turbine bypass valves and following the load with inventory, the intercooled cycle surpasses the reheat cycle to achieve maximum thermal efficiency. Based on the above research, it can be concluded that for sCO₂ Brayton cycle operating at medium and low loads for extended periods, to optimize thermal efficiency and safety performance, the intercooled cycle with inventory control should be prioritized. When the system operates at high loads for extended periods, to achieve the optimization of thermal efficiency and response speed, the reheat cycle controlled by the turbine throttle valve should be prioritized.

After analyzing and comparing different control strategies, this paper summarizes the advantages and disadvantages of different control methods, as shown in the

Table 2. Comparison of control method characteristics.

Control Method	Characteristics	Sensitivity to Operating Conditions	Impact of Ambient Temperature	Compatible Load Profile Characteristics
Inventory Control	Effectively improves cycle efficiency, but exhibits slow response to load changes; control range is limited by tank volume. Sole inventory control cannot maintain supercritical state at compressor inlet, requiring combined use with other methods.	Highly sensitive to system pressure variations; tank volume limitations become critical under extreme conditions.	Reduced pressure maintenance capability in low temperatures may compromise control stability.	Suitable for scenarios with gradual load variations and medium-to-long-term regulation needs.
Bypass Control	Enables rapid load adjustment, ideal for fast system response to load changes; regulation can be achieved solely via reliable valve control.	Performance depends heavily on valve response characteristics; high pressure differentials accelerate valve wear.	High temperatures may affect valve sealing and actuation reliability.	Appropriate for systems with frequent and sudden load changes requiring quick actuation.
Turbine Speed Control	Allows rapid load regulation while maintaining cycle efficiency; only applicable when compressor and turbine are arranged on separate shafts.	Requires high precision in speed control; split-shaft configuration is sensitive to system vibrations.	Low temperatures may impact lubrication and material contraction, affecting dynamic response.	Best suited for medium–high load ranges where both efficiency and dynamic performance are critical.
Turbine Throttling Control	Provides relatively fast load regulation but significantly reduces cycle efficiency; offers narrow control range under varying loads and cannot achieve low-load operation; may cause compressor choking.	Throttle valve position is highly sensitive to flow changes, potentially inducing surge under low pressure ratio conditions.	Efficiency losses worsen under high temperatures, with even poorer performance at low loads.	Recommended only for short-term, minor load adjustments or as a backup control strategy.

below. The turbine inlet throttle valve control method is not discussed in detail due to its low efficiency and other unfavorable reasons. In actual operation, turbine bypass can achieve better results.

3.4. Combined Control

Different control methods have their respective advantages. Thus, it is possible to compensate for the shortcomings of each by employing combined control strategies.

3.4.1. Inventory Control-Dominated Combined Control

The advantage of inventory control is that it can expand the range of system load regulation, but its disadvantage is the relatively slow response. With the deepening of research, single inventory control can no longer meet the needs of complex system control. Therefore, the application of combined control strategies based on inventory control is increasingly common [48].

In Du’s study [49], the non-design operating conditions of the system compressor under four control strategies were derived. It was found that the system efficiency and the change of the main compressor operating parameters under bypass control are similar to those under inventory control. Under low load conditions, the performance of the turbine adjusted by turbine inlet parameter control is superior to the aforementioned control schemes. Based on the compressor operating charts under various control strategies, it was concluded that the main compressor and the recompression compressor in the inventory control system have higher operating efficiency due to the simultaneous reduction of mass flow rate and pressure ratio. This led to the conclusion that the sCO₂ cycle under inventory control has higher off-design efficiency, thus proposing a series of hybrid control strategies based on inventory control. The study found that the new hybrid control method achieved an efficiency of up to 42.12% at a split ratio of 0.37, with thermodynamic performance far superior to that of using single inventory control.

3.4.2. Bypass Control-Dominated Combined Control

Although inventory control has the advantage of expanding the range of system load regulation, its slow response limits its performance in conditions that emphasize rapid load response. As mentioned in the introduction of bypass control, this strategy’s significant advantage in rapidly responding to parameter fluctuations and sudden load changes makes it better suited for such conditions. Compared to single bypass control, the use of hybrid control strategies based on bypass control shows better performance in operational efficiency [31].

In the study [34], to avoid dangerous conditions and excessive pressure fluctuations in the sCO₂ recompression Brayton cycle, a combination of different control strategies was adopted. In addition to turbine bypass control, pre-cooler bypass control was also used. Furthermore, a combination of turbine bypass control and throttle valve control was employed. The study found that the use of combined control methods effectively improved the efficiency of pre-cooler bypass control. The researchers also pointed out that, considering the slow response of turbine parameters when using only pre-cooler bypass control, coupling it with turbine bypass control can effectively increase the transient response speed, providing a new direction for the development of future combined control strategies.

3.5. Feedback Control Technology Based on PID Controllers

3.5.1. Introduction and Examples of Feedback Control Techniques

The Proportional-Integral-Derivative (PID) controller is designed to act on error signals to generate control signals. The general function of the controller is to maintain the controlled variable near its desired value. PID controller technology is mature, has a simple structure, and is widely applied in the industrial field. The output signal u(t) of the PID controller consists of three components: the proportional, integral, and derivative of the error e(t) between the desired value and the current value of the control variable [50]:

$$\mathrm{u}(\mathrm{t})=\mathrm{K}\mathrm{p}\mathrm{e}(\mathrm{t})+\mathrm{K}\mathrm{i}\int \mathrm{t} 0 \mathrm{e}(\mathrm{t})\mathrm{d}\mathsf{\tau } + \mathrm{K}\mathrm{d} \mathrm{d}\mathrm{e}(\mathrm{t}) \mathrm{d}\mathsf{\tau }$$

(1)

In the above equation, Kp represents the proportional gain, Ki represents the integral gain, and Kd represents the derivative gain. The proportional term is utilized to rapidly eliminate errors, causing the controlled variable to approach the desired value; the integral term is used to eliminate steady-state errors; and the derivative term is employed to enhance system stability [51].

In the system’s inventory control, to meet the demand for maintaining the highest system efficiency, the net load is changed by adding/removing sCO₂ mass to/from the pressurized storage tank, thereby reducing/increasing the total mass flow rate of sCO₂ in the cycle. PID control primarily serves as Inventory Management Control (IMC) for load tracking [29,52]. In the study by Liese [24], the scholar established an inventory management control with a load gain scheduling signal output. The research found that the most effective method to reduce the load is to remove the inventory from the cycle during design and maintain TIT and MCIT. The PID controller can receive the system’s actual load and TIT and MCIT and further control through the controller to maintain TIT at its maximum design value of 715 and MCIT at its minimum design value of 35. According to the Carnot principle, the maximum theoretical cycle efficiency of the system can be achieved at this point, which is defined as the ratio of the net power generated to the total heat input.

In the system’s bypass control, the system controls the system’s split control valve through the PID controller. The fractional valve opening or flow area y is defined as the ratio of the valve’s current flow area to the flow area when the valve is fully open [53]. It will depend on the valve stem position x (also known as valve stroke) and the valve flow characteristics determined by the valve’s geometry. The valve stroke position is usually determined by the actuator based on the signal from the controller, and the actuator moves the valve stem to the corresponding position according to u(t), with the valve adjusting the opening based on the valve stem position. The actuator and valve as a whole can be considered a first-order delay system with a small time constant. Based on the above operation, the PID controller completes the control execution part of the split valve [27]. The reception of the controller signal is determined by the actual control variable of the system. The scholar [6] implemented the overall control of the sCO₂ closed Brayton cycle using Simulink, where the controller employs a PID feedback control method. In the control of the bypass valve, the system compares the measured turbine inlet temperature with the set value. The bypass valve controller uses this error to generate a control signal based on the PID algorithm. The control signal is sent to the bypass valve actuator to manipulate the valve stem position, thereby controlling the opening and closing of the bypass valve, achieving the system’s bypass control.

3.5.2. Limitations of Feedback Control Techniques

PID control, while relatively straightforward to implement through simulation or experimental data fitting, often faces challenges in practical applications. Disturbances such as environmental factors, component replacement, and aging can lead to a decline in control performance, making real-time optimization and precise tracking of optimal exhaust pressure difficult to achieve. Additionally, PID control exhibits high model dependency. The design of PID controllers relies on accurate system models to determine controller parameters, which can be challenging to obtain in practice, particularly for nonlinear, time-varying, or complex systems. Studies have shown that traditional model-based control and optimization strategies struggle to obtain precise system models due to system nonlinearity, significant environmental changes, and equipment degradation, thereby limiting their performance [54].

For instance, ESC controllers take approximately 2080 s to reach a steady state under fixed conditions, while PID controllers require more than 3000 s or even longer to adjust parameters to achieve similar control effects when facing similar system changes. Additionally, PID control has poor adaptability and flexibility to the environment [55]. Scholars such as Liao [56] and Sarkar [57] have explored the correlation between system parameters like gas cooler outlet temperature, ambient temperature, and isentropic efficiency and the optimal exhaust pressure based on theoretical research. However, existing correlation models have been found to have limitations in practical applications, especially as they are greatly affected by changes in working conditions, with significant performance variations under different operating states. Cecchinato [58] believes in his assessment that the model is only applicable under specific working conditions thus its reliability must be strictly verified and evaluated before practical application.

In the study of the control performance and robustness of PID control technology, researchers have found that PID controllers may not achieve optimal control performance in some cases, especially when the system has multiple operating points or the objective function is complex. A scholar [59] found in their research that model-based methods may not yield optimal results in practical applications due to significant changes in actual system characteristics if the calibration/update of model parameters is not accurately executed. Moreover, PID controllers may require additional robustness design when facing model uncertainty and external disturbances. Hu et al. [60] proposed a hybrid ground-source heat pump self-optimization control strategy based on the ESC scheme. The study found that when using the ESC control strategy, the system’s steady-state error under fixed conditions was 0.4% (for compressor discharge pressure) and 0.3% (for COP), while the error of the PID controller without optimization would exceed 5%. It is worth noting that compared to many other static optimization methods, ESC has demonstrated better performance robustness in dealing with slowly changing processes, such as typical HVAC systems. However, compared to dynamic optimization schemes, static optimization methods generally suffer from a lack of adaptability to environmental changes and system parameter fluctuations, making it difficult to maintain optimal performance in practical applications. This limitation indicates that the robustness of static optimization schemes in dealing with complex and uncertain system dynamics is still insufficient.

4. Novel AI Integrated Control Technology

4.1. Extremum Seeking Control

In recent years, based on formula derivation or gradient optimization ideas, many scholars have proposed self-optimization control technologies centered around extremum-seeking control (ESC) [61]. Compared to traditional PID control, these new control technologies have significant advantages. They can adjust control parameters in real time to cope with external disturbances and system changes, thereby achieving dynamic tracking and optimization of the system’s optimal state, which significantly enhances the system’s robustness and adaptability [62].

4.1.1. Research Background of ESC

ESC is a self-optimization control technology that has gained considerable attention in recent years. This strategy estimates the gradient through a dither demodulation scheme, and as a dynamic gradient search method [63], ESC can find the optimal input in real time with almost no need for a model. Compared to traditional static optimization methods, ESC demonstrates significant advantages in robustness and transient response performance [64]. The principle is illustrated in Figure 8.

For transcritical refrigeration CO₂ cycle, the discharge pressure setpoint can be used as the input for the ESC controller, with the system’s Coefficient of Performance (COP) serving as the performance index. An inner-loop Proportional-Integral (PI) controller can adjust the opening of the Electronic Expansion Valve (EEV) to regulate the discharge pressure, thereby implementing the ESC method. This approach allows for the effective dynamic realization of gradient search based on a dither-demodulation scheme, enabling the search for unknown or slowly varying optimal inputs according to certain performance indicators (such as COP) [65,66]. Due to these advantages of the ESC method, it has shown great potential and broad application prospects in the dynamic load tracking and response of HVAC systems. The Hu team [67] proposed a self-optimizing control scheme based on the ESC strategy, aiming to maximize the system’s performance coefficient (COP) in real time. This method uses the discharge pressure setpoint as the input to the ESC controller and takes the system’s COP as the performance index, thus using the extremum-seeking process of COP as the feedback signal. Research has shown that the ESC strategy can successfully search for and track fixed or slowly varying optimal COP without relying on a system model. This feature allows for significant energy efficiency improvements in practical operations in a nearly model-free manner. In Rampazzo’s research [68], due to a limited understanding of certain system parameters for single-stage refrigeration CO₂ vapor compression units under transcritical refrigeration and subcritical conditions and the difficulty in developing and implementing effective model-based control technologies, the researchers decided to use a model-free approach to determine the optimal value of the high pressure in the vapor compression cycle, thereby maximizing system performance. The results indicate that in terms of air-cooled refrigeration and heat pump water heaters, the ESC method outperforms the PID control system established through a correlation model proposed by Liao et al. [69]. These results demonstrate that ESC control technology can achieve better optimization of system power consumption minimization and system performance coefficient maximization in both simulation and practical applications

Inventory control can be achieved by adjusting the CO₂ flow rate of the system to match the required electrical load, thereby regulating the import temperature and pressure of the main compressor and the import temperature of the turbine in the recompression cycle. For the solar thermal input sCO₂ Brayton cycle, which is sensitive to fluctuations in solar heat input and ambient temperature during different time periods, the control of the import temperature and pressure of the compressor and turbine is crucial. The ESC method is used to solve such problems that require high adaptability to system variable conditions. The Rajinesh team [70] proposed an extremum-seeking controller for a concentrated solar thermal technology device applied to the sCO₂ power cycle to maximize the power output of the closed Brayton cycle (CBC) when the solar heat input and cold air temperature change. This controller achieves this effect by manipulating the CO₂ mass inventory in the CBC. A relaxation variable is introduced in the ESC performance measurement to impose constraints on the import temperature and pressure of the turbine in order to protect the CBC from damage. The research results show that the proposed ESC algorithm exhibits similar or better overall performance, and the required calibration work is significantly reduced. The authors point out that to improve the load-tracking performance of the system during startup and shutdown phases, the convergence speed and robustness of the ESC method need to be further enhanced, which will involve the study of more complex ESC methods, such as Newton-like extremum search [71].

4.1.2. Principle of Operation of ESC

ESC is a real-time optimization control method that solves the problem of set-point optimization by seeking the optimal input for an unknown performance index [72]. Figure 9 presents the block diagram of extremum-seeking control based on the dither demodulation scheme [64].

The control process of ESC can be systematically and scientifically described as follows:

1.

Input Perturbation

The ESC algorithm perturbs the system by introducing a small amplitude periodic dither signal into the control input. This dither signal is typically a sine wave, with its frequency (ω) and amplitude (A) being preset parameters.

2.

Output Gradient Information

The system output (such as the performance index) is affected by the input dither, resulting in changes. According to the Taylor series expansion, the first harmonic term of the output change contains gradient information, which is the sensitivity of system performance to input parameters.

3.

High-pass Filter

The signal at the output end is processed through a high-pass filter to remove the direct current (DC) component, retaining the alternating current (AC) component that contains the gradient information.

4.

Demodulation Process

The demodulation signal has the same frequency as the dither signal but with a certain phase difference (ϕ). This process transfers the gradient information from the AC component to the DC component.

5.

Low-pass Filter

The signal after demodulation is then passed through a low-pass filter to suppress higher-order harmonics, retaining the DC component that is proportional to the gradient.

6.

Integrator

The integrator is used to close the control loop, integrating the DC component that is proportional to the gradient, thereby driving the control input to adjust towards the direction where the gradient is zero, achieving performance optimization.

7.

System Stabilizer

If the system is asymptotically stable, meaning that the system state will eventually tend towards an equilibrium point, then the gradient will be driven to zero, thereby achieving optimal performance.

After this series of optimization steps, if the system is asymptotically stable, the gradient will be driven to zero, thereby achieving optimality. In the flowchart, the scaling factor Ki is used to control the rate of convergence to the optimum and to adapt the algorithm for problems where the objective is to minimize (Ki < 0) or maximize (Ki > 0). Since the integrator may drive the input beyond its physically feasible domain, special attention must be paid to the selection of control parameters when designing this model-free algorithm, as they significantly affect control performance. In Rampazzo’s [68] study, he pointed out that the amplitude A of the sine wave needs to be carefully chosen: it must be large enough to achieve an acceptable signal-to-noise ratio in the system output at a specific dither frequency, while also being small enough to reduce the system’s steady-state error. To ensure the stability of the system and convergence to the optimal value, the frequency ω of the dither signal should be chosen to be slower than the system’s own dynamic response. Additionally, the phase shift ϕ of the demodulation signal can be determined experimentally by applying specific input signals to the system and measuring the phase difference between the input and the high-pass filter output. Care must also be taken when selecting the parameters of the high-pass filter (HPF) and low-pass filter (LPF): the cut-off frequency of the LPF should be set below ω, while that of the HPF should be even lower. Ultimately, the selection of the integral gain Ki directly affects the convergence rate of the algorithm; it should be relatively small to ensure stability, but sufficiently large to achieve an acceptable convergence speed [73]. Furthermore, the ESC design guidelines [74,75] and the guidelines provided in Cui’s [76] research indicate that the anti-winding setting should be chosen for the anti-calculation gain Ki to ensure asymptotic stability.

4.1.3. Problems in the Practical Operation of ESC

In practical systems, all actuators have physical limitations, which cause the control actions to saturate at certain points. When the integral control loop is in a saturated actuator state, integral windup is inevitable [61]. ESC is essentially an integral control loop that adjusts the gradient proportional signal. Therefore, ESC operations may suffer from integral windup under actuator saturation. For example, in Hu’s [60] study, to address the issue of integral windup in the ESC control system caused by the saturation of the effective flow area of the EEV in an air-source transcritical CO₂ heat pump system, scholars resolved this problem by adopting an anti-windup scheme in the ESC control. Similar applications of this solution method have been observed in the ESC control of air-side economizers with damper saturation and chilled water systems with fan speed saturation [77,78].

Extremum-seeking control achieves real-time control of transcritical refrigeration CO₂ cycle and avoids the complex process of modeling. However, the long convergence time during its experimental process makes it unsuitable for conditions where ambient temperature changes rapidly [79]. In the study [70], it was found that under winter conditions, the ESC controller significantly increased the CO₂ inventory between 14:30 and 15:00 to reduce the turbine inlet temperature below the target value (350 °C). However, between 15:00 and 16:00, when the ambient temperature changed dramatically, the ESC controller failed to reduce the CO₂ inventory in time to increase the turbine inlet temperature. This led to a decrease in the average electrical energy production of the system during this period from 5.8 MWh to 5.7 MWh, indicating a decline in performance, which requires further optimization. At the same time, the extremum-seeking algorithm based on the perturbation demodulation scheme has the problem of only being able to reach a local optimum. The reasons are as follows: the convergence results of ESC often depend on the initial conditions and the choice of input perturbations. If the initial state is close to a local optimum, the algorithm may converge near that point and fail to explore other potentially better solutions. Additionally, due to the limitations of gradient information and the influence of the dither signal, if the gradient information is insufficient or inaccurate, or if the amplitude or frequency of the dither signal is improperly chosen, it may lead to a misjudgment of the system’s dynamic response, causing the algorithm to miss the global optimum during the search process. Therefore, in complex multi-peak environments, the ESC control method cannot ensure the acquisition of a global optimum [80]. In the study of CO₂ heat pump water heaters [81], it was found that during certain transitional stages with drastic temperature changes, the system’s performance may experience temporary fluctuations. For example, during the process where the ambient temperature rises rapidly from −15 °C to 0 °C, the system’s COP may temporarily decrease from 3.05 to 2.95 before gradually returning to the optimal level. This temporary performance decline indicates that the stability and optimization capabilities of the ESC controller need further improvement in complex multi-peak environments. At the same time, the extremum-seeking control algorithm also has the problem of requiring the adjustment of many parameters, which increases the complexity of algorithm design and implementation. Specifically, the amplitude and frequency of the dither signal, the phase shift of the demodulation signal, the parameters of the high-pass and low-pass filters, the integral gain, and the parameters of the feedback controller all need to be finely adjusted to ensure that the algorithm can effectively extract gradient information and maintain the stability and rapid convergence of the system. This dependence on multiple parameters makes the debugging and optimization of the algorithm more difficult in practical applications, especially in dynamically changing environments, leading to potential instability in system performance and reduced response speed [82].

In summary, compared to PID control, ESC control technology demonstrates a better ability to adjust control parameters in real-time to cope with external disturbances and system changes, and it reduces the dependence on system models. However, in the face of drastic system dynamic changes and complex multi-peak environments, ESC control technology still needs further optimization and improvement. These limitations point out the challenges that ESC algorithms may face in practical applications, providing directions for future research and also offering insights for the development of new control technologies. For example, enhancing the global search capability of control systems to avoid local optimum traps, or exploring automated and intelligent parameter adjustment methods, such as machine learning-based parameter optimization techniques, to reduce human intervention and improve the adaptability and robustness of the algorithms.

4.2. Data-Driven MPC-Based Predictive Control Techniques

4.2.1. Overview of MPC Control

Model Predictive Control (MPC) is an advanced control technique that calculates control inputs by solving an optimization problem at each control step. This optimization problem is based on a process model, which predicts future behavior and considers future constraints. The objective of MPC is to minimize the cost function J, which is typically the sum of squared errors within the future prediction horizon. At each sampling instant, MPC uses the current system state information to update the prediction and re-solve the optimization problem to determine the optimal control action for the current moment. Only the first control action of this control sequence is then applied to the system, and the subsequent dynamics of the system are fed back into the MPC controller for the next optimization at the next sampling instant. This process is continuously repeated, ensuring that the control technique can adapt to changes in system dynamics and external disturbances. The framework diagram of MPC predictive control is shown in the figure. In Figure 10, y_sp represents the system’s setpoint output value, y_r is the system’s reference trajectory, u is the input value, y_m is the model output value, y_c is the predicted output value, and y is the actual output value.

The two main pillars of the MPC method are prediction and optimization. MPC has the following notable features: (1) It can systematically and flexibly handle system state and control input constraints. (2) This is a strategy centered on “optimization,” which naturally produces optimal performance to a certain extent. (3) It inherently possesses a degree of robustness, and robust MPC strategies can enhance its resilience to external disturbances, parameter uncertainties, and noise. (4) It generates and fully utilizes forward-looking predictions, thereby improving performance [83]. When dealing with complex processes that are difficult to control effectively with traditional PID controllers, one may encounter the problem of process variables failing to stabilize at the desired setpoints. This situation may arise due to insufficient tuning of PID controller parameters or excessive disturbances to the system, causing it to take a long time to reach a stable state. MPC addresses these challenges by solving a dynamic optimization problem to calculate control inputs. In this dynamic optimization problem, constraints applied to controlled and manipulated variables ensure that process parameters remain within acceptable ranges. Furthermore, compared to PID controllers, MPC can produce smaller oscillation amplitudes in response to specific disturbances, indicating that MPC has advantages in suppressing system fluctuations and improving control accuracy [84].

In other thermodynamic systems such as the ORC, many advanced control algorithms like various MPC predictive controls have been applied [85], and MPC predictive control has also found wide applications in other fields [86,87]. The general MPC approach relies on an accurate system model for prediction and optimization. However, obtaining an accurate system model is often challenging in practical applications. For advanced CO₂ cycle, due to the high nonlinearity of the system’s fundamental equations, the model of the CO₂ cycle may be too complex for these model-based control algorithms [88]. A learning model obtained through artificial neural networks can achieve satisfactory accuracy, thereby significantly reducing computational tasks. Therefore, learning-based MPC control also holds promise for development in CO₂ cycles. Machine learning methods require learning the mapping from an input dataset to an output dataset based on labeled input-output pairs. Common machine learning methods include algorithms based on artificial neural networks, Support Vector Regression (SVR) [89], etc.

4.2.2. Multivariate Regression-Based Model Predictive Control

Multivariate regression-based Model Predictive Control (MPC) is a control method that combines multivariate regression models with MPC strategies. The multivariate regression model is used to predict the system’s output, while the MPC strategy utilizes these predictions to optimize control inputs, thereby achieving precise control of the system. The “multivariate” characteristic of the multivariate regression model is primarily reflected in its ability to consider the effects of multiple independent variables on a single dependent variable simultaneously. Compared to general linear regression models, multivariate regression models have the advantage of being able to analyze the impact of multiple engineering factors (such as compressor temperature, pressure, turbine inlet temperature, pressure, and recuperator design, etc.) on system efficiency and the interrelationships among independent variables, especially in the system control of CO₂ cycles where the relationships among independent variables are complex.

The multivariate regression model first establishes a data model of the system’s input and output variables, learning the relationship between input variables (such as operating conditions, environmental parameters, etc.) and output variables (system state) through historical data. The model can be represented as [90]:

$$\mathrm{y}=\mathrm{X}\mathsf{\beta }+\mathsf{ϵ}$$

(2)

where, y is the vector of output variables; X is the matrix of input variables; β is the vector of regression coefficients, and ϵ is the error term.

Since the sCO₂ cycle is a highly nonlinear system, its performance is influenced by multiple factors, including the operating conditions of major components such as compressors, turbines, and heat exchangers. Linear models are incapable of capturing these complex nonlinear relationships, leading to insufficient prediction accuracy. Therefore, linear regression models have not been widely applied in CO₂ cycles. Instead, a series of nonlinear prediction techniques have been developed. However, the capability of multivariate regression has been preserved and further developed in nonlinear prediction techniques such as SVR [91].

4.2.3. Support Vector Regression-Based Model Predictive Control

Support Vector Regression (SVR) is based on principles similar to those of Multivariate Regression (MR), but it extends these principles by employing kernel functions to handle nonlinear relationships. SVR uses kernel functions to map input features into a high-dimensional feature space, enabling it to address nonlinear regression problems. Due to the highly nonlinear nature of the CO₂ cycle’s fundamental equations, SVR is more suitable for handling such problems compared to MR. This is particularly important for complex systems that traditional linear regression methods cannot effectively manage. Consequently, the advantages of SVR-based MPC in terms of prediction accuracy and robustness against disturbances make it superior in the control of nonlinear systems [87].

The optimization objective of SVR is to find a function f(x) = w^Tx + b, such that the prediction error for most samples does not exceed ε, while also striving to keep the model simple. Therefore, SVR solves the problem by using kernel functions to map input features into a high-dimensional feature space. In this high-dimensional space, SVR seeks the optimal hyperplane by maximizing the margin width to identify the hyperplane that best fits the training data. The objective function is found by minimizing the following function [92]:

$$\underset{w,b,\xi ,\xi \ast }{\min }{\displaystyle \frac{1}{2}}{\parallel w\parallel }^2+C\sum _{i=1}^n(\xi i+\xi i\ast )$$

(3)

The objective function of SVR is to simplify the model as much as possible while satisfying a certain level of error. Its optimization problem can be expressed as minimizing half the squared norm of the weight vector plus a penalty term for the slack variables. Specifically, ||w||^2 is used to control the complexity of the model and maintain the smoothness of the function, while C∑(ξi + ξi∗) penalizes samples that violate the ε-insensitive loss function, thereby controlling the error. Here, w is the weight vector, b is the bias term, C is the penalty parameter, and ξi and ξi∗ are the slack variables.

The constraints of SVR include:

$$yi-(w\cdot xi+b)\le ϵ+\xi i$$

(4)

$$(w\cdot xi+b)-yi\le ϵ+\xi i\ast$$

(5)

$$\xi i,\xi i\ast \ge 0$$

(6)

These constraints ensure that the difference between the model’s predicted values and the actual values remains within a certain error range, while the slack variables allow some data points to exceed this range, thereby reducing the model’s sensitivity to outliers.

Based on the determination of the objective function and constraints, to solve the aforementioned optimization problem, SVR introduces Lagrange multipliers and constructs the Lagrangian function, thereby obtaining the dual problem. The form of the dual problem is:

$$\underset{a,a\ast }{\max }\sum _{i=1}^{n}yi\left(\alpha i-\alpha i\ast \right)-ϵ\sum _{i=1}^{n}yi(\alpha i+\alpha i\ast )-{\displaystyle \frac{1}{2}}\sum _{i,j=1}^n(\alpha i-\alpha i\ast )(\alpha j-\alpha j\ast )(xi\cdot xj)$$

(7)

Under constraints:

$$\sum _{i=1}^n\left(\alpha i-\alpha i\ast \right)=0$$

(8)

$$0\le \alpha i, \alpha i\ast \le C$$

(9)

These constraints and the objective function together define the dual problem of SVR. By solving the dual problem, the final regression model can be obtained.

Within the MPC framework, the MPC predictive control system primarily establishes a nonlinear system’s predictive model and system output predictions using SVR. Additionally, the MPC system further optimizes the predictions through a rolling optimization process. Within each control cycle of MPC, the SVR model updates its predictions to reflect the latest system state, thereby achieving rolling optimization.

Nonlinear dynamic prediction control based on nonlinear dynamic prediction models has been applied in the control of transcritical CO₂ heat pumps [93]. Scholars have used the MPC method to optimize the operation of transcritical carbon dioxide air-source heat pump (ASHP) water heaters. Initially, a high-fidelity physical model was established in Dymola to generate operational data. Subsequently, a data-driven nonlinear control-oriented model for MPC was derived from the data source through dynamic system identification, maximizing the performance coefficient of the ASHP water heater under state space equations and constraints. Further evaluations were conducted under several scenarios, including fixed ambient temperature, realistic ambient temperature, and variable outlet water temperature. The study found that within 3 min, the system successfully converged the operating discharge pressure to 91.24 bar through MPC control technology, which is quite close to the calibrated actual discharge pressure of 90.92 bar, with a relative error of 0.3%. The results indicate that the MPC control technology based on nonlinear control-oriented models has strong applicability in CO₂ cycle control.

Model predictive control based on support vector regression has also been applied in other areas, such as the thermal load prediction of seasonal thermochemical energy storage systems (STES) in district heating networks. Scholars [87] have developed an MPC strategy using SVR to accurately predict thermal loads with a machine learning model, optimizing the charge and discharge cycles of the thermochemical energy storage system to reduce mismatches between heating energy supply and demand. The study found that incorporating these models into the MPC strategy can precisely predict thermal demand, enhancing the management of energy storage and distribution. This provides new insights for the load prediction control of MPC predictive control technology in CO₂ cycles.

4.2.4. Artificial Intelligence-Based Model Predictive Control

Nonlinear models based on data-driven approaches, such as SVR, offer advantages such as online output, multivariable control, and nonlinear dynamic prediction. However, their high demand for online computational power poses significant challenges, particularly in model construction and parameter optimization. To address these issues, scholars have conducted research on model predictive control based on artificial neural networks [87]. Compared to regression models, artificial intelligence models represented by artificial neural networks possess strong nonlinear fitting capabilities, allowing them to handle complex data relationships more flexibly. Moreover, the performance of data-driven models largely depends on the quality and quantity of training data; if the data are biased or insufficient, the model’s prediction accuracy will be affected [94]. Artificial intelligence models also rely on data, but through techniques such as deep learning, they can better learn complex patterns and relationships from large datasets. Unlike machine learning data-driven models that are approximate black-box models, artificial neural networks can achieve a certain degree of model interpretability through appropriate network design and feature engineering [95]. Common artificial intelligence models include artificial neural networks, particle swarm optimization, and Gaussian processes [96].

An artificial neural network (ANN) is a computational model that simulates the human brain’s neural network, consisting of an input layer, hidden layers, and an output layer. Each neuron is connected through weights, and nonlinearity is introduced through activation functions, enabling the network to learn and simulate complex function mappings. ANNs minimize prediction errors by adjusting the connection weights between neurons, a process known as training. Training data are used to guide the adjustment of network weights, while validation and test sets are employed to evaluate the model’s performance. The modeling diagram of the artificial neural network is shown in Figure 11 [97].

The general process of ANN includes steps such as data collection and preprocessing, model establishment, training and validation, testing and evaluation, and model interpretation [98]. First, before modeling, it is necessary to collect experimental data of heat exchangers, including input parameters (such as fluid flow rate, temperature, pressure, etc.) and output parameters (such as heat exchange efficiency, pressure drop, etc.), and further normalize the data to eliminate the influence of different dimensions, making the data suitable for neural network processing. Then, an ANN model with input, hidden, and output layers is constructed, and appropriate activation functions are selected. Next, the network’s predicted output is calculated through forward propagation, and the difference between the predicted and actual values is measured using a loss function. Subsequently, the gradient of the loss function with respect to the network weights is calculated using the backpropagation algorithm, and the weights are updated according to the gradient descent method. Finally, the model performance is evaluated and necessary adjustments are made to optimize the model until satisfactory accuracy is achieved. The entire process is iterative, requiring continuous adjustment and optimization to enhance the model’s predictive capability [99].

In recent years, neural networks have been widely applied in nonlinear system identification. Theoretically, neural networks can fit any nonlinear function [100]. Due to ANN’s strong ability to handle nonlinear relationships and automate feature extraction, ANN models are often used in CO₂ cycles for multi-objective performance optimization to improve the overall efficiency and performance of the system [101], or for dynamic performance prediction. In the study by Son et al. [95], scholars successfully predicted internal bottleneck phenomena in sCO₂ heat exchangers using an ANN model, significantly improving the efficiency of cycle design. Specifically, the study reduced the computational time for cycle analysis from 2.63 s using traditional methods to 0.00377 s with the ANN model, an improvement of approximately 700 times. Moreover, the model also demonstrated excellent predictive accuracy, with an error range kept within 0.001%, indicating that ANN has extremely high efficiency and accuracy in dealing with complex heat exchanger problems. The Mohanraj team [96] conducted a detailed analysis of the application of ANN in heat exchanger thermal analysis, mentioning that the ANN model had a low Mean Absolute Relative Error (MaRE) of 5.56% when predicting the heat transfer of a condenser, showing higher predictive accuracy than traditional methods. Additionally, in another study [28], the deviation between the ANN model’s predicted condenser heat rejection rate and the experimental value was within 5%, further confirming the effectiveness and accuracy of ANN in heat exchanger performance prediction. In the research by Efatinasab et al. [94], the ANN model played a significant role in the design of micro-fin tube heat exchangers. The results showed that the ANN model had an average Mean Absolute Error (MAE) of less than 4.5% when predicting the Heat Transfer Coefficient (HTC) and Friction Pressure Drop (FPD). This result not only demonstrates the advantage of the ANN model in predictive accuracy but also highlights its potential application value in heat exchanger design and performance optimization.

Although ANN models possess extremely high predictive accuracy and are widely used for multi-objective parameter optimization in CO₂ cycle, their ability to control the system’s dynamic performance is relatively poor. This is mainly due to the complexity of the algorithm’s iterative process, which involves updating positions and velocities through numerous mathematical computations [102,103]. This makes ANN’s execution speed less advantageous, making it unsuitable for real-time or near-real-time optimization problems. Zhang et al. [104] proposed a MPC method to optimize the operation of transcritical CO₂ air-source heat pump water heaters. By using data-driven models to predict future operating conditions, the MPC strategy calculates the optimal input to optimize system operations. The results indicate that with the use of ANN models, the system’s predictive error is less than 4.5%. The validation results show that the proposed predictive model, which combines physical laws and data, can successfully capture dynamic performance. However, in this study, there is a problem of computational delay in the real-time control of the ANN model, which causes a certain degree of lag in the control of the MPC system. Therefore, in actual system control, predictive models based on algorithms with fast convergence capabilities, such as the PSO algorithm, may be more applicable.

The Particle Swarm Optimization (PSO) algorithm is an optimization algorithm based on swarm intelligence, which seeks the optimal solution by simulating the social behavior of bird flocks or fish schools. Compared to ANN, PSO optimization omits the processes of data preprocessing, model structure design, training parameter configuration, and model training iteration, which are not required in PSO because it is an optimization algorithm for finding the optimal solution, not for direct prediction [105]. Moreover, PSO has the advantage of fast convergence in nonlinear real-time control. The reason why PSO can converge quickly is mainly due to the following: the particle velocity update formula in the PSO algorithm includes three parts: the inertia term, the individual historical best position (cognitive part), and the social (swarm historical best position) part. This structure allows PSO to maintain its search capability over the global solution space while also utilizing individual and social information for local search, effectively balancing the abilities of global and local searches. The Particle Swarm Optimization algorithm mainly approaches the optimal solution by continuously updating the particles’ velocities and positions according to the following formula [106].

Particle swarm velocity update formula:

$$vi(t+1)=w\cdot vi\left(t\right)+c1\cdot r1\cdot (pbesti-xi(t\left)\right)+c2\cdot r2\cdot (gbest-xi(t\left)\right)$$

(10)

where:

–

vi(t): velocity of the i-th particle at moment t.

–

w: inertia weight, used to control the influence of the previous iteration value of the particle velocity.

–

c1: Individual learning factor, which indicates how much the particle is influenced by its own experience.

–

c2: social learning factor, indicating the extent to which the particle is influenced by the experience of the group.

–

r1 and r2: randomly generated numbers in the interval [0, 1].

–

pbest i: the historical best position of the i-th particle.

–

gbest: Historical best position of the whole population.

–

xi (t): the position of the i-th particle at moment t.

This formula enables particles to explore the solution space more efficiently and find the optimal solution by combining individual and population experience.

Particle position update formula:

$$xi(t+1)=xi\left(t\right)+vi(t+1)$$

(11)

The formula adjusts the position of the particle according to the updated velocity so that the particle moves to a new position in the solution space, by constantly updating the position, the particle is able to search in the solution space and gradually approach the optimal solution.

Inertia weight adjustment formula:

$$w(t)=w\mathrm{m}\mathrm{a}\mathrm{x}-{\displaystyle \frac{w\mathrm{m}\mathrm{a}\mathrm{x}-w\mathrm{m}\mathrm{i}\mathrm{n}}{MaxIter}}\cdot t$$

(12)

where:

–

MaxIter: the maximum number of iterations of the algorithm.

The formula controls the influence of the previous iteration value of the particle’s velocity and is a key parameter in balancing global and local search capabilities. It also allows for the gradual reduction of the inertia weight as the number of iterations increases, enabling the algorithm to perform extensive global searches in the early stages and focus on local searches in the later stages to improve search accuracy. By adjusting the inertia weight, the algorithm can exhibit different search characteristics at various stages of iteration, thereby increasing the probability of finding the global optimal solution. This algorithm enables PSO to achieve global optimality while enhancing convergence speed, ultimately reaching the goal of the optimization problem.

The Yin team [106] constructed a PSO-BP neural network to predict the optimal discharge pressure in CO₂ heat pump systems, as shown in Figure 12. The study indicates that the new model has higher predictive accuracy compared to traditional correlation methods. The relative error is approximately 1.6%, while the error range of traditional methods is between 11.1% and 44.9%. This demonstrates that the CO₂ heat pump model built using the PSO-BP neural network can operate better under optimal COP conditions. Furthermore, the team experimentally validated the discharge pressure MPC control technology based on the PSO-BP algorithm [107], as shown in the predictive model based on PSO-PB. The experimental results show that the PSO-BP method has satisfactory consistency with the test data, displaying relative errors of less than 5% and 3% under different ambient temperatures. Compared to Wang’s [108] correlation prediction, the PSO-BP algorithm shows better consistency in almost all test conditions. Additionally, compared to the baseline cycle, the subcooler-based PSO-BP control cycle achieved performance improvements of over 15% and 25% under floor heating and radiator conditions, respectively.

4.3. Comparative Analysis of Different Control Technologies

Comparative Analysis

From the analysis presented above, it can be concluded, as shown in

Table 3. Comparative analysis of different control technologies.

Control Technology	Advantages	Disadvantages	Suitable Application Scenarios
PID Control	Mature technology, simple structure easy to implement and maintain, suitable for many industrial processes	Performance may be insufficient for nonlinear and time-varying systems; requires an accurate system model to design controller parameters; poor adaptability and flexibility to environmental changes	Suitable for scenarios with relatively stable system dynamics and not extremely high control accuracy requirements
Extremum-Seeking Control	Can dynamically adjust control parameters to cope with external disturbances and system changes, reducing dependence on system models; suitable for dynamic optimization and real-time performance enhancement	May have long convergence times, not suitable for conditions with rapid environmental temperature changes; may only reach local optima; many parameters, complex design and implementation	Suitable for scenarios requiring real-time optimization and performance enhancement
ANN-based MPC Control	Strong nonlinear fitting capability, able to handle complex data relationships, capable of online output of control strategies	Existence of computational delay issues, relatively weak ability to control system dynamic performance; requires a large amount of data for training	Suitable for prediction and optimization problems, especially when there is abundant data and nonlinear relationships need to be processed
PSO-optimized MPC Control	Rapid convergence capability; ability to balance global and local search; suitable for real-time control	High dependence on data quality	Suitable for real-time control scenarios requiring fast response and optimization, especially in nonlinear and dynamically changing environments

, that different control technologies in CO₂ cycle have their respective advantages and limitations, making them suitable for various application scenarios. In summary, PID control is widely adopted due to its simplicity and broad applicability; however, its performance may be constrained when faced with complex and variable systems. ESC can dynamically track the optimal state of the system but may encounter issues related to slow convergence speed and local optima. MPC based on ANN possesses strong nonlinear fitting capabilities, yet it may suffer from computational delays and weak real-time control abilities. In contrast, MPC based on PSO demonstrates advantages in real-time control due to its rapid convergence capability, but it requires more parameter adjustments and optimization. Different control technologies have distinct usage scenarios, and the selection of the most suitable control technology should be determined based on the specific requirements and characteristics of the system.

4.4. The Latest Progress and Challenges in the Application of AI-Based Model Predictive Control

Although the CO₂ cycle controlled by AI models has not been widely applied, there have been many research and application cases in similar fields. The advantages and limitations of the application of artificial intelligence control strategies provide actionable suggestions for their application in CO₂ cycles.

4.4.1. Engineering Application Analysis

AI-Based Model Predictive Control has been applied in many fields, and its unique advantages and disadvantages are also shown in the application process. Guo [109] team proposed an energy-saving model predictive control technology based on extended state Kalman filter for supercritical units. Aiming at the control problem of supercritical coal-fired power plants under the background of intermittent growth of renewable energy, an advanced control strategy combining artificial intelligence model modeling, extended state Kalman filter (ESKF) and energy-saving model predictive control was proposed. A neural network model was developed to capture the dynamic behavior of the power plant. Compared with the traditional RNN and Gru models, the test RMSE decreased by 14.0% (compared with Gru) and 64.3% (compared with RNN), and the average relative error (MRE) index increased by 39.5% and 67.9%. However, the development and integration of this model has brought great challenges. The MPC integration architecture coupled with the artificial self energy model needs to handle state estimation, predictive control and energy optimization at the same time. At the same time, the memory limit of edge devices (usually <4 gb) restricts the deployment of complex models in industrial applications, and requires a large number of characteristic working condition data for model training. The study found that the supercritical unit needs at least 200 continuous time steps (about 6.7 min) to capture the dynamic coupling effect. At the same time, the long-time thermal inertia (up to 3–6 h) of the supercritical coal-fired power plant is also one of the problems to be solved urgently.

In the application of hydraulic transmission technology of walking machinery, fan team [110] summarized the application of hydrostatic transmission system in walking machinery, including pump controlled hydraulic system, secondary regulation hydraulic system and adaptive regulation system, and analyzed the control strategies of torque control, fuzzy control and optimization control. Among them, the application of artificial intelligence technology has played a great role in promoting. In the wheel loader, the particle swarm optimization algorithm improves the transmission efficiency by 2.06%; In the front-end loader, the fuel consumption of the three degree of freedom control is 34% lower than that of the two degree of freedom control, and the overall transmission efficiency is relatively improved by 20–30%. At the same time, the digital hydraulic technology combined with high-speed on-off valve and neural network can expand the pressure regulation range, with an efficiency of more than 80%, and reduce the energy loss by 60% compared with the traditional load sensing system. However, the variability of the industrial environment puts forward higher requirements for the robustness of the model predictive controller, so it also puts forward higher requirements for its multimodal data to adapt to the fitting of various working conditions, and it is required to deal with multi-scale parameters such as pressure, flow, temperature and so on at the same time.

For the application of artificial intelligence control of lithium bromide water two-stage absorption heat converter, aveledo [111] team compared the performance of two AI technologies, fuzzy logic and artificial neural network, in predicting the maximum product of cop and total temperature rise. The study found that the RMSE of mev2 predicted by 30 neuron ANN model was 6.23 × 10⁻¹, MBE was 3.88 × 10⁻¹, R ² = 1.000, which was more accurate than fuzzy logic. At the same time, the optimal evaporator flow is calculated by AI to maximize the product of cop and GTL, which verifies the applicability of AI in nonlinear thermal systems. However, a high-precision artificial intelligence prediction model also needs the support of high-quality data. The research shows that the accurate annotation of cop requires professional thermal testing, and the cost of a single sample is about $50–80. This is not a small expenditure. In the process of model training, the full sensor deployment generates 2.4 mb/s data flow, accounting for 35% of the industrial Ethernet bandwidth. The data and computing requirements of AI control technology are still the problems to be solved at present. The research conclusion provides a reliable reference value for the application of AI technology in CO₂ cycle.

In the application of artificial intelligence to predict the heating and cooling of ground source heat pumps, Naveed Ahmed’s team [112] evaluated the predictive performance of four AI learning models (LSTM, BD-LSTM, GRU, CNN) across datasets featuring borehole heat exchangers (BHEx) of varying quality. They also analyzed the impact of data quality on model accuracy. The study revealed that LSTM-based models excel in time series prediction but come with high computational costs. High-quality training data required for these models should possess a high sampling frequency (≤15 min), a low outlier rate (≤5%), and complete seasonal characteristics to keep the prediction error within 1%. A comparison between a 15-min sampling frequency and a 60-min sampling frequency showed that a decrease in frequency results in a 6.17% increase in MAPE, significantly diminishing the model’s predictive performance. Notably, BD-LSTM’s training time on an NVIDIA T4 is eight times that of CNN, indicating its high computational demand. In scenarios demanding high real-time performance, CNN with a high sampling frequency can serve as a viable compromise.

The research further found that AI technology has achieved millisecond-level real-time control in complex engineering systems such as power plants and heat pumps, with key performance indicators generally improving by over 30%, validating its practical worth in industrial settings. Future research directions should focus on developing more efficient data cleaning methods and hybrid AI architectures to strike a balance between computational costs and prediction accuracy, while enhancing the model’s robustness in cases of missing data [113]. The next chapter will delve into the current limitations and prospects of AI model applications in greater detail.

4.4.2. Limitations and Prospects of AI-Based Model Predictive Control in Applications

Combined with the application cases of the above artificial intelligence in various fields, we can draw the advantages and limitations of the application of artificial intelligence, as shown in

Table 4. Comparative Analysis of Advantages and Limitations in AI-Based Model Predictive Control Across Key Dimensions.

Characteristic Dimension	Common Advantages	Common Challenges	Representative Performance Indicators
Performance Enhancement	>30% improvement in key performance indicators	Poor model interpretability	20–60% energy efficiency improvement
Control Precision	Millisecond-level real-time control capability	Stringent real-time requirements	Latency requirement < 500 ms
Nonlinear Processing	Effective resolution of complex nonlinear relationships	High maintenance costs	Annual maintenance cost ~$15,000/system
Data Requirements	Multi-objective collaborative optimization	High-quality data dependency (“data hunger”)	Data acquisition cost $10K–15K/system
Computational Resources	Edge-cloud collaborative deployment architecture	Substantial computing power demands	BD-LSTM training time = 8 × CNN (NVIDIA T4 platform)

, Upon summarizing, it is evident that artificial intelligence model predictive control technology exhibits significant performance breakthroughs and high control accuracy across various fields. However, it correspondingly necessitates a demand for high-quality data and substantial computational power for data fitting and analysis. This poses heightened demands on the system’s computational capacity and real-time performance.

At the same time, in existing research, the application of artificial intelligence models remains constrained by several widely recognized limitations, which can be categorized into the following key areas: high dependency on data quality and scale; significant uncertainty in model convergence processes; reliance on large-scale datasets for training; lack of systematic model validation methodologies; dependence on trial-and-error operations for network structure optimization; inadequate supervision mechanisms during training, particularly in data selection and training methodology; complexity in constructing causal inference capabilities; absence of unified standards for determining hidden layer neuron quantities; undefined optimization criteria for training algorithms; and the need for further development of efficient and robust training and verification algorithms [114].

Based on this analysis of the limitations in AI model applications, mitigation strategies and technical pathways can be developed across three primary levels. At the model level, hybrid modeling approaches can be adopted, integrating data-driven models with physical models. This involves applying data-driven models to localized areas of strong nonlinearity within an overarching physical model framework, thereby reducing overall prediction errors. Additionally, implementing knowledge distillation via model simplification and employing joint training of multiple AI models represent viable solutions [115].

From a data perspective [116], enhanced data acquisition strategies can be employed to reduce sampling intervals, thereby improving prediction accuracy. Leveraging digital twin models to supplement experimental data with simulation datasets can enhance data completeness, covering 99.6% of extreme operating conditions. For scenarios with limited sample sizes, transfer learning can significantly enhance predictive performance in small-sample settings, with pre-trained models improving outcomes by 55% for datasets containing ≤1000 data points.

On the computational front, addressing the substantial computational demands of AI model training requires innovative solutions. These include edge-cloud collaboration, which involves decomposing AI models into cloud-based training and edge-based inference components, as well as employing techniques such as quantization compression and hardware acceleration to optimize inference speeds and reduce training power consumption.

Table 5. Analysis of Future Breakthrough Directions in AI-Based Model Predictive Control.

Breakthrough Direction	Core Concept	Technical Characteristics	Application Value	Development Goal
Physics-Embedded AI	Encoding physical laws as model prior knowledge	Integration of first principles & data-driven approaches	Enhance model extrapolation capability and generalizability	Reduce dependency on large training datasets
Self-Healing Systems	Automatic detection of data drift & triggered model updates	Online monitoring + adaptive adjustment	Reduce system maintenance costs and manual intervention	Achieve fully automated model lifecycle management
Energy Efficiency Optimization	Reducing energy consumption of AI control systems themselves	Lightweight models + efficient inference	Improve overall system energy efficiency ratio	Achieve self-energy consumption < 0.1%
Few-Shot Learning	Addressing scarce fault data problems	Meta-learning + transfer learning	Reduce data collection costs and annotation requirements	Train usable models with <100 samples
Physics-Informed Fusion	Encoding physical laws into neural network architectures	Combination of hard constraints + soft constraints	Enhance model interpretability and reliability	Improve cross-condition generalization by 20–30%
Self-Explaining AI	Developing interpretable fault diagnosis models	Visualized decision paths + uncertainty quantification	Meet industrial safety certification requirements	Pass SIL3/ISO certification
Lifelong Learning	Systems continuously adapt to equipment changes	Incremental learning + catastrophic forgetting avoidance	Adapt to equipment aging and environmental changes	Extend effective model lifecycle by 50%
Digital Twin	High-fidelity virtual models support AI training	Multi-physics simulation + real-time data interaction	Reduce field debugging time and risks	Reduce onsite debugging costs by 40–60%

summarizes these strategic directions, providing a roadmap for advancing AI-Based Model Predictive Control technologies.

5. Analysis of the Application of New Control Technologies

The control technology of the advanced CO₂ cycle has made significant progress with certain real-time optimisation performance [117]. However, in view of the limitations of the existing control technology and the lack of control strategies in certain application scenarios, future research can further advance the development of system control technology in the following directions.

5.1. Thermal Management of New Energy Vehicles

The application of CO₂ heat pump air conditioning technology in pure electric vehicles can effectively address the issue of reduced driving range caused by low-temperature heating, thereby enhancing the thermal management efficiency of electric vehicles [118]. Additionally, MPC technology can optimize the CO₂ heat pump air conditioning system in electric vehicles, achieving a balance between energy efficiency and passenger comfort [119]. The transcritical refrigeration CO₂ cycle demonstrates superior performance in low-temperature environments, and appropriate control strategies can further improve the energy efficiency of this system, which is crucial for alleviating the problem of decreased driving range that new energy vehicles face in winter. In response to the interplay between cabin temperature control and energy efficiency in automotive air conditioning systems, MPC control technology can be employed to adjust multiple parameters such as exhaust pressure, outlet temperature, and cabin temperature, thereby achieving real-time multi-objective optimization of cabin temperature and COP [120]. Through the optimization of these control strategies, not only can the development of new energy vehicle technology be promoted, but the broader application of renewable energy can also be facilitated.

5.2. Solar Thermal Power Generation

Due to the fluctuating and intermittent nature of solar thermal power plants, the system is prone to deviate from the design point [118]. Therefore, studying the dynamic characteristics of sCO₂ solar power plants under non-design conditions and developing rapid control strategies are crucial for ensuring reliable and stable operation under parameter fluctuations [119]. By integrating machine learning algorithms and genetic algorithms, multi-objective performance optimization and thermodynamic analysis of solar-driven supercritical CO₂ power cycles can be conducted. This integrated approach can optimize the CO₂ cycle in solar thermal power generation systems, enhancing system efficiency and economic feasibility [120]. Additionally, combining machine learning models with MPC can serve as an alternative model in process control and optimization, improving the application effectiveness of MPC in solar thermal power generation systems. In summary, the application prospects of novel MPC control technology in CO₂ cycle for solar thermal power generation are broad, particularly in improving system efficiency, optimizing energy utilization, integrating renewable energy, and enhancing system performance.

5.3. Aerospace Field

During the operation of ships [121,122] and hypersonic vehicles, their engines generate a significant amount of waste heat, and the aerodynamic heat produced by the intense friction between the vehicle’s shell and air also has a certain impact on their operation [123]. At the same time, the fuel used by ships and aircraft contains abundant cold energy. By applying transcritical refrigeration CO₂ cycle and combining them with artificial intelligence methods, it is possible to predict aerodynamic heat load and adjust the system’s operating state to cope with scenarios where the energy of aerodynamic heat sources fluctuates dramatically. This strategy can not only recover the engine heat and aerodynamic heat for system power generation but also utilize the cold energy of the fuel for refrigeration and system thermal protection, which helps to improve energy utilization efficiency and reduce resource waste.

6. Conclusions and Prospects

6.1. Conclusions

This paper comprehensively reviews the research progress of advanced CO₂ cycle in terms of characteristic selection, control strategies, technical applications, and system optimization. Initially, the paper categorizes typical configurations of advanced CO₂ cycles and delves into their thermal efficiency and application scenarios. Subsequently, the paper meticulously analyzes and compares the system performance under various control strategies, aiming to identify the advantages and limitations of each strategy in enhancing thermal efficiency and response speed, and to determine their most suitable application fields. Finally, the paper introduces novel control technologies such as MPC based on artificial intelligence models and explores their application prospects in CO₂ cycle systems, particularly in improving system performance and optimizing energy conversion. Through these analyses, this paper provides a comprehensive perspective for the design, optimization, and control of CO₂ cycle and points out directions for future research. The main conclusions drawn from the study are as follows:

(1)

A multifaceted study has been conducted on the main variable load control strategies for CO₂ cycles, including inventory control, bypass control, turbine speed control, and turbine throttling control. The first three control methods have been extensively studied and can serve as the primary variable load control strategies. Additionally, the safety characteristics of the system during startup and shutdown conditions should be controlled. Each control strategy has its own advantages and disadvantages, and their applicable ranges vary. Therefore, adopting different control strategies for different conditions or combining multiple control methods is necessary to ensure the safe, stable, efficient, and flexible operation of the unit.

(2)

After analyzing the existing problems in CO₂ cycle control, various novel control technologies for CO₂ cycles have been studied and analyzed, including extremum seeking control, MPC control based on ANN models, and MPC control optimized by PSO. A comparative analysis has been conducted on aspects such as the system’s model dependency, real-time optimization capability, and implementation difficulty. The results indicate that PID control is simple to establish and low in cost, but it is easily affected by environmental factors and changes in system components, leading to reduced control performance. Real-time control technologies represented by extremum seeking control can track system parameters such as maximum thermal efficiency in real-time, but the long optimization process results in extended convergence times for the control system. Model predictive control systems can achieve real-time optimization and rapid convergence, showing promising development prospects. Additionally, new-generation artificial intelligence model control technologies represented by PSO-optimized MPC control possess rapid convergence capabilities and a balance between global and local search capabilities, making them more suitable for new application scenarios with high real-time control requirements.

(3)

A comprehensive exposition of MPC control technology has been provided, including its control theory, methods, applicable scope, strengths, and weaknesses. The actual application status of control strategies in fields such as new energy vehicles, solar thermal power generation, and aerospace has been discussed. By integrating the development trends in the energy sector and artificial intelligence methods, the development direction of control strategies for advanced CO₂ cycle has been explored, and ideas for the practical application of artificial intelligence model predictive control in these systems have been proposed.

(4)

In the existing research, the application of AI-Based Model Predictive Control still faces a series of widely recognized bottleneck problems, mainly including the following aspects: high dependence on data quality and scale; Relying on large-scale data sets for training; Lack of systematic model validation methodology; Network structure optimization still depends on a large number of trial and error operations; The supervision mechanism in the training process is not perfect.

6.2. Prospects

Current research on CO₂ cycle control technology remains limited, with most systems relying on PID controllers and single control strategies. These approaches exhibit significant shortcomings in terms of control accuracy and system response speed. To address these challenges, future research should focus on the following directions:

(1)

Integration of Advanced Control Algorithms: Develop hybrid control frameworks that combine multiple strategies, such as ESC and MPC, to enhance system adaptability and precision.

(2)

Reinforcement Learning-Based Adaptive MPC: Explore the application of artificial intelligence, particularly reinforcement learning, to create adaptive MPC frameworks that dynamically adjust to system changes and optimize performance in real time.

(3)

Digital Twin Integration: Combine digital twin technology with MPC to enable real-time optimization and predictive performance analysis of CO₂ cycles, thereby improving operational efficiency and reliability under varying conditions.

(4)

AI-Driven Model Learning: Leverage advancements in AI to accurately learn CO₂ cycle dynamics, enabling MPC-based control systems that achieve unprecedented levels of precision and efficiency.

With the rapid progress of AI and computational methods, these directions hold significant potential for advancing CO₂ cycle control technology toward smarter, more adaptive, and higher-performing systems. In the future research, we need to solve the limited technical path of AI in engineering application from the three levels of model, data and calculation. At the model level, the accuracy and efficiency are improved through hybrid modeling (physical+ data driven), knowledge distillation and multi model collaboration; At the data level, high-frequency acquisition, digital twinning and transfer learning are used to enhance data quality and coverage; The computing level relies on edge cloud collaboration, quantitative compression and hardware acceleration to achieve efficient reasoning and energy optimization. These directions jointly promote the reliability, real-time and applicability of AI model in predictive control.