A method for controlling temperature balance of a high-temperature superconducting magnet using circulating cold helium

Abstract

This invention provides a temperature balancing control method for high-temperature superconducting magnets using circulating cold helium gas. This method, which belongs to the field of superconducting magnet temperature control, includes: calculating the difference between the target temperature and the actual measured temperature to obtain the steady-state error for each magnet group; constructing a mathematical model of the cold helium cooling system based on singular perturbation theory; clarifying control objectives to improve the overall performance and stability of the cold helium cooling system; and constructing a singular perturbation control model based on the variable domain Smith. This invention can improve the response speed and accuracy of the cold helium cooling system.

Description

Technical Field

[0001] The invention belongs to the field of superconducting magnet temperature control, and in particular relates to a temperature balancing control method for a high-temperature superconducting magnet using circulating cold helium gas. Background Art

[0002] The stable operation of superconducting synchronous condensers (SCs) relies on an efficient cooling system, particularly in cryogenic environments. The efficiency and stability of cooling technology directly impact the performance of superconducting materials. The superconducting magnets in high-temperature superconducting synchronous condensers typically operate at temperatures around 20 K, requiring the system to maintain stable operation within a cryogenic vacuum vessel. To ensure the timely removal of heat generated by the superconducting magnets and other key components during operation, an efficient and precise cooling solution is required. Superconducting rotor magnets generate multiple thermal loads during operation, primarily including heat leakage from the superconducting magnets themselves, heat leakage through the cooling system due to radiation or conduction, and heat loss from the electrical conductors. These thermal loads must be removed through effective cooling methods to ensure the superconducting magnets maintain the required cryogenic temperature. Currently, refrigeration systems that circulate cold helium are widely used to cool superconducting magnets, leveraging the excellent thermophysical properties of cold helium at low temperatures to provide a stable cryogenic environment.

[0003] Due to its excellent cryogenic properties, cold helium plays a key role in temperature control in high-temperature superconducting magnets. A cold helium circulation system enables efficient heat exchange and maintains the temperature of the superconducting magnet within a stable operating range. Numerous studies at home and abroad have demonstrated that cold helium circulation cooling technology can effectively address the temperature control issues of superconducting magnets and exhibits broad prospects in various application scenarios. For example, Dai Yijun et al. employed a closed cold helium circulation cooling system in their research. Using the lumped parameter method to analyze the rotor cooling process, they successfully reduced the rotor temperature to 25 K, achieving a significant technological breakthrough. Shi Zhengjun et al., targeting the cooling needs of the 10 Mvar high-temperature superconducting synchronous condenser of the China Southern Power Grid, designed a circulating cold helium low-temperature thermal management system in a vacuum environment. Test results demonstrated that the system can stably control the rotor temperature at 22.4 K, verifying its feasibility and stability in practical engineering.

[0004] Although cold helium circulation cooling technology has achieved certain application in superconducting motors and phase regulators, several challenges remain, particularly regarding temperature control. Superconducting magnets can experience temperature imbalances under varying operating conditions, which can affect their superconducting performance and system stability. Therefore, further improving the uniformity of cold helium flow and the balance of temperature distribution is crucial for enhancing cooling system performance. Optimizing parameters such as cold helium flow rate, pressure, and flow path remains a research hotspot.

[0005] In summary, although the existing cooling technology has solved the temperature control problem of superconducting magnets to a certain extent, in practical applications, how to achieve more precise and balanced temperature regulation is still the key to improving the performance of the cooling system of superconducting synchronous condensers. Summary of the Invention

[0006] To solve the above technical problems, the present invention provides a temperature balance control method for a high-temperature superconducting magnet using circulating cold helium. By optimizing the flow and cooling parameters of the cold helium, the temperature balance and system stability of the superconducting magnet are further improved to meet the higher performance requirements of future superconducting synchronous condensers.

[0007] In order to achieve the above object, the present invention adopts the following technical solutions:

[0008] A method for controlling temperature balance of a high-temperature superconducting magnet by circulating cold helium gas comprises the following steps:

[0009] Step 1: Calculate the difference between the target temperature and the actual measured temperature to obtain the steady-state error of each set of magnets;

[0010] Step 2. Based on singular perturbation theory, a mathematical model of the cold helium cooling system is constructed. By decomposing the cold helium cooling system into a fast subsystem and a slow subsystem, the fast subsystem controls the rapid adjustment of the cold helium flow rate and the long-term stability of the magnet temperature, respectively. The fast subsystem is responsible for rapid response when the steady-state error is large, while the slow subsystem ensures that the entire system maintains temperature equilibrium over a long period of time.

[0011] Step 3: Define control objectives, including maximizing total heat dissipation, minimizing total pressure drop, and optimizing time. By optimizing these objectives, overheating can be avoided while reducing energy consumption and rapidly responding to temperature fluctuations, ensuring that the superconducting magnet always maintains a temperature range of 30 ± 1.5 K, ultimately improving the overall performance and stability of the cold helium cooling system.

[0012] Step 4. Construct a singular perturbation control model based on the variable domain Smith to ensure that the cold helium cooling system can flexibly adjust control parameters under different operating conditions to cope with local hot spots and temperature fluctuations. At the same time, use the Smith predictor to compensate for sensor delays and actuator response lags in the cold helium cooling system, thereby improving the response speed and accuracy of the cold helium cooling system.

[0013] Beneficial effects:

[0014] 1. The present invention provides a temperature balancing control method for a high-temperature superconducting magnet based on the Smith singular perturbation control model with a variable universe. By precisely adjusting the flow rate, pressure, and temperature of cold helium gas, the present invention can effectively address the instability caused by local hot spots and achieve precise control of the temperature of the superconducting magnet, thereby avoiding quenching caused by excessively high or low temperatures and ensuring the stability and safety of the superconducting system during operation.

[0015] 2. The present invention introduces a temperature control strategy for fast and slow subsystems and uses singular perturbation theory to perform hierarchical management of the cold helium cooling system. The fast subsystem can quickly respond to local hot spots, adjust the flow and pressure of cold helium, and promptly eliminate transient temperature deviations; the slow subsystem handles the distribution and stability of the overall temperature field, ensuring temperature balance among multiple magnets, thereby avoiding quench problems caused by excessive temperature differences.

[0016] 3. This invention combines variable universe control with fuzzy adaptive PID control, dynamically adjusting control parameters based on real-time steady-state error. When the steady-state error is small, the system maintains low control sensitivity, reducing energy consumption. When the steady-state error increases, the system automatically increases sensitivity, rapidly adjusting the cold helium flow and pressure to maximize heat dissipation and minimize system pressure drop, thereby improving cooling efficiency and reducing energy consumption.

[0017] 4. This invention incorporates the Smith Predictor for time delay compensation, effectively resolving the time delay issue between the temperature sensor and the actuator. By predicting system dynamics in advance, the Smith Predictor can compensate for delays during the temperature adjustment process, thereby improving system response speed, avoiding temperature oscillations and overshoot caused by hysteresis, and ensuring precise regulation of the cold helium flow rate. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 This is a flow chart of the temperature balance control method for a high-temperature superconducting magnet using circulating cold helium gas according to the present invention;

[0019] Figure 2 Detailed steps of the improved singular perturbation theory control strategy;

[0020] Figure 3 Schematic diagram of the Smith predictive PID controller structure. DETAILED DESCRIPTION

[0021] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only intended to illustrate the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0022] The proposed method for temperature balancing control of high-temperature superconducting magnets using circulating cold helium gas effectively addresses the temperature balancing issue in actual operation of high-temperature superconducting magnets by combining variable domain control with a Smith predictor. In the cold helium circulation system, local hot spots are first addressed by precisely controlling the cold helium flow and pressure, and the magnet temperature is adjusted in real time to ensure that all magnets operate stably within a temperature range of 30±1.5K. By combining the coordinated control of the fast and slow subsystems with a steady-state error feedback mechanism, the method enables dynamic adjustment under different operating conditions, ensuring maximum cooling efficiency and optimized energy efficiency.

[0023] The cold helium circulation system employs a variable universe control strategy, adaptively adjusting control parameters based on steady-state errors. This ensures low energy consumption and stability during normal operation with small temperature differences. In the event of localized hotspots with large temperature differences, the cold helium flow rate is rapidly increased to eliminate them. A Smith predictor compensates for delays between sensors and actuators within the system, ensuring timely and accurate control responses.

[0024] The cold helium circulation system also features a redundant design and fault response mechanism. Through real-time monitoring and temperature sensor feedback, it can promptly respond to sudden faults, ensuring stable operation of the cold helium circulation system and consistent magnet temperature. This invention boasts strong adaptability and robustness, effectively addressing issues such as uneven temperature distribution and excessive temperature fluctuations in the cold helium cooling system in practical applications, ensuring that high-temperature superconducting magnets maintain optimal operating conditions under various operating conditions.

[0025] like Figure 1 As shown, the temperature balance control method of a high-temperature superconducting magnet circulating cold helium in an embodiment of the present invention includes the following steps:

[0026] Step 1: Calculate the steady-state error, including:

[0027] Step 1.1. Build a temperature accuracy model:

[0028] The temperature accuracy model includes the calculation and feedback mechanism of temperature error, the dynamic change of temperature distribution, and the coupling relationship between cold helium flow and temperature regulation. These features work together to ensure that the system can accurately monitor and adjust the temperature of the superconducting magnet to keep it within the target range:

[0029] In a high-temperature superconducting condenser, six magnets comprise 18 sets of double-panel coils. Each magnet set must maintain a temperature range of 30 ± 1.5 K during operation. Because magnet temperature directly impacts superconducting performance, any single double-panel coil temperature exceeding this precision range could cause a partial quench (destruction of the superconducting state) in the magnet, compromising the stability and safety of the entire system. Therefore, precisely controlling the temperature uniformity of each magnet set and maintaining consistent thermal equilibrium across the coils are key tasks of the cold helium cooling system. A temperature accuracy model, designed to ensure both uniformity and precision, was implemented to monitor and regulate the temperatures of the six magnets and 18 double-panel coils.

[0030] Step 1.2 describes the error state of the temperature field:

[0031] Define the real-time temperature of the i-th double-pancake coil as , the target temperature is The steady-state error is defined by the following formula: :

[0032] (1)

[0033] In order to accurately control the temperature of each set of magnets and each coil in the entire cooling system, an index To represent each specific coil, the range is , indicating the Hedi A double-pancake coil, t represents time.

[0034] The steady-state error needs to satisfy:

[0035] (2)

[0036] Step 1.3 Control the uniformity of the temperature gradient:

[0037] In order to avoid the quench problem caused by uneven temperature of different coils, it is defined that represents the actual temperature of the th double-pancake coil, represents the actual temperature of the th double-pancake coil, and the maximum temperature difference is It can be expressed as:

[0038] (3)

[0039] in, That is, the real-time temperature of the i-th group of magnets is ; That is, the real-time temperature of the jth group of magnets.

[0040] Control objectives ensure:

[0041] (4)

[0042] Step 2. Construct a mathematical model of the system based on singular perturbation theory. By decomposing the system into fast and slow subsystems, the present invention can independently control the rapid adjustment of the cold helium flow rate and the long-term stability of the magnet temperature. The fast subsystem is responsible for rapid response when large steady-state errors occur, while the slow subsystem ensures that the entire system maintains temperature equilibrium over a long period of time.

[0043] The cold helium cooling system exhibits multi-timescale characteristics: temperature distribution changes slowly, while flow and pressure vary rapidly. To effectively address this multi-timescale phenomenon, singular perturbation theory is employed to divide the system into two subsystems: a fast subsystem responsible for responding to local hot spots in real time, rapidly eliminating transient temperature deviations by adjusting the cold helium flow and pressure. The slow subsystem focuses on long-term temperature balance, managing the overall temperature field distribution and stability, ensuring consistent temperatures across multiple magnet groups during operation.

[0044] The mathematical models of the fast and slow systems are:

[0045] (5)

[0046] (6)

[0047] in, is a slow-changing variable, namely the temperature state, is a fast-changing variable, namely the flow rate and pressure state of helium, is the time scale ratio, satisfying . Represents a slowly changing variable The rate of change of, that is, the change of the temperature field in the system. Represents fast-changing variables The rate of change. and They represent the functions of the slow system and the fast system respectively, and t represents time.

[0048] Construct the temperature control equation of the fast and slow subsystems:

[0049] Based on the existing singular perturbation theory, the temperature dynamic equation can be described as:

[0050] (7)

[0051] in, It is Real-time temperature of the group magnet; is the cooling coefficient related to the cold helium flow rate; is the generation coefficient describing the local hotspot; is the local heat generated inside the magnet. The target temperature is .

[0052] Construct the governing equations for the fast subsystem:

[0053] The generation of local hot spots is instantaneous and unpredictable, requiring the fast subsystem to respond quickly to prevent the temperature from exceeding the range of 30 ± 1.5 K. The fast subsystem mainly achieves rapid heat dissipation by adjusting the flow and pressure of cold helium.

[0054] In the tachyon system, the flow rate and pressure of the cold helium need to be adjusted dynamically to ensure that the steady-state error is quickly eliminated:

[0055] (8)

[0056] in, It is cold helium in the Flow rate in group magnets; It is the control input of the cold helium pump

[0057] This equation ensures that the temperature changes of all coils remain within the allowed accuracy range by adjusting the flow rate of each magnet flow channel in real time.

[0058] Construct the control equations of the slow subsystem:

[0059] While the fast subsystem can quickly smooth out short-term temperature fluctuations, steady-state temperature control requires a slow subsystem to achieve global temperature balance. The slow subsystem focuses on the long-term evolution of temperature, which is relevant to the system's overall heat transfer and distribution. It can better handle temperature balance across multiple sets of magnets and their coils.

[0060] Considering the long-term heat conduction and temperature balance of the system, the temperature dynamics of the slow subsystem can be described as:

[0061] (9)

[0062] in, It is Heat capacity of the magnet group; is the thermal conductivity, describing the heat exchange rate between magnets; It is Real-time temperature of the group magnet; It is the outside world Thermal power of the magnet group; It refers to the heat taken away by the cold helium.

[0063] In practical applications, the fast and slow subsystems must work together to ensure the stability and high-precision temperature control of the entire cooling system. The fast subsystem rapidly adjusts the flow of cold helium when a local hotspot appears, ensuring that the hotspot temperature quickly returns to the target range. The slow subsystem balances the temperature between each magnet group, ensuring long-term temperature consistency and stability across the entire system.

[0064] The synergy between the two can be achieved in the following ways: the fast subsystem compresses local temperature fluctuations into a controllable range; the slow subsystem refines the cold helium adjustment strategy according to the global temperature distribution, so that the temperature tends to be balanced among different magnets.

[0065] Step 3. Define control objectives, including maximizing total heat dissipation, minimizing total pressure drop, and optimizing time. By optimizing these objectives, overheating can be avoided while reducing energy consumption and rapidly responding to temperature fluctuations, ensuring that the superconducting magnet always maintains a temperature range of 30 ± 1.5 K, ultimately improving the overall performance and stability of the cold helium cooling system.

[0066] Step 3.1 Maximize total heat dissipation:

[0067] Heat dissipation Calculated as:

[0068] (10)

[0069] in, is the mass flow rate of cold helium, is the specific heat capacity of cold helium. The temperature of the cold helium gas entering the cooling system (inlet temperature), that is, the temperature of the cold helium gas at the inlet of the cooling equipment; It is the temperature of the cold helium gas flowing out of the cooling system (outlet temperature), that is, the temperature of the cold helium gas at the outlet of the cooling system.

[0070] Step 3.2 Minimize the total pressure drop:

[0071] Pressure drop of cold helium Calculated as:

[0072] (11)

[0073] in, is the friction coefficient, is the pipe length, is the pipe diameter.

[0074] Step 3.3: Time optimal control:

[0075] The time optimization goal is:

[0076] (12)

[0077] in, is the steady-state error, is the rate of change of error. Indicates the total time during the optimization process.

[0078] Step 4: Figure 2 As shown, a singular perturbation control model based on variable domain Smith is constructed:

[0079] When modeling the cold helium cooling system, the dynamic characteristics of local hot spots, as well as potential sensor delays, actuator response lags, and controller parameter switching issues, must be considered. To this end, a variable universe control approach and Smith predictor combined with singular perturbation theory were employed to construct the model.

[0080] Step 4.1. Derive variable domain control:

[0081] The core idea of ​​variable universe control is to dynamically adjust controller parameters based on changes in system state (such as the magnitude of temperature errors or the presence of local hot spots). In this way, the system can achieve optimal control under different operating conditions.

[0082] In high-temperature superconducting phase shifters, local hot spots vary over time, so control parameters need to be dynamically adjusted under different conditions. To achieve this, variable universe control is introduced to adaptively adjust control parameters.

[0083] Design of variable domain controller: In control systems, steady-state error It is used as the basis for adjusting the controller parameters. According to different steady-state error ranges, the controller gain , , Need to change dynamically. Specific settings are:

[0084] When the temperature difference When , the system remains in normal operation, parameter adjustment is slow, and the controller sensitivity is low;

[0085] When the temperature difference When the system enters the hotspot state, the cold helium flow needs to be increased rapidly. The controller is highly sensitive and the parameters change rapidly.

[0086] When the temperature difference is too large or too small, the system enters an overcooling or abnormal state, and the controller needs to reduce the flow rate to avoid overcooling or wasting energy.

[0087] Formula derivation of variable domain controller: First define the steady-state error :

[0088] (13)

[0089] Derivative of the steady-state error:

[0090] (14)

[0091] In order to achieve adaptive control, Indicates adjusting the response speed of the system. Indicates the elimination of steady-state error, Indicates that the controller gain is used to reduce the overshoot of the system. , , All are errors Function:

[0092] = (15)

[0093] (16)

[0094] (17)

[0095] in, is the critical value of the error, which determines the threshold for switching the control parameters. is the sensitivity of the control parameter adjustment. Indicates low proportional gain. When the system temperature error is small or stable, a lower proportional gain is used to reduce the response to temperature fluctuations and reduce over-regulation. Indicates low integral gain. Low integral gain allows the system to avoid excessive integration of small errors accumulated over a long period of time when the error is small, thereby improving the stability of the system; Indicates low differential gain. Low differential gain means that in a stable state, the system responds more smoothly to temperature changes, reducing the risk of overshoot and oscillation. Indicates high proportional gain, which can quickly respond to temperature errors, quickly adjust the flow to eliminate local hot spots, and increase the sensitivity of the control system to temperature errors; Indicates high integral gain, which can accelerate the elimination of steady-state errors in the system and prevent the temperature from deviating from the target value for a long time. It is suitable for the rapid elimination of local hot spots. It means that high differential gain can respond to temperature changes more sensitively, quickly suppress temperature fluctuations and overshoot, and thus avoid further increase in hot spot temperature. In the Smith predictor, Is the gain parameter of the system, K usually refers to one of the three gain parameters, namely proportional gain , integral gain or differential gain ,These gain parameters determine how the controller responds to temperature error, the rate of error change, and the accumulation of error.

[0096] The control input formula of the variable universe PID controller is:

[0097] (18)

[0098] in, represents the control input, which is the control signal for the flow rate of cold helium, and regulates the cooling system to achieve the target temperature; It represents the integral part of the steady-state error function, which calculates the accumulated steady-state error from the start of the system to the current moment. It plays the role of eliminating the steady-state error in the integral term, ensuring that the system maintains accurate temperature during long-term operation. is the integrating variable, which varies in the time range [0.t].

[0099] In practical applications, excessive parameter switching may lead to control discontinuity or oscillation problems, especially when the system frequently switches between different states. To avoid this phenomenon, it is possible to consider introducing a hysteresis control strategy, that is, adding an appropriate hysteresis interval when switching between states to ensure smooth system switching. Therefore, the hysteresis region is introduced. To define the boundary of parameter switching. We can get:

[0100] (19)

[0101] (20)

[0102] (twenty one)

[0103] Apply variable universe control to singularly perturbed models:

[0104] In the singular perturbation model, the fast subsystem describes the rapid changes in the flow and pressure of cold helium gas, and its control input is is generated by the variable domain controller. The state equation of the fast subsystem is:

[0105] (twenty two)

[0106] in, Represents the time scale factor, which is the ratio of the fast and slow time scales. Indicates the gain coefficient of flow regulation, Indicates that cold helium gas The flow rate in the magnetic coil, Indicates the target flow rate of cold helium gas.

[0107] For each set of magnets or cooling circuits, the control objective of the fast subsystem is to respond to local hot spots by adjusting the flow of cold helium and maintain the target temperature. It is obtained through the variable domain controller as follows:

[0108] (twenty three)

[0109] Substituting this formula into the state equation of the fast subsystem, we obtain the dynamic change equation of the cold helium flow rate:

[0110] (twenty four)

[0111] This equation describes how the cold helium flow rate can be adaptively adjusted to cope with changes in steady-state error through variable universe control.

[0112] Step 4.2. Obtain the time delay compensation formula of the Smith Predictor in the system:

[0113] like Figure 3 As shown in the figure, in a cold helium cooling system, there may be a certain time delay between the temperature sensor and the actuator. The temperature data obtained by the sensor is not real-time, which can lead to a response lag in the control system. This requires an estimated dynamic mathematical model of the controlled object. A compensator is connected in parallel with the controlled object to form a new controller. To enable the controller to initiate action in advance, the controlled variable is fed back to the input in advance. This reduces or even eliminates the oscillation and overshoot caused by the lag effect in the original system, thereby improving system stability.

[0114] Figure 3 middle, is the expected value of the system; is the measured value of the system; is the controller transfer function; It is a pure lag link; It is the mathematical model of the system; is the lag time; is the mathematical model of Smith's predictor variable; the function of Smith's predictor compensation is expressed as ;in is a complex variable used in Laplace transform to represent the frequency response of the system.

[0115] The model derivation process of Smith estimator is:

[0116] The core idea of ​​the Smith predictor is to separate the delayed model from the non-delayed model and predict the future output through the model to achieve delay compensation. It is given by the following formula:

[0117] (25)

[0118] in, is the reference input of the system (target temperature); It is the model estimate of the system, representing the output of the system without delay; is the actual output of the system with delay; It's a time delay. It is the transfer function of the controller, which represents the response of the controller to the input signal in the control system.

[0119] The state space representation of the Smith predictor is:

[0120] In actual engineering, the state of the system can be described by the state space model. Assume that the state equation of the system is:

[0121] (26)

[0122] (27)

[0123] in, Represents the state variables of the system; Represents the output of the system; represents the control input; A, B, C, and D are the state matrix, input matrix, output matrix, and transfer matrix of the system respectively;

[0124] Delayed output for:

[0125] (28)

[0126] The Smith predictor uses a delay-free model of the system To predict the system output and perform delay compensation. The estimated output is:

[0127] (29)

[0128] in, Represents the estimated value of the system output, that is, through predicted system outputs; It represents the estimated value of the control input, that is, the estimated control signal; is an estimated state variable that represents a predicted or estimated value of the system state.

[0129] Control Input The formula is:

[0130] (30)

[0131] The complete governing equations for the fast subsystem combined with the Smith predictor are:

[0132] When combined with the Smith predictor, the state equation of the fast subsystem becomes:

[0133] (31)

[0134] in, is the reference temperature; Compensate for the delay predicted by the Smith predictor.

[0135] This equation describes how to compensate for time delays using the Smith Predictor, ensuring that the flow rate of cold helium responds promptly and is not affected by these delays.

[0136] Step 4.3. Build the overall control framework of the HTS condenser helium cooling system:

[0137] After combining the variable domain controller and the Smith predictor, the overall control framework of the system is as follows:

[0138] The fast subsystem is used to quickly adjust the cold helium flow:

[0139] (32)

[0140] The variable domain controller generates control input by adaptively adjusting PID parameters :

[0141] (33)

[0142] The Smith Predictor compensates for system time delays to ensure prompt response.

Claims

1. A method for controlling the temperature balance of a high-temperature superconducting magnet using circulating cold helium, characterized in that: The steps include: Step 1: Calculate the difference between the target temperature and the actual measured temperature to obtain the steady-state error of each set of magnets; Step 2: Based on singular perturbation theory, a mathematical model of the cold helium cooling system is constructed. By decomposing the cold helium cooling system into a fast subsystem and a slow subsystem, the fast subsystem controls the rapid adjustment of the cold helium flow rate and the long-term stability of the magnet temperature, respectively. The fast subsystem is responsible for quickly responding to large steady-state errors, while the slow subsystem ensures that the entire system maintains temperature equilibrium over a long period of time. Step 3: Define control objectives, including maximizing total heat dissipation, minimizing total pressure drop, and optimizing time. By optimizing these objectives, overheating can be avoided while reducing energy consumption and rapidly responding to temperature fluctuations, ensuring that the superconducting magnet always maintains a temperature range of 30 ± 1.5 K, ultimately improving the overall performance and stability of the cold helium cooling system. Step 4. Construct a singular perturbation control model based on the variable domain Smith to ensure that the cold helium cooling system can flexibly adjust control parameters under different operating conditions to cope with local hot spots and temperature fluctuations. At the same time, use the Smith predictor to compensate for sensor delays and actuator response lags in the cold helium cooling system, thereby improving the response speed and accuracy of the cold helium cooling system.

2. The method for controlling temperature balance of a high-temperature superconducting magnet using circulating cold helium according to claim 1, wherein: The step 1 comprises: Step 1.1: Build a temperature accuracy model. This model includes the temperature error calculation and feedback mechanism, the dynamic changes in temperature distribution, and the coupling relationship between cold helium flow and temperature regulation. This ensures accurate monitoring and adjustment of the superconducting magnet temperature to keep it within the target range. Step 1.2 describes the error state of the temperature field: Define the real-time temperature of the i-th double-pancake coil as , the target temperature is ; The steady-state error is defined by the following formula : （1） Among them, the index To represent each specific coil, the range is , indicating the Hedi A double-pancake coil, t represents time; The steady-state error satisfies: （2） Step 1.3 Control the uniformity of the temperature gradient: Maximum temperature difference Expressed as: （3） in, Indicates the The actual temperature of the double-pancake coil, Indicates the The actual temperature of the double-pancake coil; Ensure maximum temperature difference satisfy: （4）。 3. The method for controlling temperature balance of a high-temperature superconducting magnet using circulating cold helium according to claim 1, wherein: In step 2, the mathematical models of the fast system and the slow system are respectively: （5） （6） in, is a slow-changing variable, namely the temperature state, is a fast-changing variable, namely the flow rate and pressure state of helium, is the time scale ratio, satisfying ; Represents a slowly changing variable The rate of change of , that is, the change of the temperature field in the system; Represents fast-changing variables rate of change; and Then they represent the functions of the slow system and the fast system respectively, and t represents time; The temperature control equation for constructing the fast and slow subsystem is: Based on the existing singular perturbation theory, the temperature control equation is described as: （7） in, It is Real-time temperature of the double-pancake coil; is the cooling coefficient related to the cold helium flow rate; is the generation coefficient describing the local hotspot; is the local heat generated inside the magnet; the target temperature is ; The control equation of the fast subsystem is constructed as: （8） in, It is cold helium in the Flow rate in a double-pancake coil; It is the control input of the cold helium pump; The control equation of the slow subsystem is constructed as: （9） in, It is The heat capacity of a double-pancake coil; is the thermal conductivity, describing the heat exchange rate between magnets; It is Real-time temperature of the double-pancake coil; It is the outside world Thermal power of a double-pancake coil; It refers to the heat taken away by the cold helium.

4. The method for controlling temperature balance of a high-temperature superconducting magnet using circulating cold helium gas according to claim 3, wherein: The coordination of the control equations of the fast subsystem and the slow subsystem is achieved in the following way: the fast subsystem compresses local temperature fluctuations into a controllable range; the slow subsystem refines the regulation strategy of the cold helium gas according to the global temperature distribution, so that the temperature tends to be balanced among different magnets.

5. The method for controlling temperature balance of a high-temperature superconducting magnet using circulating cold helium gas according to claim 1, wherein: The step 3 comprises: Step 3.1 Maximize total heat dissipation: Heat dissipation Calculated as: （10） in, is the mass flow rate of cold helium, is the specific heat capacity of cold helium, The temperature of the cold helium gas entering the cooling system, that is, the inlet temperature, is the temperature of the cold helium gas at the inlet of the cooling equipment; The temperature of the cold helium gas flowing out of the cooling system, that is, the outlet temperature, is the temperature of the cold helium gas at the outlet of the cooling system; Step 3.2 Minimize the total pressure drop: Pressure drop of cold helium Calculated as: （11） in, is the friction coefficient, is the pipe length, is the pipe diameter; Step 3.3: Time optimal control: The time optimization goal is: （12） in, is the steady-state error, is the error rate of change; Indicates the total time during the optimization process.

6. The method for controlling temperature balance of a high-temperature superconducting magnet using circulating cold helium gas according to claim 1, wherein: The step 4 comprises: Step 4.1 Derive the specific formula of variable domain control: In a control system, the steady-state error It is used as the basis for adjusting the controller parameters. According to different steady-state error ranges, the controller gain 、 、 Dynamic changes, specific settings are: When the steady-state error When , the system remains in normal operation, parameter adjustment is slow, and the controller sensitivity is low; When the steady-state error When the system enters the hotspot state, the cold helium flow needs to be increased rapidly. The controller is highly sensitive and the parameters change rapidly. When the temperature difference is too large or too small, the system enters an overcooling or abnormal state, and the controller needs to reduce the flow rate to avoid overcooling or wasting energy; First, define the steady-state error : （13） Derivative of the steady-state error: （14） In order to achieve adaptive control, Indicates adjusting the response speed of the system. Indicates the elimination of steady-state error, Indicates that the controller gain is used to reduce the overshoot of the system. 、 、 Both are steady-state errors Function: = （15） （16） （17） in, is the critical value of the error, which determines the threshold for switching the control parameters. is the sensitivity of control parameter adjustment; represents the proportional gain; represents the integral gain; represents the differential gain; Indicates high proportional gain; Indicates high integral gain; This means that a high differential gain can respond more sensitively to temperature changes.

7. The method for controlling temperature balance of a high-temperature superconducting magnet using circulating cold helium gas according to claim 6, wherein: The control input formula of the variable universe PID controller is: （18） in, represents the control input, which is the control signal for the flow rate of cold helium, and regulates the cooling system to achieve the target temperature; Represents the integral part of the steady-state error function, which calculates the steady-state error accumulation from the start of the system to the current moment. is the integral variable, which changes in the time range [0, t]; Introducing hysteresis regions To define the boundary of parameter switching, we get: （19） （20） （21） Apply variable universe control to singularly perturbed models: In the singular perturbation model, the fast subsystem describes the rapid changes in the flow rate and pressure of cold helium gas, and its control input is is generated by the variable domain controller, and the state equation of the fast subsystem is: （22） in, represents the time scale factor, which is the ratio of the fast and slow time scales. Indicates the gain coefficient of flow regulation, Indicates that cold helium gas The flow rate in the magnetic coil, Indicates the target flow rate of cold helium; For each set of magnets or cooling circuits, the control objective of the fast subsystem is to respond to local hot spots by adjusting the flow of cold helium and maintain the target temperature; the control input It is obtained through the variable domain controller as follows: （23） Substituting this formula into the state equation of the fast subsystem, we obtain the dynamic change equation of the cold helium flow rate: （24）。 8. The method for controlling temperature balance of a high-temperature superconducting magnet using circulating cold helium gas according to claim 7, wherein: The step 4 further comprises: Step 4.2 obtains the time delay compensation formula of the Smith Predictor in the system. The model derivation process of the Smith Predictor is as follows: Control Input It is given by the following formula: （25） in, is the reference input of the system (target temperature); It is the model estimate of the system, representing the output of the system without delay; is the actual output of the system with delay; It is a time delay; is the transfer function of the controller, which represents the response of the controller to the input signal in the control system; The state space representation of the Smith predictor is: In actual engineering, the state of the system can be described by the state space model. Assume that the state equation of the system is: （26） （27） in, Represents the state variables of the system; Represents the output of the system; represents the control input; A, B, C, and D are the state matrix, input matrix, output matrix, and transfer matrix of the system respectively; Delayed output for: （28） The Smith predictor uses a delay-free model of the system To predict the output of the system and perform delay compensation, the estimated output is: （29） in, Represents the estimated value of the system output, that is, through predicted system outputs; It represents the estimated value of the control input, that is, the estimated control signal; is an estimated state variable, used to represent the predicted or estimated value of the system state; Control Input The formula is: （30） The complete governing equations for the fast subsystem combined with the Smith predictor are: When combined with the Smith predictor, the state equation of the fast subsystem becomes: （31） in, is the reference temperature; Compensate for the delay predicted by the Smith predictor.

9. The method for controlling temperature balance of a high-temperature superconducting magnet using circulating cold helium gas according to claim 8, wherein: The step 4 further comprises: Step 4.3: Combine the variable domain controller and Smith predictor to build the overall control framework of the HTS CMOS helium cooling system: The fast subsystem is used to quickly adjust the cold helium flow: （32） The variable domain controller generates control input by adaptively adjusting PID parameters : （33） The Smith Predictor compensates for system time delays to ensure prompt response.

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