CN105160047A

CN105160047A - Resistive-type superconducting fault current limiter digital modeling and simulation method based on YBCO superconducting tape

Info

Publication number: CN105160047A
Application number: CN201510204556.7A
Authority: CN
Inventors: 张志丰; 郭腾炫; 杨嘉彬; 邱清泉; 丘明; 靖立伟; 刘怡; 马韬; 许熙; 魏明磊; 孙辰军; 戴少涛; 张国民
Original assignee: Institute of Electrical Engineering of CAS; China Electric Power Research Institute Co Ltd CEPRI; State Grid Hebei Electric Power Co Ltd; State Grid Corp of China SGCC
Current assignee: Institute of Electrical Engineering of CAS; China Electric Power Research Institute Co Ltd CEPRI; State Grid Hebei Electric Power Co Ltd; State Grid Corp of China SGCC
Priority date: 2015-04-27
Filing date: 2015-04-27
Publication date: 2015-12-16
Anticipated expiration: 2035-04-27
Also published as: CN105160047B

Abstract

A digital modeling and simulation method for resistive superconducting current limiter based on YBCO superconducting tape, in establishing the equivalent structure model of YBCO superconducting tape, the equivalent circuit model of YBCO superconducting tape, the heat conduction model of YBCO superconducting tape Based on the model and the circuit model of the resistive superconducting current limiter, the line current and the superconducting strip current are calculated according to the circuit parameters of the given current limiter, the structural parameters of the superconducting strip and the initial operating conditions; according to the YBCO superconducting The heat conduction model of the conductive strip calculates the temperature of the superconducting strip; calculates the resistance of the superconducting strip according to the equivalent circuit model of the YBCO superconducting strip, the temperature and current of the superconducting strip; calculates the resistance of the superconducting strip according to the resistive superconducting current limiter The circuit model is used to calculate the line current and superconducting strip current, and realize the simulation calculation of the resistive superconducting current limiter. The invention can be used for superconducting current limiters with multiple units and complex structures.

Description

Digital modeling and simulation method of resistive superconducting current limiter based on YBCO superconducting strip

技术领域technical field

本发明涉及一种超导限流器数字建模仿真方法。The invention relates to a digital modeling simulation method for a superconducting current limiter.

背景技术Background technique

随着国民经济的快速发展，社会对电力的需求不断增加，带动了电力系统的不断发展，单机和发电厂容量、变电所容量、城市和工业中心负荷不断增加，就使得电力系统之间互联，各级电网中的短路电流水平不断提高，短路故障对电力系统及其相连的电气设备的破坏性也越来越大。然而，大电网的暂态稳定性问题比较突出，其中最重要的原因之一是由于常规电力技术缺乏行之有效的短路故障电流限制技术。With the rapid development of the national economy, the society's demand for electricity continues to increase, which drives the continuous development of the power system. The capacity of single machines and power plants, the capacity of substations, and the load of cities and industrial centers continue to increase, which makes the interconnection of power systems , the level of short-circuit current in the power grid at all levels is increasing continuously, and the destructiveness of short-circuit faults to the power system and its connected electrical equipment is also increasing. However, the transient stability of large power grids is more prominent, and one of the most important reasons is that conventional power technology lacks effective short-circuit fault current limiting technology.

电阻型超导限流器是一种具有远大发展前途的限流器，它利用超导体从超导态到正常态的转变，即从无阻状态到有阻状态的变化来限流，具有自动故障辨识和自动响应故障的特点。电阻型超导限流器的限流故障是非线性变化过程，其变化过程涉及到电磁场和热场的相互耦合。对于电阻型超导限流器的数字建模仿真方法，在文献“ThermalandElectricalAnalysisofCoatedConductorUnderACOver-Current(IEEETRANSACTIONSONAPPLIEDSUPERCONDUCTIVITY,VOL.17,NO.2,JUNE2007)”中有过YBCO超导带材的电磁场和热场相互耦合关系进了建模研究，分析了YBCO超导带材在经历超导态、磁通流阻态和正常态过程中，所产生的电阻及超导的温度等。然而，该数字建模仿真方法简单地把YBCO超导带材在超导态、磁通流阻态和正常态的电阻分段处理，使得在故障限流的暂态过程中，建模无法平滑过渡，经常产生奇异点，函数无法收敛；同时，由于参数多，给分析问题带来了极大的不便。The resistive superconducting current limiter is a current limiter with great development prospects. It uses the transition of the superconductor from the superconducting state to the normal state, that is, the change from the non-resistance state to the resistance state to limit the current. It has automatic fault identification. and automatic response to failure features. The current limiting fault of the resistive superconducting current limiter is a nonlinear change process, and the change process involves the mutual coupling of the electromagnetic field and the thermal field. For the digital modeling and simulation method of resistive superconducting current limiter, in the literature "Thermal and Electrical Analysis of Coated Conductor Under AC Over-Current (IEEE TRANS ACTION SONAPPLIED SUPER CONDUCTIVITY, VOL.17, NO.2, JUNE2007)", there is mutual coupling of electromagnetic field and thermal field of YBCO superconducting strip The relationship is entered into modeling research, and the resistance and superconducting temperature of YBCO superconducting strips are analyzed in the process of experiencing superconducting state, magnetic flux resistance state and normal state. However, this digital modeling and simulation method simply divides the resistance of the YBCO superconducting strip in the superconducting state, the magnetic flux flow resistance state and the normal state, so that the modeling cannot be smooth during the transient process of fault current limitation. Transition often produces singular points, and the function cannot converge; at the same time, due to the large number of parameters, it brings great inconvenience to the analysis problem.

从横截面来看，基于YBCO超导带材从上到下呈层状结构，分为6层，上表面铜层，依次为银层、YBCO层、缓冲层、哈氏合金基底层和下表面的铜层，如图1所示。当超导带材处于超导态时，YBCO层的电阻为零，电流通过YBCO层而导通；当超导带材处于正常态时，YBCO层因失超而产生电阻，电流超导带材的各层中按电阻分配而导通。From the perspective of cross section, based on the YBCO superconducting strip, it has a layered structure from top to bottom and is divided into 6 layers. The copper layer on the upper surface is followed by a silver layer, a YBCO layer, a buffer layer, a Hastelloy base layer and a lower surface. copper layer, as shown in Figure 1. When the superconducting tape is in the superconducting state, the resistance of the YBCO layer is zero, and the current is turned on through the YBCO layer; when the superconducting tape is in a normal state, the YBCO layer produces resistance due to quenching, and the current superconducting tape Each layer of each layer is turned on according to the distribution of resistance.

对于基于YBCO超导带材的电阻型超导限流器，不可避免地需要进行超导体从超导态、磁通流阻态和正常态的电磁场和热场的耦合关系分析，只有准确分析此过程，才能准确计算出YBCO超导带材所产生的电阻，从而实现限流器的设计。For the resistive superconducting current limiter based on YBCO superconducting tape, it is inevitable to analyze the coupling relationship between the electromagnetic field and thermal field of the superconductor from the superconducting state, the magnetic flux resistance state and the normal state. Only by accurately analyzing this process , in order to accurately calculate the resistance produced by the YBCO superconducting strip, so as to realize the design of the current limiter.

发明内容Contents of the invention

本发明的目的是克服已有技术的不足，提出一种基于YBCO超导带材的电阻型超导限流器数字建模仿真方法。本发明不但能够保证建模的准确性，而且避免了超导限流器暂态分析过程中的函数收敛问题，提高了系统仿真效率。The purpose of the invention is to overcome the deficiencies of the prior art, and propose a digital modeling and simulation method for a resistive superconducting current limiter based on a YBCO superconducting strip. The invention not only can ensure the accuracy of modeling, but also avoids the function convergence problem in the transient analysis process of the superconducting current limiter, and improves the system simulation efficiency.

本发明采用的技术方案：The technical scheme adopted in the present invention:

本发明在建立YBCO超导带材等效结构模型、YBCO超导带材等效电路模型、YBCO超导带材热传导模型和电阻型超导限流器电路模型的基础上，根据给定电阻型超导限流器的电路参数、超导带材的结构参数和初始运行条件，在超导无感线圈电阻R_sc为零的前提下，计算线路电流的初始值和超导带材电流的初始值；根据YBCO超导带材的热传导模型计算超导带材的温度；根据YBCO超导带材等效电路模型、超导带材温度和电流，计算超导带材的电阻；根据电阻型超导限流器的电路模型，计算并反馈线路电流和超导带材电流，实现电阻型超导限流器的建模仿真。In the present invention, on the basis of establishing the equivalent structure model of YBCO superconducting strip, the equivalent circuit model of YBCO superconducting strip, the heat conduction model of YBCO superconducting strip and the circuit model of resistive superconducting current limiter, according to the given resistive type The circuit parameters of the superconducting current limiter, the structural parameters and initial operating conditions of the superconducting strip, on the premise that the resistance R _sc of the superconducting non-inductive coil is zero, calculate the initial value of the line current and the initial value of the superconducting strip current value; calculate the temperature of the superconducting strip according to the heat conduction model of the YBCO superconducting strip; calculate the resistance of the superconducting strip according to the equivalent circuit model of the YBCO superconducting strip, the temperature and current of the superconducting strip; The circuit model of the conductive current limiter is calculated and fed back to the line current and the superconducting strip current to realize the modeling and simulation of the resistive superconducting current limiter.

本发明建模仿真方法的具体步骤如下：The concrete steps of modeling simulation method of the present invention are as follows:

步骤1.建立YBCO超导带材等效结构模型；Step 1. Establish the equivalent structure model of YBCO superconducting tape;

建立YBCO超导带材等效结构模型时，为便于分析YBCO超导带材各层在失超过程中的电阻情况，采用以下简化方法：When establishing the equivalent structural model of the YBCO superconducting strip, in order to facilitate the analysis of the resistance of each layer of the YBCO superconducting strip during the quenching process, the following simplified method is adopted:

因YBCO超导带材的缓冲层的厚度很小，电阻很大，可以忽略。同时，又因YBCO超导带材的上表面铜层和下表面铜层的物理特性相同，可以合二为一进行分析。因此，所建立的YBCO超导带材等效结构模型简化分为4层：表面铜层、银层、YBCO层和哈氏合金基底层。表面铜层包括上表面铜层和下表面铜层。Because the thickness of the buffer layer of the YBCO superconducting tape is very small, the resistance is very large and can be ignored. At the same time, because the physical properties of the copper layer on the upper surface and the copper layer on the lower surface of the YBCO superconducting strip are the same, they can be combined into one for analysis. Therefore, the established equivalent structure model of YBCO superconducting strip is simplified and divided into four layers: surface copper layer, silver layer, YBCO layer and Hastelloy base layer. The surface copper layer includes an upper surface copper layer and a lower surface copper layer.

步骤2.建立电阻型超导限流器的电路模型，给定电阻型超导限流器的电路参数：线路等效电感L_s、线路等效电阻r和负载电阻R_load，以及交流电源电压U_s，在超导无感线圈电阻R_sc等于零的前提下，计算线路电流初始值I₀(t)和超导带材电流初始值I_s0(t)；Step 2. Establish the circuit model of the resistive superconducting current limiter, given the circuit parameters of the resistive superconducting current limiter: line equivalent inductance L _s , line equivalent resistance r and load resistance R _load , and AC power supply voltage U _s , on the premise that the resistance R _sc of the superconducting non-inductive coil is equal to zero, calculate the initial value of the line current I ₀ (t) and the initial value of the superconducting strip current I _s0 (t);

电阻型超导限流器包括交流电源U_s、线路等效感抗X、线路等效电阻r、断路器Br、超导无感线圈R_sc和负载电阻R_load。交流电源U_s、线路等效感抗X、线路等效电阻r、断路器Br、超导无感线圈电阻R_sc和负载电阻R_load依次串联，交流电源U_s和负载电阻R_load的一端接地；断路器Br和超导无感线圈电阻R_sc连接在第一连接点A上，超导无感线圈电阻R_sc和负载电阻R_load连接在第二连接点B上。The resistive superconducting current limiter includes an AC power supply U _s , an equivalent inductive reactance X of a line, an equivalent resistance r of a line, a circuit breaker Br, a superconducting non-inductive coil R _sc and a load resistance R _load . AC power supply U _s , line equivalent inductance X, line equivalent resistance r, circuit breaker Br, superconducting non-inductive coil resistance R _sc and load resistance R _load are connected in series in sequence, and one end of AC power supply U _s and load resistance R _load is grounded The circuit breaker Br and the superconducting non-inductive coil resistance R _sc are connected to the first connection point A, and the superconducting non-inductive coil resistance R _sc and the load resistance R _load are connected to the second connection point B.

根据全电路欧姆定律，当超导无感线圈电阻R_sc为零时，电阻型超导限流器的电流与电压关系为：According to Ohm's law of the whole circuit, when the resistance R _sc of the superconducting non-inductive coil is zero, the relationship between the current and voltage of the resistive superconducting current limiter is:

U_s＝I₀X+I₀r+I₀R_load(1)U _s ＝I ₀ X+I ₀ r+I ₀ R _load (1)

式中：R_load为负载电阻，I₀为线路电流，U_s为交流电源电压，X为线路等效感抗，r为线路等效电阻。In the formula: R _load is the load resistance, I ₀ is the line current, U _s is the AC power supply voltage, X is the equivalent inductive reactance of the line, and r is the equivalent resistance of the line.

其中，交流电源电压U_s、线路等效感抗X分别表示为：Among them, the AC power supply voltage U _s and the equivalent inductive reactance X of the line are respectively expressed as:

${U u}_{s the s} ((t t)) = = \sqrt{22} {U u}_{00} cos cos ((22 πft πft)) - - - - - - ((22))$

X＝j2πfL_s(3)X＝j2πfL _s (3)

式中，U₀为交流电源电压有效值，f为交流电源频率，L_s为线路等效电感，j为虚函数符号。In the formula, U ₀ is the effective value of the AC power supply voltage, f is the frequency of the AC power supply, L _s is the equivalent inductance of the line, and j is the symbol of the virtual function.

线路电流的初始值为：The initial value of the line current is:

${I I}_{00} ((t t)) = = \frac{\sqrt{22} {U u}_{00}}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {R R}_{load load}))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((44))$

超导无感线圈的电阻为R_sc(T)，超导无感线圈一般由m根超导带材并联组成，m≥1，根据分流定律：The resistance of the superconducting non-inductive coil is R _sc (T), and the superconducting non-inductive coil is generally composed of m superconducting strips connected in parallel, m≥1, according to the shunt law:

I_s0(t)＝I₀(t)/m(5)I _s0 (t) = I ₀ (t)/m(5)

式中，I₀(t)和I_s0(t)分别为线路电流初始值和超导带材的电流初始值。In the formula, I ₀ (t) and I _s0 (t) are the initial value of the line current and the current initial value of the superconducting strip, respectively.

步骤3.建立YBCO超导带材的热传导模型，给定超导带材的结构参数：超导带材的宽度w、厚度d、长度l_e，所述的厚度d包括表面铜层厚度d₁、银层的厚度d₂、YBCO层的厚度d₃和哈氏合金基底层厚度d₄；给定超导无感线圈的初始运行条件：工作温度Top，计算超导带材的温度T；Step 3. Establish the heat conduction model of the YBCO superconducting strip, given the structural parameters of the superconducting strip: the width w, thickness d, and length l _e of the superconducting strip, and the thickness d includes the surface copper layer thickness d ₁ , the thickness d 2 of the silver layer, the thickness d ₃ of the YBCO layer and the thickness d ₄ of the Hastelloy base layer _; given the initial operating conditions of the superconducting non-inductive coil: working temperature Top, calculate the temperature T of the superconducting strip;

(1)YBCO超导带材直接在液氮浸泡环境中冷却，根据热平衡方程，沿超导带材的长度方向，一维热传导方程为：(1) The YBCO superconducting strip is directly cooled in a liquid nitrogen immersion environment. According to the heat balance equation, along the length direction of the superconducting strip, the one-dimensional heat conduction equation is:

${v v}_{cm cm} ((T T)) {C C}_{cm cm} ((T T)) \frac{&PartialD; &PartialD; T T}{&PartialD; &PartialD; t t} = = \frac{&PartialD; &PartialD;}{&PartialD; &PartialD; t t} [[{K K}_{cm cm} ((T T)) \frac{&PartialD; &PartialD; T T}{&PartialD; &PartialD; x x}]] + + {g g}_{j j} ((T T)) - - {W W}_{cool cool} ((T T)) - - - - - - ((66))$

式中，K_cm(T)为热传导系数，V_cm和C_cm分别为YBCO超导带材的密度与比热容，g_j(T)和W_cool(T)分别是超导带材的焦耳热和散失热量。In the formula, K _cm (T) is the thermal conductivity, V _cm and C _cm are the density and specific heat capacity of the YBCO superconducting tape, g _j (T) and W _cool (T) are the Joule heat and Loss of heat.

YBCO超导带材的密度V_cm与比热容C_cm满足式：The density V _cm and specific heat capacity C _cm of YBCO superconducting tape satisfy the formula:

${v v}_{cm cm} {C C}_{cm cm} = = {v v}_{11} {C C}_{11} \frac{{d d}_{11}}{d d} + + {v v}_{22} {C C}_{22} \frac{{d d}_{22}}{d d} + + {v v}_{33} {C C}_{33} \frac{{d d}_{33}}{d d} {v v}_{44} {C C}_{44} \frac{{d d}_{44}}{d d} - - - - - - ((77))$

其中_，ν₁,ν₂,ν₃,ν₄分别为表面铜层、银层、YBCO层和哈氏合金基底层的密度。d₁为表面铜层厚度，即上表面铜层和下表面铜层的厚度之和，d₂为银层的厚度，d₃为YBCO层的厚度，d₄为哈氏合金基底层的厚度，d为YBCO超导带材的厚度。C₁，C₂，C₃，C₄分别为表面铜层、银层、YBCO层和哈氏合金基底层的比热容。各种密度和比热容均可查阅手册获得，比如《超导电力技术基础》(科学出版社，2011年)。Wherein _, ν ₁ , ν ₂ , ν ₃ , and ν ₄ are the densities of the surface copper layer, silver layer, YBCO layer and Hastelloy base layer, respectively. d1 is the thickness _of the surface copper layer, that is, the sum of the thicknesses of the upper surface copper layer and the lower surface copper layer, _d2 is the thickness of the silver layer, _d3 is the thickness of the YBCO layer, _d4 is the thickness of the Hastelloy base layer, d is the thickness of the YBCO superconducting tape. C ₁ , C ₂ , C ₃ , and C ₄ are the specific heat capacities of the surface copper layer, silver layer, YBCO layer and Hastelloy base layer, respectively. Various densities and specific heat capacities can be obtained by consulting handbooks, such as "Basics of Superconducting Power Technology" (Science Press, 2011).

${K K}_{cm cm} = = {K K}_{11} \frac{{d d}_{11}}{d d} + + {K K}_{22} \frac{{d d}_{22}}{d d} + + {K K}_{33} \frac{{d d}_{33}}{d d} + + {K K}_{44} \frac{{d d}_{44}}{d d} - - - - - - ((88))$

其中，K₁，K₂，K₃，K₄分别为表面铜层，银层，YBCO层和哈氏合金基底层的热导率。各种材料的热导率均可查阅手册获得，比如《超导电力技术基础》(科学出版社，2011年)。Among them, K ₁ , K ₂ , K ₃ , and K ₄ are the thermal conductivity of the surface copper layer, silver layer, YBCO layer and Hastelloy base layer, respectively. The thermal conductivity of various materials can be obtained by consulting handbooks, such as "Basics of Superconducting Power Technology" (Science Press, 2011).

(2)根据焦耳定律，超导带材的焦耳热(2) According to Joule's law, the Joule heat of the superconducting strip

${g g}_{j j} ((T T)) = = {&Integral; &Integral;}_{00}^{t t} {U u}_{S S} ((t t)) {I I}_{S S} ((t t)) dt dt - - - - - - ((99))$

其中，U_S(t)为超导带材的电压，I_S(t)为超导带材的电流。Among them, U _S (t) is the voltage of the superconducting strip, and I _S (t) is the current of the superconducting strip.

(3)超导带材浸泡在液氮中，根据经验公式，散失热量完全由液氮带走，超导带材的散失热量W_cool(T)为：(3) The superconducting strip is immersed in liquid nitrogen. According to the empirical formula, the heat loss is completely taken away by the liquid nitrogen. The heat loss W _cool (T) of the superconducting strip is:

W_cool(T)＝hA(T)(10)W _cool (T) = hA (T) (10)

其中，A为超导带材与液氮的接触面积，即超导带材的表面积；h为液氮传热系数，与超导带材和液氮的温度差ΔT(T-Top)有关，根据实验可知，液氮传热系数h的取值。对应于不同的温差ΔT(T-Top)，热传递过程有对流、核沸腾、过渡态和膜沸腾4种状态，不同的状态对应不同的液氮传热系数h。液氮传热系数h的与拟合结果：Among them, A is the contact area between the superconducting tape and liquid nitrogen, that is, the surface area of the superconducting tape; h is the heat transfer coefficient of liquid nitrogen, which is related to the temperature difference ΔT(T-Top) between the superconducting tape and liquid nitrogen, According to the experiment, it can be known that the value of the heat transfer coefficient h of liquid nitrogen. Corresponding to different temperature differences ΔT(T-Top), the heat transfer process has four states: convection, nucleate boiling, transition state and film boiling, and different states correspond to different heat transfer coefficients h of liquid nitrogen. The fitting result of liquid nitrogen heat transfer coefficient h:

$h h = = \{\begin{matrix} 0.091011 0.091011 * * ΔT ΔT + + 0.089888 0.089888 & ((ΔT ΔT \leq \leq 1010)) \\ 0.5 0.5 * * ΔT ΔT - - 44 & ((1010 < < ΔT ΔT \leq \leq 3131)) \\ - - 0.069 0.069 * * ΔT ΔT + + 2.259 2.259 & ((3131 < < ΔT ΔT \leq \leq 600600)) \\ 0.002833 0.002833 * * ΔT ΔT & ((ΔT ΔT > > 600600)) \end{matrix} - - - - - - ((1111))$

步骤4.建立YBCO超导带材等效电路模型，依据超导带材温度T和超导带材电流I_s(t)，计算YBCO超导带材的电阻r_sc；Step 4. set up the YBCO superconducting strip equivalent circuit model, according to the superconducting strip temperature T and the superconducting strip current I _s (t), calculate the resistance r _sc of the YBCO superconducting strip;

YBCO超导带材的等效电路为4个电阻并联结构。第一电阻r₁为表面铜层电阻，第二电阻r₂为银层的电阻，第三电阻r₃为YBCO层的电阻，第四电阻r₄为哈氏合金基底层的电阻。The equivalent circuit of the YBCO superconducting tape is a parallel structure of four resistors. The first resistance r ₁ is the resistance of the surface copper layer, the second resistance r ₂ is the resistance of the silver layer, the third resistance r ₃ is the resistance of the YBCO layer, and the fourth resistance r ₄ is the resistance of the Hastelloy base layer.

根据所建立的YBCO超导带材等效电路模型，并根据超导带材温度和电流，根据欧姆定律和电路原理，计算YBCO超导带材的等效电路的电阻：According to the established YBCO superconducting strip equivalent circuit model, and according to the temperature and current of the superconducting strip, according to Ohm's law and circuit principle, calculate the resistance of the equivalent circuit of the YBCO superconducting strip:

(1)第一电阻r₁为表面铜层电阻，为铜材料制作。第一电阻r₁是超导带材温度T的函数。根据欧姆定律可得：(1) The first resistor r ₁ is the resistance of the surface copper layer, which is made of copper material. The first resistance r ₁ is a function of the temperature T of the superconducting strip. According to Ohm's law:

${r r}_{11} ((T T)) = = {ρ ρ}_{11} ((T T)) \frac{{l l}_{e e}}{{wd wd}_{11}} - - - - - - ((1212))$

其中，ρ₁(T)为铜的电阻率，是超导带材温度T的函数，w为YBCO超导带材的宽度，d₁为表面铜层厚度，即上表面铜层和下表面铜层的厚度之和，l_e为超导带材的长度。Among them, ρ ₁ (T) is the resistivity of copper, which is a function of the temperature T of the superconducting strip, w is the width of the YBCO superconducting strip, and d ₁ is the thickness of the surface copper layer, that is, the upper surface copper layer and the lower surface copper layer The sum of the thicknesses of the layers, l _e is the length of the superconducting strip.

(2)第二电阻r₂为银层电阻，为银材料制作。第二电阻r₂是超导带材温度T的函数。根据欧姆定律可得：(2) The second resistor r ₂ is a silver layer resistor made of silver material. The second resistance r ₂ is a function of the temperature T of the superconducting strip. According to Ohm's law:

${r r}_{22} ((T T)) = = {ρ ρ}_{22} ((T T)) \frac{{l l}_{e e}}{{wd wd}_{22}} - - - - - - ((1313))$

其中，ρ₂(T)为银的电阻率，w为YBCO超导带材的宽度，d₂为银层的厚度，l_e为超导带材的长度。Among them, ρ ₂ (T) is the resistivity of silver, w is the width of the YBCO superconducting tape, d ₂ is the thickness of the silver layer, l _e is the length of the superconducting tape.

(3)第三电阻r₃为YBCO层电阻。根据超导带材的电阻变化规律，第三电阻r₃的电阻率ρ₃(T)是超导带材温度T的函数，通过对YBCO超导带材的特性测试，并采用数值拟合的方法可得第三电阻r₃的电阻率ρ₃(T)：(3) The third resistor r ₃ is the resistance of the YBCO layer. According to the change law of the resistance of the superconducting strip, the resistivity ρ ₃ (T) of the third resistor r ₃ is a function of the temperature T of the superconducting strip, through the characteristic test of the YBCO superconducting strip, and using the numerical fitting method method can obtain the resistivity ρ ₃ (T) of the third resistor r ₃ :

ρ₃(T)＝ρ₃₁(T)+ρ₃₂(T)(14)ρ ₃ (T) = ρ ₃₁ (T) + ρ ₃₂ (T) (14)

其中，ρ₃₁(T)和ρ₃₂(T)由分段函数拟合而成：Among them, ρ ₃₁ (T) and ρ ₃₂ (T) are fitted by piecewise functions:

${ρ ρ}_{3131} ((T T)) = = \{\begin{matrix} 00 & ((J J < < {J J}_{C C} ((T T)))) \\ {E E.}_{00} {((J J / / Jc Jc ((T T)) - - 11))}^{n no 11} / / J J & ((J J > > {J J}_{C C} ((T T)) \end{matrix} - - - - - - ((1515))$

${ρ ρ}_{3232} ((T T)) = = \{\begin{matrix} 00 & ((J J < < {γJ γJ}_{C C} ((T T)))) \\ {E E.}_{00} {((J J / / Jc Jc ((T T)) - - 11))}^{n no 22} / / J J & ((J J > > {γJ γJ}_{C C} ((T T)) \end{matrix} - - - - - - ((1616))$

其中，Jc(T)是超导带材临界电流密度Jc随温度T变化的函数：Among them, Jc(T) is a function of the critical current density Jc of the superconducting strip with temperature T:

J_c(T)＝J_c0[(T_c-T)/(T_c-T_op)]^1.5(17)J _c (T)＝J _c0 [(T _c -T)/(T _c -T _op )] ^1.5 (17)

其中，Jc为超导带材临界电流密度。J_c0＝2×10⁶A/cm²，为77K下的临界电流密度；Tc＝92K，为YBCO的临界温度，Top为工作温度，在液氮池中为77K。参数n₁＝3；n₂＝20；γ＝2，均为根据超导带材的特性而得到的拟合参数。针对ρ₃₁(T)和ρ₃₂(T)为零的情况，在实际计算过程中，可以取为一个非常小的数据，如10^-19Ω··cm等，避免运算出错。Among them, Jc is the critical current density of the superconducting strip. J _c0 =2×10 ⁶ A/cm ² , which is the critical current density at 77K; Tc=92K, which is the critical temperature of YBCO, and Top is the working temperature, which is 77K in the liquid nitrogen pool. Parameters n ₁ =3; n ₂ =20; γ=2 are fitting parameters obtained according to the characteristics of the superconducting strip. For the case where ρ ₃₁ (T) and ρ ₃₂ (T) are zero, in the actual calculation process, they can be taken as a very small value, such as 10 ^-19 Ω··cm, etc., to avoid calculation errors.

根据欧姆定律，超导带材的电流Is表示为：According to Ohm's law, the current Is of the superconducting strip is expressed as:

I_S(T)＝J(T)/(dw)(18) _IS (T)＝J(T)/(dw)(18)

式中：d为超导带材的厚度，w为YBCO超导带材的宽度。Where: d is the thickness of the superconducting tape, and w is the width of the YBCO superconducting tape.

其中，超导带材的厚度d表示为：Among them, the thickness d of the superconducting tape is expressed as:

d＝(d₁+d₂+d₃+d₄)(19)d=(d ₁ +d ₂ +d ₃ +d ₄ )(19)

式中，d₁为表面铜层厚度，即上表面铜层和下表面铜层的厚度之和，d₂为银层的厚度，d₃为YBCO层的厚度，d₄为哈氏合金基底层的厚度。In the formula, d1 is the thickness _of the surface copper layer, that is, the sum of the thicknesses of the upper surface copper layer and the lower surface copper layer, _d2 is the thickness of the silver layer, _d3 is the thickness of the YBCO layer, and _d4 is the Hastelloy base layer thickness of.

YBCO层的第三电阻r₃，根据欧姆定律可得：The third resistance r ₃ of the YBCO layer can be obtained according to Ohm's law:

${r r}_{33} ((T T,, {I I}_{s the s})) = = {ρ ρ}_{33} ((T T)) \frac{{l l}_{e e}}{{wd wd}_{33}} - - - - - - ((2020))$

式中：d₃为YBCO层厚度，w为YBCO超导带材的宽度，I_s为超导带材电流，l_e为超导带材的长度，T为超导带材的温度。Where: d ₃ is the thickness of the YBCO layer, w is the width of the YBCO superconducting tape, I _s is the current of the superconducting tape, l _e is the length of the superconducting tape, and T is the temperature of the superconducting tape.

(4)第四电阻r₄是哈氏合金基底层电阻，第四电阻r₄是超导带材温度T的函数。根据欧姆定律可得：(4) The fourth resistance r ₄ is the resistance of the Hastelloy base layer, and the fourth resistance r ₄ is a function of the temperature T of the superconducting strip. According to Ohm's law:

${r r}_{44} ((T T)) = = {ρ ρ}_{44} ((T T)) \frac{{l l}_{e e}}{{wd wd}_{44}} - - - - - - ((21 twenty one))$

式中，ρ₄(T)为银的电阻率，w为YBCO超导带材的宽度，d₄为哈氏合金基底层的厚度。In the formula, ρ ₄ (T) is the resistivity of silver, w is the width of the YBCO superconducting strip, and d ₄ is the thickness of the Hastelloy base layer.

(5)按照YBCO超导带材等效电路的电阻并联结构，根据全电路欧姆定律，超导带材的电阻r_sc为：(5) According to the resistance parallel structure of the YBCO superconducting strip equivalent circuit, according to the whole circuit Ohm's law, the resistance r _sc of the superconducting strip is:

${r r}_{sc sc} ((T T,, {I I}_{s the s})) = = \frac{11}{\frac{11}{{r r}_{11}} + + \frac{11}{{r r}_{22}} + + \frac{11}{{r r}_{33}} + + \frac{11}{{r r}_{44}}} - - - - - - ((22 twenty two))$

式中：r₁、r₂、r₃、r₄分别为第一电阻r₁、第二电阻r₂、第三电阻r₃、第四电阻r₄的阻值，I_s为超导带材电流，T为超导带材的温度。In the formula: r ₁ , r ₂ , r ₃ , and r ₄ are the resistance values of the first resistor r ₁ , the second resistor r ₂ , the third resistor r ₃ , and the fourth resistor r ₄ respectively, and I _s is the superconducting tape current, and T is the temperature of the superconducting strip.

在电网稳态运行时，超导带材工作在超导态，即J<J_C(T)时，超导带材的YBCO层的电阻为零，电流都通过YBCO层而导通，不会对电网造成电压降。当电网发生故障时，电网电流增大，超导带材失超而产生电阻，电流将在超导带材的各层之间分配。When the power grid is running in a steady state, the superconducting strip works in the superconducting state, that is, when J<J _C (T), the resistance of the YBCO layer of the superconducting strip is zero, and the current is conducted through the YBCO layer, which will not cause a voltage drop on the grid. When the power grid fails, the grid current increases, and the superconducting tape quenches to generate resistance, and the current will be distributed among the layers of the superconducting tape.

步骤5.根据电阻型超导限流器的电路模型、超导带材的电阻r_sc，计算线路电流I(t)和超导带材电流I_s(t)，输出线路电流I(t)和超导带材电流I_s(t)，并反馈超导带材的温度T给步骤3、反馈超导带材电流I_s(t)给步骤4，实现系统循环建模和仿真；Step 5. Calculate the line current I(t) and the superconducting strip current I _s (t) according to the circuit model of the resistive superconducting current limiter and the resistance r _sc of the superconducting strip, and output the line current I(t) and the superconducting strip current I _s (t), and feed back the temperature T of the superconducting strip to step 3, and feed back the superconducting strip current I _s (t) to step 4 to realize system cycle modeling and simulation;

根据全电路欧姆定律，电阻型超导限流器的电流与电压关系为：According to Ohm's law of the whole circuit, the relationship between the current and voltage of the resistive superconducting current limiter is:

U_s＝IX+Ir+IR_sc+IR_load(23)U _s ＝IX+Ir+IR _sc +IR _load (23)

式中：R_load为负载电阻，R_sc为超导无感线圈电阻，I为线路电流，U_s为交流电源电压，X为线路等效感抗，r为线路等效电阻。In the formula: R _load is the load resistance, R _sc is the superconducting non-inductive coil resistance, I is the line current, U _s is the AC power supply voltage, X is the equivalent inductance of the line, and r is the equivalent resistance of the line.

${U u}_{s the s} ((t t)) = = \sqrt{22} {U u}_{00} cos cos ((22 πft πft)) - - - - - - ((24 twenty four))$

X＝j2πfL_s(25)X＝j2πfL _s (25)

R_sc(T)＝r_sc(T)/m(26)R _sc (T) = r _sc (T)/m(26)

I(t)＝mI_s(t)(27)I(t) = mI _s (t) (27)

式中，r_sc(T)为超导带材的电阻、I(t)和I_s(t)分别为线路电流和超导带材的电流。In the formula, r _sc (T) is the resistance of the superconducting tape, I(t) and I _s (t) are the line current and the current of the superconducting tape, respectively.

根据全电路欧姆定律，把式(24)-(27)代入式(23)，得到电阻型超导限流器的电流与电压关系：According to Ohm's law of the whole circuit, substituting equations (24)-(27) into equation (23), the relationship between current and voltage of the resistive superconducting current limiter is obtained:

$I I ((t t)) = = \frac{\sqrt{22} {U u}_{00}}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {r r}_{sc sc} / / m m + + {R R}_{load load}))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((2828))$

其中，相角θ为：Among them, the phase angle θ is:

$tg tg ((θ θ)) = = \frac{22 πf πf {L L}_{s the s}}{r r + + {r r}_{sc sc} / / m m + + {R R}_{load load}} - - - - - - ((2929))$

因此，通过超导带材的电流：Therefore, the current through the superconducting strip:

${I I}_{s the s} ((t t)) = = \frac{\sqrt{22} {U u}_{00} / / m m}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {r r}_{sc sc} / / m m + + {R R}_{load load}))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((3030))$

超导带材的电流峰值为：The peak current of the superconducting strip is:

${I I}_{sM sM} ((t t)) = = \frac{\sqrt{22} {U u}_{00} / / m m}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {r r}_{sc sc} / / m m + + {R R}_{load load}))}^{22}}} - - - - - - ((3131))$

式中，U₀为交流电源电压有效值，f为交流电源频率，L_s为线路等效电感，m为超导带材的并联根数，r为线路等效电阻，R_load为负载电阻，r_sc为超导带材的电阻。In the formula, U ₀ is the effective value of the AC power supply voltage, f is the frequency of the AC power supply, L _s is the equivalent inductance of the line, m is the number of parallel connections of superconducting strips, r is the equivalent resistance of the line, R _load is the load resistance, r _sc is the electrical resistance of the superconducting strip.

在电网稳态时，电阻型超导限流器工作在超导态，超导无感线圈的电阻为R_sc(T)＝0，根据式(30)，线路电流为：In the steady state of the power grid, the resistive superconducting current limiter works in the superconducting state, and the resistance of the superconducting non-inductive coil is R _sc (T) = 0. According to formula (30), the line current is:

$I I ((t t)) = = \frac{\sqrt{22} {U u}_{00}}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {R R}_{load load}))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((3232))$

在电网发生短路故障时，负载电阻R_load减小为零，同时，电阻型超导限流器工作的正常态而产生电阻，因此，根据式(23)，线路电流为：When a short-circuit fault occurs in the power grid, the load resistance R _load decreases to zero, and at the same time, the resistive superconducting current limiter works in a normal state to generate resistance. Therefore, according to formula (23), the line current is:

$I I ((t t)) = = \frac{\sqrt{22} {U u}_{00}}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {r r}_{sc sc} / / m m))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((3333))$

式中，U₀为交流电源电压有效值，f为交流电源频率，L_s为线路等效电感，m为超导带材的并联根数，r为线路等效电阻，R_load为负载电阻，r_sc为超导带材的电阻。线路等效电感L_s和线路等效电阻r表示了电网短路故障的程度，而超导无感线圈的电阻r_sc/m则体现了电阻型超导限流器的限流能力，当电阻r_sc/m增大时，线路电流I(t)就减小。In the formula, U ₀ is the effective value of the AC power supply voltage, f is the frequency of the AC power supply, L _s is the equivalent inductance of the line, m is the number of parallel connections of superconducting strips, r is the equivalent resistance of the line, R _load is the load resistance, r _sc is the electrical resistance of the superconducting strip. The line equivalent inductance L _s and the line equivalent resistance r indicate the degree of short-circuit faults in the power grid, while the resistance r _sc /m of the superconducting non-inductive coil reflects the current limiting capability of the resistive superconducting current limiter. When the resistance r As _sc /m increases, the line current I(t) decreases.

本发明的主要优点：Main advantage of the present invention:

1.本发明通过对YBCO超导带材结构和电路的简化和等效，使得超导带材在过流失超过程中的暂态电阻变化关系更加明确，便于计算。1. By simplifying and equivalenting the structure and circuit of the YBCO superconducting strip, the present invention makes the relationship of the transient resistance change of the superconducting strip in the process of overcurrent leakage more clear and easy to calculate.

2.本发明通过建立YBCO超导带材的热传导模型，准确分析了限流暂态过程中，超导带材温度变化的原因和热扩散的特征。2. The present invention accurately analyzes the cause of the temperature change of the superconducting strip and the characteristics of thermal diffusion in the current-limiting transient process by establishing the heat conduction model of the YBCO superconducting strip.

3.本发明给出了YBCO超导带材的电路和热传导建模所需的多种参数，使得超导带材建模更加简单。3. The present invention provides a variety of parameters required for modeling the electric circuit and heat conduction of the YBCO superconducting tape, making the modeling of the superconducting tape simpler.

4.本发明所建立的电阻型超导限流器的电路的数字模型，从超导带材的结构和热传导模型出发，解决了超导限流器在限流过程中的复杂的电磁场-热场耦合问题。4. The digital model of the circuit of the resistive superconducting current limiter set up by the present invention, from the structure of the superconducting tape and the heat conduction model, solves the complex electromagnetic field-thermal problem of the superconducting current limiter in the current limiting process. field coupling problem.

5.本发明所建立的电阻型超导限流器的电路的数字模型，使得建模方法简单化，为包含电阻型超导限流器的限流技术的建模和控制，提供了一种简单、易于操作的方法。5. The digital model of the circuit of the resistive superconducting current limiter established by the present invention simplifies the modeling method, and provides a kind of modeling and control for the current limiting technology comprising the resistive superconducting current limiter Simple, easy-to-operate method.

附图说明Description of drawings

图1为YBCO超导带材结构图；Fig. 1 is the structural diagram of YBCO superconducting tape;

图2为本发明的电阻型超导限流器的仿真计算流程图；Fig. 2 is the flow chart of simulation calculation of the resistive superconducting current limiter of the present invention;

图3为本发明YBCO超导带材等效结构模型；Fig. 3 is the equivalent structure model of YBCO superconducting strip material of the present invention;

图4为本发明的电阻型超导限流器电路原理图；Fig. 4 is the circuit schematic diagram of the resistive superconducting current limiter of the present invention;

图5为本发明的液氮传热系数与带材温度和运行温度的温差的关系图；Fig. 5 is the relation figure of the temperature difference of liquid nitrogen heat transfer coefficient and strip temperature and operating temperature of the present invention;

图6为本发明YBCO超导带材等效电路模型。Fig. 6 is the equivalent circuit model of the YBCO superconducting tape of the present invention.

具体实施方式Detailed ways

下面结合附图和具体实施例对本发明作进一步说明。The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

本发明包括建立YBCO超导带材等效结构模型、等效电路模型和热传导模型、电阻型超导限流器的电路原理的数字建模仿真等步骤，如图2所示。The invention includes the steps of establishing the equivalent structure model of the YBCO superconducting strip, the equivalent circuit model, the heat conduction model, and the digital modeling and simulation of the circuit principle of the resistive superconducting current limiter, as shown in FIG. 2 .

本发明的具体步骤如下：Concrete steps of the present invention are as follows:

因YBCO超导带材的缓冲层的厚度很小，电阻很大，可以忽略。同时，又因YBCO超导带材的上表面铜层和下表面铜层的物理特性相同，可以合二为一进行分析。因此，所建立的YBCO超导带材等效结构模型简化分为4层：表面铜层、银层、YBCO层和哈氏合金基底层。表面铜层包括上表面铜层和下表面铜层，如图3所示。Because the thickness of the buffer layer of the YBCO superconducting tape is very small, the resistance is very large and can be ignored. At the same time, because the physical properties of the copper layer on the upper surface and the copper layer on the lower surface of the YBCO superconducting strip are the same, they can be combined into one for analysis. Therefore, the established equivalent structure model of YBCO superconducting strip is simplified and divided into four layers: surface copper layer, silver layer, YBCO layer and Hastelloy base layer. The surface copper layer includes an upper surface copper layer and a lower surface copper layer, as shown in FIG. 3 .

步骤2.建立电阻型超导限流器的电路模型，给定电阻型超导限流器的电路参数：线路等效电感L_s、线路等效电阻r和负载电阻R_load，以及交流电源电压U_s，在超导无感线圈电阻Rsc等于零的前提下，计算线路电流初始值I₀(t)和超导带材电流初始值I_s0(t)；Step 2. Establish the circuit model of the resistive superconducting current limiter, given the circuit parameters of the resistive superconducting current limiter: line equivalent inductance L _s , line equivalent resistance r and load resistance R _load , and AC power supply voltage U _s , on the premise that the resistance Rsc of the superconducting non-inductive coil is equal to zero, calculate the initial value of the line current I ₀ (t) and the initial value of the superconducting strip current I _s0 (t);

电阻型超导限流器的电路原理图如图4所示，包括交流电源U_s、线路等效感抗X、线路等效电阻r、断路器Br、超导无感线圈R_sc和负载电阻R_load。交流电源U_s、线路等效感抗X、线路等效电阻r、断路器Br、超导无感线圈电阻R_sc和负载电阻R_load依次串联，交流电源U_s和负载电阻R_load的一端接地；断路器Br和超导无感线圈电阻R_sc连接在第一连接点A上，超导无感线圈电阻R_sc和负载电阻R_load连接在第二连接点B上。The circuit schematic diagram of a resistive superconducting current limiter is shown in Figure 4, including AC power supply U _s , line equivalent inductive reactance X, line equivalent resistance r, circuit breaker Br, superconducting non-inductive coil R _sc and load resistance R _load . AC power supply U _s , line equivalent inductance X, line equivalent resistance r, circuit breaker Br, superconducting non-inductive coil resistance R _sc and load resistance R _load are connected in series in sequence, and one end of AC power supply U _s and load resistance R _load is grounded The circuit breaker Br and the superconducting non-inductive coil resistance R _sc are connected to the first connection point A, and the superconducting non-inductive coil resistance R _sc and the load resistance R _load are connected to the second connection point B.

U_s＝I₀X+I₀r+I₀R_load(1)U _s ＝I ₀ X+I ₀ r+I ₀ R _load (1)

X＝j2πfL_s(3)X＝j2πfL _s (3)

线路电流的初始值为：The initial value of the line current is:

超导无感线圈的电阻为R_sc(T)，超导无感线圈一般由m(m≥1)根超导带材并联组成，根据分流定律：The resistance of the superconducting non-inductive coil is R _sc (T). The superconducting non-inductive coil is generally composed of m (m≥1) superconducting strips connected in parallel. According to the shunt law:

I_s0(t)＝I₀(t)/m(5)I _s0 (t) = I ₀ (t)/m(5)

步骤3.建立YBCO超导带材的热传导模型，给定超导带材的结构参数：超导带材的宽度w、厚度d和长度l_e，所述的后度d包括了包括表面铜层厚度d₁、银层的厚度d₂、YBCO层的厚度d₃和哈氏合金基底层厚度d₄；给定超导无感线圈的初始运行条件：工作温度Top，计算超导带材的温度T；Step 3. Set up the heat conduction model of YBCO superconducting tape, given the structural parameters of superconducting tape: the width w, thickness d and length l _e of superconducting tape, described back degree d includes including surface copper layer Thickness d ₁ , silver layer thickness d ₂ , YBCO layer thickness d ₃ and Hastelloy base layer thickness d ₄ ; given the initial operating conditions of the superconducting non-inductive coil: operating temperature Top, calculate the temperature of the superconducting strip T;

其中，ν₁,ν₂,ν₃,ν₄分别为表面铜层、银层、YBCO层和哈氏合金基底层的密度。d₁为表面铜层厚度，即上表面铜层和下表面铜层的厚度之和，d₂为银层的厚度，d₃为YBCO层的厚度，d₄为哈氏合金基底层的厚度，d为YBCO超导带材的厚度。C₁，C₂，C₃，C₄分别为表面铜层、银层、YBCO层和哈氏合金基底层的比热容。各种密度和比热容均可查阅手册获得。Wherein, ν ₁ , ν ₂ , ν ₃ , and ν ₄ are the densities of the surface copper layer, silver layer, YBCO layer and Hastelloy base layer, respectively. d1 is the thickness _of the surface copper layer, that is, the sum of the thicknesses of the upper surface copper layer and the lower surface copper layer, _d2 is the thickness of the silver layer, _d3 is the thickness of the YBCO layer, _d4 is the thickness of the Hastelloy base layer, d is the thickness of the YBCO superconducting tape. C ₁ , C ₂ , C ₃ , and C ₄ are the specific heat capacities of the surface copper layer, silver layer, YBCO layer and Hastelloy base layer, respectively. Various densities and specific heat capacities are available in the handbook.

其中，K₁，K₂，K₃，K₄分别为表面铜层，银层，YBCO层和哈氏合金基底层的热导率。各种材料的热导率均可查阅手册获得。Among them, K ₁ , K ₂ , K ₃ , and K ₄ are the thermal conductivity of the surface copper layer, silver layer, YBCO layer and Hastelloy base layer, respectively. The thermal conductivity of various materials can be found in handbooks.

W_cool(T)＝hA(T)(10)W _cool (T) = hA (T) (10)

其中，A为超导带材与液氮的接触面积，即超导带材的表面积；h为液氮传热系数，与超导带材和液氮的温度差ΔT(T-Top)有关，根据实验可知，液氮传热系数h的取值，如图5所示。对应于不同的温差ΔT(T-Top)，热传递过程有对流、核沸腾、过渡态和膜沸腾4种状态，不同的状态对应不同的液氮传热系数h。液氮传热系数h的与拟合结果：Among them, A is the contact area between the superconducting tape and liquid nitrogen, that is, the surface area of the superconducting tape; h is the heat transfer coefficient of liquid nitrogen, which is related to the temperature difference ΔT(T-Top) between the superconducting tape and liquid nitrogen, According to the experiment, the value of the heat transfer coefficient h of liquid nitrogen is shown in Figure 5. Corresponding to different temperature differences ΔT(T-Top), the heat transfer process has four states: convection, nucleate boiling, transition state and film boiling, and different states correspond to different heat transfer coefficients h of liquid nitrogen. The fitting result of liquid nitrogen heat transfer coefficient h:

YBCO超导带材的等效电路为4个电阻并联结构。第一电阻r₁为表面铜层电阻，第二电阻r₂为银层的电阻，第三电阻r₃为YBCO层的电阻，第四电阻r₄为哈氏合金基底层的电阻，如图6所示。The equivalent circuit of the YBCO superconducting tape is a parallel structure of four resistors. The first resistance r ₁ is the resistance of the surface copper layer, the second resistance r ₂ is the resistance of the silver layer, the third resistance r ₃ is the resistance of the YBCO layer, and the fourth resistance r ₄ is the resistance of the Hastelloy base layer, as shown in Figure 6 shown.

其中，Jc(T)是超导带材临界电流密度Jc随温度T的函数：Among them, Jc(T) is the function of the critical current density Jc of the superconducting strip with temperature T:

I_S(T)＝J(T)/(dw)(18) _IS (T)＝J(T)/(dw)(18)

d＝(d₁+d₂+d₃+d₄)(19)d=(d ₁ +d ₂ +d ₃ +d ₄ )(19)

式中，ρ₄(T)为银的电阻率w为YBCO超导带材的宽度，d₄为哈氏合金基底层的厚度。In the formula, ρ ₄ (T) is the resistivity of silver, w is the width of the YBCO superconducting strip, and d ₄ is the thickness of the Hastelloy base layer.

步骤5.根据电阻型超导限流器的电路模型、超导带材的电阻r_sc，计算线路电流I(t)和超导带材电流I_s(t)；输出线路电流I(t)和超导带材电流I_s(t)，并反馈超导带材的温度T给步骤3、反馈超导带材电流I_s(t)给步骤4，实现系统循环建模和仿真。Step 5. According to the circuit model of the resistive superconducting current limiter, the resistance r _sc of the superconducting strip, calculate the line current I (t) and the superconducting strip current I _s (t); output the line current I (t) and the superconducting strip current I _s (t), and feed back the temperature T of the superconducting strip to step 3, and feed back the superconducting strip current I _s (t) to step 4 to realize system cycle modeling and simulation.

U_s＝IX+Ir+IR_sc+IR_load(23)U _s ＝IX+Ir+IR _sc +IR _load (23)

X＝j2πfL_s(25)X＝j2πfL _s (25)

R_sc(T)＝r_sc(T)/m(26)R _sc (T) = r _sc (T)/m(26)

I(t)＝mI_s(t)(27)I(t) = mI _s (t) (27)

其中，相角θ为：Among them, the phase angle θ is:

式中，U₀为交流电源电压有效值，f为交流电源频率，L_s为线路等效电感，m为超导带材的并联根数，r为线路等效电阻，R_load为负载电阻，r_sc为超导带材的电阻。线路等效电感L_s和线路等效电阻r表示了电网短路故障的程度，而超导无感线圈的电阻r_sc/m则体现了电阻型超导限流器的限流能力，当电阻r_sc/m增大时，线路电流I(t)减小。In the formula, U ₀ is the effective value of the AC power supply voltage, f is the frequency of the AC power supply, L _s is the equivalent inductance of the line, m is the number of parallel connections of superconducting strips, r is the equivalent resistance of the line, R _load is the load resistance, r _sc is the electrical resistance of the superconducting strip. The line equivalent inductance L _s and the line equivalent resistance r indicate the degree of short-circuit faults in the power grid, while the resistance r _sc /m of the superconducting non-inductive coil reflects the current limiting capability of the resistive superconducting current limiter. When the resistance r When _sc /m increases, the line current I(t) decreases.

本发明基于YBCO超导带材的电阻型超导限流器数字建模仿真方法所构造的超导带材结构模型和超导带材等效电路模型，简化了超导带材的结构，全面描述了超导带材在由超导态转化为正常态过程的电阻变化规律；通过YBCO超导带材的热传导建模的建立，分析了超导带材各个组分在不同温度下的比热容、电导率等的变化规律，以及带材的散热情况；通过电阻型超导限流器的电路模型的建立，提供了简单而有效的电路和限流器模型和分析方法。通过建模提供了一种有效的基于YBCO超导带材的电阻型超导限流器的故障分析方法，提高了工作效率。通过本发明所提供的建模方法，不仅解决了基于YBCO超导带材的电阻型超导限流器精确分析方法，而且，也可用于多单元和复杂结构的超导限流器的研究，推动了超导限流器在电网中的应用。The present invention is based on the superconducting strip structure model and the superconducting strip equivalent circuit model constructed by the digital modeling and simulation method of the resistive superconducting current limiter of the YBCO superconducting strip, which simplifies the structure of the superconducting strip and is comprehensive Describes the change law of the resistance of the superconducting tape in the process of transforming from the superconducting state to the normal state; through the establishment of the heat conduction modeling of the YBCO superconducting tape, the specific heat capacity of each component of the superconducting tape at different temperatures, The change law of electrical conductivity, etc., and the heat dissipation of the strip; through the establishment of the circuit model of the resistive superconducting current limiter, a simple and effective circuit and current limiter model and analysis method are provided. An effective fault analysis method of the resistive superconducting current limiter based on the YBCO superconducting strip is provided through modeling, and the working efficiency is improved. The modeling method provided by the present invention not only solves the precise analysis method of the resistive superconducting current limiter based on the YBCO superconducting strip, but also can be used for the research of the superconducting current limiter of multi-unit and complex structure, Promote the application of superconducting current limiter in power grid.

Claims

1. A resistance type superconducting current limiter digital modeling simulation method based on YBCO superconducting strip material, it is characterized in that: described modeling simulation method establishes YBCO superconducting strip material equivalent structure model, YBCO superconducting strip Based on the equivalent circuit model of the material, the heat conduction model of the YBCO superconducting strip and the circuit model of the resistive superconducting current limiter, according to the circuit parameters of the given resistive superconducting current limiter, the structural parameters of the superconducting strip and the initial Operating conditions, under the premise that the superconducting non-inductive coil resistance R _sc is zero, calculate the initial value of the line current and the initial value of the superconducting strip current; calculate the temperature of the superconducting strip according to the heat conduction model of the YBCO superconducting strip ; Calculate the resistance of the superconducting strip according to the equivalent circuit model of the YBCO superconducting strip, the temperature and current of the superconducting strip; calculate and feed back the line current and the superconducting strip according to the circuit model of the resistive superconducting current limiter Current, to realize the modeling and simulation of resistive superconducting current limiter.

2. according to the resistance type superconducting current limiter digital modeling simulation method based on YBCO superconducting strip material according to claim 1, it is characterized in that: the concrete steps of described modeling simulation method are as follows:

Step 1. Establish the equivalent structure model of YBCO superconducting tape;

When establishing the equivalent structural model of the YBCO superconducting tape, in order to facilitate the analysis of the resistance of each layer of the YBCO superconducting tape during the quenching process, the thickness of the buffer layer of the YBCO superconducting tape is ignored, and the upper layer of the YBCO superconducting tape is The surface copper layer and the lower surface copper layer are combined into one for analysis; therefore, the established equivalent structural model of the YBCO superconducting strip is simplified into 4 layers: surface copper layer, silver layer, YBCO layer and Hastelloy base layer ; The surface copper layer includes an upper surface copper layer and a lower surface copper layer;

Step 2. Establish the circuit model of the resistive superconducting current limiter, given the circuit parameters of the resistive superconducting current limiter: line equivalent inductance L _s , line equivalent resistance r and load resistance R _load , and AC power supply voltage U _s , on the premise that the resistance R _sc of the superconducting non-inductive coil is equal to zero, calculate the initial value of the line current I ₀ (t) and the initial value of the superconducting strip current I _s0 (t);

Resistive superconducting current limiter includes AC power supply U _s , line equivalent inductance X, line equivalent resistance r, circuit breaker Br, superconducting non-inductive coil R _sc and load resistance R _load ; AC power supply U _s , line, etc. Effective inductive reactance X, line equivalent resistance r, circuit breaker Br, superconducting non-inductive coil resistance R _sc and load resistance R _load are connected in series in sequence, one end of AC power supply U _s and load resistance R _load is grounded; circuit breaker Br and superconducting The non-inductive coil resistance R _sc is connected to the first connection point A, and the superconducting non-inductive coil resistance R _sc and the load resistance R _load are connected to the second connection point B;

According to Ohm's law of the whole circuit, when the resistance R _sc of the superconducting non-inductive coil is zero, the relationship between the current and voltage of the resistive superconducting current limiter is:

U _s ＝I ₀ X+I ₀ r+I ₀ R _load (1)

In the formula: R _load is the load resistance, I ₀ is the line current, U _s is the AC power supply voltage, X is the equivalent inductance of the line, and r is the equivalent resistance of the line;

Among them, the AC power supply voltage U _s and the equivalent inductive reactance X of the line are respectively expressed as:

{U u}_{s the s} ((t t)) = = \sqrt{22} {U u}_{00} cos cos ((22 πft πft)) - - - - - - ((22))

X＝j2πfL _s (3)

In the formula, U ₀ is the effective value of the AC power supply voltage, f is the frequency of the AC power supply, L _s is the equivalent inductance of the line, and j is the symbol of the virtual function;

The initial value of the line current is:

{I I}_{00} ((t t)) = = \frac{\sqrt{22} {U u}_{00}}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {R R}_{load load}))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((44))

The resistance of the superconducting non-inductive coil is R _sc (T), and the superconducting non-inductive coil is generally composed of m superconducting strips connected in parallel, m≥1, according to the shunt law:

I _s0 (t) = I ₀ (t)/m(5)

In the formula, I ₀ (t) is the initial value of the line current, and I _s0 (t) is the initial value of the current of the superconducting strip;

Step 3. Establish the heat conduction model of the YBCO superconducting tape, given the structural parameters of the superconducting tape: the width w of the superconducting tape, including the thickness d _{1 of the surface copper layer, the thickness d 2} _of the silver layer, and the thickness d ₃ of the YBCO layer , the thickness d of Hastelloy base layer thickness d ₄ , and the length l _e , given the initial operating conditions of the superconducting non-inductive coil: working temperature Top, calculate the temperature T of the superconducting strip;

(1) The YBCO superconducting strip is directly cooled in a liquid nitrogen immersion environment. According to the heat balance equation, along the length direction of the superconducting strip, the one-dimensional heat conduction equation is:

{v v}_{cm cm} ((T T)) {C C}_{cm cm} ((T T)) \frac{&PartialD; &PartialD; T T}{&PartialD; &PartialD; t t} = = \frac{&PartialD; &PartialD;}{&PartialD; &PartialD; t t} [[{K K}_{cm cm} ((T T)) \frac{&PartialD; &PartialD; T T}{&PartialD; &PartialD; x x}]] + + {g g}_{j j} ((T T)) - - {W W}_{cool cool} ((T T)) - - - - - - ((66))

In the formula, K _cm (T) is the thermal conductivity, V _cm and C _cm are the density and specific heat capacity of the YBCO superconducting tape, g _j (T) and W _cool (T) are the Joule heat and loss of heat;

The density V _cm and specific heat capacity C _cm of YBCO superconducting tape satisfy the formula:

{v v}_{cm cm} {C C}_{cm cm} = = {v v}_{11} {C C}_{11} \frac{{d d}_{11}}{d d} + + {v v}_{22} {C C}_{22} \frac{{d d}_{22}}{d d} + + {v v}_{33} {C C}_{33} \frac{{d d}_{33}}{d d} + + {v v}_{44} {C C}_{44} \frac{{d d}_{44}}{d d} - - - - - - ((77))

Among them, ν ₁ , ν ₂ , ν ₃ , ν ₄ are the densities of the surface copper layer, silver layer, YBCO layer and Hastelloy base layer respectively, d ₁ is the thickness of the surface copper layer, and is the thickness of the upper surface copper layer and the lower surface copper layer. The sum of the thicknesses of the layers, d ₂ is the thickness of the silver layer, d ₃ is the thickness of the YBCO layer, d ₄ is the thickness of the Hastelloy base layer, d is the thickness of the YBCO superconducting strip; C ₁ , C ₂ , C ₃ and C ₄ are the specific heat capacities of the surface copper layer, silver layer, YBCO layer and Hastelloy base layer respectively;

{K K}_{cm cm} = = {K K}_{11} \frac{{d d}_{11}}{d d} + + {K K}_{22} \frac{{d d}_{22}}{d d} + + {K K}_{33} \frac{{d d}_{33}}{d d} + + {K K}_{44} \frac{{d d}_{44}}{d d} - - - - - - ((88))

Among them, K ₁ , K ₂ , K ₃ , and K ₄ are the thermal conductivity of the surface copper layer, silver layer, YBCO layer and Hastelloy base layer, respectively;

(2) According to Joule's law, the Joule heat of the superconducting strip:

{g g}_{j j} ((T T)) = = {&Integral; &Integral;}_{00}^{t t} {U u}_{S S} ((t t)) {I I}_{S S} ((t t)) dt dt - - - - - - ((99))

Wherein, U _S (t) is the voltage of the superconducting strip, I _S (t) is the electric current of the superconducting strip;

(3) The superconducting strip is immersed in liquid nitrogen. According to the empirical formula, the heat loss is completely taken away by the liquid nitrogen. The heat loss W _cool (T) of the superconducting strip is:

W _cool (T) = hA (T) (10)

Among them, A is the contact area between the superconducting tape and liquid nitrogen, that is, the surface area of the superconducting tape; h is the heat transfer coefficient of liquid nitrogen, which is related to the temperature difference ΔT(T-Top) between the superconducting tape and liquid nitrogen, According to the experiment, the value of the heat transfer coefficient h of liquid nitrogen; corresponding to different temperature differences ΔT(T-Top), the heat transfer process has four states: convection, nucleate boiling, transition state and film boiling, and different states correspond to different states. Liquid nitrogen heat transfer coefficient h; the fitting result of liquid nitrogen heat transfer coefficient h:

h h = = \{\begin{matrix} 0.091011 0.091011 * * ΔT ΔT + + 0.089888 0.089888 & ((ΔT ΔT \leq \leq 1010)) \\ 0.5 0.5 * * ΔT ΔT - - 44 & ((1010 < < ΔT ΔT \leq \leq 3131)) \\ - - 0.069 0.069 * * ΔT ΔT + + 2.259 2.259 & ((3131 < < ΔT ΔT \leq \leq 600600)) \\ 0.002833 0.002833 * * ΔT ΔT & ((ΔT ΔT > > 600600)) \end{matrix} - - - - - - ((1111))

Step 4. set up the YBCO superconducting strip equivalent circuit model, according to the superconducting strip temperature T and the superconducting strip current I _s (t), calculate the resistance r _sc of the YBCO superconducting strip;

The equivalent circuit of the YBCO superconducting tape is a parallel structure of 4 resistors; the first resistor r ₁ is the resistance of the surface copper layer, the second resistor r ₂ is the resistance of the silver layer, the third resistor r ₃ is the resistance of the YBCO layer, and the third resistor r 3 is the resistance of the YBCO layer. Four resistance r ₄ is the resistance of Hastelloy base layer;

According to the established YBCO superconducting strip equivalent circuit model, and according to the temperature and current of the superconducting strip, according to Ohm's law and circuit principle, calculate the resistance of the equivalent circuit of the YBCO superconducting strip:

(1) The first resistance r ₁ is the resistance of the surface copper layer, which is made of copper material; the first resistance r ₁ is a function of the temperature T of the superconducting strip; according to Ohm's law:

{r r}_{11} ((T T)) = = {ρ ρ}_{11} ((T T)) \frac{{l l}_{e e}}{{wd wd}_{11}} - - - - - - ((1212))

Among them, ρ ₁ (T) is the resistivity of copper, which is a function of the temperature T of the superconducting strip, w is the width of the YBCO superconducting strip, and d ₁ is the thickness of the surface copper layer, that is, the upper surface copper layer and the lower surface copper layer The sum of the thicknesses of the layers, l _e is the length of the superconducting strip;

(2) The second resistance r ₂ is a silver layer resistance, which is made of silver material; the second resistance r ₂ is a function of the temperature T of the superconducting strip; according to Ohm's law:

{r r}_{22} ((T T)) = = {ρ ρ}_{22} ((T T)) \frac{{l l}_{e e}}{{wd wd}_{22}} - - - - - - ((1313))

Wherein, ρ ₂ (T) is the resistivity of silver, w is the width of the YBCO superconducting tape, d ₂ is the thickness of the silver layer, l _e is the length of the superconducting tape;

(3) The third resistor r ₃ is the resistance of the YBCO layer; according to the resistance change law of the superconducting strip, the resistivity ρ ₃ (T) of the third resistor r ₃ is a function of the temperature T of the superconducting strip, and the YBCO superconducting The characteristic test of the conductive strip material, and the resistivity ρ ₃ (T) of the third resistor r ₃ can be obtained by numerical fitting method:

ρ ₃ (T) = ρ ₃₁ (T) + ρ ₃₂ (T) (14)

Among them, ρ ₃₁ (T) and ρ ₃₂ (T) are fitted by piecewise functions:

{ρ ρ}_{3131} ((T T)) = = \{\begin{matrix} 00 & ((J J < < {J J}_{C C} ((T T)))) \\ {E E.}_{00} {((J J / / Jc Jc ((T T)) - - 11))}^{n no 11} / / J J & ((J J > > {J J}_{C C} ((T T)) \end{matrix} - - - - - - ((1515))

{ρ ρ}_{3232} ((T T)) = = \{\begin{matrix} 00 & ((J J < < γ γ {J J}_{C C} ((T T)))) \\ {E E.}_{00} {((J J / / Jc Jc ((T T)) - - 11))}^{n no 22} / / J J & ((J J > > {γJ γJ}_{C C} ((T T)) \end{matrix} - - - - - - ((1616))

Among them, Jc(T) is the function of the critical current density Jc of the superconducting strip with temperature T:

J _c (T)＝J _c0 [(T _c -T)/(T _c -T _op )] ^1.5 (17)

Among them, Jc is the critical current density of the superconducting strip; J _c0 = 2×10 ⁶ A/cm ² , which is the critical current density at 77K; Tc = 92K, which is the critical temperature of YBCO, and Top is the working temperature. 77K in the pool; parameters n ₁ =3; n ₂ =20; γ=2, all are fitting parameters obtained according to the characteristics of the superconducting strip; according to Ohm’s law, the current Is of the superconducting strip is expressed as:

_IS (T)＝J(T)/(dw)(18)

Where: d is the thickness of the superconducting tape, w is the width of the YBCO superconducting tape;

Among them, the thickness d of the superconducting tape is expressed as:

d=(d ₁ +d ₂ +d ₃ +d ₄ )(19)

In the formula, d1 is the thickness _of the surface copper layer, that is, the sum of the thicknesses of the upper surface copper layer and the lower surface copper layer, _d2 is the thickness of the silver layer, _d3 is the thickness of the YBCO layer, and _d4 is the Hastelloy base layer thickness of;

The third resistance r ₃ of the YBCO layer can be obtained according to Ohm's law:

{r r}_{33} ((T T,, {I I}_{s the s})) = = {ρ ρ}_{33} ((T T)) \frac{{l l}_{e e}}{{wd wd}_{33}} - - - - - - ((2020))

In the formula: d ₃ is the thickness of the YBCO layer, w is the width of the YBCO superconducting strip, I _s is the superconducting strip current, l _e is the length of the superconducting strip, and T is the temperature of the superconducting strip;

(4) The fourth resistance _r4 is the Hastelloy base layer resistance, and the fourth resistance _r4 is a function of the temperature T of the superconducting strip, which can be obtained according to Ohm's law:

{r r}_{44} ((T T)) = = {ρ ρ}_{44} ((T T)) \frac{{l l}_{e e}}{{wd wd}_{44}} - - - - - - ((21 twenty one))

In the formula, ρ ₄ (T) is the resistivity of silver, w is the width of the YBCO superconducting strip, and d ₄ is the thickness of the Hastelloy base layer;

(5) According to the resistance parallel structure of the YBCO superconducting strip equivalent circuit, according to the whole circuit Ohm's law, the resistance r _sc of the superconducting strip is:

{r r}_{sc sc} ((T T,, {I I}_{s the s})) = = \frac{11}{\frac{11}{{r r}_{11}} + + \frac{11}{{r r}_{22}} + + \frac{11}{{r r}_{33}} + + \frac{11}{{r r}_{44}}} - - - - - - ((22 twenty two))

In the formula: r ₁ , r ₂ , r ₃ , and r ₄ are the resistance values of the first resistor r ₁ , the second resistor r ₂ , the third resistor r ₃ , and the fourth resistor r ₄ respectively, and I _s is the superconducting tape Current, T is the temperature of the superconducting strip;

When the power grid is running in a steady state, the superconducting strip works in the superconducting state, that is, when J<J _C (T), the resistance of the YBCO layer of the superconducting strip is zero, and the current is conducted through the YBCO layer, which will not Cause voltage drop to the grid; when the grid fails, the grid current increases, the superconducting strip quenches and generates resistance, and the current will be distributed among the layers of the superconducting strip;

Step 5. According to the circuit model of the resistive superconducting current limiter, the resistance r _sc of the superconducting strip, calculate the line current I (t) and the superconducting strip current I _s (t); output the line current I (t) and the superconducting strip current I _s (t), and feed back the temperature T of the superconducting strip to step 3, and feed back the superconducting strip current I _s (t) to step 4 to realize system cycle modeling and simulation;

According to Ohm's law of the whole circuit, the relationship between the current and voltage of the resistive superconducting current limiter is:

U _s ＝IX+Ir+IR _sc +IR _load (23)

In the formula: R _load is the load resistance, R _sc is the superconducting non-inductive coil resistance, I is the line current, U _s is the AC power supply voltage, X is the equivalent inductance of the line, and r is the equivalent resistance of the line;

{U u}_{s the s} ((t t)) = = \sqrt{22} {U u}_{00} cos cos ((22 πft πft)) - - - - - - ((24 twenty four))

X＝j2πfL _s (25)

R _sc (T) = r _sc (T)/m(26)

I(t) = mI _s (t) (27)

where r _sc (T) is the resistance of the superconducting strip, I( _t ) and Is (t) are the line current and the current of the superconducting strip, respectively;

According to Ohm's law of the whole circuit, substituting equations (24)-(27) into equation (23), the relationship between current and voltage of the resistive superconducting current limiter is obtained:

I I ((t t)) = = \frac{\sqrt{22} {U u}_{00}}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {r r}_{sc sc} / / m m + + {R R}_{load load}))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((2828))

Among them, the phase angle θ is:

tg tg ((θ θ)) = = \frac{22 πf πf {L L}_{s the s}}{r r + + {r r}_{sc sc} / / m m + + {R R}_{load load}} - - - - - - ((2929))

Therefore, the current through the superconducting strip:

{I I}_{s the s} ((t t)) = = \frac{\sqrt{22} {U u}_{00} / / m m}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {r r}_{sc sc} / / m m + + {R R}_{load load}))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((3030))

The peak current of the superconducting strip is:

{I I}_{sM sM} ((t t)) = = \frac{\sqrt{22} {U u}_{00} / / m m}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {r r}_{sc sc} / / m m + + {R R}_{load load}))}^{22}}} - - - - - - ((3131))

In the formula, U ₀ is the effective value of the AC power supply voltage, f is the frequency of the AC power supply, L _s is the equivalent inductance of the line, m is the number of parallel connections of superconducting strips, r is the equivalent resistance of the line, R _load is the load resistance, r _sc is the resistance of the superconducting strip;

In the steady state of the power grid, the resistive superconducting current limiter works in the superconducting state, and the resistance of the superconducting non-inductive coil is R _sc (T) = 0. According to formula (30), the line current is:

I I ((t t)) = = \frac{\sqrt{22} {U u}_{00}}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {R R}_{load load}))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((3232))

When a short-circuit fault occurs in the power grid, the load resistance R _load decreases to zero, and at the same time, the resistive superconducting current limiter works in a normal state to generate resistance. Therefore, according to formula (23), the line current is:

I I ((t t)) = = \frac{\sqrt{22} {U u}_{00}}{\sqrt{{((22 πf πf {L L}_{s the s}))}^{22} + + {((r r + + {r r}_{sc sc} / / m m))}^{22}}} cos cos ((22 πft πft + + θ θ)) - - - - - - ((3333))

In the formula, U ₀ is the effective value of the AC power supply voltage, f is the frequency of the AC power supply, L _s is the equivalent inductance of the line, m is the parallel number of superconducting strips, r is the equivalent resistance of the line, R _load is the load resistance, r _sc is the resistance of the superconducting strip; the line equivalent inductance L _s and the line equivalent resistance r indicate the degree of short-circuit fault in the power grid, while the resistance r _sc /m of the superconducting non-inductive coil reflects the resistance superconducting limit The current limiting capability of the current device, when the resistance r _sc /m increases, the line current I(t) decreases.