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

Abstract

The invention relates to a resistive-type superconducting fault current limiter digital modeling and simulation method based on a YBCO superconducting tape. On the basis of establishing a YBCO superconducting tape equivalent structure model, a YBCO superconducting tape equivalent circuit model, a YBCO superconducting tape thermal conduction model and a resistive-type superconducting fault current limiter circuit model, the line current and the superconducting tape current are calculated according to given circuit parameters of the current limiter, and structure parameters and initial operation conditions of the superconducting tape; the temperature of the superconducting tape is calculated according to the YBCO superconducting tape thermal conduction model; the resistance of the superconducting tape is calculated according to the YBCO superconducting tape equivalent circuit model, the temperature of the superconducting tape, and the superconducting tape current; and the line current and the superconducting tape current are calculated according to the resistive-type superconducting fault current limiter circuit model, and simulating calculation of the resistive-type superconducting fault current limiter is achieved. The resistive-type superconducting fault current limiter digital modeling and simulation method based on the YBCO superconducting tape can be used for superconducting fault current limiters with multiple units and complex structures.

Description

Based on the resistive superconducting restrictor digital modeling emulation mode of YBCO superconducting tape

Technical field

The present invention relates to a kind of superconductive current limiter digital modeling emulation mode.

Background technology

Along with the fast development of national economy, society constantly increases the demand of electric power, drive the development of electric system, unit and station capacity, substation capacity, city and industrial center load constantly increase, just make between electric system interconnected, short circuit current level in electrical network at different levels improves constantly, and the destructiveness of short trouble to electric system and connected electrical equipment thereof is also increasing.But the transient stability sex chromosome mosaicism of bulk power grid is more outstanding, wherein one of most important reason is because conventional electric power technology lacks effective short-circuit current restriction technologies.

Resistive superconducting restrictor is a kind of restrictor with long-range development prospect, and it utilizes the transformation of superconductor from superconducting state to normal state, namely carrying out current limliting from without hindrance state to there being the change of resistance state, having automatic fault identification and the feature from dynamic response fault.The current limliting fault of resistive superconducting restrictor is non-linear course, and its change procedure relates to intercoupling of electromagnetic field and thermal field.For the digital modeling emulation mode of resistive superconducting restrictor, in document " ThermalandElectricalAnalysisofCoatedConductorUnderACOver-Current (IEEETRANSACTIONSONAPPLIEDSUPERCONDUCTIVITY; VOL.17; NO.2; JUNE2007) ", there were the electromagnetic field of YBCO superconducting tape and the thermal field relation of intercoupling to enter Modeling Research, analyze YBCO superconducting tape in experience superconducting state, magnetic flux flow resistance state and normal state process, the resistance produced and the temperature etc. of superconduction.But this digital modeling emulation mode is simply the resistor section process of YBCO superconducting tape in superconducting state, magnetic flux flow resistance state and normal state, and make in the transient state process of failure current limit, modeling cannot seamlessly transit, and often produces singular point, and function cannot be restrained; Meanwhile, because parameter is many, bring great inconvenience to problem analysis.

From xsect, based on YBCO superconducting tape from top to bottom in layer structure, be divided into 6 layers, upper surface layers of copper, be followed successively by the layers of copper of silver layer, YBCO layer, cushion, Hastelloy basalis and lower surface, as shown in Figure 1.When superconducting tape is in superconducting state, the resistance of YBCO layer is zero, electric current conducting by YBCO layer; When superconducting tape is in normal state, YBCO layer produces resistance because of quench, presses resistance and distribute and conducting in each layer of current superconducting band.

For the resistive superconducting restrictor based on YBCO superconducting tape, inevitably need to carry out the electromagnetic field of superconductor from superconducting state, magnetic flux flow resistance state and normal state and the coupled relation analysis of thermal field, only has this process of accurate analysis, accurately could calculate the resistance that YBCO superconducting tape produces, thus realize the design of restrictor.

Summary of the invention

The object of the invention is the deficiency overcoming prior art, propose a kind of resistive superconducting restrictor digital modeling emulation mode based on YBCO superconducting tape.The present invention not only can ensure the accuracy of modeling, and avoids the function convergence problem in superconductive current limiter transient analysis process, improves system emulation efficiency.

The technical solution used in the present invention:

The present invention is on the basis setting up YBCO superconducting tape Equivalent Structure Model for Calculating, YBCO superconducting tape equivalent-circuit model, YBCO superconducting tape heat conduction model and resistive superconducting current limiter circuit model, according to the given circuit parameter of resistive superconducting restrictor, the structural parameters of superconducting tape and initial operating condition, at superconduction noninductive coil resistance R _scbe under the prerequisite of zero, the initial value of computational scheme electric current and the initial value of superconducting tape electric current; The temperature of superconducting tape is calculated according to the heat conduction model of YBCO superconducting tape; According to YBCO superconducting tape equivalent-circuit model, superconducting tape temperature and electric current, calculate the resistance of superconducting tape; According to the circuit model of resistive superconducting restrictor, calculate and feedback line electric current and superconducting tape electric current, realize the modeling and simulating of resistive superconducting restrictor.

The concrete steps of modeling and simulating method of the present invention are as follows:

Step 1. sets up YBCO superconducting tape Equivalent Structure Model for Calculating;

When setting up YBCO superconducting tape Equivalent Structure Model for Calculating, for ease of analyzing the resistance situation of each layer of YBCO superconducting tape in quench process, adopt following short-cut method:

Because the thickness of the cushion of YBCO superconducting tape is very little, resistance is very large, can ignore.Meanwhile, the upper surface layers of copper again because of YBCO superconducting tape is identical with the physical characteristics of lower surface layers of copper, can unite two into one and analyze.Therefore, the YBCO superconducting tape Equivalent Structure Model for Calculating simplification of setting up is divided into 4 layers: surperficial layers of copper, silver layer, YBCO layer and Hastelloy basalis.Surface layers of copper comprises upper surface layers of copper and lower surface layers of copper.

Step 2. sets up the circuit model of resistive superconducting restrictor, the circuit parameter of given resistive superconducting restrictor: line equivalent inductance L _s, line equivalent resistance r and pull-up resistor R _load, and AC supply voltage U _s, at superconduction noninductive coil resistance R _scunder null prerequisite, computational scheme electric current initial value I ₀(t) and superconducting tape electric current initial value I _s0(t);

Resistive superconducting restrictor comprises AC power U _s, line equivalent induction reactance X, line equivalent resistance r, isolating switch Br, superconduction noninductive coil R _scwith pull-up resistor R _load.AC power U _s, line equivalent induction reactance X, line equivalent resistance r, isolating switch Br, superconduction noninductive coil resistance R _scwith pull-up resistor R _loadconnect successively, AC power U _swith pull-up resistor R _loadone end ground connection; Isolating switch Br and superconduction noninductive coil resistance R _scbe connected on the first tie point A, superconduction noninductive coil resistance R _scwith pull-up resistor R _loadbe connected on the second tie point B.

According to whole circuit law of ohm, as superconduction noninductive coil resistance R _scwhen being zero, the electric current of resistive superconducting restrictor and voltage relationship are:

U _s＝I ₀X+I ₀r+I ₀R _load(1)

In formula: R _loadfor pull-up resistor, I ₀for line current, U _sfor AC supply voltage, X is line equivalent induction reactance, and r is line equivalent resistance.

Wherein, AC supply voltage U _s, line equivalent induction reactance X is expressed as:

$U_s\left(t\right)=\sqrt{2}{U}_0\cos \left(2\mathrm{\πft}\right)---\left(2\right)$

X＝j2πfL _s(3)

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, j is Virtual Function symbol.

The initial value of line current is:

$I_0\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\πf}{L}_s\right)}^2+{(r+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\πft}+\mathrm{\θ})---\left(4\right)$

The resistance of superconduction noninductive coil is R _sc(T), superconduction noninductive coil is generally composed in parallel by m root superconducting tape, m>=1, according to shunting law:

I _s0(t)＝I ₀(t)/m(5)

In formula, I ₀(t) and I _s0t () is respectively the electric current initial value of line current initial value and superconducting tape.

Step 3. sets up the heat conduction model of YBCO superconducting tape, the structural parameters of given superconducting tape: the width w of superconducting tape, thickness d, length l _e, described thickness d comprises surface copper layer thickness d ₁, silver layer thickness d ₂, YBCO layer thickness d ₃with Hastelloy base layer thickness d ₄; The initial operating condition of given superconduction noninductive coil: work temperature op, calculates the temperature T of superconducting tape;

(1) YBCO superconducting tape directly cools in immersing in liquid nitrogen environment, and according to thermal balance equation, along the length direction of superconducting tape, One-dimensional Heat Conduction Equation is:

$v_{\mathrm{cm}}\left(T\right)C_{\mathrm{cm}}\left(T\right)\frac{\∂T}{\∂t}=\frac{\∂}{\∂t}\left[K_{\mathrm{cm}}\left(T\right)\frac{\∂T}{\∂x}\right]+g_j\left(T\right)-W_{\mathrm{cool}}\left(T\right)---\left(6\right)$

In formula, K _cm(T) be heat-conduction coefficient, V _cmand C _cmbe respectively density and the specific heat capacity of YBCO superconducting tape, g _jand W (T) _cool(T) be Joule heat and the dissipated heat of superconducting tape respectively.

The density V of YBCO superconducting tape _cmwith specific heat capacity C _cmmeet formula:

$v_{\mathrm{cm}}{C}_{\mathrm{cm}}=v_1{C}_1\frac{d_1}{d}+v_2{C}_2\frac{d_2}{d}+v_3{C}_3\frac{d_3}{d}{v}_4{C}_4\frac{d_4}{d}---\left(7\right)$

Wherein _,ν ₁, ν ₂, ν ₃, ν ₄be respectively the density of surperficial layers of copper, silver layer, YBCO layer and Hastelloy basalis.D ₁for surface copper layer thickness, i.e. the thickness sum of upper surface layers of copper and lower surface layers of copper, d ₂for the thickness of silver layer, d ₃for the thickness of YBCO layer, d ₄for the thickness of Hastelloy basalis, d is the thickness of YBCO superconducting tape.C ₁, C ₂, C ₃, C ₄be respectively the specific heat capacity of surperficial layers of copper, silver layer, YBCO layer and Hastelloy basalis.Various density and specific heat capacity all can be consulted handbook and be obtained, such as " Superconducting Power Technology basis " (Science Press, 2011).

$K_{\mathrm{cm}}=K_1\frac{d_1}{d}+K_2\frac{d_2}{d}+K_3\frac{d_3}{d}+K_4\frac{d_4}{d}---\left(8\right)$

Wherein, K ₁, K ₂, K ₃, K ₄be respectively surperficial layers of copper, silver layer, the thermal conductivity of YBCO layer and Hastelloy basalis.The thermal conductivity of various material all can be consulted handbook and be obtained, such as " Superconducting Power Technology basis " (Science Press, 2011).

(2) according to Joule law, the Joule heat of superconducting tape

$g_j\left(T\right)={\∫}_0^t{U}_S\left(t\right)I_S\left(t\right)\mathrm{dt}---\left(9\right)$

Wherein, U _st voltage that () is superconducting tape, I _st electric current that () is superconducting tape.

(3) superconducting tape is immersed in liquid nitrogen, rule of thumb formula, and dissipated heat is taken away by liquid nitrogen completely, the dissipated heat W of superconducting tape _cool(T) be:

W _cool(T)＝hA(T)(10)

Wherein, A is the contact area of superconducting tape and liquid nitrogen, i.e. the surface area of superconducting tape; H is liquid nitrogen heat transfer coefficient, relevant, experimentally known with the temperature difference Δ T (T-Top) of superconducting tape and liquid nitrogen, the value of liquid nitrogen heat transfer coefficient h.Corresponding to different temperature difference T (T-Top), heat transfer process has convection current, nuclear boiling, transition state and film boiling 4 kinds of states, the liquid nitrogen heat transfer coefficient h that different states is corresponding different.Liquid nitrogen heat transfer coefficient be h's and fitting result:

$h=\left\{\begin{array}{cc}0.091011*\mathrm{\&Delta;T}+0.089888& (\mathrm{\&Delta;T}\&le;10)\\ 0.5*\mathrm{\&Delta;T}-4& (10<\mathrm{\&Delta;T}\&le;31)\\ -0.069*\mathrm{\&Delta;T}+2.259& (31<\mathrm{\&Delta;T}\&le;600)\\ 0.002833*\mathrm{\&Delta;T}& (\mathrm{\&Delta;T}>600)\end{array}\right.---\left(11\right)$

Step 4. sets up YBCO superconducting tape equivalent-circuit model, according to superconducting tape temperature T and superconducting tape electric current I _st (), calculates the resistance r of YBCO superconducting tape _sc;

The equivalent electrical circuit of YBCO superconducting tape is 4 resistor coupled in parallel structures.First resistance r ₁for surperficial layers of copper resistance, the second resistance r ₂for the resistance of silver layer, the 3rd resistance r ₃for the resistance of YBCO layer, the 4th resistance r ₄for the resistance of Hastelloy basalis.

According to set up YBCO superconducting tape equivalent-circuit model, and according to superconducting tape temperature and electric current, according to Ohm law and circuit theory, calculate the resistance of the equivalent electrical circuit of YBCO superconducting tape:

(1) first resistance r ₁for surperficial layers of copper resistance, for copper product makes.First resistance r ₁it is the function of superconducting tape temperature T.Can obtain according to Ohm law:

$r_1\left(T\right)={\mathrm{\&rho;}}_1\left(T\right)\frac{l_e}{{\mathrm{wd}}_1}---\left(12\right)$

Wherein, ρ ₁(T) being the resistivity of copper, is the function of superconducting tape temperature T, and w is the width of YBCO superconducting tape, d ₁for surface copper layer thickness, i.e. the thickness sum of upper surface layers of copper and lower surface layers of copper, l _efor the length of superconducting tape.

(2) second resistance r ₂for silver layer resistance, for ag material makes.Second resistance r ₂it is the function of superconducting tape temperature T.Can obtain according to Ohm law:

$r_2\left(T\right)={\mathrm{\&rho;}}_2\left(T\right)\frac{l_e}{{\mathrm{wd}}_2}---\left(13\right)$

Wherein, ρ ₂(T) be silver-colored resistivity, w is the width of YBCO superconducting tape, d ₂for the thickness of silver layer, l _efor the length of superconducting tape.

(3) the 3rd resistance r ₃for YBCO layer resistance.According to the resistance changing law of superconducting tape, the 3rd resistance r ₃electricalresistivityρ ₃(T) be the function of superconducting tape temperature T, by the characteristic test to YBCO superconducting tape, and adopt the method for numerical fitting can obtain the 3rd resistance r ₃electricalresistivityρ ₃(T):

ρ ₃(T)＝ρ ₃₁(T)+ρ ₃₂(T)(14)

Wherein, ρ ₃₁and ρ (T) ₃₂(T) formed by piecewise function matching:

${\mathrm{\&rho;}}_{31}\left(T\right)=\left\{\begin{array}{cc}0& (J<J_C\left(T\right))\\ E_0{(J/\mathrm{Jc}\left(T\right)-1)}^{n1}/J& (J>J_C\left(T\right)\end{array}\right.---\left(15\right)$

${\mathrm{\&rho;}}_{32}\left(T\right)=\left\{\begin{array}{cc}0& (J<{\mathrm{\&gamma;J}}_C\left(T\right))\\ E_0{(J/\mathrm{Jc}\left(T\right)-1)}^{n2}/J& (J>{\mathrm{\&gamma;J}}_C\left(T\right)\end{array}\right.---\left(16\right)$

Wherein, Jc (T) is the function that superconduction belt material critical current density Jc changes with temperature T:

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

Wherein, Jc is superconduction belt material critical current density.J _c0=2 × 10 ⁶a/cm ², be the critical current density under 77K; Tc=92K, be the critical temperature of YBCO, Top is working temperature, is 77K in liquid nitrogen bath.Parameter n ₁=3; n ₂=20; γ=2, are the characteristic according to superconducting tape and the fitting parameter obtained.For ρ ₃₁and ρ (T) ₃₂(T) be zero situation, in actual computation process, very little data can be taken as, as 10 ^-19Ω cm etc., avoid computing to make mistakes.

According to Ohm law, the electric current I s of superconducting tape is expressed as:

I _S(T)＝J(T)/(dw)(18)

In formula: d is the thickness of superconducting tape, w is the width of YBCO superconducting tape.

Wherein, the thickness d of superconducting tape is expressed as:

d＝(d ₁+d ₂+d ₃+d ₄)(19)

In formula, d ₁for surface copper layer thickness, i.e. the thickness sum of upper surface layers of copper and lower surface layers of copper, d ₂for the thickness of silver layer, d ₃for the thickness of YBCO layer, d ₄for the thickness of Hastelloy basalis.

3rd resistance r of YBCO layer ₃, can obtain according to Ohm law:

$r_3(T,I_s)={\mathrm{\&rho;}}_3\left(T\right)\frac{l_e}{{\mathrm{wd}}_3}---\left(20\right)$

In formula: d ₃for YBCO layer thickness, w is the width of YBCO superconducting tape, I _sfor superconducting tape electric current, l _efor the length of superconducting tape, T is the temperature of superconducting tape.

(4) the 4th resistance r ₄hastelloy basalis resistance, the 4th resistance r ₄it is the function of superconducting tape temperature T.Can obtain according to Ohm law:

$r_4\left(T\right)={\mathrm{\&rho;}}_4\left(T\right)\frac{l_e}{{\mathrm{wd}}_4}---\left(21\right)$

In formula, ρ ₄(T) be silver-colored resistivity, w is the width of YBCO superconducting tape, d ₄for the thickness of Hastelloy basalis.

(5) according to the resistor coupled in parallel structure of YBCO superconducting tape equivalent electrical circuit, according to whole circuit law of ohm, the resistance r of superconducting tape _scfor:

$r_{\mathrm{sc}}(T,I_s)=\frac{1}{\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4}}---\left(22\right)$

In formula: r ₁, r ₂, r ₃, r ₄be respectively the first resistance r ₁, the second resistance r ₂, the 3rd resistance r ₃, the 4th resistance r ₄resistance, I _sfor superconducting tape electric current, T is the temperature of superconducting tape.

When Power System Steady-state runs, superconducting tape is operated in superconducting state, i.e. J<J _c(T) time, the resistance of the YBCO layer of superconducting tape is zero, and electric current all passes through YBCO layer and conducting, can not cause voltage drop to electrical network.When the grid collapses, power network current increases, superconducting tape quench and produce resistance, and electric current distributes between each layer of superconducting tape.

The circuit model of step 5. according to resistive superconducting restrictor, the resistance r of superconducting tape _sc, computational scheme electric current I (t) and superconducting tape electric current I _s(t), outlet line electric current I (t) and superconducting tape electric current I _s(t), and the temperature T feeding back superconducting tape is to step 3, feedback superconducting tape electric current I _st (), to step 4, realizes systemic circulation modeling and simulation;

According to whole circuit law of ohm, the electric current of resistive superconducting restrictor and voltage relationship are:

U _s＝IX+Ir+IR _sc+IR _load(23)

In formula: R _loadfor pull-up resistor, R _scfor superconduction noninductive coil resistance, I is line current, U _sfor AC supply voltage, X is line equivalent induction reactance, and r is line equivalent resistance.

Wherein, AC supply voltage U _s, line equivalent induction reactance X is expressed as:

$U_s\left(t\right)=\sqrt{2}{U}_0\cos \left(2\mathrm{\&pi;ft}\right)---\left(24\right)$

X＝j2πfL _s(25)

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, j is Virtual Function symbol.

The resistance of superconduction noninductive coil is R _sc(T), superconduction noninductive coil is generally composed in parallel by m root superconducting tape, m>=1, according to shunting law:

R _sc(T)＝r _sc(T)/m(26)

I(t)＝mI _s(t)(27)

In formula, r _sc(T) be resistance, the I (t) and I of superconducting tape _st () is respectively the electric current of line current and superconducting tape.

According to whole circuit law of ohm, wushu (24)-(27) substitute into formula (23), obtain electric current and the voltage relationship of resistive superconducting restrictor:

$I\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(28\right)$

Wherein, phase angle theta is:

$\mathrm{tg}\left(\mathrm{\&theta;}\right)=\frac{2\mathrm{\&pi;f}{L}_s}{r+r_{\mathrm{sc}}/m+R_{\mathrm{load}}}---\left(29\right)$

Therefore, by the electric current of superconducting tape:

$I_s\left(t\right)=\frac{\sqrt{2}{U}_0/m}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(30\right)$

The current peak of superconducting tape is:

$I_{\mathrm{sM}}\left(t\right)=\frac{\sqrt{2}{U}_0/m}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m+R_{\mathrm{load}})}^2}}---\left(31\right)$

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, m is the radical in parallel of superconducting tape, and r is line equivalent resistance, R _loadfor pull-up resistor, r _scfor the resistance of superconducting tape.

When Power System Steady-state, resistive superconducting restrictor is operated in superconducting state, and the resistance of superconduction noninductive coil is R _sc(T)=0, according to formula (30), line current is:

$I\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(32\right)$

When electrical network is short-circuited fault, pull-up resistor R _loadbe reduced to zero, meanwhile, the normal state of resistive superconducting restrictor work and produce resistance, therefore, according to formula (23), line current is:

$I\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m)}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(33\right)$

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, m is the radical in parallel of superconducting tape, and r is line equivalent resistance, R _loadfor pull-up resistor, r _scfor the resistance of superconducting tape.Line equivalent inductance L _sthe degree of short circuit malfunction is illustrated with line equivalent resistance r, and the resistance r of superconduction noninductive coil _sc/ m then embodies the current limiting capacity of resistive superconducting restrictor, as resistance r _scwhen/m increases, line current I (t) just reduces.

Major advantage of the present invention:

1. the present invention is by the simplification of YBCO superconducting tape structure and circuit and equivalence, makes the transient state resistance variations relation of superconducting tape in overcurrent quench process definitely, is convenient to calculate.

2. the present invention is by setting up the heat conduction model of YBCO superconducting tape, in accurate analysis current limliting transient state process, and the reason of superconducting tape temperature variation and the feature of thermal diffusion.

3. The present invention gives the circuit of YBCO superconducting tape and the many kinds of parameters needed for heat transfer modeling, make superconducting tape modeling simpler.

4. the digital model of the circuit of resistive superconducting restrictor set up of the present invention, from structure and the heat conduction model of superconducting tape, solves the electromagnetic field-thermal field coupled problem of the complexity of superconductive current limiter in current limliting process.

5. the digital model of the circuit of resistive superconducting restrictor set up of the present invention, makes modeling method simplify, for comprising the modeling and control of the current limiting technique of resistive superconducting restrictor, provide a kind of simple, be easy to the method that operates.

Accompanying drawing explanation

Fig. 1 is YBCO superconducting tape structural drawing;

Fig. 2 is the simulation calculation flow process figure of resistive superconducting restrictor of the present invention;

Fig. 3 is YBCO superconducting tape Equivalent Structure Model for Calculating of the present invention;

Fig. 4 is resistive superconducting current limiter circuit schematic diagram of the present invention;

Fig. 5 is the graph of a relation of the temperature difference of liquid nitrogen heat transfer coefficient of the present invention and strip temperature and running temperature;

Fig. 6 is YBCO superconducting tape equivalent-circuit model of the present invention.

Embodiment

Below in conjunction with the drawings and specific embodiments, the invention will be further described.

The present invention includes the step such as digital modeling emulation of circuit theory setting up YBCO superconducting tape Equivalent Structure Model for Calculating, equivalent-circuit model and heat conduction model, resistive superconducting restrictor, as shown in Figure 2.

Concrete steps of the present invention are as follows:

Step 1. sets up YBCO superconducting tape Equivalent Structure Model for Calculating;

When setting up YBCO superconducting tape Equivalent Structure Model for Calculating, for ease of analyzing the resistance situation of each layer of YBCO superconducting tape in quench process, adopt following short-cut method:

Because the thickness of the cushion of YBCO superconducting tape is very little, resistance is very large, can ignore.Meanwhile, the upper surface layers of copper again because of YBCO superconducting tape is identical with the physical characteristics of lower surface layers of copper, can unite two into one and analyze.Therefore, the YBCO superconducting tape Equivalent Structure Model for Calculating simplification of setting up is divided into 4 layers: surperficial layers of copper, silver layer, YBCO layer and Hastelloy basalis.Surface layers of copper comprises upper surface layers of copper and lower surface layers of copper, as shown in Figure 3.

Step 2. sets up the circuit model of resistive superconducting restrictor, the circuit parameter of given resistive superconducting restrictor: line equivalent inductance L _s, line equivalent resistance r and pull-up resistor R _load, and AC supply voltage U _s, under the null prerequisite of superconduction noninductive coil resistance Rsc, computational scheme electric current initial value I ₀(t) and superconducting tape electric current initial value I _s0(t);

The circuit theory diagrams of resistive superconducting restrictor as shown in Figure 4, comprise AC power U _s, line equivalent induction reactance X, line equivalent resistance r, isolating switch Br, superconduction noninductive coil R _scwith pull-up resistor R _load.AC power U _s, line equivalent induction reactance X, line equivalent resistance r, isolating switch Br, superconduction noninductive coil resistance R _scwith pull-up resistor R _loadconnect successively, AC power U _swith pull-up resistor R _loadone end ground connection; Isolating switch Br and superconduction noninductive coil resistance R _scbe connected on the first tie point A, superconduction noninductive coil resistance R _scwith pull-up resistor R _loadbe connected on the second tie point B.

According to whole circuit law of ohm, as superconduction noninductive coil resistance R _scwhen being zero, the electric current of resistive superconducting restrictor and voltage relationship are:

U _s＝I ₀X+I ₀r+I ₀R _load(1)

In formula: R _loadfor pull-up resistor, I ₀for line current, U _sfor AC supply voltage, X is line equivalent induction reactance, and r is line equivalent resistance.

Wherein, AC supply voltage U _s, line equivalent induction reactance X is expressed as:

$U_s\left(t\right)=\sqrt{2}{U}_0\cos \left(2\mathrm{\&pi;ft}\right)---\left(2\right)$

X＝j2πfL _s(3)

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, j is Virtual Function symbol.

The initial value of line current is:

$I_0\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(4\right)$

The resistance of superconduction noninductive coil is R _sc(T), superconduction noninductive coil is generally composed in parallel by m (m>=1) root superconducting tape, according to shunting law:

I _s0(t)＝I ₀(t)/m(5)

In formula, I ₀(t) and I _s0t () is respectively the electric current initial value of line current initial value and superconducting tape.

Step 3. sets up the heat conduction model of YBCO superconducting tape, the structural parameters of given superconducting tape: the width w of superconducting tape, thickness d and length l _e, described rear degree d includes and comprises surface copper layer thickness d ₁, silver layer thickness d ₂, YBCO layer thickness d ₃with Hastelloy base layer thickness d ₄; The initial operating condition of given superconduction noninductive coil: work temperature op, calculates the temperature T of superconducting tape;

(1) YBCO superconducting tape directly cools in immersing in liquid nitrogen environment, and according to thermal balance equation, along the length direction of superconducting tape, One-dimensional Heat Conduction Equation is:

$v_{\mathrm{cm}}\left(T\right)C_{\mathrm{cm}}\left(T\right)\frac{\&PartialD;T}{\&PartialD;t}=\frac{\&PartialD;}{\&PartialD;t}\left[K_{\mathrm{cm}}\left(T\right)\frac{\&PartialD;T}{\&PartialD;x}\right]+g_j\left(T\right)-W_{\mathrm{cool}}\left(T\right)---\left(6\right)$

In formula, K _cm(T) be heat-conduction coefficient, V _cmand C _cmbe respectively density and the specific heat capacity of YBCO superconducting tape, g _jand W (T) _cool(T) be Joule heat and the dissipated heat of superconducting tape respectively.

The density V of YBCO superconducting tape _cmwith specific heat capacity C _cmmeet formula:

$v_{\mathrm{cm}}{C}_{\mathrm{cm}}=v_1{C}_1\frac{d_1}{d}+v_2{C}_2\frac{d_2}{d}+v_3{C}_3\frac{d_3}{d}{v}_4{C}_4\frac{d_4}{d}---\left(7\right)$

Wherein, ν ₁, ν ₂, ν ₃, ν ₄be respectively the density of surperficial layers of copper, silver layer, YBCO layer and Hastelloy basalis.D ₁for surface copper layer thickness, i.e. the thickness sum of upper surface layers of copper and lower surface layers of copper, d ₂for the thickness of silver layer, d ₃for the thickness of YBCO layer, d ₄for the thickness of Hastelloy basalis, d is the thickness of YBCO superconducting tape.C ₁, C ₂, C ₃, C ₄be respectively the specific heat capacity of surperficial layers of copper, silver layer, YBCO layer and Hastelloy basalis.Various density and specific heat capacity all can be consulted handbook and be obtained.

$K_{\mathrm{cm}}=K_1\frac{d_1}{d}+K_2\frac{d_2}{d}+K_3\frac{d_3}{d}+K_4\frac{d_4}{d}---\left(8\right)$

Wherein, K ₁, K ₂, K ₃, K ₄be respectively surperficial layers of copper, silver layer, the thermal conductivity of YBCO layer and Hastelloy basalis.The thermal conductivity of various material all can be consulted handbook and be obtained.

(2) according to Joule law, the Joule heat of superconducting tape

$g_j\left(T\right)={\&Integral;}_0^t{U}_S\left(t\right)I_S\left(t\right)\mathrm{dt}---\left(9\right)$

Wherein, U _st voltage that () is superconducting tape, I _st electric current that () is superconducting tape.

(3) superconducting tape is immersed in liquid nitrogen, rule of thumb formula, and dissipated heat is taken away by liquid nitrogen completely, the dissipated heat W of superconducting tape _cool(T) be:

W _cool(T)＝hA(T)(10)

Wherein, A is the contact area of superconducting tape and liquid nitrogen, i.e. the surface area of superconducting tape; H is liquid nitrogen heat transfer coefficient, relevant, experimentally known with the temperature difference Δ T (T-Top) of superconducting tape and liquid nitrogen, the value of liquid nitrogen heat transfer coefficient h, as shown in Figure 5.Corresponding to different temperature difference T (T-Top), heat transfer process has convection current, nuclear boiling, transition state and film boiling 4 kinds of states, the liquid nitrogen heat transfer coefficient h that different states is corresponding different.Liquid nitrogen heat transfer coefficient be h's and fitting result:

$h=\left\{\begin{array}{cc}0.091011*\mathrm{\&Delta;T}+0.089888& (\mathrm{\&Delta;T}\&le;10)\\ 0.5*\mathrm{\&Delta;T}-4& (10<\mathrm{\&Delta;T}\&le;31)\\ -0.069*\mathrm{\&Delta;T}+2.259& (31<\mathrm{\&Delta;T}\&le;600)\\ 0.002833*\mathrm{\&Delta;T}& (\mathrm{\&Delta;T}>600)\end{array}\right.---\left(11\right)$

Step 4. sets up YBCO superconducting tape equivalent-circuit model, according to superconducting tape temperature T and superconducting tape electric current I _st (), calculates the resistance r of YBCO superconducting tape _sc;

The equivalent electrical circuit of YBCO superconducting tape is 4 resistor coupled in parallel structures.First resistance r ₁for surperficial layers of copper resistance, the second resistance r ₂for the resistance of silver layer, the 3rd resistance r ₃for the resistance of YBCO layer, the 4th resistance r ₄for the resistance of Hastelloy basalis, as shown in Figure 6.

According to set up YBCO superconducting tape equivalent-circuit model, and according to superconducting tape temperature and electric current, according to Ohm law and circuit theory, calculate the resistance of the equivalent electrical circuit of YBCO superconducting tape:

(1) first resistance r ₁for surperficial layers of copper resistance, for copper product makes.First resistance r ₁it is the function of superconducting tape temperature T.Can obtain according to Ohm law:

$r_1\left(T\right)={\mathrm{\&rho;}}_1\left(T\right)\frac{l_e}{{\mathrm{wd}}_1}---\left(12\right)$

Wherein, ρ ₁(T) being the resistivity of copper, is the function of superconducting tape temperature T, and w is the width of YBCO superconducting tape, d ₁for surface copper layer thickness, i.e. the thickness sum of upper surface layers of copper and lower surface layers of copper, l _efor the length of superconducting tape.

(2) second resistance r ₂for silver layer resistance, for ag material makes.Second resistance r ₂it is the function of superconducting tape temperature T.Can obtain according to Ohm law:

$r_2\left(T\right)={\mathrm{\&rho;}}_2\left(T\right)\frac{l_e}{{\mathrm{wd}}_2}---\left(13\right)$

Wherein, ρ ₂(T) be silver-colored resistivity, w is the width of YBCO superconducting tape, d ₂for the thickness of silver layer, l _efor the length of superconducting tape.

(3) the 3rd resistance r ₃for YBCO layer resistance.According to the resistance changing law of superconducting tape, the 3rd resistance r ₃electricalresistivityρ ₃(T) be the function of superconducting tape temperature T, by the characteristic test to YBCO superconducting tape, and adopt the method for numerical fitting can obtain the 3rd resistance r ₃electricalresistivityρ ₃(T):

ρ ₃(T)＝ρ ₃₁(T)+ρ ₃₂(T)(14)

Wherein, ρ ₃₁and ρ (T) ₃₂(T) formed by piecewise function matching:

${\mathrm{\&rho;}}_{31}\left(T\right)=\left\{\begin{array}{cc}0& (J<J_C\left(T\right))\\ E_0{(J/\mathrm{Jc}\left(T\right)-1)}^{n1}/J& (J>J_C\left(T\right)\end{array}\right.---\left(15\right)$

${\mathrm{\&rho;}}_{32}\left(T\right)=\left\{\begin{array}{cc}0& (J<{\mathrm{\&gamma;J}}_C\left(T\right))\\ E_0{(J/\mathrm{Jc}\left(T\right)-1)}^{n2}/J& (J>{\mathrm{\&gamma;J}}_C\left(T\right)\end{array}\right.---\left(16\right)$

Wherein, Jc (T) is the function of superconduction belt material critical current density Jc with temperature T:

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

Wherein, Jc is superconduction belt material critical current density.J _c0=2 × 10 ⁶a/cm ², be the critical current density under 77K; Tc=92K, be the critical temperature of YBCO, Top is working temperature, is 77K in liquid nitrogen bath.Parameter n ₁=3; n ₂=20; γ=2, are the characteristic according to superconducting tape and the fitting parameter obtained.For ρ ₃₁and ρ (T) ₃₂(T) be zero situation, in actual computation process, very little data can be taken as, as 10 ^-19Ω cm etc., avoid computing to make mistakes.

According to Ohm law, the electric current I s of superconducting tape is expressed as:

I _S(T)＝J(T)/(dw)(18)

In formula: d is the thickness of superconducting tape, w is the width of YBCO superconducting tape.

Wherein, the thickness d of superconducting tape is expressed as:

d＝(d ₁+d ₂+d ₃+d ₄)(19)

In formula, d ₁for surface copper layer thickness, i.e. the thickness sum of upper surface layers of copper and lower surface layers of copper, d ₂for the thickness of silver layer, d ₃for the thickness of YBCO layer, d ₄for the thickness of Hastelloy basalis.

3rd resistance r of YBCO layer ₃, can obtain according to Ohm law:

$r_3(T,I_s)={\mathrm{\&rho;}}_3\left(T\right)\frac{l_e}{{\mathrm{wd}}_3}---\left(20\right)$

In formula: d ₃for YBCO layer thickness, w is the width of YBCO superconducting tape, I _sfor superconducting tape electric current, l _efor the length of superconducting tape, T is the temperature of superconducting tape.

(4) the 4th resistance r ₄hastelloy basalis resistance, the 4th resistance r ₄it is the function of superconducting tape temperature T.Can obtain according to Ohm law:

$r_4\left(T\right)={\mathrm{\&rho;}}_4\left(T\right)\frac{l_e}{{\mathrm{wd}}_4}---\left(21\right)$

In formula, ρ ₄(T) be the width that silver-colored resistivity w is YBCO superconducting tape, d ₄for the thickness of Hastelloy basalis.

(5) according to the resistor coupled in parallel structure of YBCO superconducting tape equivalent electrical circuit, according to whole circuit law of ohm, the resistance r of superconducting tape _scfor:

$r_{\mathrm{sc}}(T,I_s)=\frac{1}{\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4}}---\left(22\right)$

In formula: r ₁, r ₂, r ₃, r ₄be respectively the first resistance r ₁, the second resistance r ₂, the 3rd resistance r ₃, the 4th resistance r ₄resistance, I _sfor superconducting tape electric current, T is the temperature of superconducting tape.

When Power System Steady-state runs, superconducting tape is operated in superconducting state, i.e. J<J _c(T) time, the resistance of the YBCO layer of superconducting tape is zero, and electric current all passes through YBCO layer and conducting, can not cause voltage drop to electrical network.When the grid collapses, power network current increases, superconducting tape quench and produce resistance, and electric current distributes between each layer of superconducting tape.

The circuit model of step 5. according to resistive superconducting restrictor, the resistance r of superconducting tape _sc, computational scheme electric current I (t) and superconducting tape electric current I _s(t); Outlet line electric current I (t) and superconducting tape electric current I _s(t), and the temperature T feeding back superconducting tape is to step 3, feedback superconducting tape electric current I _st (), to step 4, realizes systemic circulation modeling and simulation.

According to whole circuit law of ohm, the electric current of resistive superconducting restrictor and voltage relationship are:

U _s＝IX+Ir+IR _sc+IR _load(23)

In formula: R _loadfor pull-up resistor, R _scfor superconduction noninductive coil resistance, I is line current, U _sfor AC supply voltage, X is line equivalent induction reactance, and r is line equivalent resistance.

Wherein, AC supply voltage U _s, line equivalent induction reactance X is expressed as:

$U_s\left(t\right)=\sqrt{2}{U}_0\cos \left(2\mathrm{\&pi;ft}\right)---\left(24\right)$

X＝j2πfL _s(25)

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, j is Virtual Function symbol.

The resistance of superconduction noninductive coil is R _sc(T), superconduction noninductive coil is generally composed in parallel by m root superconducting tape, m>=1, according to shunting law:

R _sc(T)＝r _sc(T)/m(26)

I(t)＝mI _s(t)(27)

In formula, r _sc(T) be resistance, the I (t) and I of superconducting tape _st () is respectively the electric current of line current and superconducting tape.

According to whole circuit law of ohm, wushu (24)-(27) substitute into formula (23), obtain electric current and the voltage relationship of resistive superconducting restrictor:

$I\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(28\right)$

Wherein, phase angle theta is:

$\mathrm{tg}\left(\mathrm{\&theta;}\right)=\frac{2\mathrm{\&pi;f}{L}_s}{r+r_{\mathrm{sc}}/m+R_{\mathrm{load}}}---\left(29\right)$

Therefore, by the electric current of superconducting tape:

$I_s\left(t\right)=\frac{\sqrt{2}{U}_0/m}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(30\right)$

The current peak of superconducting tape is:

$I_{\mathrm{sM}}\left(t\right)=\frac{\sqrt{2}{U}_0/m}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m+R_{\mathrm{load}})}^2}}---\left(31\right)$

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, m is the radical in parallel of superconducting tape, and r is line equivalent resistance, R _loadfor pull-up resistor, r _scfor the resistance of superconducting tape.

When Power System Steady-state, resistive superconducting restrictor is operated in superconducting state, and the resistance of superconduction noninductive coil is R _sc(T)=0, according to formula (30), line current is:

$I\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(32\right)$

When electrical network is short-circuited fault, pull-up resistor R _loadbe reduced to zero, meanwhile, the normal state of resistive superconducting restrictor work and produce resistance, therefore, according to formula (23), line current is:

$I\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m)}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(33\right)$

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, m is the radical in parallel of superconducting tape, and r is line equivalent resistance, R _loadfor pull-up resistor, r _scfor the resistance of superconducting tape.Line equivalent inductance L _sthe degree of short circuit malfunction is illustrated with line equivalent resistance r, and the resistance r of superconduction noninductive coil _sc/ m then embodies the current limiting capacity of resistive superconducting restrictor, as resistance r _scwhen/m increases, line current I (t) reduces.

The superconducting tape structural model that the resistive superconducting restrictor digital modeling emulation mode that the present invention is based on YBCO superconducting tape constructs and superconducting tape equivalent-circuit model, simplify the structure of superconducting tape, describe superconducting tape at the resistance changing law being converted into normal state process by superconducting state comprehensively; By the foundation of the heat transfer modeling of YBCO superconducting tape, analyze the Changing Pattern of each component of superconducting tape specific heat capacity at different temperatures, conductivity etc., and the heat dispersal situations of band; By the foundation of the circuit model of resistive superconducting restrictor, provide simple and effective circuit and restrictor model and analytical approach.Provided by modeling a kind of effectively based on the failure analysis methods of the resistive superconducting restrictor of YBCO superconducting tape, improve work efficiency.By modeling method provided by the present invention, not only solve the resistive superconducting restrictor Accurate Analysis method based on YBCO superconducting tape, and, also can be used for the research of the superconductive current limiter of multiple-unit and labyrinth, promote the application of superconductive current limiter in electrical network.

Claims (2)

1. the resistive superconducting restrictor digital modeling emulation mode based on YBCO superconducting tape, it is characterized in that: described modeling and simulating method is on the basis setting up YBCO superconducting tape Equivalent Structure Model for Calculating, YBCO superconducting tape equivalent-circuit model, YBCO superconducting tape heat conduction model and resistive superconducting current limiter circuit model, according to the given circuit parameter of resistive superconducting restrictor, the structural parameters of superconducting tape and initial operating condition, at superconduction noninductive coil resistance R _scbe under the prerequisite of zero, the initial value of computational scheme electric current and the initial value of superconducting tape electric current; The temperature of superconducting tape is calculated according to the heat conduction model of YBCO superconducting tape; According to YBCO superconducting tape equivalent-circuit model, superconducting tape temperature and electric current, calculate the resistance of superconducting tape; According to the circuit model of resistive superconducting restrictor, calculate and feedback line electric current and superconducting tape electric current, realize the modeling and simulation of resistive superconducting restrictor.

2., according to the resistive superconducting restrictor digital modeling emulation mode based on YBCO superconducting tape according to claim 1, it is characterized in that: the concrete steps of described modeling and simulating method are as follows:

Step 1. sets up YBCO superconducting tape Equivalent Structure Model for Calculating;

When setting up YBCO superconducting tape Equivalent Structure Model for Calculating, for ease of analyzing the resistance situation of each layer of YBCO superconducting tape in quench process, ignore the thickness of YBCO superconducting strip cushion, the upper surface layers of copper of YBCO superconducting tape and lower surface layers of copper are united two into one and analyzes; Therefore, the YBCO superconducting tape Equivalent Structure Model for Calculating simplification of setting up is divided into 4 layers: surperficial layers of copper, silver layer, YBCO layer and Hastelloy basalis; Surface layers of copper comprises upper surface layers of copper and lower surface layers of copper;

Step 2. sets up the circuit model of resistive superconducting restrictor, the circuit parameter of given resistive superconducting restrictor: line equivalent inductance L _s, line equivalent resistance r and pull-up resistor R _load, and AC supply voltage U _s, at superconduction noninductive coil resistance R _scunder null prerequisite, computational scheme electric current initial value I ₀(t) and superconducting tape electric current initial value I _s0(t);

Resistive superconducting restrictor comprises AC power u _s , line equivalent induction reactance X, line equivalent resistance r, isolating switch Br, superconduction noninductive coil R _scwith pull-up resistor R _load; AC power U _s, line equivalent induction reactance X, line equivalent resistance r, isolating switch Br, superconduction noninductive coil resistance R _scwith pull-up resistor R _loadconnect successively, AC power U _swith pull-up resistor R _loadone end ground connection; Isolating switch Br and superconduction noninductive coil resistance R _scbe connected on the first tie point A, superconduction noninductive coil resistance R _scwith pull-up resistor R _loadbe connected on the second tie point B;

According to whole circuit law of ohm, as superconduction noninductive coil resistance R _scwhen being zero, the electric current of resistive superconducting restrictor and voltage relationship are:

U _s＝I ₀X+I ₀r+I ₀R _load(1)

In formula: R _loadfor pull-up resistor, I ₀for line current, U _sfor AC supply voltage, X is line equivalent induction reactance, and r is line equivalent resistance;

Wherein, AC supply voltage U _s, line equivalent induction reactance X is expressed as:

$U_s\left(t\right)=\sqrt{2}{U}_0\cos \left(2\mathrm{\&pi;ft}\right)---\left(2\right)$

X＝j2πfL _s(3)

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, j is Virtual Function symbol;

Line current initial value is:

$I_0\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(4\right)$

The resistance of superconduction noninductive coil is R _sc(T), superconduction noninductive coil is generally composed in parallel by m root superconducting tape, m>=1, according to shunting law:

I _s0(t)＝I ₀(t)/m(5)

In formula, I ₀t () is line current initial value, I _s0t electric current initial value that () is superconducting tape;

Step 3. sets up the heat conduction model of YBCO superconducting tape, the structural parameters of given superconducting tape: the width w of superconducting tape, comprise surface copper layer thickness d ₁, silver thickness d ₂, YBCO layer thickness d ₃, Hastelloy base layer thickness d ₄thickness d, and length l _e, the initial operating condition of given superconduction noninductive coil: work temperature op, calculates the temperature T of superconducting tape;

(1) YBCO superconducting tape directly cools in immersing in liquid nitrogen environment, and according to thermal balance equation, along the length direction of superconducting tape, One-dimensional Heat Conduction Equation is:

$v_{\mathrm{cm}}\left(T\right)C_{\mathrm{cm}}\left(T\right)\frac{\&PartialD;T}{\&PartialD;t}=\frac{\&PartialD;}{\&PartialD;t}\left[K_{\mathrm{cm}}\left(T\right)\frac{\&PartialD;T}{\&PartialD;x}\right]+g_j\left(T\right)-W_{\mathrm{cool}}\left(T\right)---\left(6\right)$

In formula, K _cm(T) be heat-conduction coefficient, V _cmand C _cmbe respectively density and the specific heat capacity of YBCO superconducting tape, g _jand W (T) _cool(T) be Joule heat and the dissipated heat of superconducting tape respectively;

The density V of YBCO superconducting tape _cmwith specific heat capacity C _cmmeet formula:

$v_{\mathrm{cm}}{C}_{\mathrm{cm}}=v_1{C}_1\frac{d_1}{d}+v_2{C}_2\frac{d_2}{d}+v_3{C}_3\frac{d_3}{d}+v_4{C}_4\frac{d_4}{d}---\left(7\right)$

Wherein, ν ₁, ν ₂, ν ₃, ν ₄be respectively the density of surperficial layers of copper, silver layer, YBCO layer and Hastelloy basalis, d ₁for surface copper layer thickness, be the thickness sum of upper surface layers of copper and lower surface layers of copper, d ₂for the thickness of silver layer, d ₃for the thickness of YBCO layer, d ₄for the thickness of Hastelloy basalis, d is the thickness of YBCO superconducting tape; C ₁, C ₂, C ₃, C ₄be respectively the specific heat capacity of surperficial layers of copper, silver layer, YBCO layer and Hastelloy basalis;

$K_{\mathrm{cm}}=K_1\frac{d_1}{d}+K_2\frac{d_2}{d}+K_3\frac{d_3}{d}+K_4\frac{d_4}{d}---\left(8\right)$

Wherein, K ₁, K ₂, K ₃, K ₄be respectively surperficial layers of copper, silver layer, the thermal conductivity of YBCO layer and Hastelloy basalis;

(2) according to Joule law, the Joule heat of superconducting tape:

$g_j\left(T\right)={\&Integral;}_0^t{U}_S\left(t\right)I_S\left(t\right)\mathrm{dt}---\left(9\right)$

Wherein, U _st voltage that () is superconducting tape, I _st electric current that () is superconducting tape;

(3) superconducting tape is immersed in liquid nitrogen, rule of thumb formula, and dissipated heat is taken away by liquid nitrogen completely, the dissipated heat W of superconducting tape _cool(T) be:

W _cool(T)＝hA(T)(10)

Wherein, A is the contact area of superconducting tape and liquid nitrogen, i.e. the surface area of superconducting tape; H is liquid nitrogen heat transfer coefficient, relevant, experimentally known with the temperature difference Δ T (T-Top) of superconducting tape and liquid nitrogen, the value of liquid nitrogen heat transfer coefficient h; Corresponding to different temperature difference T (T-Top), heat transfer process has convection current, nuclear boiling, transition state and film boiling 4 kinds of states, the liquid nitrogen heat transfer coefficient h that different states is corresponding different; Liquid nitrogen heat transfer coefficient be h's and fitting result:

$h=\left\{\begin{array}{cc}0.091011*\mathrm{\&Delta;T}+0.089888& (\mathrm{\&Delta;T}\&le;10)\\ 0.5*\mathrm{\&Delta;T}-4& (10<\mathrm{\&Delta;T}\&le;31)\\ -0.069*\mathrm{\&Delta;T}+2.259& (31<\mathrm{\&Delta;T}\&le;600)\\ 0.002833*\mathrm{\&Delta;T}& (\mathrm{\&Delta;T}>600)\end{array}\right.---\left(11\right)$

Step 4. sets up YBCO superconducting tape equivalent-circuit model, according to superconducting tape temperature T and superconducting tape electric current I _st (), calculates the resistance r of YBCO superconducting tape _sc;

The equivalent electrical circuit of YBCO superconducting tape is 4 resistor coupled in parallel structures; First resistance r ₁for surperficial layers of copper resistance, the second resistance r ₂for the resistance of silver layer, the 3rd resistance r ₃for the resistance of YBCO layer, the 4th resistance r ₄for the resistance of Hastelloy basalis;

According to set up YBCO superconducting tape equivalent-circuit model, and according to superconducting tape temperature and electric current, according to Ohm law and circuit theory, calculate the resistance of the equivalent electrical circuit of YBCO superconducting tape:

(1) first resistance r ₁for surperficial layers of copper resistance, for copper product makes; First resistance r ₁it is the function of superconducting tape temperature T; Can obtain according to Ohm law:

$r_1\left(T\right)={\mathrm{\&rho;}}_1\left(T\right)\frac{l_e}{{\mathrm{wd}}_1}---\left(12\right)$

Wherein, ρ ₁(T) being the resistivity of copper, is the function of superconducting tape temperature T, and w is the width of YBCO superconducting tape, d ₁for surface copper layer thickness, i.e. the thickness sum of upper surface layers of copper and lower surface layers of copper, l _efor the length of superconducting tape;

(2) second resistance r ₂for silver layer resistance, for ag material makes; Second resistance r ₂it is the function of superconducting tape temperature T; Can obtain according to Ohm law:

$r_2\left(T\right)={\mathrm{\&rho;}}_2\left(T\right)\frac{l_e}{{\mathrm{wd}}_2}---\left(13\right)$

Wherein, ρ ₂(T) be silver-colored resistivity, w is the width of YBCO superconducting tape, d ₂for the thickness of silver layer, l _efor the length of superconducting tape;

(3) the 3rd resistance r ₃for YBCO layer resistance; According to the resistance changing law of superconducting tape, the 3rd resistance r ₃electricalresistivityρ ₃(T) be the function of superconducting tape temperature T, by the characteristic test to YBCO superconducting tape, and adopt the method for numerical fitting can obtain the 3rd resistance r ₃electricalresistivityρ ₃(T):

ρ ₃(T)＝ρ ₃₁(T)+ρ ₃₂(T)(14)

Wherein, ρ ₃₁and ρ (T) ₃₂(T) formed by piecewise function matching:

${\mathrm{\&rho;}}_{31}\left(T\right)=\left\{\begin{array}{cc}0& (J<J_C\left(T\right))\\ E_0{(J/\mathrm{Jc}\left(T\right)-1)}^{n1}/J& (J>J_C\left(T\right)\end{array}\right.---\left(15\right)$

${\mathrm{\&rho;}}_{32}\left(T\right)=\left\{\begin{array}{cc}0& (J<\mathrm{\&gamma;}{J}_C\left(T\right))\\ E_0{(J/\mathrm{Jc}\left(T\right)-1)}^{n2}/J& (J>{\mathrm{\&gamma;J}}_C\left(T\right)\end{array}\right.---\left(16\right)$

Wherein, Jc (T) is the function of superconduction belt material critical current density Jc with temperature T:

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

Wherein, Jc is superconduction belt material critical current density; J _c0=2 × 10 ⁶a/cm ², be the critical current density under 77K; Tc=92K, be the critical temperature of YBCO, Top is working temperature, is 77K in liquid nitrogen bath; Parameter n ₁=3; n ₂=20; γ=2, are the characteristic according to superconducting tape and the fitting parameter obtained; According to Ohm law, the electric current I s of superconducting tape is expressed as:

I _S(T)＝J(T)/(dw)(18)

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

Wherein, the thickness d of superconducting tape is expressed as:

d＝(d ₁+d ₂+d ₃+d ₄)(19)

In formula, d ₁for surface copper layer thickness, i.e. the thickness sum of upper surface layers of copper and lower surface layers of copper, d ₂for the thickness of silver layer, d ₃for the thickness of YBCO layer, d ₄for the thickness of Hastelloy basalis;

3rd resistance r of YBCO layer ₃, can obtain according to Ohm law:

$r_3(T,I_s)={\mathrm{\&rho;}}_3\left(T\right)\frac{l_e}{{\mathrm{wd}}_3}---\left(20\right)$

In formula: d ₃for YBCO layer thickness, w is the width of YBCO superconducting tape, I _sfor superconducting tape electric current, l _efor the length of superconducting tape, T is the temperature of superconducting tape;

(4) the 4th resistance r ₄hastelloy basalis resistance, the 4th resistance r ₄be the function of superconducting tape temperature T, can obtain according to Ohm law:

$r_4\left(T\right)={\mathrm{\&rho;}}_4\left(T\right)\frac{l_e}{{\mathrm{wd}}_4}---\left(21\right)$

In formula, ρ ₄(T) be the width that silver-colored resistivity w is YBCO superconducting tape, d ₄for the thickness of Hastelloy basalis;

(5) according to the resistor coupled in parallel structure of YBCO superconducting tape equivalent electrical circuit, according to whole circuit law of ohm, the resistance r of superconducting tape _scfor:

$r_{\mathrm{sc}}(T,I_s)=\frac{1}{\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4}}---\left(22\right)$

In formula: r ₁, r ₂, r ₃, r ₄be respectively the first resistance r ₁, the second resistance r ₂, the 3rd resistance r ₃, the 4th resistance r ₄resistance, I _sfor superconducting tape electric current, T is the temperature of superconducting tape;

When Power System Steady-state runs, superconducting tape is operated in superconducting state, i.e. J<J _c(T) time, the resistance of the YBCO layer of superconducting tape is zero, and electric current all passes through YBCO layer and conducting, can not cause voltage drop to electrical network; When the grid collapses, power network current increases, superconducting tape quench and produce resistance, and electric current will distribute between each layer of superconducting tape;

The circuit model of step 5. according to resistive superconducting restrictor, the resistance r of superconducting tape _sc, computational scheme electric current I (t) and superconducting tape electric current I _s(t); Outlet line electric current I (t) and superconducting tape electric current I _s(t), and the temperature T feeding back superconducting tape is to step 3, feedback superconducting tape electric current I _st (), to step 4, realizes systemic circulation modeling and simulation;

According to whole circuit law of ohm, the electric current of resistive superconducting restrictor and voltage relationship are:

U _s＝IX+Ir+IR _sc+IR _load(23)

In formula: R _loadfor pull-up resistor, R _scfor superconduction noninductive coil resistance, I is line current, U _sfor AC supply voltage, X is line equivalent induction reactance, and r is line equivalent resistance;

Wherein, AC supply voltage U _s, line equivalent induction reactance X is expressed as:

$U_s\left(t\right)=\sqrt{2}{U}_0\cos \left(2\mathrm{\&pi;ft}\right)---\left(24\right)$

X＝j2πfL _s(25)

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, j is Virtual Function symbol;

The resistance of superconduction noninductive coil is R _sc(T), superconduction noninductive coil is generally composed in parallel by m root superconducting tape, m>=1, according to shunting law:

R _sc(T)＝r _sc(T)/m(26)

I(t)＝mI _s(t)(27)

In formula, r _sc(T) be resistance, the I (t) and I of superconducting tape _st () is respectively the electric current of line current and superconducting tape;

According to whole circuit law of ohm, wushu (24)-(27) substitute into formula (23), obtain electric current and the voltage relationship of resistive superconducting restrictor:

$I\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(28\right)$

Wherein, phase angle theta is:

$\mathrm{tg}\left(\mathrm{\&theta;}\right)=\frac{2\mathrm{\&pi;f}{L}_s}{r+r_{\mathrm{sc}}/m+R_{\mathrm{load}}}---\left(29\right)$

Therefore, by the electric current of superconducting tape:

$I_s\left(t\right)=\frac{\sqrt{2}{U}_0/m}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(30\right)$

The current peak of superconducting tape is:

$I_{\mathrm{sM}}\left(t\right)=\frac{\sqrt{2}{U}_0/m}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m+R_{\mathrm{load}})}^2}}---\left(31\right)$

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, m is the radical in parallel of superconducting tape, and r is line equivalent resistance, R _loadfor pull-up resistor, r _scfor the resistance of superconducting tape;

When Power System Steady-state, resistive superconducting restrictor is operated in superconducting state, and the resistance of superconduction noninductive coil is R _sc(T)=0, according to formula (30), line current is:

$I\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+R_{\mathrm{load}})}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(32\right)$

When electrical network is short-circuited fault, pull-up resistor R _loadbe reduced to zero, meanwhile, the normal state of resistive superconducting restrictor work and produce resistance, therefore, according to formula (23), line current is:

$I\left(t\right)=\frac{\sqrt{2}{U}_0}{\sqrt{{\left(2\mathrm{\&pi;f}{L}_s\right)}^2+{(r+r_{\mathrm{sc}}/m)}^2}}\cos (2\mathrm{\&pi;ft}+\mathrm{\&theta;})---\left(33\right)$

In formula, U ₀for AC supply voltage effective value, f is ac power frequency, L _sfor line equivalent inductance, m is the radical in parallel of superconducting tape, and r is line equivalent resistance, R _loadfor pull-up resistor, r _scfor the resistance of superconducting tape; Line equivalent inductance L _sthe degree of short circuit malfunction is illustrated with line equivalent resistance r, and the resistance r of superconduction noninductive coil _sc/ m then embodies the current limiting capacity of resistive superconducting restrictor, as resistance r _scwhen/m increases, line current I (t) reduces.