CN105678017A - Frequency domain analysis based DFIG (Doubly-fed Induction Generator) crowbar resistance setting constraint computing method - Google Patents

Abstract

The invention proposes a frequency domain analysis based DFIG (Doubly-fed Induction Generator) crowbar resistance setting constraint computing method. The solving and superposition of a zero state response and a zero input response of a stator/rotor current after triggering a crowbar circuit are realized in a frequency domain through Laplace transformation. A time domain solution of a stator/rotor transient current is calculated by adopting Laplace inverse transformation. A line voltage upper limit after triggering the crowbar circuit under the condition of different rotational speeds and different crowbar resistance is calculated by analyzing physical characteristics of an analytic expression obtained by solving the stator/rotor current in the frequency domain and the time domain, a constraint condition of a crowbar resistance value is obtained according to the value of the upper limit, and a rotor current transformer is guaranteed to be reliably bypassed during crowbar circuit triggering.

Description

The DFIG crow bar resistance resolved based on frequency domain is adjusted constraint computational methods

Technical field

The present invention relates to the DFIG crow bar resistance resolved based on frequency domain to adjust constraint computational methods, belong to Electrified Transmission applied technical field.

Background technology

Crowbar circuit achieves the double-fed wind power generator (Doubly-fedInductionGenerator is called for short DFIG) rotor current transformer bypass protection in low voltage crossing process. When adopting this guard method to be controlled; rotor current transformer loses the control to rotor current; transient characterisitics depend entirely on the parameter of crowbar circuit, and the resistance of adjusting of crow bar resistance is that double-fed wind power generator realizes low voltage crossing and recovers the key factor of AC excitation in good time. It is the technical bottleneck realizing this parameter tuning that rotor transient current during crowbar circuit triggering resolves.

Current double-fed generator crow bar resistance setting method mainly has 3 class methods:

1) according to generator parameter and grid voltage sags characteristic, parameter tuning is realized by low voltage crossing Electromagnetic Simulation. Owing to the physical process of the method is indefinite, it is necessary to emulate respectively under various working, assess the cost height.

2), when ignoring double-fed generator fixed rotor resistance, resolved by the rotor transient current during triggering, for guaranteeing that low voltage crossing process rotor transient current and DC bus-bar voltage are not past limit value, it is achieved crow bar resistance is adjusted. There is error in the method, and the limits of error is difficult to determine, is only capable of providing reference to engineer applied.

3) by rotor transient time constant, analyze the winding maximum transient electric current in low voltage crossing process in the time domain, but the transient DC electric current of rotor is approximate DC, it is actually and rotates with angular frequency more slowly, and error producing cause is not clear, it is necessary to incorporation engineering experience carries out parameter adjustment.

As can be seen here, currently, double-fed wind generator crow bar resistance is adjusted the method relying on experiment explorations, engineering experience, emulation or approximate calculation more, the problem that the reliability that said method all exists that cost height, cycle length, resistance setting range be indefinite, rotor current transformer is bypassed after crowbar circuit triggering is difficult to ensure that.

Put before this, resolved by double-fed wind power generator stator and rotor current in crowbar circuit trigger process, calculate the transient process crowbar circuit line upper voltage limit under different rotating speeds, different crow bar resistance condition, obtain the constraints of crow bar resistance setting valve, for the reliable bypass of rotor current transformer during guaranteeing crowbar circuit triggering, improve generating set low-voltage reliability and reduce crowbar circuit parameter designing difficulty, equal significance.

Summary of the invention

The present invention is directed to above-mentioned technical barrier, it is proposed to a kind of rotor transient current computational methods based on Laplace conversion and inverse transformation, this analytic method is applicable to arbitrarily fall the line voltage balance of the degree of depth or imbalance fault. Converted by Laplace, in frequency domain, realize solving and superposition of stator and rotor current zero state response and zero input response after crowbar circuit triggers. Adopt Laplace inverse transformation, solve the time solution of rotor transient current.

By analyzing the stator and rotor current physical characteristic solving gained analytical expression at frequency domain and time domain, calculate the line upper voltage limit after crowbar circuit under different rotating speeds, different crow bar resistance condition triggers, the constraints of crow bar resistance is obtained, it is ensured that rotor current transformer is reliably bypassed during crowbar circuit triggers according to this higher limit.

For solving above-mentioned technical problem, the present invention provides a kind of DFIG crow bar resistance resolved based on frequency domain to adjust constraint computational methods, it is characterized in that, parameter defines:

Rotor voltage in stator stationary coordinate system,Rotor voltage in rotor rotating coordinate system

Stator voltage in stator stationary coordinate system

Stator current in stator stationary coordinate system,Rotor current in stator stationary coordinate system

Stator current in rotor rotating coordinate system,Rotor current in rotor rotating coordinate system

Stator magnetic linkage in stator stationary coordinate system,Rotor flux in stator stationary coordinate system

R_sStator resistance, R_rRotor loop all-in resistance

R_rwRotor windings resistance, R_cRotor crow bar resistance

L_ssStator leakage inductance, L_rsRotor leakage inductance

L_mRotor mutual inductance, L_sStator inductance, L_rInductor rotor

L_rrThe self-induction through air gap produced by rotor windings

N_rk_NrStator effective turn

N_sk_NsRotor effective turn

K winding conversion factor

ω₁Line voltage synchronous rotational speed, ω_rRotor electric rotating angular velocity

θ_rAngle between stator A phase winding and rotor a phase winding

U_dcFour-quadrant phase current transformer DC bus-bar voltage setting value

p_nPower generator electrode logarithm

θ_sp0Grid voltage sags moment positive sequence voltage, θ_sn0The initial phase angle of grid voltage sags moment negative sequence voltage

The Vector Mode of positive sequence voltage after grid voltage sags,The Vector Mode of negative sequence voltage after grid voltage sags

Plural number is asked for the operator of real part by Re

Plural number is asked for the operator of imaginary part by Im;

Parameter subscript defines:

→ space vector

S stator coordinate,

R rotor coordinate

' through winding convert after numerical value;

Parameter subscript defines:

The biphase rest frame α axle of α stator,

The biphase rest frame β axle of β stator

S stator,

R rotor;

Comprise the following steps:

1) the stator and rotor space vector of voltage equation in stator stationary coordinate system and rotor rotating coordinate system is respectively as follows:

$\stackrel{\→}{U_s^s}=R_s\stackrel{\→}{I_s^s}+\frac{d\stackrel{\→}{{\mathrm{\ψ}}_s^s}}{dt}$ (formula 1a)

$\stackrel{\→}{U_r^r}=R_r\stackrel{\→}{I_r^r}+\frac{d\stackrel{\→}{{\mathrm{\ψ}}_r^r}}{dt}$ (formula 1b)

When the rotor number of phases is identical, by winding conversion factorWithCarry out winding reduction according to (formula 1b) to obtain:

$\stackrel{\→}{U_r^{r^{\′}}}={R_r}^{\′}\stackrel{\→}{I_r^{r^{\′}}}+{L_{rr}}^{\′}\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}+{L_{r\mathrm{\δ}}}^{\′}\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}+{L_m}^{\′}\frac{d\stackrel{\→}{I_s^{r^{\′}}}}{dt}$ (formula 2)

Through winding reduction, L_rr' equal to L_m'; Calculated by (formula 2):

$\stackrel{\→}{U_r^{r^{\′}}}={R_r}^{\′}\stackrel{\→}{I_r^{r^{\′}}}+{L_m}^{\′}(\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}+\frac{d\stackrel{\→}{I_s^r}}{dt})+{L_{r\mathrm{\δ}}}^{\′}\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}$ (formula 3)

In stator stationary coordinate system, (formula 3) is expressed as:

$\stackrel{\→}{U_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}={R_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}+{L_m}^{\′}\frac{d(\stackrel{\→}{I_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}+\stackrel{\→}{I_s^s}{e}^{-{\mathrm{j\θ}}_r})}{dt}+{L_{r\mathrm{\δ}}}^{\′}\frac{d\left(\stackrel{\→}{I_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}\right)}{dt}$ (formula 4)

The rotor voltage equation in the biphase rest frame of stator is obtained by (formula 4):

$\stackrel{\→}{U_r^{s^{\′}}}={R_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}+{L_r}^{\′}\frac{d\stackrel{\→}{I_r^{s^{\′}}}}{dt}+{L_m}^{\′}\frac{d\stackrel{\→}{I_s^s}}{dt}-{\mathrm{j\ω}}_r{L_m}^{\′}\stackrel{\→}{I_s^s}-{\mathrm{j\ω}}_r{L_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}$ (formula 5)

Consider grid voltage sags t₀The electric current initial value in moment, is carried out Laplace by (formula 5) and converts:

$I_r^{s^{\′}}=\frac{U_r^{s^{\′}}+{L_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}\left(t_0\right)+{L_m}^{\′}\stackrel{\→}{I_s^s}\left(t_0\right)-(s-{\mathrm{j\ω}}_r){L_m}^{\′}{I}_s^s}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}$ (formula 6)

(formula 6) is brought into:

${\mathrm{\ψ}}_s^s=L_s{I}_s^s+{L_m}^{\′}\[\frac{U_r^{s^{\′}}+{L_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}\left(t_0\right)+{L_m}^{\′}\stackrel{\→}{I_s^s}\left(t_0\right)-(s-{\mathrm{j\ω}}_r){L_m}^{\′}{I}_s^s}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\]$ (formula 7)

Space vector expression formula after grid voltage sags is:

$\stackrel{\→}{U_s^s}=\left|\stackrel{\→}{U_{spm}}\right|e^{j({\mathrm{\ω}}_{1}t+{\mathrm{\θ}}_{sp0})}+\left|\stackrel{\→}{U_{snm}}\right|e^{j(-{\mathrm{\ω}}_{1}t+{\mathrm{\θ}}_{sn0})}$ (formula 8)

(formula 8) carries out Laplace conversion can obtain:

$U_s^s=\frac{\left|\stackrel{\→}{U_{spm}}\right|}{s-{\mathrm{j\ω}}_1}{e}^{{\mathrm{j\θ}}_{sp0}}+\frac{\left|\stackrel{\→}{U_{snm}}\right|}{s+{\mathrm{j\ω}}_1}{e}^{{\mathrm{j\θ}}_{sn0}}$ (formula 9)

Consider grid voltage sags t₀The magnetic linkage initial value in moment, carries out Laplace to (formula 1a) and converts:

$U_s^s=R_s{I}_s^s+{\mathrm{s\ψ}}_s^s-\stackrel{\→}{{\mathrm{\ψ}}_s^s}\left(t_0\right)$ (formula 10)

2) Stator transient Current calculation

After grid voltage sags, trigger crowbar circuit and rotor windings is carried out short circuit, by zero state response and zero input response, the transient current of stator is calculated;

(formula 7), (formula 9) are substituted into (formula 10),Expression formula in a frequency domain is:

$I_s^s=I_{s1}^s+I_{s2}^s$ (formula 11)

In formulaWithRespectively Stator transient electric current is in the zero state response of stator stationary coordinate system and zero input response, and expression formula is respectively as follows:

$I_{s1}^s=\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s+{\mathrm{j\ω}}_1)\left|U_{spm}\right|e^{{\mathrm{j\θ}}_{sp0}}+\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s-{\mathrm{j\ω}}_1)\left|U_{snm}\right|e^{{\mathrm{j\θ}}_{sn0}}}{(s^2+{{\mathrm{\ω}}_1}^2)A_1}=\frac{{\mathrm{NUM}}_1\left(s\right)}{{\mathrm{DEN}}_1\left(s\right)}$ (formula 12)

$I_{s2}^s=\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\]\stackrel{\→}{{\mathrm{\ψ}}_s^s}\left(t_0\right)-{{\mathrm{sL}}_m}^{\′}\stackrel{\→}{{\mathrm{\ψ}}_r^{s^{\′}}}\left(t_0\right)}{A_1}=\frac{{\mathrm{NUM}}_2\left(s\right)}{{\mathrm{DEN}}_2\left(s\right)}$ (formula 13)

A in formula₁=s²L_r'L_s-s²L_m'²+sL_r'R_s+sR_r'L_s-sjω_rL_r'L_s+sjω_rL_m'²-jω_rL_r'R_s+R_r'R_s; NUM₁(s) and NUM₂S replacement molecule that () is expression formula, DEN₁(s) and DEN₂S replacement denominator that () is expression formula;

Calculate NUM₁(s)/DEN₁S four limits of () expression formula are:

$\left\{\begin{array}{c}{a}_1={\mathrm{j\ω}}_1\\ a_2=-{\mathrm{j\ω}}_1\\ a_3=\frac{-R_s{L_r}^{\′}-{R_r}^{\′}{L}_s+L_s{L_r}^{\′}{\mathrm{j\ω}}_r-{\mathrm{j\ω}}_r{L_m}^{\′2}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}+\frac{\sqrt{{(R_s{L_r}^{\′}+{R_r}^{\′}{L}_s-L_s{L_r}^{\′}{\mathrm{j\ω}}_r+{\mathrm{j\ω}}_r{L_m}^{\′2})}^2-4(L_s{L_r}^{\′}-{L_m}^{\′2})(R_s{R_r}^{\′}-{\mathrm{j\ω}}_r{R}_s{L_r}^{\′})}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}\\ a_4=\frac{-R_s{L_r}^{\′}-{R_r}^{\′}{L}_s+L_s{L_r}^{\′}{\mathrm{j\ω}}_r-{\mathrm{j\ω}}_r{L_m}^{\′2}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}-\frac{\sqrt{{(R_s{L_r}^{\′}+{R_r}^{\′}{L}_s-L_s{L_r}^{\′}{\mathrm{j\ω}}_r+{\mathrm{j\ω}}_r{L_m}^{\′2})}^2-4(L_s{L_r}^{\′}-{L_m}^{\′2})(R_s{R_r}^{\′}-{\mathrm{j\ω}}_r{R}_s{L_r}^{\′})}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}\end{array}\right.$ (formula 14)

Calculate NUM₂(s)/DEN₂S two limits of () expression formula are:

$\left\{\begin{array}{c}{b}_1=a_3\\ b_2=a_4\end{array}\right.$ (formula 15)

To DEN₁(s) and DEN₂S () carries out derivation, obtain dDEN₁(a_n)/ds and dDEN₂(b_n)/ds; Due to Re (a₁)=Re (a₂)=0, and Re (a₃)、Re(a₄)、Re(b₁) and Re (b₂) for non-zero, in the biphase rest frame of stator, grid voltage sags can be obtained by inverse Laplace transform, crowbar circuit trigger after the expression formula of Stator transient electric current be:

$\stackrel{\→}{I_{s\_transient}}\left(t\right)=\underset{n=3,4}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_1\left(a_n\right)}{{\mathrm{dDEN}}_1\left(a_n\right)/ds}{e}^{a_{n}t}+\underset{n=1,2}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_2\left(b_n\right)}{{\mathrm{dDEN}}_2\left(b_n\right)/ds}{e}^{b_{n}t}$ (formula 16)

-1/Re(a₃)、-1/Re(a₄)、-1/Re(b₁) and-1/Re (b₂) damping time constant of respectively each transient state component;

3) rotor transient current calculates

WillWithSubstituting into (formula 6), obtain grid voltage sags, trigger after rotor windings carries out short circuit by crowbar circuit, rotor transient current is in the zero state response of the biphase rest frame of stator and zero input response:

$I_{r1}^{s^{\′}}=-\frac{(s-{\mathrm{j\ω}}_r){L_m}^{\′}}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}{I}_{s1}^s=\frac{{\mathrm{NUM}}_3\left(s\right)}{{\mathrm{DEN}}_3\left(s\right)}$ (formula 17)

$I_{r2}^{s^{\′}}=\frac{\stackrel{\→}{{\mathrm{\ψ}}_r^{s^{\′}}}\left(t_0\right)-(s-{\mathrm{j\ω}}_r){L_m}^{\′}{I}_{s2}^s}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}=\frac{{\mathrm{NUM}}_4\left(s\right)}{{\mathrm{DEN}}_4\left(s\right)}$ (formula 18)

In formula, NUM₃(s) and NUM₄S replacement molecule that () is expression formula, DEN₃(s) and DEN₄S replacement denominator that () is expression formula;

Five limit expression formulas be:

$\left\{\begin{array}{c}{c}_1=-({R_r}^{\′}/{L_r}^{\′})+j{\mathrm{\ω}}_r\\ c_2=a_1\\ e_3=a_2\\ c_4=a_3\\ c_5=a_4\end{array}\right.$ (formula 21)

Three limit expression formulas be:

$\left\{\begin{array}{c}{d}_1=c_1=-({R_r}^{\′}/{L_r}^{\′})+{\mathrm{j\ω}}_r\\ d_2=a_3\\ d_3=a_4\end{array}\right.$ (formula 22)

By to DEN₃(s) and DEN₄S () derivation, obtains dDEN₃(c_n)/ds and dDEN₄(d_n)/ds; Due to Re (c₂)=Re (c₃)=0, and Re (c₁)、Re(c₄)、Re(c₅)、Re(d₁)、Re(d₂) and Re (d₃) for non-zero, in the biphase rest frame of stator, obtain grid voltage sags by inverse Laplace transform, crowbar circuit trigger after rotor expression formula steady, transient current be respectively as follows:

$\stackrel{\→}{I_r^{s^{\′}}}\left(t\right){|}_{t\→\mathrm{\∞}}=\underset{n=2,3}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{c_{n}t}$ (formula 23a)

$\stackrel{\→}{I_{r\_transient}^{s^{\′}}}\left(t\right)=\underset{n=1,4,5}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{c_{n}t}+\underset{n=1,2,3}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_4\left(d_n\right)}{{\mathrm{dDEN}}_4\left(d_n\right)/ds}{e}^{d_{n}t}$ (formula 23b)

-1/Re(c₁),-1/Re(c₄),-1/Re(c₅),-1/Re(d₁),-1/Re(d₂) and-1/Re (d₃) damping time constant of respectively each transient state component;

4) crow bar resistance is adjusted to retrain and is set

If the double-fed generator electric angle range of speeds is ω_r∈[K₁ω₁,K₂ω₁], wherein K₁∈ (0,1] and K₂∈ [1,2); With ω_r∈[K₁ω₁,K₂ω₁] for constraints, build with the x function ω being variable_r(n_x)=K₁ω₁+ 0.314x, wherein x be nonnegative integer (x=0,1,2,3 ...), with ω_r(n_x)∈[K₁ω₁,K₂ω₁] for constraints, obtain by ω_r(n_x) the sequence ω that constitutes_se; Making y is R_c' independent variable, constructor R_c'(m_y)=10^-3Y, wherein y be nonnegative integer (y=0,1,2,3 ...);

When line voltage falls, when rotor-side converter is bypassed by crow bar, it is assumed that c₁、a₃And a₄Real part is zero, and transient current is not decayed, (formula 23a), (formula 23b) obtain rotor current expression formula and be:

$\stackrel{\→}{I_{r\_no\_decay}^{s^{\′}}}\left(t\right)=\underset{n=2,3}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{c_{n}t}+\underset{n=1,4,5}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{\mathrm{Im}\left(c_n\right)t}+\underset{n=1,2,3}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_4\left(d_n\right)}{{\mathrm{dDEN}}_4\left(d_n\right)/ds}{e}^{\mathrm{Im}\left(d_n\right)t}$ (formula 24)

According to R_c'(m_y)=10^-3Y, with y=0 for initial value, when the parameter of electric machine is known, if R_c'=R_c'(m₀), R_c'(m₀) resistance value when representing y=0, and by sequence ω_seAll elements gradually one by one substitute into (formula 24), ask forCycle expression formula, obtainsAt maximum when different rotating speeds of the peak value of a cycle internal moldConverted by winding, obtain ${\left|\stackrel{\→}{I_{r\_no\_decay}^s}\right|}_{T\_\max }\left(m_0\right)=\frac{1}{k}{\left|\stackrel{\→}{I_{r\_no\_decay}^{s^{\′}}}\right|}_{T\_\max }\left(m_0\right),$ ByCalculating obtains crowbar circuit line voltage peak higher limit:

$U_{LL\_max}\left(m_0\right)=\frac{2\sqrt{6}}{3k}{\left|\stackrel{\→}{I_{r\_no\_decay}^{s^{\′}}}\right|}_{T\_max}\left(m_0\right){R_c}^{\′}\left(m_0\right)$ (formula 25)

It is incremented by y, repeats above procedure, obtain U_{LL_max}(m₁), U_{LL_max}(m₂) ...;

In y increasing process, work as U_{LL_max}(m_V)≥U_dc, when namely crowbar circuit line voltage peak higher limit is be more than or equal to four quadrant convertor DC bus-bar voltage setting value during y=V, stops calculating, convert R through winding_c=k²·R_c'(m_V-1) it is set as that crow bar resistance is adjusted the upper limit;With R_c∈[0,k²·R_c'(m_V-1)] for constraints, it is ensured that after crowbar circuit triggers, rotor current transformer is reliably bypassed.

Step 3) in,WithExpression expands into:

$I_{r1}^{s^{\′}}=-\frac{(s-{\mathrm{j\ω}}_r){L_m}^{\′}}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\·\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s+{\mathrm{j\ω}}_1)\left|U_{spm}\right|e^{{\mathrm{j\θ}}_{sp0}}+\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s-{\mathrm{j\ω}}_1)\left|U_{snm}\right|e^{{\mathrm{j\θ}}_{sn0}}}{(s^2+{{\mathrm{\ω}}_1}^2)\{+s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s\}}$ (formula 19)

$\begin{array}{c}{I}_{r2}^{s^{\′}}=\frac{\stackrel{\→}{{{\mathrm{\ψ}}_r}^{s^{\′}}}\left(t_0\right)}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\·\frac{s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s}{s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s}\\ +\frac{-(s-{\mathrm{j\ω}}_r){L_m}^{\′}}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\·\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\]\stackrel{\→}{{{\mathrm{\ψ}}_s}^s}\left(t_0\right)-{{\mathrm{sL}}_m}^{\′}\stackrel{\→}{{\mathrm{\ψ}}_r^{s^{\′}}}\left(t_0\right)}{s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s}\end{array}$ (formula 20).

The beneficial effect that the present invention reaches:

The present invention passes through frequency-domain calculations, it is achieved that double-fed wind power generator in low voltage crossing process crowbar circuit trigger after stator and rotor current resolve. Based on this analytical expression; set by the upper limit of crowbar circuit line voltage (i.e. rotor current transformer output point terminal voltage); give the Operations of Interva Constraint condition that crow bar resistance is adjusted; after guaranteeing crowbar circuit triggering, rotor current transformer is reliably bypassed, and significantly improves the protection safety in low voltage crossing process of the rotor current transformer.

Accompanying drawing explanation

Fig. 1 is double-fed wind power generator rotor crowbar circuit schematic diagram.

Detailed description of the invention

Below in conjunction with accompanying drawing, the invention will be further described. Following example are only for clearly illustrating technical scheme, and can not limit the scope of the invention with this.

Symbol definition:

Rotor voltage in stator stationary coordinate system, the rotor voltage in rotor rotating coordinate system

Stator voltage in stator stationary coordinate system

Stator current in stator stationary coordinate system, the rotor current in stator stationary coordinate system

Stator current in rotor rotating coordinate system, the rotor current in rotor rotating coordinate system

Stator magnetic linkage in stator stationary coordinate system, the rotor flux in stator stationary coordinate system

R_s,R_rStator resistance, rotor loop all-in resistance

R_rw,R_cRotor windings resistance, rotor crow bar resistance

L_ss,L_rsStator leakage inductance, rotor leakage inductance

L_m,L_s,L_rRotor mutual inductance, stator inductance, inductor rotor

L_rrThe self-induction through air gap produced by rotor windings

N_rk_NrStator effective turn

N_sk_NsRotor effective turn

K winding conversion factor

ω₁,ω_rLine voltage synchronous rotational speed, rotor electric rotating angular velocity

θ_rAngle between stator A phase winding and rotor a phase winding

U_dcFour-quadrant phase current transformer DC bus-bar voltage setting value

p_nPower generator electrode logarithm

θ_sp0,θ_sn0The initial phase angle of grid voltage sags moment positive sequence voltage and negative sequence voltage

The Vector Mode of positive sequence voltage and negative sequence voltage after grid voltage sags

Plural number is asked for the operator of real part by Re

Plural number is asked for the operator of imaginary part by Im

Symbol subscript defines:

→ space vector

S, r stator coordinate, rotor coordinate

' through winding convert after numerical value

Symbol subscript defines:

The biphase rest frame α axle of α, β stator, the biphase rest frame β axle of stator

S, r stator, rotor

Frequency domain variable

1) the stator and rotor space vector of voltage equation in stator stationary coordinate system and rotor rotating coordinate system is respectively as follows:

$\stackrel{\→}{U_s^s}=R_s\stackrel{\→}{I_s^s}+\frac{d\stackrel{\→}{{\mathrm{\ψ}}_s^s}}{dt}$ (formula 1a)

$\stackrel{\→}{U_r^r}=R_r\stackrel{\→}{I_r^r}+\frac{d\stackrel{\→}{{\mathrm{\ψ}}_r^r}}{dt}$ (formula 1b)

When the rotor number of phases is identical, by winding reduction coefficientWithCarry out winding reduction according to (formula 1b) to obtain:

$\stackrel{\→}{U_r^{r^{\′}}}={R_r}^{\′}\stackrel{\→}{I_r^{r^{\′}}}+{L_{rr}}^{\′}\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}+{L_{r\mathrm{\δ}}}^{\′}\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}+{L_m}^{\′}\frac{d\stackrel{\→}{I_s^{r^{\′}}}}{dt}$ (formula 2)

Through winding reduction, L_rr' equal to L_m'. Be can be calculated by (formula 2):

$\stackrel{\→}{U_r^{r^{\′}}}={R_r}^{\′}\stackrel{\→}{I_r^{r^{\′}}}+{L_m}^{\′}(\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}+\frac{d\stackrel{\→}{I_s^r}}{dt})+{L_{r\mathrm{\δ}}}^{\′}\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}$ (formula 3)

In stator stationary coordinate system, (formula 3) is represented by:

$\stackrel{\→}{U_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}={R_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}+{L_m}^{\′}\frac{d(\stackrel{\→}{I_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}+\stackrel{\→}{I_s^s}{e}^{-{\mathrm{j\θ}}_r})}{dt}+{L_{r\mathrm{\δ}}}^{\′}\frac{d\left(\stackrel{\→}{I_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}\right)}{dt}$ (formula 4)

The rotor voltage equation in the biphase rest frame of stator can be obtained by (formula 4):

$\stackrel{\→}{U_r^{s^{\′}}}={R_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}+{L_r}^{\′}\frac{d\stackrel{\→}{I_r^{s^{\′}}}}{dt}+{L_m}^{\′}\frac{d\stackrel{\→}{I_s^s}}{dt}-{\mathrm{j\ω}}_r{L_m}^{\′}\stackrel{\→}{I_s^s}-{\mathrm{j\ω}}_r{L_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}$ (formula 5)

Consider the rotary inertia that double-fed fan motor unit is bigger, at low voltage crossing transient process rotor rotating speed approximately constant. Consider grid voltage sags t₀The electric current initial value in moment, (formula 5) carrying out Laplace conversion can obtain:

$I_r^{s^{\′}}=\frac{U_r^{s^{\′}}+{L_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}\left(t_0\right)+{L_m}^{\′}\stackrel{\→}{I_s^s}\left(t_0\right)-(s-{\mathrm{j\ω}}_r){L_m}^{\′}{I}_s^s}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}$ (formula 6)

(formula 6) is brought intoCan obtain:

${\mathrm{\ψ}}_s^s=L_s{I}_s^s+{L_m}^{\′}\[\frac{U_r^{s^{\′}}+{L_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}\left(t_0\right)+{L_m}^{\′}\stackrel{\→}{I_s^s}\left(t_0\right)-(s-{\mathrm{j\ω}}_r){L_m}^{\′}{I}_s^s}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\]$ (formula 7)

Space vector expression formula after grid voltage sags is:

$\stackrel{\→}{U_s^s}=\left|\stackrel{\→}{U_{spm}}\right|e^{j({\mathrm{\ω}}_{1}t+{\mathrm{\θ}}_{sp0})}+\left|\stackrel{\→}{U_{snm}}\right|e^{j(-{\mathrm{\ω}}_{1}t+{\mathrm{\θ}}_{sn0})}$ (formula 8)

(formula 8) carries out Laplace conversion can obtain:

$U_s^s=\frac{\left|\stackrel{\→}{U_{spm}}\right|}{s-{\mathrm{j\ω}}_1}{e}^{{\mathrm{j\θ}}_{sp0}}+\frac{\left|\stackrel{\→}{U_{snm}}\right|}{s+{\mathrm{j\ω}}_1}{e}^{{\mathrm{j\θ}}_{sn0}}$ (formula 9)

Consider grid voltage sags t₀The magnetic linkage initial value in moment, (formula 1a) carries out Laplace conversion can be obtained:

$U_s^s=R_s{I}_s^s+{\mathrm{s\ψ}}_s^s-\stackrel{\→}{{\mathrm{\ψ}}_s^s}\left(t_0\right)$ (formula 10)

2) Stator transient Current calculation.

Fig. 1 show double-fed wind power generator rotor crowbar circuit schematic diagram, S in figure_crowbarSwitch, R is triggered for crowbar circuit_cFor crow bar resistance. After grid voltage sags, trigger crowbar circuit and rotor windings is carried out short circuitBy zero state response and zero input response, the transient current of stator is calculated.

(formula 7), (formula 9) are substituted into (formula 10), can obtainExpression formula in a frequency domain is:

$I_s^s=I_{s1}^s+I_{s2}^s$ (formula 11)

In formulaFor zero state response,For zero input response, expression formula is respectively as follows:

$I_{s1}^s=\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s+{\mathrm{j\ω}}_1)\left|\stackrel{\→}{U_{spm}}\right|e^{{\mathrm{j\θ}}_{sp0}}+\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s-{\mathrm{j\ω}}_1)\left|\stackrel{\→}{U_{snm}}\right|e^{{\mathrm{j\θ}}_{sn0}}}{(s^2+{{\mathrm{\ω}}_1}^2)A_1}=\frac{{\mathrm{NUM}}_1\left(s\right)}{{\mathrm{DEN}}_1\left(s\right)}$ (formula 12)

$I_{s2}^s=\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\]\stackrel{\→}{{\mathrm{\ψ}}_s^s}\left(t_0\right)-{{\mathrm{sL}}_m}^{\′}\stackrel{\→}{{\mathrm{\ψ}}_r^{s^{\′}}}\left(t_0\right)}{A_1}=\frac{{\mathrm{NUM}}_2\left(s\right)}{{\mathrm{DEN}}_2\left(s\right)}$ (formula 13)

A in formula₁=s²L_r'L_s-s²L_m'²+sL_r'R_s+sR_r'L_s-sjω_rL_r'L_s+sjω_rL_m'²-jω_rL_r'R_s+R_r'R_s。NUM₁(s) and NUM₂S replacement molecule that () is expression formula, DEN₁(s) and DEN₂S replacement denominator that () is expression formula.

Calculate NUM₁(s)/DEN₁S four limits of () expression formula are:

$\left\{\begin{array}{c}{a}_1={\mathrm{j\ω}}_1\\ a_2=-{\mathrm{j\ω}}_1\\ a_3=\frac{-R_s{L_r}^{\′}-{R_r}^{\′}{L}_s+L_s{L_r}^{\′}{\mathrm{j\ω}}_r-{\mathrm{j\ω}}_r{L_m}^{\′2}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}+\frac{\sqrt{{(R_s{L_r}^{\′}+{R_r}^{\′}{L}_s-L_s{L_r}^{\′}{\mathrm{j\ω}}_r+{\mathrm{j\ω}}_r{L_m}^{\′2})}^2-4(L_s{L_r}^{\′}-{L_m}^{\′2})(R_s{R_r}^{\′}-{\mathrm{j\ω}}_r{R}_s{L_r}^{\′})}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}\\ a_4=\frac{-R_s{L_r}^{\′}-{R_r}^{\′}{L}_s+L_s{L_r}^{\′}{\mathrm{j\ω}}_r-{\mathrm{j\ω}}_r{L_m}^{\′2}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}-\frac{\sqrt{{(R_s{L_r}^{\′}+{R_r}^{\′}{L}_s-L_s{L_r}^{\′}{\mathrm{j\ω}}_r+{\mathrm{j\ω}}_r{L_m}^{\′2})}^2-4(L_s{L_r}^{\′}-{L_m}^{\′2})(R_s{R_r}^{\′}-{\mathrm{j\ω}}_r{R}_s{L_r}^{\′})}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}\end{array}\right.$ (formula 14)

Calculate NUM₂(s)/DEN₂S two limits of () expression formula are:

$\left\{\begin{array}{c}{b}_1=a_3\\ b_2=a_4\end{array}\right.$ (formula 15)

To DEN₁(s) and DEN₂S () carries out derivation, obtain dDEN₁(a_n)/ds and dDEN₂(b_n)/ds. Due to Re (a₁)=Re (a₂)=0, and Re (a₃)、Re(a₄)、Re(b₁) and Re (b₂) for non-zero, in the biphase rest frame of stator, grid voltage sags can be obtained by inverse Laplace transform, crowbar circuit trigger after the expression formula of Stator transient electric current be:

$\stackrel{\→}{I_{s\_transient}}\left(t\right)=\underset{n=3,4}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_1\left(a_n\right)}{{\mathrm{dDEN}}_1\left(a_n\right)/ds}{e}^{a_{n}t}+\underset{n=1,2}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_2\left(b_n\right)}{{\mathrm{dDEN}}_2\left(b_n\right)/ds}{e}^{b_{n}t}$ (formula 16)

Visible :-1/Re (a₃)、-1/Re(a₄)、-1/Re(b₁) and-1/Re (b₂) damping time constant of respectively each transient state component.

3) rotor transient current calculates.

WillWithSubstitute into (formula 6), obtain grid voltage sags, trigger crowbar circuit rotor windings is carried out short circuitAfter, rotor transient current is in the zero state response of the biphase rest frame of stator and zero input response:

$I_{r1}^{s^{\′}}=-\frac{(s-{\mathrm{j\ω}}_r){L_m}^{\′}}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}{I}_{s1}^s=\frac{{\mathrm{NUM}}_3\left(s\right)}{{\mathrm{DEN}}_3\left(s\right)}$ (formula 17)

$I_{r2}^{s^{\′}}=\frac{\stackrel{\→}{{\mathrm{\ψ}}_r^{s^{\′}}}\left(t_0\right)-(s-{\mathrm{j\ω}}_r){L_m}^{\′}{I}_{s2}^s}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}=\frac{{\mathrm{NUM}}_4\left(s\right)}{{\mathrm{DEN}}_4\left(s\right)}$ (formula 18)

In formula, NUM₃(s) and NUM₄S replacement molecule that () is expression formula, DEN₃(s) and DEN₄S replacement denominator that () is expression formula.For zero state response,For zero input response, its expression is deployable is:

$I_{r1}^{s^{\′}}=-\frac{(s-{\mathrm{j\ω}}_r){L_m}^{\′}}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\·\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s+{\mathrm{j\ω}}_1)\left|\stackrel{\→}{U_{spm}}\right|e^{{\mathrm{j\θ}}_{sp0}}+\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s-{\mathrm{j\ω}}_1)\left|\stackrel{\→}{U_{snm}}\right|e^{{\mathrm{j\θ}}_{sn0}}}{(s^2+{{\mathrm{\ω}}_1}^2)\{+s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s\}}$ (formula 19)

$\begin{array}{c}{I}_{r2}^{s^{\′}}=\frac{\stackrel{\→}{{{\mathrm{\ψ}}_r}^{s^{\′}}}\left(t_0\right)}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\·\frac{s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s}{s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s}\\ +\frac{-(s-{\mathrm{j\ω}}_r){L_m}^{\′}}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\·\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\]\stackrel{\→}{{{\mathrm{\ψ}}_s}^s}\left(t_0\right)-{{\mathrm{sL}}_m}^{\′}\stackrel{\→}{{\mathrm{\ψ}}_r^{s^{\′}}}\left(t_0\right)}{s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s}\end{array}$ (formula 20)

Five limit expression formulas be:

$\left\{\begin{array}{c}{c}_1=-({R_r}^{\′}/{L_r}^{\′})+j{\mathrm{\ω}}_r\\ c_2=a_1\\ c_3=a_2\\ c_4=a_3\\ c_5=a_4\end{array}\right.$ (formula 21)

Three limit expression formulas be:

$\left\{\begin{array}{c}{d}_1=c_1=-({R_r}^{\′}/{L_r}^{\′})+j{\mathrm{\ω}}_r\\ d_2=a_3\\ d_3=a_4\end{array}\right.$ (formula 22)

By to DEN₃(s) and DEN₄S () derivation, obtains dDEN₃(c_n)/ds and dDEN₄(d_n)/ds. Due to Re (c₂)=Re (c₃)=0, and Re (c₁)、Re(c₄)、Re(c₅)、Re(d₁)、Re(d₂) and Re (d₃) for non-zero, in the biphase rest frame of stator, grid voltage sags can be obtained by inverse Laplace transform, crowbar circuit trigger after rotor expression formula steady, transient current be respectively as follows:

$\stackrel{\→}{I_r^{s^{\′}}}\left(t\right){|}_{t\→\mathrm{\∞}}=\underset{n=2,3}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{c_{n}t}$ (formula 23a)

$\stackrel{\→}{I_{r\_transient}^{s^{\′}}}\left(t\right)=\underset{n=1,4,5}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{c_{n}t}+\underset{n=1,2,3}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_4\left(d_n\right)}{{\mathrm{dDEN}}_4\left(d_n\right)/ds}{e}^{d_{n}t}$ (formula 23b)

Visible ,-1/Re (c₁),-1/Re(c₄),-1/Re(c₅),-1/Re(d₁),-1/Re(d₂) and-1/Re (d₃) damping time constant of respectively each transient state component.

4) crow bar resistance is adjusted to retrain and is set.

Known ω₁=100 π, if double-fed generator angular rate ranges for ω_r∈[K₁ω₁,K₂ω₁], wherein K₁∈ (0,1] and K₂∈ [1,2). Rotor loop all-in resistance R when line voltage falls, when rotor-side converter is bypassed by crow bar, after winding is converted_r' include the rotor windings resistance R after winding is converted_rw' and through winding convert after crow bar resistance R_c', i.e. R_r'=R_rw'+R_c'. Making x is ω_rIndependent variable, constructor ω_r(n_x)=K₁ω₁+ 0.314x, wherein x be nonnegative integer (x=0,1,2,3 ...), with ω_r(n_x)∈[K₁ω₁,K₂ω₁] for constraints, obtain by ω_r(n_x) the sequence ω that constitutes_se. Making y is R_c' independent variable, constructor R_c'(m_y)=10^-3Y, wherein y be nonnegative integer (y=0,1,2,3 ...).

When line voltage falls, when rotor-side converter is bypassed by crow bar, it is assumed that c₁、a₃And a₄Real part is zero, and transient current is not decayed, (formula 23a), (formula 23b) can obtain rotor current expression formula and be:

$\stackrel{\→}{I_{r\_no\_decay}^{s^{\′}}}\left(t\right)=\underset{n=2,3}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{c_{n}t}+\underset{n=1,4,5}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{\mathrm{Im}\left(c_n\right)t}+\underset{n=1,2,3}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_4\left(d_n\right)}{{\mathrm{dDEN}}_4\left(d_n\right)/ds}{e}^{\mathrm{Im}\left(d_n\right)t}$ (formula 24)

According to R_c'(m_y)=10^-3Y, with y=0 for initial value, when the parameter of electric machine is known, if R_c'=R_c'(m₀), R_c'(m₀) resistance value when representing y=0, and by sequence ω_seAll elements gradually one by one substitute into (formula 24), ask forCycle expression formula, obtainsAt maximum when different rotating speeds of the peak value of a cycle internal moldConverted by winding, obtain ${\left|\stackrel{\→}{I_{r\_no\_decay}^s}\right|}_{T\_\max }\left(m_0\right)=\frac{1}{k}{\left|\stackrel{\→}{I_{r\_no\_decay}^{s^{\′}}}\right|}_{T\_\max }\left(m_0\right),$ ByCalculating obtains crowbar circuit line voltage peak higher limit:

$U_{LL\_max}\left(m_0\right)=\frac{2\sqrt{6}}{3k}{\left|\stackrel{\→}{I_{r\_no\_decay}^{s^{\′}}}\right|}_{T\_max}\left(m_0\right){R_c}^{\′}\left(m_0\right)$ (formula 25)

It is incremented by y, repeats above procedure, obtain U_{LL_max}(m₁), U_{LL_max}(m₂) ....

In y increasing process, work as U_{LL_max}(m_V)≥U_dc, when namely crowbar circuit line voltage peak higher limit is be more than or equal to four quadrant convertor DC bus-bar voltage setting value during y=V, stop above-mentioned calculating, convert R through winding_c=k²·R_c'(m_V-1) it is set as that crow bar resistance is adjusted the upper limit. With R_c∈[0,k²·R_c'(m_V-1)] for constraints, it is ensured that after crowbar circuit triggers, rotor current transformer is reliably bypassed.

The above is only the preferred embodiment of the present invention; it should be pointed out that, for those skilled in the art, under the premise without departing from the technology of the present invention principle; can also making some improvement and deformation, these improve and deformation also should be regarded as protection scope of the present invention.

Claims (2)

1. the DFIG crow bar resistance resolved based on frequency domain is adjusted constraint computational methods, it is characterized in that,

Parameter defines:

Rotor voltage in stator stationary coordinate system,Rotor voltage in rotor rotating coordinate system

Stator voltage in stator stationary coordinate system

Stator current in stator stationary coordinate system,Rotor current in stator stationary coordinate system

Stator current in rotor rotating coordinate system,Rotor current in rotor rotating coordinate system

Stator magnetic linkage in stator stationary coordinate system,Rotor flux in stator stationary coordinate system

R_sStator resistance, R_rRotor loop all-in resistance

R_rwRotor windings resistance, R_cRotor crow bar resistance

L_ssStator leakage inductance, L_rsRotor leakage inductance

L_mRotor mutual inductance, L_sStator inductance, L_rInductor rotor

L_rrThe self-induction through air gap produced by rotor windings

N_rk_NrStator effective turn

N_sk_NsRotor effective turn

K winding conversion factor

ω₁Line voltage synchronous rotational speed, ω_rRotor electric rotating angular velocity

θ_rAngle between stator A phase winding and rotor a phase winding

U_dcFour-quadrant phase current transformer DC bus-bar voltage setting value

p_nPower generator electrode logarithm

θ_sp0Grid voltage sags moment positive sequence voltage, θ_sn0The initial phase angle of grid voltage sags moment negative sequence voltage

The Vector Mode of positive sequence voltage after grid voltage sags,The Vector Mode of negative sequence voltage after grid voltage sags

Plural number is asked for the operator of real part by Re

Plural number is asked for the operator of imaginary part by Im;

Parameter subscript defines:

→ space vector

S stator coordinate,

R rotor coordinate

' through winding convert after numerical value;

Parameter subscript defines:

The biphase rest frame α axle of α stator,

The biphase rest frame β axle of β stator

S stator,

R rotor;

Comprise the following steps:

1) the stator and rotor space vector of voltage equation in stator stationary coordinate system and rotor rotating coordinate system is respectively as follows:

$\stackrel{\→}{U_s^s}=R_s\stackrel{\→}{I_s^s}+\frac{d\stackrel{\→}{{\mathrm{\ψ}}_s^s}}{dt}$ (formula 1a)

$\stackrel{\→}{U_r^r}=R_r\stackrel{\→}{I_r^r}+\frac{d\stackrel{\→}{{\mathrm{\ψ}}_r^r}}{dt}$ (formula 1b)

When the rotor number of phases is identical, by winding conversion factor $\frac{I_r^{s^{\′}}}{I_r^s}=\frac{I_r^{r^{\′}}}{I_r^r}=\frac{N_r{k}_{Nr}}{N_s{k}_{Ns}}=k$ With $\frac{U_r^{s^{\′}}}{U_r^s}=\frac{U_r^{r^{\′}}}{U_r^r}=\frac{N_s{k}_{Ns}}{N_r{k}_{Nr}}=\frac{1}{k},$ Carry out winding reduction according to (formula 1b) to obtain:

$\stackrel{\→}{U_r^{r^{\′}}}={R_r}^{\′}\stackrel{\→}{I_r^{r^{\′}}}+{L_{rr}}^{\′}\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}+{L_{r\mathrm{\δ}}}^{\′}\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}+{L_m}^{\′}\frac{d\stackrel{\→}{I_s^{r^{\′}}}}{dt}$ (formula 2)

Through winding reduction, L_rr' equal to L_m'; Calculated by (formula 2):

$\stackrel{\→}{U_r^{r^{\′}}}={R_r}^{\′}\stackrel{\→}{I_r^{r^{\′}}}+{L_m}^{\′}(\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}+\frac{d\stackrel{\→}{I_s^r}}{dt})+{L_{r\mathrm{\δ}}}^{\′}\frac{d\stackrel{\→}{I_r^{r^{\′}}}}{dt}$ (formula 3)

In stator stationary coordinate system, (formula 3) is expressed as:

$\stackrel{\→}{U_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}={R_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}+{L_m}^{\′}\frac{d(\stackrel{\→}{I_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}+\stackrel{\→}{I_s^s}{e}^{-{\mathrm{j\θ}}_r})}{dt}+{L_{r\mathrm{\δ}}}^{\′}\frac{d\left(\stackrel{\→}{I_r^{s^{\′}}}{e}^{-{\mathrm{j\θ}}_r}\right)}{dt}$ (formula 4)

The rotor voltage equation in the biphase rest frame of stator is obtained by (formula 4):

$\stackrel{\→}{U_r^{s^{\′}}}={R_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}+{L_r}^{\′}\frac{d\stackrel{\→}{I_r^{s^{\′}}}}{dt}+{L_m}^{\′}\frac{d\stackrel{\→}{I_s^s}}{dt}-{\mathrm{j\ω}}_r{L_m}^{\′}\stackrel{\→}{I_s^s}-{\mathrm{j\ω}}_r{L_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}$ (formula 5)

Consider grid voltage sags t₀The electric current initial value in moment, is carried out Laplace by (formula 5) and converts:

$I_r^{s^{\′}}=\frac{U_r^{s^{\′}}+{L_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}\left(t_0\right)+{L_m}^{\′}\stackrel{\→}{I_s^s}\left(t_0\right)-(s-{\mathrm{j\ω}}_r){L_m}^{\′}{I}_s^s}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}$ (formula 6)

(formula 6) is brought into ${\mathrm{\ψ}}_s^s=L_s{I}_s^s+{L_m}^{\′}{I}_r^{s^{\′}}$ :

${\mathrm{\ψ}}_s^s=L_s{I}_s^s+{L_m}^{\′}\[\frac{U_r^{s^{\′}}+{L_r}^{\′}\stackrel{\→}{I_r^{s^{\′}}}\left(t_0\right)+{L_m}^{\′}\stackrel{\→}{I_s^s}\left(t_0\right)-(s-{\mathrm{j\ω}}_r){L_m}^{\′}{I}_s^s}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\]$ (formula 7)

Space vector expression formula after grid voltage sags is:

$\stackrel{\→}{U_s^s}=\left|\stackrel{\→}{U_{spm}}\right|e^{j({\mathrm{\ω}}_{1}t+{\mathrm{\θ}}_{sp0})}+\left|\stackrel{\→}{U_{snm}}\right|e^{j(-{\mathrm{\ω}}_{1}t+{\mathrm{\θ}}_{sn0})}$ (formula 8)

(formula 8) carries out Laplace conversion can obtain:

$U_s^s=\frac{\left|\stackrel{\→}{U_{spm}}\right|}{s-{\mathrm{j\ω}}_1}{e}^{{\mathrm{j\θ}}_{sp0}}+\frac{\left|\stackrel{\→}{U_{snm}}\right|}{s+{\mathrm{j\ω}}_1}{e}^{{\mathrm{j\θ}}_{sn0}}$ (formula 9)

Consider grid voltage sags t₀The magnetic linkage initial value in moment, carries out Laplace to (formula 1a) and converts:

$U_s^s=R_s{I}_s^s+{\mathrm{s\ψ}}_s^s-\stackrel{\→}{{\mathrm{\ψ}}_s^s}\left(t_0\right)$ (formula 10)

2) Stator transient Current calculation

After grid voltage sags, trigger crowbar circuit and rotor windings is carried out short circuit, by zero state response and zero input response, the transient current of stator is calculated;

(formula 7), (formula 9) are substituted into (formula 10),Expression formula in a frequency domain is:

$I_s^s=I_{s1}^s+I_{s2}^s$ (formula 11)

In formulaWithRespectively Stator transient electric current is in the zero state response of stator stationary coordinate system and zero input response, and expression formula is respectively as follows:

$I_{s1}^s=\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s+{\mathrm{j\ω}}_1)\left|U_{spm}\right|e^{{\mathrm{j\θ}}_{sp0}}+\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s-{\mathrm{j\ω}}_1)\left|U_{snm}\right|e^{{\mathrm{j\θ}}_{sn0}}}{(s^2+{{\mathrm{\ω}}_1}^2)A_1}=\frac{{\mathrm{NUM}}_1\left(s\right)}{{\mathrm{DEN}}_1\left(s\right)}$ (formula 12)

$I_{s2}^s=\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\]\stackrel{\→}{{\mathrm{\ψ}}_s^s}\left(t_0\right)-{{\mathrm{sL}}_m}^{\′}\stackrel{\→}{{\mathrm{\ψ}}_r^{s^{\′}}}\left(t_0\right)}{A_1}=\frac{{\mathrm{NUM}}_1\left(s\right)}{{\mathrm{DEN}}_1\left(s\right)}$ (formula 13)

A in formula₁=s²L_r'L_s-s²L_m'²+sL_r'R_s+sR_r'L_s-sjω_rL_r'L_s+sjω_rL_m'²-jω_rL_r'R_s+R_r'R_s;NUM₁(s) and NUM₂S replacement molecule that () is expression formula, DEN₁(s) and DEN₂S replacement denominator that () is expression formula;

Calculate NUM₁(s)/DEN₁S four limits of () expression formula are:

$\left\{\begin{array}{c}{a}_1={\mathrm{j\ω}}_1\\ a_2=-{\mathrm{j\ω}}_1\\ a_3=\frac{-R_s{L_r}^{\′}-{R_r}^{\′}{L}_s+L_s{L_r}^{\′}{\mathrm{j\ω}}_r-{\mathrm{j\ω}}_r{L_m}^{\′2}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}+\frac{\sqrt{{(R_s{L_r}^{\′}+{R_r}^{\′}{L}_s-L_s{L_r}^{\′}{\mathrm{j\ω}}_r+{\mathrm{j\ω}}_r{L_m}^{\′2})}^2-4(L_s{L_r}^{\′}-{L_m}^{\′2})(R_s{R_r}^{\′}-{\mathrm{j\ω}}_r{R}_s{L_r}^{\′})}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}\\ a_4=\frac{-R_s{L_r}^{\′}-{R_r}^{\′}{L}_s+L_s{L_r}^{\′}{\mathrm{j\ω}}_r-{\mathrm{j\ω}}_r{L_m}^{\′2}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}-\frac{\sqrt{{(R_s{L_r}^{\′}+{R_r}^{\′}{L}_s-L_s{L_r}^{\′}{\mathrm{j\ω}}_r+{\mathrm{j\ω}}_r{L_m}^{\′2})}^2-4(L_s{L_r}^{\′}-{L_m}^{\′2})(R_s{R_r}^{\′}-{\mathrm{j\ω}}_r{R}_s{L_r}^{\′})}}{2(L_s{L_r}^{\′}-{L_m}^{\′2})}\end{array}\right.$ (formula 14)

Calculate NUM₂(s)/DEN₂S two limits of () expression formula are:

$\left\{\begin{array}{c}{b}_1=a_3\\ b_2=a_4\end{array}\right.$ (formula 15)

To DEN₁(s) and DEN₂S () carries out derivation, obtain dDEN₁(a_n)/ds and dDEN₂(b_n)/ds; Due to Re (a₁)=Re (a₂)=0, and Re (a₃)、Re(a₄)、Re(b₁) and Re (b₂) for non-zero, in the biphase rest frame of stator, grid voltage sags can be obtained by inverse Laplace transform, crowbar circuit trigger after the expression formula of Stator transient electric current be:

$\stackrel{\→}{I_{s\_transient}}\left(t\right)=\underset{n=3,4}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_1\left(a_n\right)}{{\mathrm{dDEN}}_1\left(a_n\right)/ds}{e}^{a_{n}t}+\underset{n=1,2}{\mathrm{\Σ}}\frac{{\mathrm{NUM}}_2\left(b_n\right)}{{\mathrm{dDEN}}_2\left(b_n\right)/ds}{e}^{b_{n}t}$ (formula 16)

-1/Re(a₃)、-1/Re(a₄)、-1/Re(b₁) and-1/Re (b₂) damping time constant of respectively each transient state component;

3) rotor transient current calculates

WillWithSubstituting into (formula 6), obtain grid voltage sags, trigger after rotor windings carries out short circuit by crowbar circuit, rotor transient current is in the zero state response of the biphase rest frame of stator and zero input response:

$I_{r1}^{s^{\′}}=-\frac{(s-{\mathrm{j\ω}}_r){L_m}^{\′}}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}{I}_{s1}^s=\frac{{\mathrm{NUM}}_3\left(s\right)}{{\mathrm{DEN}}_3\left(s\right)}$ (formula 17)

$I_{r2}^{s^{\′}}=\frac{\stackrel{\→}{{\mathrm{\ψ}}_r^{s^{\′}}}\left(t_0\right)-(s-{\mathrm{j\ω}}_r){L_m}^{\′}{I}_{s2}^s}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}=\frac{{\mathrm{NUM}}_4\left(s\right)}{{\mathrm{DEN}}_4\left(s\right)}$ (formula 18)

In formula, NUM₃(s) and NUM₄S replacement molecule that () is expression formula, DEN₃(s) and DEN₄S replacement denominator that () is expression formula;

Five limit expression formulas be:

$\left\{\begin{array}{c}{c}_1=-({R_r}^{\′}/{L_r}^{\′})+j{\mathrm{\ω}}_r\\ c_2=a_1\\ c_3=a_2\\ c_4=a_3\\ c_5=a_4\end{array}\right.$ (formula 21)

Three limit expression formulas be:

$\left\{\begin{array}{c}{d}_1=c_1=-({R_r}^{\′}/{L_r}^{\′})+j{\mathrm{\ω}}_r\\ d_2=a_3\\ d_3=a_4\end{array}\right.$ (formula 22)

By to DEN₃(s) and DEN₄S () derivation, obtains dDEN₃(c_n)/ds and dDEN₄(d_n)/ds; Due to Re (c₂)=Re (c₃)=0, and Re (c₁)、Re(c₄)、Re(c₅)、Re(d₁)、Re(d₂) and Re (d₃) for non-zero, in the biphase rest frame of stator, obtain grid voltage sags by inverse Laplace transform, crowbar circuit trigger after rotor expression formula steady, transient current be respectively as follows:

$\stackrel{\→}{I_r^{s^{\′}}}\left(t\right){|}_{t\→\mathrm{\∞}}=\underset{n=2,3}{\Σ}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)}{e}^{c_{n}t}$ (formula 23a)

$\stackrel{\→}{I_{r\_transient}^{s^{\′}}}\left(t\right)=\underset{n=1,4,5}{\Σ}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{c_{n}t}+\underset{n=1,2,3}{\Σ}\frac{{\mathrm{NUM}}_4\left(d_n\right)}{{\mathrm{dDEN}}_4\left(d_n\right)/ds}{e}^{d_{n}t}$ (formula 23b)

-1/Re(c₁),-1/Re(c₄),-1/Re(c₅),-1/Re(d₁),-1/Re(d₂) and-1/Re (d₃) damping time constant of respectively each transient state component;

4) crow bar resistance is adjusted to retrain and is set

If the double-fed generator electric angle range of speeds is ω_r∈[K₁ω₁,K₂ω₁], wherein K₁∈ (0,1] and K₂∈ [1,2); With ω_r∈[K₁ω₁,K₂ω₁] for constraints, build with the x function ω being variable_r(n_x)=K₁ω₁+ 0.314x, wherein x be nonnegative integer (x=0,1,2,3 ...), with ω_r(n_x)∈[K₁ω₁,K₂ω₁] for constraints, obtain by ω_r(n_x) the sequence ω that constitutes_se; Making y is R_c' independent variable, constructor R_c'(m_y)=10^-3Y, wherein y be nonnegative integer (y=0,1,2,3 ...);

When line voltage falls, when rotor-side converter is bypassed by crow bar, it is assumed that c₁、a₃And a₄Real part is zero, and transient current is not decayed, (formula 23a), (formula 23b) obtain rotor current expression formula and be:

$\stackrel{\→}{I_{r\_no\_decay}^{s^{\′}}}\left(t\right)=\underset{n=2,3}{\Σ}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{c_{n}t}+\underset{n=1,4,5}{\Σ}\frac{{\mathrm{NUM}}_3\left(c_n\right)}{{\mathrm{dDEN}}_3\left(c_n\right)/ds}{e}^{\mathrm{Im}\left(c_n\right)t}+\underset{n=1,2,3}{\Σ}\frac{{\mathrm{NUM}}_4\left(d_n\right)}{{\mathrm{dDEN}}_4\left(d_n\right)/ds}{e}^{\mathrm{Im}\left(d_n\right)t}$ (formula 24)

According to R_c'(m_y)=10^-3Y, with y=0 for initial value, when the parameter of electric machine is known, if R_c'=R_c'(m₀), R_c'(m₀) resistance value when representing y=0, and by sequence ω_seAll elements gradually one by one substitute into (formula 24), ask forCycle expression formula, obtainsAt maximum when different rotating speeds of the peak value of a cycle internal moldConverted by winding, obtain $|\stackrel{\→}{I_{r\_no\_decay}^s}{|}_{T\_\max }\left(m_0\right)=\frac{1}{k}|\stackrel{\→}{I_{r\_no\_decay}^{s^{\′}}}{|}_{T\_max}\left(m_0\right),$ ByCalculating obtains crowbar circuit line voltage peak higher limit:

$U_{LL\_max}\left(m_0\right)=\frac{2\sqrt{6}}{3k}|\stackrel{\→}{I_{r\_no\_decay}^{s^{\′}}}{|}_{T\_max}\left(m_0\right){R_c}^{\′}\left(m_0\right)$ (formula 25)

It is incremented by y, repeats above procedure, obtain U_{LL_max}(m₁), U_{LL_max}(m₂) ...;

In y increasing process, work as U_{LL_max}(m_V)≥U_dc, when namely crowbar circuit line voltage peak higher limit is be more than or equal to four quadrant convertor DC bus-bar voltage setting value during y=V, stops calculating, convert R through winding_c=k²·R_c'(m_V-1) it is set as that crow bar resistance is adjusted the upper limit; With R_c∈[0,k²·R_c'(m_V-1)] for constraints, it is ensured that after crowbar circuit triggers, rotor current transformer is reliably bypassed.

2. the DFIG crow bar resistance resolved based on frequency domain according to claim 1 is adjusted constraint computational methods, it is characterized in that, step 3) in,WithExpression expands into:

$I_{r1}^{s^{\′}}=-\frac{(s-{\mathrm{j\ω}}_r){L_m}^{\′}}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\·\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s+{\mathrm{j\ω}}_1)\left|\right|U_{spm}|e^{{\mathrm{j\θ}}_{sp0}}+\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\](s-{\mathrm{j\ω}}_1)|U_{snm}|e^{{\mathrm{j\θ}}_{sn0}}}{(s^2+{{\mathrm{\ω}}_1}^2)\{+s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s\}}$ (formula 19)

$\begin{array}{c}{I}_{r2}^{s^{\′}}=\frac{\stackrel{\→}{{{\mathrm{\ψ}}_r}^{s^{\′}}}\left(t_0\right)}{{R_s}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\·\frac{s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_s}^{\′}{L}_r-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s}{s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_s}^{\′}{L}_r-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s}\\ +\frac{-(s-{\mathrm{j\ω}}_r){L_m}^{\′}}{{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}}\·\frac{\[{R_r}^{\′}+(s-{\mathrm{j\ω}}_r){L_r}^{\′}\]\stackrel{\→}{{{\mathrm{\ψ}}_s}^s}\left(t_0\right)-{{\mathrm{sL}}_m}^{\′}\stackrel{\→}{{\mathrm{\ψ}}_r^{s^{\′}}}\left(t_0\right)}{s^2{L_r}^{\′}{L}_s-s^2{L_m}^{\′2}+{{\mathrm{sL}}_r}^{\′}{R}_s+{{\mathrm{sR}}_r}^{\′}{L}_s-{\mathrm{sj\ω}}_r{L_r}^{\′}{L}_s+{\mathrm{sj\ω}}_r{L_m}^{\′2}-{\mathrm{j\ω}}_r{L_r}^{\′}{R}_s+{R_r}^{\′}{R}_s}\end{array}$ (formula 20).