CN103995245B - Fault judgment method of stator and rotor current signal detection system of doubly-fed wind generator - Google Patents

Abstract

The invention provides a fault judgment method of a stator and rotor current signal detection system of a doubly-fed wind generator, and belongs to the field of electrical control. The method is used for solving the problem that after the stator and rotor current signal detection system of a doubly-fed wind power system breaks down, reliability of the whole doubly-fed wind power system is influenced. According to the fault judgment method of the stator and rotor current signal detection system of the doubly-fed wind generator, a current state equation is deduced on the basis of voltage and flux linkage equations of the doubly-fed wind generator, and a stator and rotor current observer is created according to the model reference self-adaptation theory; errors are output by analyzing state variables of the observer, and observed values of all phases of a current are compared with actual values to find specific sources of faults of a sensor. The method is based on the model reference self-adaptation theory, the observed values of the current do not depend on corresponding sampling values of the sensor, and rapidness and accuracy of fault judgment are improved. The method is suitable for fault judgment of the stator and rotor current signal detection system of the doubly-fed wind generator.

Description

The fault judgment method of double-fed wind power generator stator and rotor current signal detection system

Technical field

The invention belongs to field of electrical control.

Background technology

The rotor of double feedback electric engine is all connected with electrical network, can not only be used for generator operation, also can be used as motor running. When using as wind-driven generator, double feedback electric engine stator winding is generally joined directly together with electrical network, rotor windings pass through current transformer with Electrical network is connected, and current transformer provides the alternating current power supply of amplitude, frequency, phase-adjustable for rotor, as shown in Figure 1.Double-fed wind generating Unit belongs to vscf generator group, has range of operation width, wind energy conversion efficiency height, stable output power and mechanical stress Low advantage, and pass through vector and vector oriented control technology, enable stator active power of output and reactive power Independent uneoupled control, improve the motility of power system and dynamic and static stability, be the wind-force being most widely used at present Generating set type.

With wind energy turbine set generating capacity increase and marine wind electric field popularization, to dual feedback wind power generation system can There is higher requirement by property.The reliability of double-fed wind power system depends on signal detection system, control system and electrical system Deng stable operation.But because the signal detection system of wind power system includes sensor sample, signal condition and ad sampling And the intermediate link such as correlation signal transmission, during this results in double-fed wind power system long-term work, inevitably occur The signal detection system failure problems such as electric current, voltage.Therefore the presence or absence of failure judgement and position are of crucial importance, and only quick Determine whether fault occurs, the accurate location of fault and replace the stability that fault-signal could strengthen wind power system in time.

Content of the invention

The present invention is whole in order to solve to affect after the stator and rotor current signal detection system of double-fed wind power system breaks down The problem of the reliability of individual system, now provides the breakdown judge side of double-fed wind power generator stator and rotor current signal detection system Method.

The fault judgment method of double-fed wind power generator stator and rotor current signal detection system, the method includes:

Step one: gather the stator voltage sig u of double-fed generator respectively_{s_abc}, stator current signal i_{s_abc}, rotor electricity Stream signal i_{r_abc}With rotor voltage control signal u_{r_dq}；

Step 2: to stator voltage sig u_{s_abc}Carry out clark conversion, obtain stator voltage letter under static two phase coordinate systems Number u_{s_αβ}, to stator voltage sig u under this static two phase coordinate system_{s_αβ}Carry out park conversion, it is fixed under synchronous rotating frame to obtain Sub- voltage signal u_{s_dq}；

To stator current signal i_{s_abc}Carry out clark conversion, obtain the stator current signal under static two phase coordinate systems i_{s_αβ}, to stator current signal i under this static two phase coordinate system_{s_αβ}Carry out park conversion, obtain stator under synchronous rotating frame Current signal i_{s_dq}；

To rotor current signal i_{r_abc}Carry out clark conversion, obtain static two phase coordinate system lower rotor part current signal i_{r_αβ}, To this static two phase coordinate system lower rotor part current signal i_{r_αβ}Carry out park conversion, obtain synchronous rotating frame lower rotor part electric current Signal i_{r_dq}；

Step 3: by stator current signal i under synchronous rotating frame_{s_dq}With synchronous rotating frame lower rotor part electric current letter Number i_{r_dq}As state variable, by stator voltage sig u under synchronous rotating frame_{s_dq}With rotor voltage control signal u_{r_dq}Make For rotor input variable, according to double feedback electric engine voltage equation and flux linkage equations, set up stator and rotor current discrete state equations；

Step 4: using rotor discrete time state equation and with reference to reduced order observer design theory, it is fixed to set up respectively Electron current observer and rotor current observer,

Obtain stator current observation under synchronous rotating frame using stator current observer

Obtain synchronous rotating frame lower rotor part current observation using rotor current observer

Step 5: stator current predictive value i ' is obtained according to stator current predictive equation_{s_dq}, according to rotor current prediction side Journey obtains rotor current predictive value i '_{r_dq}, according to stator current signal i under synchronous rotating frame_{s_dq}, synchronous rotating frame Lower rotor part current signal i_{r_dq}, stator current predictive value i ' under synchronous rotating frame_{s_dq}With synchronous rotating frame lower rotor part electricity Stream predictive value i '_{r_dq}, obtain stator current signal detecting system fault difference r_esPoor with rotor current signal detecting system fault Value r_er；

Step 6: by stator current signal detecting system fault difference r_esWith rotor current signal detecting system fault difference r_erDiffer from, obtain difference information r_c, according to the stator current predictive value i ' under synchronous rotating frame_{s_dq}And synchronously rotating reference frame Rotor current predictive value i ' under system_{r_dq}Determine difference information r with the steady-state error of corresponding actual value_cJudgment threshold+δ and- η, and by difference information r_cIt is compared with judgment threshold+δ and-η respectively,

Work as r_cDuring ＞+δ, then the stator current signal detecting system of double-fed wind power generator breaks down, and then executes step Rapid seven；

Work as r_cDuring ＜-η, then the rotor current signal detecting system of double-fed wind power generator breaks down, and then executes step Rapid eight；

Step 7: by stator current observation under synchronous rotating frameCarry out anti-coordinate transform, obtain static three Stator a phase current observation under phase coordinate systemWith stator b phase current observationTo gather fixed under static three phase coordinate systems simultaneously Sub- a phase current actual value i_saWith stator b phase current actual value i_sb；

WillWith stator a phase current threshold value δ_saIt is compared, whenMore than stator a phase current threshold value δ_saWhen, Then stator a phase current signal detecting system breaks down；Wherein, stator a phase current threshold value δ_saFor determining under static three phase coordinate systems Sub- a phase current observationWith stator a phase current actual value i under static three phase coordinate systems_saThe maximum of steady-state error；

WillWith stator b phase current threshold value δ_sbIt is compared, whenMore than stator b phase current threshold value δ_sbWhen, Then stator b phase current signal detecting system breaks down；Wherein, stator b phase current threshold value δ_sbFor determining under static three phase coordinate systems Sub- b phase current observationWith stator b phase current actual value i under static three phase coordinate systems_sbThe maximum of steady-state error；

Step 8: by the rotor current observation under synchronous rotating frameCarry out anti-coordinate transform, obtain static three Rotor a phase current observation under phase coordinate systemWith rotor b phase current observationGather under static three phase coordinate systems simultaneously Rotor a phase current actual value i_raWith rotor b phase current actual value i_rb；

WillWith rotor a phase current threshold value δ_raIt is compared, whenMore than rotor a phase current threshold value δ_raWhen, Then rotor a phase current signal detecting system breaks down；Wherein, rotor a phase current threshold value δ_raFor lower turn of static three phase coordinate systems Sub- a phase current observationWith static three phase coordinate system lower rotor part a phase current actual value i_raThe maximum of steady-state error；

WillWith rotor b phase current threshold value δ_rbIt is compared, whenMore than rotor b phase current threshold value δ_rbWhen, Then rotor b phase current signal detecting system breaks down；Wherein, rotor b phase current threshold value δ_rbFor lower turn of static three phase coordinate systems Sub- b phase current observationWith static three phase coordinate system lower rotor part b phase current actual value i_raThe maximum of steady-state error.

The fault judgment method of double-fed wind power generator stator and rotor current signal detection system of the present invention, fault is sentenced Break signal comes from the mould of the state variable deviation of observer model and actual measurement, can quickly obtain during signal detection system fault Breakdown judge signal；Stator and rotor current observation can be used for controlling as redundant signals during current signal detecting system fault System processed, provides basis for faults-tolerant control during signal detection system fault, can real-time monitored double-fed wind power generator surely turn Electron current actual value, increases the reliability of double-fed wind power system；And stator and rotor current observation respectively with stator and rotor electric current believe Separate between the measured value that number detecting system obtains, improve the accuracy of current observer.The present invention is suitable for double Feedback stator and rotor of wind power generator current signal detecting system carries out breakdown judge.

Brief description

Fig. 1 is double-fed wind power generator control system block diagram in actual applications；

Fig. 2 is the principle schematic setting up stator current observer described in specific embodiment four；

Fig. 3 is the principle schematic setting up rotor current observer described in specific embodiment five；

Fig. 4 is the principle schematic in step 6, current signal detecting system fault being judged；

Fig. 5 is the principle schematic in step 7, stator current signal detecting system fault being judged；

Fig. 6 is the principle schematic in step 8, rotor current signal detecting system fault being judged.

Specific embodiment

Specific embodiment one: illustrate present embodiment with reference to Fig. 4,5 and 6, the double-fed wind described in present embodiment The fault judgment method of power generator stator and rotor current signal detection system, the method includes:

Step one: gather the stator voltage sig u of double-fed generator respectively_{s_abc}, stator current signal i_{s_abc}, rotor electricity Stream signal i_{r_abc}With rotor voltage control signal u_{r_dq}；

Step 2: to stator voltage sig u_{s_abc}Carry out clark conversion, obtain stator voltage letter under static two phase coordinate systems Number u_{s_αβ}, to stator voltage sig u under this static two phase coordinate system_{s_αβ}Carry out park conversion, it is fixed under synchronous rotating frame to obtain Sub- voltage signal u_{s_dq}；

To stator current signal i_{s_abc}Carry out clark conversion, obtain the stator current signal under static two phase coordinate systems i_{s_αβ}, to stator current signal i under this static two phase coordinate system_{s_αβ}Carry out park conversion, obtain stator under synchronous rotating frame Current signal i_{s_dq}；

To rotor current signal i_{r_abc}Carry out clark conversion, obtain static two phase coordinate system lower rotor part current signal i_{r_αβ}, To this static two phase coordinate system lower rotor part current signal i_{r_αβ}Carry out park conversion, obtain synchronous rotating frame lower rotor part electric current Signal i_{r_dq}；

Step 3: by stator current signal i under synchronous rotating frame_{s_dq}With synchronous rotating frame lower rotor part electric current letter Number i_{r_dq}As state variable, by stator voltage sig u under synchronous rotating frame_{s_dq}With rotor voltage control signal u_{r_dq}Make For rotor input variable, according to double feedback electric engine voltage equation and flux linkage equations, set up stator and rotor current discrete state equations；

Step 4: using rotor discrete time state equation and with reference to reduced order observer design theory, it is fixed to set up respectively Electron current observer and rotor current observer,

Obtain stator current observation under synchronous rotating frame using stator current observer

Obtain synchronous rotating frame lower rotor part current observation using rotor current observer

Step 5: stator current predictive value i ' is obtained according to stator current predictive equation_{s_dq}, according to rotor current prediction side Journey obtains rotor current predictive value i '_{r_dq}, according to stator current signal i under synchronous rotating frame_{s_dq}, synchronous rotating frame Lower rotor part current signal i_{r_dq}, stator current predictive value i ' under synchronous rotating frame_{s_dq}With synchronous rotating frame lower rotor part electricity Stream predictive value i '_{r_dq}, obtain stator current signal detecting system fault difference r_esPoor with rotor current signal detecting system fault Value r_er；

Step 6: by stator current signal detecting system fault difference r_esWith rotor current signal detecting system fault difference r_erDiffer from, obtain difference information r_c, according to the stator current predictive value i ' under synchronous rotating frame_{s_dq}And synchronously rotating reference frame Rotor current predictive value i ' under system_{r_dq}Determine difference information r with the steady-state error of corresponding actual value_cJudgment threshold+δ and- η, and by difference information r_cIt is compared with judgment threshold+δ and-η respectively,

Work as r_cDuring ＞+δ, then the stator current signal detecting system of double-fed wind power generator breaks down, and then executes step Rapid seven；

Work as r_cDuring ＜-η, then the rotor current signal detecting system of double-fed wind power generator breaks down, and then executes step Rapid eight；

Step 7: by stator current observation under synchronous rotating frameCarry out anti-coordinate transform, obtain static three Stator a phase current observation under phase coordinate systemWith stator b phase current observationTo gather fixed under static three phase coordinate systems simultaneously Sub- a phase current actual value i_saWith stator b phase current actual value i_sb；

WillWith stator a phase current threshold value δ_saIt is compared, whenMore than stator a phase current threshold value δ_saWhen, Then stator a phase current signal detecting system breaks down；Wherein, stator a phase current threshold value δ_saFor determining under static three phase coordinate systems Sub- a phase current observationWith stator a phase current actual value i under static three phase coordinate systems_saThe maximum of steady-state error；

WillWith stator b phase current threshold value δ_sbIt is compared, whenMore than stator b phase current threshold value δ_sbWhen, Then stator b phase current signal detecting system breaks down；Wherein, stator b phase current threshold value δ_sbFor determining under static three phase coordinate systems Sub- b phase current observationWith stator b phase current actual value i under static three phase coordinate systems_sbThe maximum of steady-state error；

Step 8: by the rotor current observation under synchronous rotating frameCarry out anti-coordinate transform, obtain static three Rotor a phase current observation under phase coordinate systemWith rotor b phase current observationGather under static three phase coordinate systems simultaneously Rotor a phase current actual value i_raWith rotor b phase current actual value i_rb；

WillWith rotor a phase current threshold value δ_raIt is compared, whenMore than rotor a phase current threshold value δ_raWhen, Then rotor a phase current signal detecting system breaks down；Wherein, rotor a phase current threshold value δ_raFor lower turn of static three phase coordinate systems Sub- a phase current observationWith static three phase coordinate system lower rotor part a phase current actual value i_raThe maximum of steady-state error；

WillWith rotor b phase current threshold value δ_rbIt is compared, whenMore than rotor b phase current threshold value δ_rbWhen, Then rotor b phase current signal detecting system breaks down；Wherein, rotor b phase current threshold value δ_rbFor lower turn of static three phase coordinate systems Sub- b phase current observationWith static three phase coordinate system lower rotor part b phase current actual value i_raThe maximum of steady-state error.

In present embodiment step one, described stator voltage sig u_{s_abc}By the stator electricity of double-fed generator stator winding Pressure signal detection system obtains；Described stator current signal i_{s_abc}Obtained by stator current signal detecting system；Described rotor Current signal i_{r_abc}Obtained by rotor current signal detecting system；Described rotor voltage control signal u_{r_dq}By rotor unsteady flow Device pwm control module obtains.

In step 6, r_cBe a positive and negative undulate quantity, take its stable state to be+δ to the maximum, minimum-η as threshold value, in fault Under the conditions of the r that obtains_cIt is possible to judge stator failure or rotor fault not in the range of this.

Specific embodiment two: present embodiment is to the double-fed wind power generator rotor described in specific embodiment one The fault judgment method of current signal detecting system is described further, in present embodiment, double feedback electric engine described in step 3 Voltage equation is:

$\left\{\begin{array}{c}{u}_{sd}={\mathrm{d\ψ}}_{sd}-{\mathrm{\ω}}_1{\mathrm{\ψ}}_{sq}-r_s{i}_{sd}\\ u_{sq}={\mathrm{d\ψ}}_{sq}+{\mathrm{\ω}}_1{\mathrm{\ψ}}_{sd}-r_s{i}_{sq}\end{array}\right.,\left\{\begin{array}{c}{u}_{rd}={\mathrm{d\ψ}}_{rd}-{\mathrm{\ω}}_2{\mathrm{\ψ}}_{rq}+r_r{i}_{rd}\\ u_{rq}={\mathrm{d\ψ}}_{rq}+{\mathrm{\ω}}_2{\mathrm{\ψ}}_{rd}+r_r{i}_{rq}\end{array}\right.,$

Wherein, r_sFor stator resistance, r_rFor rotor resistance；u_sdFor the stator voltage component of d axle under synchronous rotating frame, u_sqStator voltage component for q axle under synchronous rotating frame；u_rdFor the rotor voltage component of d axle under synchronous rotating frame, u_rqRotor voltage component for q axle under synchronous rotating frame；i_sdFor the stator current components of d axle under synchronous rotating frame, i_sqStator current components for q axle under synchronous rotating frame；i_rdFor the rotor current component of d axle under synchronous rotating frame, i_rqRotor current component for q axle under synchronous rotating frame；ψ_sdFor the stator magnetic linkage component of d axle under synchronous rotating frame, ψ_sqStator magnetic linkage component for q axle under synchronous rotating frame；ψ_rdFor the rotor flux component of d axle under synchronous rotating frame, ψ_rqRotor flux component for q axle under synchronous rotating frame；ω₁For stator synchronization angular frequency, ω₂For rotor electric angle frequency；d =d/dt is differential operator；

Described flux linkage equations are:

$\left\{\begin{array}{c}{\mathrm{\ψ}}_{sd}=-l_s{i}_{sd}+l_m{i}_{rd}\\ {\mathrm{\ψ}}_{sq}=-l_s{i}_{sq}+l_m{i}_{rq}\end{array}\right.,\left\{\begin{array}{c}{\mathrm{\ψ}}_{rd}=l_r{i}_{rd}-l_m{i}_{sd}\\ {\mathrm{\ψ}}_{rq}=l_r{i}_{rq}-l_m{i}_{sq}\end{array}\right.,$

Wherein, l_sFor stator inductance, l_rFor inductor rotor, l_mMutual inductance for stator and rotor.

Specific embodiment three: present embodiment is to the double-fed wind power generator rotor described in specific embodiment two The fault judgment method of current signal detecting system is described further, in present embodiment, the electricity of rotor described in step 3 Stream discrete state equations are:

I (k+1)=a_di(k)+b_du(k)

In formula, i (k)=[i_sdi_sqi_rdi_rq]^tFor state variable, u (k)=[u_sdu_squ_rdu_rq]^tFor input variable； K is positive integer；a_dFor state equation state variable coefficient matrix, b_dFor state equation input variable coefficient matrix, and have:

$a_d=\left[\begin{array}{cccc}1-\frac{r_s{t}_s}{{\mathrm{\σl}}_s}& ({\mathrm{\ω}}_1+{\mathrm{p\ω}}_m(\frac{1}{\mathrm{\σ}}-1\left)\right)t_s& -\frac{r_r{l}_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& -\frac{{\mathrm{p\ω}}_m{l}_m{t}_s}{{\mathrm{\σl}}_s}\\ -\[{\mathrm{\ω}}_1+{\mathrm{p\ω}}_m\left(\frac{1}{\mathrm{\σ}}-1\right)\]t_s& 1-\frac{r_s{t}_s}{{\mathrm{\σl}}_s}& \frac{{\mathrm{p\ω}}_m{l}_m{t}_s}{{\mathrm{\σl}}_s}& -\frac{r_r{l}_m{t}_s}{{\mathrm{\σl}}_r{l}_s}\\ -\frac{r_s{l}_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& \frac{{\mathrm{p\ω}}_m{l}_m{t}_s}{{\mathrm{\σl}}_r}& 1-\frac{r_r{t}_s}{{\mathrm{\σl}}_r}& ({\mathrm{\ω}}_1-\frac{{\mathrm{p\ω}}_m}{\mathrm{\σ}})t_s\\ -\frac{{\mathrm{p\ω}}_m{l}_m{t}_s}{{\mathrm{\σl}}_r}& -\frac{r_s{l}_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& -({\mathrm{\ω}}_1-\frac{{\mathrm{p\ω}}_m}{\mathrm{\σ}})t_s& 1-\frac{r_r{t}_s}{{\mathrm{\σl}}_r}\end{array}\right],$

$b_d=\left[\begin{array}{cccc}-\frac{t_s}{{\mathrm{\σl}}_s}& 0& \frac{l_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& 0\\ 0& -\frac{t_s}{{\mathrm{\σl}}_s}& 0& \frac{l_m{t}_s}{{\mathrm{\σl}}_r{l}_s}\\ -\frac{l_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& 0& \frac{t_s}{{\mathrm{\σl}}_r}& 0\\ 0& -\frac{l_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& 0& \frac{t_s}{{\mathrm{\σl}}_r}\end{array}\right],$

Wherein,Represent leakage inductance coefficient, t_sFor the sampling time, p is double feedback electric engine number of pole-pairs, ω_mFor son Machinery rotation angular frequency.

For realizing energy converting between mechanical, stator and rotor rotating excitation field should keep geo-stationary it may be assumed that stator synchronization angular frequency ω₁, rotor electric angle frequencies omega₂With rotor machinery rotation angular frequency_mBetween three, relation is ω₁=p ω_m±ω₂；Take rotor Electric current { i_sdi_sqi_rdi_rqIt is state variable, with rotor voltage { u_sdu_squ_rdu_rqIt is input variable, set up rotor Electric current discrete state equations are: i (k+1)=a_di(k)+b_du(k).

Specific embodiment four: illustrate present embodiment with reference to Fig. 2, present embodiment is to specific embodiment three The fault judgment method of described double-fed wind power generator stator and rotor current signal detection system is described further, this embodiment party In formula, the method setting up stator current observer described in step 4 is:

State variable i (k) of stator and rotor current discrete state equations is divided into part i not directly measuring_s(k) and can Part i measured directly_r(k), and stator and rotor current discrete state equations are rewritten as block form:

Wherein,

$a_{d12}=\left[\begin{array}{cc}-\frac{r_r{l}_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& -\frac{{\mathrm{p\ω}}_m{l}_m{t}_s}{{\mathrm{\σl}}_s}\\ \frac{{\mathrm{p\ω}}_m{l}_m{t}_s}{{\mathrm{\σl}}_s}& -\frac{r_r{l}_m{t}_s}{{\mathrm{\σl}}_r{l}_s}\end{array}\right],$

$a_{d21}=\left[\begin{array}{cc}-\frac{r_s{l}_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& \frac{{\mathrm{p\ω}}_m{l}_m{t}_s}{{\mathrm{\σl}}_r}\\ -\frac{{\mathrm{p\ω}}_m{l}_m{t}_s}{{\mathrm{\σl}}_r}& -\frac{r_s{l}_m{t}_s}{{\mathrm{\σl}}_r{l}_s}\end{array}\right],a_{d22}=\left[\begin{array}{cc}1-\frac{r_r{t}_s}{{\mathrm{\σl}}_r}& ({\mathrm{\ω}}_1-\frac{{\mathrm{p\ω}}_m}{\mathrm{\σ}})t_s\\ -({\mathrm{\ω}}_1-\frac{{\mathrm{p\ω}}_m}{\mathrm{\σ}})t& 1-\frac{r_r{t}_s}{{\mathrm{\σl}}_r}\end{array}\right],$

$b_{ds}=\left[\begin{array}{cccc}-\frac{t_s}{{\mathrm{\σl}}_s}& 0& -\frac{l_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& 0\\ 0& -\frac{t_s}{{\mathrm{\σl}}_s}& 0& \frac{l_m{t}_s}{{\mathrm{\σl}}_r{l}_s}\end{array}\right],b_{dr}=\left[\begin{array}{cccc}-\frac{l_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& 0& \frac{t_s}{{\mathrm{\σl}}_r}& 0\\ 0& -\frac{l_m{t}_s}{{\mathrm{\σl}}_r{l}_s}& 0& \frac{t_s}{{\mathrm{\σl}}_r}\end{array}\right],$

y_rK () is output vector, c is output matrix and c=[0 01 1], i_s(k+1) it is immeasurability partial stator electricity The state equation of stream, i_r(k+1) it is the state equation that can measure part rotor current；

The state equation that part rotor current can be measured is write as the form of output equation:

i_r(k+1)-a_d22i_r(k)-b_drU (k)=a_d21i_s(k)

The state equation of immeasurability partial stator electric current is:

i_s(k+1)=a_d11i_s(k)+a_d12i_r(k)+b_dsu(k)

Stator and rotor current discrete state equations are rewritten as by following form according to above formula:

$\left\{\begin{array}{c}{i}_s(k+1)=a_{d11}{i}_s\left(k\right)+\[\begin{array}{cc}{a}_{d12}& b_{ds}\end{array}\]\left[\begin{array}{c}{i}_r\left(k\right)\\ u\left(k\right)\end{array}\right]\\ y_1\left(k\right)=a_{d21}{i}_s\left(k\right)\end{array}\right.$

Wherein y₁K () is the first output vector, this output vector partly to be represented with measuring:

y₁(k)=i_r(k+1)-a_d22i_r(k)-b_dru(k)

According to full order observer design theory, obtain the full rank stator observer model of new state equation subsystem:

$\left\{\begin{array}{c}{\hat{i}}_s(k+1)=(a_{d11}-k_s{a}_{d21}){\hat{i}}_s\left(k\right)+\[\begin{array}{cc}{a}_{d12}& b_{ds}\end{array}\]\left[\begin{array}{c}{i}_r\left(k\right)\\ u\left(k\right)\end{array}\right]+k_s{y}_1\left(k\right)\\ {\hat{y}}_1\left(k\right)=a_{d21}{\hat{i}}_s\left(k\right)\end{array}\right.$

Wherein, k_sFor the first observer feedback gain matrix.

Specific embodiment five: illustrate present embodiment with reference to Fig. 3, present embodiment is to specific embodiment four The fault judgment method of described double-fed wind power generator stator and rotor current signal detection system is described further, this embodiment party In formula, the method setting up rotor current observer described in step 4 is:

State variable i (k) of stator and rotor current discrete state equations is divided into part i not directly measuring_r(k) and can Part measured directly is i_s(k), and stator and rotor current discrete state equations are rewritten as:

The state equation that part stator current can be measured is rewritten as:

i_s(k+1)-a_d11i_s(k)-b_dsU (k)=a_d12i_r(k)

The state equation of immeasurability part rotor electric current is

i_r(k+1)=a_d22i_r(k)+a_d21i_s(k)+b_dru(k)

Stator and rotor current discrete state equations are rewritten as by following form according to above formula:

$\left\{\begin{array}{c}{i}_r(k+1)=a_{d22}{i}_s\left(k\right)+\[\begin{array}{cc}{a}_{d21}& b_{dr}\end{array}\]\left[\begin{array}{c}{i}_s\left(k\right)\\ u\left(k\right)\end{array}\right]\\ y_2\left(k\right)=a_{d12}{i}_r\left(k\right)\end{array}\right.$

Wherein y₂K () is the second output vector, this output vector partly to be represented with measuring:

y₂(k)=i_s(k+1)-a_d11i_s(k)-b_dsu(k)

According to full order observer design theory, obtain the full rank rotor observer model of new state equation subsystem:

$\left\{\begin{array}{c}{\hat{i}}_r(k+1)=(a_{d22}-k_r{a}_{d12}){\hat{i}}_r\left(k\right)+\[\begin{array}{cc}{a}_{d12}& b_{dr}\end{array}\]\left[\begin{array}{c}{i}_s\left(k\right)\\ u\left(k\right)\end{array}\right]+k_r{y}_2\left(k\right)\\ {\hat{y}}_2\left(k\right)=a_{d12}{\hat{i}}_r\left(k\right)\end{array}\right.$

Wherein, k_rFor the second observer feedback gain matrix.

Specific embodiment six: present embodiment is to the double-fed wind power generator rotor described in specific embodiment five The fault judgment method of current signal detecting system is described further, in present embodiment, according to stator described in step 5 Current forecasting equation obtains stator current predictive value i '_{s_dq}, rotor current predictive value is obtained according to rotor current predictive equation i′_{r_dq}Method be:

Stator and rotor voltage and current according to the kt moment is predicted to the stator and rotor electric current of (k+1) t, prediction side Cheng Wei:

I ' [(k+1) t]=a_di(kt)+b_du(kt)

Wherein, i '=[i '_sdi′_sqi′_rdi′_rq]^tPredictive value for stator and rotor electric current；T is system communication cycle.

Specific embodiment seven: present embodiment is to the double-fed wind power generator rotor described in specific embodiment six The fault judgment method of current signal detecting system is described further, in present embodiment, stator current letter described in step 5 Number detecting system fault difference r_esWith rotor current signal detecting system fault difference r_erIt is respectively as follows:

$\left\{\begin{array}{c}{r}_{es}=\sqrt{{(i_{sd}^{\′}\[(k+1)t\]-i_{sd}\[(k+1)t\])}^2+{(i_{sq}^{\′}\[(k+1)t\]-i_{sq}\[(k+1)t\])}^2}\\ r_{er}=\sqrt{{(i_{rd}^{\′}\[(k+1)t\]-i_{rd}\[(k+1)t\])}^2+{(i_{sq}^{\′}\[(k+1)t\]-i_{rq}\[(k+1)t\])}^2}\end{array}\right.$

In formula, i_sdFor the stator current components of d axle under synchronous rotating frame, i_sqFor q axle under synchronous rotating frame Stator current components；i_rdFor the rotor current component of d axle under synchronous rotating frame, i_rqFor q axle under synchronous rotating frame Rotor current component；i′_sdFor the stator current predictive value component of d axle under synchronous rotating frame, i '_sqFor synchronously rotating reference frame The lower q axle stator current predictive value component of system；i′_rdFor the rotor current predictive value component of d axle under synchronous rotating frame, i '_rqWith The rotor current predictive value component of q axle under step rotating coordinate system.

In present embodiment, described stator current signal detecting system fault difference r_esFor stator current observation and stator The mould of current actual value difference；Rotor current signal detecting system fault difference r_erReal with rotor current for rotor current observation The mould of actual value difference.

Specific embodiment eight: present embodiment is to the double-fed wind power generator rotor described in specific embodiment seven The fault judgment method of current signal detecting system is described further, in present embodiment, described difference information r_cFor:

r_c=r_es-r_er.

Claims (7)

1. the fault judgment method of double-fed wind power generator stator and rotor current signal detection system is it is characterised in that the method bag Include:

Step one: gather the stator voltage sig u of double-fed wind power generator respectively_{s_abc}, stator current signal i_{s_abc}, rotor current Signal i_{r_abc}With rotor voltage control signal u_{r_dq}；

Step 2: to stator voltage sig u_{s_abc}Carry out clark conversion, obtain stator voltage sig under static two phase coordinate systems u_{s_αβ}, to stator voltage sig u under this static two phase coordinate system_{s_αβ}Carry out park conversion, obtain stator under synchronous rotating frame Voltage signal u_{s_dq}；

To stator current signal i_{s_abc}Carry out clark conversion, obtain the stator current signal i under static two phase coordinate systems_{s_αβ}, right Stator current signal i under this static two phase coordinate system_{s_αβ}Carry out park conversion, obtain stator current letter under synchronous rotating frame Number i_{s_dq}；

To rotor current signal i_{r_abc}Carry out clark conversion, obtain static two phase coordinate system lower rotor part current signal i_{r_αβ}, to this Static two phase coordinate system lower rotor part current signal i_{r_αβ}Carry out park conversion, obtain synchronous rotating frame lower rotor part current signal i_{r_dq}；

Step 3: by stator current signal i under synchronous rotating frame_{s_dq}With synchronous rotating frame lower rotor part current signal i_{r_dq}As state variable, by stator voltage sig u under synchronous rotating frame_{s_dq}With rotor voltage control signal u_{r_dq}As Rotor input variable, according to double-fed wind power generator voltage equation and flux linkage equations, sets up stator and rotor current discrete state side Journey；

Step 4: using rotor discrete time state equation and with reference to reduced order observer design theory, set up stator electricity respectively Stream observer and rotor current observer,

Obtain stator current observation under synchronous rotating frame using stator current observer

Obtain synchronous rotating frame lower rotor part current observation using rotor current observer

Step 5: stator current predictive value i ' is obtained according to stator current predictive equation_{s_dq}, obtained according to rotor current predictive equation Obtain rotor current predictive value i '_{r_dq}, according to stator current signal i under synchronous rotating frame_{s_dq}, lower turn of synchronous rotating frame Electron current signal i_{r_dq}, stator current predictive value i ' under synchronous rotating frame_{s_dq}Pre- with synchronous rotating frame lower rotor part electric current Measured value i '_{r_dq}, obtain stator current signal detecting system fault difference r_esWith rotor current signal detecting system fault difference r_er；

Step 6: by stator current signal detecting system fault difference r_esWith rotor current signal detecting system fault difference r_erMake Difference, obtains difference information r_c, according to the stator current predictive value i ' under synchronous rotating frame_{s_dq}Under synchronous rotating frame Rotor current predictive value i '_{r_dq}Determine difference information r with the steady-state error of corresponding actual value_cJudgment threshold+δ and-η, and By difference information r_cIt is compared with judgment threshold+δ and-η respectively,

Work as r_cDuring ＞+δ, then the stator current signal detecting system of double-fed wind power generator breaks down, then execution step seven；

Work as r_c< during-η, then the rotor current signal detecting system of double-fed wind power generator breaks down, then execution step eight；

Step 7: by stator current observation under synchronous rotating frameCarry out anti-coordinate transform, obtain static three phase coordinates The lower stator a phase current observation of systemWith stator b phase current observationGather stator a phase under static three phase coordinate systems simultaneously Current actual value i_saWith stator b phase current actual value i_sb；

WillWith stator a phase current threshold value δ_saIt is compared, whenMore than stator a phase current threshold value δ_saWhen, then fixed Sub- a phase current signal detecting system breaks down；Wherein, stator a phase current threshold value δ_saFor stator a phase under static three phase coordinate systems Current observationWith stator a phase current actual value i under static three phase coordinate systems_saThe maximum of steady-state error；

WillWith stator b phase current threshold value δ_sbIt is compared, whenMore than stator b phase current threshold value δ_sbWhen, then fixed Sub- b phase current signal detecting system breaks down；Wherein, stator b phase current threshold value δ_sbFor stator b phase under static three phase coordinate systems Current observationWith stator b phase current actual value i under static three phase coordinate systems_sbThe maximum of steady-state error；

Step 8: by the rotor current observation under synchronous rotating frameCarry out anti-coordinate transform, obtain static three-phase and sit Rotor a phase current observation under mark systemWith rotor b phase current observationGather static three phase coordinate system lower rotor parts simultaneously A phase current actual value i_raWith rotor b phase current actual value i_rb；

WillWith rotor a phase current threshold value δ_raIt is compared, whenMore than rotor a phase current threshold value δ_raWhen, then turn Sub- a phase current signal detecting system breaks down；Wherein, rotor a phase current threshold value δ_raFor static three phase coordinate system lower rotor part a phases Current observationWith static three phase coordinate system lower rotor part a phase current actual value i_raThe maximum of steady-state error；

WillWith rotor b phase current threshold value δ_rbIt is compared, whenMore than rotor b phase current threshold value δ_rbWhen, then turn Sub- b phase current signal detecting system breaks down；Wherein, rotor b phase current threshold value δ_rbFor static three phase coordinate system lower rotor part b phases Current observationWith static three phase coordinate system lower rotor part b phase current actual value i_raThe maximum of steady-state error.

2. the fault judgment method of double-fed wind power generator stator and rotor current signal detection system according to claim 1, It is characterized in that, double-fed wind power generator voltage equation described in step 3 is:

$\{\begin{array}{c}{\mathrm{\&psi;}}_{sd}={\mathrm{d\&psi;}}_{sd}-{\mathrm{\&omega;}}_1{\mathrm{\&psi;}}_{sq}-r_s{i}_{sd}\\ {\mathrm{\&psi;}}_{sq}={\mathrm{d\&psi;}}_{sq}+{\mathrm{\&omega;}}_1{\mathrm{\&psi;}}_{sd}-r_s{i}_{sq}\end{array},\left\{\begin{array}{c}{u}_{rd}={\mathrm{d\&psi;}}_{rd}-{\mathrm{\&omega;}}_2{\mathrm{\&psi;}}_{rq}+r_r{i}_{rd}\\ u_{rq}={\mathrm{d\&psi;}}_{rq}+{\mathrm{\&omega;}}_2{\mathrm{\&psi;}}_{rd}+r_r{i}_{rq}\end{array}\right.,$

Wherein, r_sFor stator resistance, r_rFor rotor resistance；u_sdFor the stator voltage component of d axle under synchronous rotating frame, u_sqFor The stator voltage component of q axle under synchronous rotating frame；u_rdFor the rotor voltage component of d axle under synchronous rotating frame, u_rqFor The rotor voltage component of q axle under synchronous rotating frame；i_sdFor the stator current components of d axle under synchronous rotating frame, i_sqFor The stator current components of q axle under synchronous rotating frame；i_rdFor the rotor current component of d axle under synchronous rotating frame, i_rqFor The rotor current component of q axle under synchronous rotating frame；ψ_sdFor the stator magnetic linkage component of d axle under synchronous rotating frame, ψ_sqFor The stator magnetic linkage component of q axle under synchronous rotating frame；ψ_rdFor the rotor flux component of d axle under synchronous rotating frame, ψ_rqFor The rotor flux component of q axle under synchronous rotating frame；ω₁For stator synchronization angular frequency, ω₂For rotor electric angle frequency；D=d/ Dt is differential operator；

Described flux linkage equations are:

$\{\begin{array}{c}{\mathrm{\&psi;}}_{sd}=-l_s{i}_{sd}+l_m{i}_{rd}\\ {\mathrm{\&psi;}}_{sq}=-l_s{i}_{sq}+l_m{i}_{rq}\end{array},\left\{\begin{array}{c}{\mathrm{\&psi;}}_{rd}=l_r{i}_{rd}-l_m{i}_{sd}\\ {\mathrm{\&psi;}}_{rq}=l_r{i}_{rq}-l_m{i}_{sq}\end{array}\right.,$

Wherein, l_sFor stator inductance, l_rFor inductor rotor, l_mMutual inductance for stator and rotor.

3. the fault judgment method of double-fed wind power generator stator and rotor current signal detection system according to claim 2, It is characterized in that, stator and rotor current discrete state equations described in step 3 is:

I (k+1)=a_di(k)+b_du(k)

In formula, i (k)=[i_sdi_sqi_rdi_rq]^tFor state variable, u (k)=[u_sdu_squ_rdu_rq]^tFor input variable；K is Positive integer；a_dFor state equation state variable coefficient matrix, b_dFor state equation input variable coefficient matrix, and have:

$a_d=\left[\begin{array}{cccc}1-\frac{r_s{t}_s}{{\mathrm{\&sigma;l}}_s}& ({\mathrm{\&omega;}}_1+{\mathrm{p\&omega;}}_m(\frac{1}{\mathrm{\&sigma;}}-1\left)\right)t_s& -\frac{r_r{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& -\frac{{\mathrm{p\&omega;}}_m{l}_m{t}_s}{{\mathrm{\&sigma;l}}_s}\\ -\&lsqb;{\mathrm{\&omega;}}_1+{\mathrm{p\&omega;}}_m\left(\frac{1}{\mathrm{\&sigma;}}-1\right)\&rsqb;t_s& 1-\frac{r_s{t}_s}{{\mathrm{\&sigma;l}}_s}& \frac{{\mathrm{p\&omega;}}_m{l}_m{t}_s}{{\mathrm{\&sigma;l}}_s}& -\frac{r_r{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}\\ -\frac{r_s{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& \frac{{\mathrm{p\&omega;}}_m{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r}& 1-\frac{r_r{t}_s}{{\mathrm{\&sigma;l}}_r}& ({\mathrm{\&omega;}}_1-\frac{{\mathrm{p\&omega;}}_m}{\mathrm{\&sigma;}})t_s\\ -\frac{{\mathrm{p\&omega;}}_m{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r}& -\frac{r_s{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& -({\mathrm{\&omega;}}_1-\frac{{\mathrm{p\&omega;}}_m}{\mathrm{\&sigma;}})t_s& 1-\frac{r_r{t}_s}{{\mathrm{\&sigma;l}}_r}\end{array}\right],$

$b_d=\left[\begin{array}{cccc}-\frac{t_s}{{\mathrm{\&sigma;l}}_s}& 0& \frac{l_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& 0\\ 0& -\frac{t_s}{{\mathrm{\&sigma;l}}_s}& 0& \frac{l_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}\\ -\frac{l_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& 0& \frac{t_s}{{\mathrm{\&sigma;l}}_r}& 0\\ 0& -\frac{l_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& 0& \frac{t_s}{{\mathrm{\&sigma;l}}_r}\end{array}\right],$

Wherein,Represent leakage inductance coefficient, t_sFor the sampling time, p is double-fed wind power generator number of pole-pairs, ω_mFor Handset tool rotates angular frequency.

4. the fault judgment method of double-fed wind power generator stator and rotor current signal detection system according to claim 3, It is characterized in that, the method setting up stator current observer described in step 4 is:

State variable i (k) of stator and rotor current discrete state equations is divided into the stator current state equation i of immeasurability part_s (k) and the rotor current state equation i that part can be measured_r(k), and stator and rotor current discrete state equations are rewritten as piecemeal shape Formula:

Wherein,

$a_{d21}=\left[\begin{array}{cc}-\frac{r_s{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& \frac{{\mathrm{p\&omega;}}_m{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r}\\ -\frac{{\mathrm{p\&omega;}}_m{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r}& -\frac{r_s{l}_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}\end{array}\right],a_{d22}=\left[\begin{array}{cc}1-\frac{r_r{t}_s}{{\mathrm{\&sigma;l}}_r}& ({\mathrm{\&omega;}}_1-\frac{{\mathrm{p\&omega;}}_m}{\mathrm{\&sigma;}})t_s\\ -({\mathrm{\&omega;}}_1-\frac{{\mathrm{p\&omega;}}_m}{\mathrm{\&sigma;}})t& 1-\frac{r_r{t}_s}{{\mathrm{\&sigma;l}}_r}\end{array}\right],$

$b_{ds}=\left[\begin{array}{cccc}-\frac{t_s}{{\mathrm{\&sigma;l}}_s}& 0& \frac{l_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& 0\\ 0& -\frac{t_s}{{\mathrm{\&sigma;l}}_s}& 0& \frac{l_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}\end{array}\right],b_{dr}=\left[\begin{array}{cccc}-\frac{l_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& 0& \frac{t_s}{{\mathrm{\&sigma;l}}_r}& 0\\ 0& -\frac{l_m{t}_s}{{\mathrm{\&sigma;l}}_r{l}_s}& 0& \frac{t_s}{{\mathrm{\&sigma;l}}_r}\end{array}\right],$

y_rK () is output vector, c is output matrix and c=[0 01 1], i_s(k+1) it is immeasurability partial stator electric current State equation, i_r(k+1) it is the state equation that can measure part rotor current；

The state equation that part rotor current can be measured is write as the form of output equation:

i_r(k+1)-a_d22i_r(k)-b_drU (k)=a_d21i_s(k)

The state equation of immeasurability partial stator electric current is:

i_s(k+1)=a_d11i_s(k)+a_d12i_r(k)+b_dsu(k)

Stator and rotor current discrete state equations are rewritten as by following form according to above formula:

$\left\{\begin{array}{c}{i}_s(k+1)=a_{d11}{i}_s\left(k\right)+\&lsqb;\begin{array}{cc}{a}_{d12}& b_{ds}\end{array}\&rsqb;\left[\begin{array}{c}{i}_r\left(k\right)\\ u\left(k\right)\end{array}\right]\\ y_1\left(k\right)=a_{d21}{i}_s\left(k\right)\end{array}\right.$

Wherein y₁K () is the first output vector, this output vector partly to be represented with measuring:

y₁(k)=i_r(k+1)-a_d22i_r(k)-b_dru(k)

According to full order observer design theory, obtain the full rank stator observer model of new state equation subsystem:

$\left\{\begin{array}{c}{\hat{i}}_s(k+1)=(a_{d11}-k_s{a}_{d21}){\hat{i}}_s\left(k\right)+\&lsqb;\begin{array}{cc}{a}_{d12}& b_{ds}\end{array}\&rsqb;\left[\begin{array}{c}{i}_r\left(k\right)\\ u\left(k\right)\end{array}\right]+k_s{y}_1\left(k\right)\\ {\hat{y}}_1\left(k\right)=a_{d21}{\hat{i}}_s\left(k\right)\end{array}\right.$

Wherein, k_sFor the first observer feedback gain matrix.

5. the fault judgment method of double-fed wind power generator stator and rotor current signal detection system according to claim 4, It is characterized in that, the method setting up rotor current observer described in step 4 is:

State variable i (k) of stator and rotor current discrete state equations is divided into the rotor current state equation i of immeasurability part_r (k) and the stator current state equation i that part can be measured_s(k), and stator and rotor current discrete state equations are rewritten as:

The state equation that part stator current can be measured is rewritten as:

i_s(k+1)-a_d11i_s(k)-b_dsU (k)=a_d12i_r(k)

The state equation of immeasurability part rotor electric current is

i_r(k+1)=a_d22i_r(k)+a_d21i_s(k)+b_dru(k)

Stator and rotor current discrete state equations are rewritten as by following form according to above formula:

$\left\{\begin{array}{c}{i}_r(k+1)=a_{d22}{i}_s\left(k\right)+\&lsqb;\begin{array}{cc}{a}_{d21}& b_{dr}\end{array}\&rsqb;\left[\begin{array}{c}{i}_s\left(k\right)\\ u\left(k\right)\end{array}\right]\\ y_2\left(k\right)=a_{d12}{i}_r\left(k\right)\end{array}\right.$

Wherein y₂K () is the second output vector, this output vector partly to be represented with measuring:

y₂(k)=i_s(k+1)-a_d11i_s(k)-b_dsu(k)

According to full order observer design theory, obtain the full rank rotor observer model of new state equation subsystem:

$\left\{\begin{array}{c}{\hat{i}}_r(k+1)=(a_{d22}-k_r{a}_{d12}){\hat{i}}_r\left(k\right)+\&lsqb;\begin{array}{cc}{a}_{d21}& b_{dr}\end{array}\&rsqb;\left[\begin{array}{c}{i}_s\left(k\right)\\ u\left(k\right)\end{array}\right]+k_r{y}_2\left(k\right)\\ {\hat{y}}_2\left(k\right)=a_{d12}{\hat{i}}_r\left(k\right)\end{array}\right.$

Wherein, k_rFor the second observer feedback gain matrix.

6. the fault judgment method of double-fed wind power generator stator and rotor current signal detection system according to claim 5, It is characterized in that, described in step 5, stator current predictive value i ' is obtained according to stator current predictive equation_{s_dq}, according to rotor electricity Stream predictive equation obtains rotor current predictive value i '_{r_dq}Method be:

Stator and rotor voltage and current according to the kt moment is predicted to the stator and rotor electric current of (k+1) t, predictive equation For:

I ' [(k+1) t]=a_di(kt)+b_du(kt)

Wherein, i '=[i '_sdi′_sqi′_rdi′_rq]^tPredictive value for stator and rotor electric current；T is system communication cycle；

i′_sdFor the stator current predictive value component of d axle under synchronous rotating frame, i '_sqFor q axle stator under synchronous rotating frame Current forecasting value component；i′_rdFor the rotor current predictive value component of d axle under synchronous rotating frame, i '_rqSynchronously rotating reference frame The rotor current predictive value component of the lower q axle of system.

7. the fault judgment method of double-fed wind power generator stator and rotor current signal detection system according to claim 6, It is characterized in that, stator current signal detecting system fault difference r described in step 5_esWith rotor current signal detecting system fault Difference r_erIt is respectively as follows:

$\left\{\begin{array}{c}{r}_{es}=\sqrt{{(i_{sd}^{\&prime;}\&lsqb;(k+1)t\&rsqb;-i_{sd}\&lsqb;(k+1)t\&rsqb;)}^2+{(i_{sq}^{\&prime;}\&lsqb;(k+1)t\&rsqb;-i_{sq}\&lsqb;(k+1)t\&rsqb;)}^2}\\ r_{er}=\sqrt{{(i_{rd}^{\&prime;}\&lsqb;(k+1)t\&rsqb;-i_{rd}\&lsqb;(k+1)t\&rsqb;)}^2+{(i_{rq}^{\&prime;}\&lsqb;(k+1)t\&rsqb;-i_{rq}\&lsqb;(k+1)t\&rsqb;)}^2}\end{array}\right..$