CN102253266B  Linear filterbased harmonic current detection method in threephase fourwire system  Google Patents
Linear filterbased harmonic current detection method in threephase fourwire system Download PDFInfo
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 CN102253266B CN102253266B CN 201110109158 CN201110109158A CN102253266B CN 102253266 B CN102253266 B CN 102253266B CN 201110109158 CN201110109158 CN 201110109158 CN 201110109158 A CN201110109158 A CN 201110109158A CN 102253266 B CN102253266 B CN 102253266B
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Abstract
The invention discloses a linear filterbased harmonic current detection method in a threephase fourwire system. According to the method, a structural characteristic of the threephase fourwire system is considered; according to the idea of an adaptive notch filter (ANF), a twodimensional linear sine filter is obtained after rotation transformation on the basis of a minimum variance principle and a gradient decent method; fundamental currents of all phases are respectively separated from an actual measured current by the twodimensional linear sine filter; and then a symmetrical fundamental wave positive sequence component is obtained by utilizing a modified instantaneous symmetric component calculation method; after calculation of the fundamental wave positive sequence component with the original actual measured current, a harmonic component of a load current, a fundamental negative sequence component and a zero sequence component are drawn off. Compared with the prior art, a technique employed in the linear filterbased harmonic current detection method in the threephase fourwire system provided of the invention enables the method to have the following advantages that: the method is simple with a small amount of computation and it is not needed to carry out complicated rotation transformation and trigonometric function calculation; the method has a good realtime performance and high detection precision; and a harmonic current and a reactive current and a neutral current in the threephase fourwire system can be detected on the condition of unsymmetrical nonlinear load and nonideal voltage.
Description
Technical field
The present invention relates to field of power, exactly refer to a kind of threephase fourwire system harmonic current detecting method based on linear filter.
Background technology
At present, along with development and the application of modern power electronics technology, it has been topmost harmonic pollution source in electrical network that power electronic equipment becomes, the safety that is seriously threatening electrical network and consumer.The harmonic wave control of electric system has been subjected to the concern of numerous Chinese scholars.At present, the research that China controls Harmonious Waves in Power Systems mainly concentrates in the threephase threewire system system, although some researchs are also arranged in threephase fourwire system, still has many problems to await furtheing investigate.
In lowvoltage distribution system, the threephase fourwire power supply mode is topmost power supply mode, is used widely in industry spot and covil construction.Symmetrical threephase nonlinear load produces positive sequence harmonic and negative phasesequence harmonic wave, but be connected to the singlephase nonlinear load in threephase fourwire system, as being widely used of the household electrical appliance such as personal computer, business automation equipment, convertible frequency airconditioner, mains lighting supply and uninterrupted power source etc., to produce a large amount of neutral harmonic currents in electrical network, although their separate unit power is less, but huge because of its quantity, the harmonic pollution that brings is extremely serious.Neutral harmonic current (being mainly 3 subharmonic currents) is stack mutually on center line, the large electric current up to 1.73 times of phase currents may occur, and this just makes the Harmonic Conditions in current in middle wire extremely serious.The center line harmonic current makes current in middle wire substantially exceed its design load on the one hand, easily causes the center line fault; Make on the other hand transformer overheated, cause insulation breakdown, also can cause simultaneously the center line potential to ground to promote, have great importance so the harmonic wave in threephase fourwire system and threephase imbalance are compensated.
It is the prerequisite of harmonic wave control that quick harmonic wave in threephase fourwire system detects.At present, main detection method is based on fast Fourier transform (FFT) with based on the large class of instantaneous reactive power theory two of red wooden Thai language.Adopt fft algorithm can detect rapidly the harmonic wave of analyzed signal, but need the sampled data of oneperiod due to fft algorithm, therefore have larger timedelay, and computing is comparatively complicated, realtime is relatively poor.The instantaneous reactive power theory that red wooden Thai language proposes, for detecting, the quick harmonic wave of electric system provides theoretical foundation, greatly promoted the development of active power filtering practical research, but asymmetric or when containing harmonic wave when supply voltage, the precision that harmonic wave detects will be greatly affected, mainly be applicable to directly to apply in threephase fourwire system in the threephase threewire system system of balance.For this reason, Chinese scholars is improved instantaneous reactive power theory, and it is generalized in threephase fourwire system.In addition, utilize wavelet analysis to carry out harmonic wave and detect, although obtained better effects, its operand is larger, and realtime is relatively poor, and the selection of wavelet basis simultaneously is also a relatively hard problem.
Summary of the invention
For defects, the technical matters that the present invention solves is to provide a kind of threephase fourwire system harmonic current detecting method based on linear filter, method is simple, operand is little, need not to carry out complicated rotational transform and the calculating of trigonometric function, realtime is good, and accuracy of detection is high, can detect harmonic current, reactive current, current in middle wire in threephase fourwire system under unbalanced nonlinear loads and nonideal voltage condition.
In order to solve above technical matters, the threephase fourwire system harmonic current detecting method based on linear filter provided by the invention comprises the steps:
(a) utilize the design feature of threephase fourwire system, regard as mutually its each three independently singlephase, the unbalanced threephase load current i
_{a}, i
_{b}, i
_{c}, each uses a twodimensional linear wave filter to extract fundamental current; If the fundamental frequency of electrical network is f, its angular frequency is ω, and the electric power signal of wave filter input is i (t), has:
State variable is chosen as x (t)=[x
_{1}(t) x
_{2}(t)]
^{T}:
The estimated value of i (t)
Can be expressed as:
$\hat{i}\left(t\right)={x}_{1}\left(t\right)\·\mathrm{sin}\mathrm{\ωt}+{x}_{2}\left(t\right)\·\mathrm{cos}\mathrm{\ωt}\left(3\right)$
The evaluated error of signal equals:
$e\left(t\right)=i\left(t\right)\hat{i}\left(t\right)\left(4\right)$
According to minimum variance thought, the definition cost function is:
$J\left(x\right)=0.5{e}^{2}\left(t\right)=0.5{(i\left(t\right)\hat{i}\left(t\right))}^{2}\left(5\right)$
According to gradient descent method, have:
Matrix Ν regulates x (t) and converges to the speed of optimal value and the stability of algorithm.Make μ
_{1}, μ
_{2}＞0, select:
Thus, can obtain the periodic coefficient linear dynamic system:
Wherein:
$\stackrel{\·}{x}={\left[\begin{array}{cc}{\stackrel{\·}{x}}_{1}\left(t\right)& {\stackrel{\·}{x}}_{2}\left(t\right)\end{array}\right]}^{T},$ x＝[x
_{1}(t)?x
_{2}(t)]
^{T}
Do rotational transform, order
Known the estimated value of i (t) by formula (1) and formula (3)
For:
With formula (9) substitution formula (8), have:
Make μ
_{1}=μ
_{2}=μ, η=ω obtains the twodimensional linear filtering algorithm:
Wherein: y (t)=[y
_{1}(t) y
_{2}(t)]
^{T}Be state variable; μ, η are the parameter of wave filter, all greater than zero real constant; y
_{1}(t) be wave filter output, in its expression input signal, contained angular frequency is the sinusoidal component of ω, is also the fundametal compoment of input signal, y
_{2}(t) be and y
_{1}(t) quadrature and leading homogenous frequency signal;
If starting condition y
_{1}(0)=0, y
_{2}(0)=0 is got i (t) and is input, y
_{1}(t) be output, the transport function of twodimensional linear wave filter is
Make s=j ω, have:
Its amplitude versus frequency characte and phasefrequency characteristic are respectively:
When getting μ=0.5, during η=ω=314rad/s, have:
(b) unbalanced fundamental current i
_{af}, i
_{bf}, i
_{cf}Calculate by instantaneous symmetrical components, obtain symmetrical fundamental positive sequence current component i
_{afp}, i
_{bfp}, i
_{cfp}If the unbalanced threephase load current instantaneous value is:
Wherein: i
_{a}(t), i
_{b}(t), i
_{c}(t) be respectively the threephase current instantaneous value; I
_{am}, I
_{bm}, I
_{cm}Be respectively the amplitude of threephase current;
Be respectively the initial phase of threephase current;
If with threephase current i
_{a}(t), i
_{b}(t), i
_{c}(t) corresponding rotating phasor is respectively
Have:
Known phasor by formula (20)
The imaginary part coefficient be the threephase current instantaneous value, as long as can determine its real part, just can determine these rotating phasors; If the instantaneous value of a phase current can be expressed as:
Also namely:
Therefore have:
Again because:
Can get after arrangement:
In like manner can get:
In formula (25)～(27) substitution formula (20), consist of instantaneous rotating phasor, unbalanced threephase load electric current positivesequence component value is:
Wherein: Im{) for asking the imaginarypart operation of plural number; A=e
^{J2 π/3}, a
^{2}=e
^{j2 π/3}, 1+a+a
^{2}=0;
(c) with former load current computing after, can isolate harmonic current components i
_{ah}, i
_{bh}, i
_{ch}
Threephase fourwire system harmonic current detecting method based on linear filter provided by the invention, design feature for threephase fourwire system, thought according to adaptive notch filter (ANF), according to minimum variance principle and Gradient Descent method, obtain the twodimensional linear sinusoidal filter through rotational transform, separation of phases fundamental current from measured current respectively, recycle improved instantaneous symmetrical components computing method, obtain symmetrical fundamental positive sequence, with the harmonic component and firstharmonic negative sequence component and the zerosequence component that extract load current after former measured current computing.Compared with prior art, threephase fourwire system harmonic current detecting method based on linear filter provided by the invention, method is simple, operand is little, need not to carry out complicated rotational transform and the calculating of trigonometric function, realtime is good, and accuracy of detection is high, can detect harmonic current, reactive current, current in middle wire in threephase fourwire system under unbalanced nonlinear loads and nonideal voltage condition.
Description of drawings
Fig. 1 is based on the detection principle schematic of the threephase fourwire system harmonic current detecting method of linear filter in the present invention;
Fig. 2 is the harmonic wave testing result figure under desirable principal voltage in embodiment 1;
Fig. 3 is that in embodiment 1, the current in middle wire under desirable principal voltage is eliminated and reactiveload compensation figure as a result;
Fig. 4 is the harmonic wave testing result figure under nonequilibrium principal voltage in embodiment 2;
Fig. 5 is that in embodiment 2, the current in middle wire under nonequilibrium principal voltage is eliminated and reactiveload compensation figure as a result;
Fig. 6 is the harmonic wave testing result figure under balance distortion principal voltage in embodiment 3;
Fig. 7 is that in embodiment 3, the current in middle wire under balance distortion principal voltage is eliminated and reactiveload compensation figure as a result;
Fig. 8 is the harmonic wave testing result figure under nonequilibrium distortion principal voltage in embodiment 4;
Fig. 9 is that in embodiment 4, the current in middle wire under nonequilibrium distortion principal voltage is eliminated and reactiveload compensation figure as a result.
Embodiment
For those skilled in the art can understand technical scheme provided by the present invention better, set forth below in conjunction with specific embodiment.
See also Fig. 1, this figure is based on the detection principle schematic of the threephase fourwire system harmonic current detecting method of linear filter in the present invention
Threephase fourwire system harmonic current detecting method based on linear filter provided by the invention comprises the steps:
(a) utilize the design feature of threephase fourwire system, regard as mutually its each three independently singlephase, the unbalanced threephase load current i
_{a}, i
_{b}, i
_{c}, each uses a twodimensional linear wave filter to extract fundamental current; If the fundamental frequency of electrical network is f, its angular frequency is ω, and the electric power signal of wave filter input is i (t), has:
State variable is chosen as x (t)=[x
_{1}(t) x
_{2}(t)]
^{T}:
The estimated value of i (t)
Can be expressed as:
$\hat{i}\left(t\right)={x}_{1}\left(t\right)\·\mathrm{sin}\mathrm{\ωt}+{x}_{2}\left(t\right)\·\mathrm{cos}\mathrm{\ωt}\left(3\right)$
The evaluated error of signal equals:
$e\left(t\right)=i\left(t\right)\hat{i}\left(t\right)\left(4\right)$
According to minimum variance thought, the definition cost function is:
$J\left(x\right)=0.5{e}^{2}\left(t\right)=0.5{(i\left(t\right)\hat{i}\left(t\right))}^{2}\left(5\right)$
According to gradient descent method, have:
Matrix Ν regulates x (t) and converges to the speed of optimal value and the stability of algorithm, makes μ
_{1}, μ
_{2}＞0, select:
Thus, can obtain the periodic coefficient linear dynamic system:
Wherein:
$\stackrel{\·}{x}={\left[\begin{array}{cc}{\stackrel{\·}{x}}_{1}\left(t\right)& {\stackrel{\·}{x}}_{2}\left(t\right)\end{array}\right]}^{T},$ x＝[x
_{1}(t)?x
_{2}(t)]
^{T}
Do rotational transform, order
Known the estimated value of i (t) by formula (1) and formula (3)
For:
With formula (9) substitution formula (8), have:
Make μ
_{1}=μ
_{2}=μ, η=ω obtains the twodimensional linear filtering algorithm:
Wherein: y (t)=[y
_{1}(t) y
_{2}(t)]
^{T}Be state variable; μ, η are the parameter of wave filter, all greater than zero real constant; y
_{1}(t) be wave filter output, in its expression input signal, contained angular frequency is the sinusoidal component of ω, is also the fundametal compoment of input signal, y
_{2}(t) be and y
_{1}(t) quadrature and leading homogenous frequency signal;
If starting condition y
_{1}(0)=0, y
_{2}(0)=0 is got i (t) and is input, y
_{1}(t) be output, the transport function of twodimensional linear wave filter is
Make s=j ω, have:
Its amplitude versus frequency characte and phasefrequency characteristic are respectively:
When getting μ=0.5, during η=ω=314rad/s, have:
(b) unbalanced fundamental current i
_{af}, i
_{bf}, i
_{cf}Calculate by instantaneous symmetrical components, obtain symmetrical fundamental positive sequence current component i
_{afp}, i
_{bfp}, i
_{cfp}If the unbalanced threephase load current instantaneous value is:
Wherein: i
_{a}(t), i
_{b}(t), i
_{c}(t) be respectively the threephase current instantaneous value; I
_{am}, I
_{bm}, I
_{cm}Be respectively the amplitude of threephase current;
Be respectively the initial phase of threephase current;
If with threephase current i
_{a}(t), i
_{b}(t), i
_{c}(t) corresponding rotating phasor is respectively
Have:
Known phasor by formula (20)
The imaginary part coefficient be the threephase current instantaneous value, as long as can determine its real part, just can determine these rotating phasors; If the instantaneous value of a phase current can be expressed as:
Also namely:
Therefore have:
Again because:
Can get after arrangement:
In like manner can get:
In formula (25)～(27) substitution formula (20), consist of instantaneous rotating phasor, unbalanced threephase load electric current positivesequence component value is:
Wherein: Im{} is the imaginarypart operation of asking plural number; A=e
^{J2 π/3}, a
^{2}=e
^{j2 π/3}, 1+a+a
^{2}=0;
(c) with former load current computing after, can isolate harmonic current components i
_{ah}, i
_{bh}, i
_{ch}
In order to verify the threephase fourwire system harmonic current detecting method based on linear filter provided by the invention, use the electric system tool box of MatlabSimulink to set up the threephase fourwire system electric power system model of four brachium pontis, adopt the load of three phase rectifier bridgetype as the threephase load of symmetry, two singlephase bridge rectification loads contain the uneven phase current and the current in middle wire that contains zerosequence component of harmonic current and reactive current as nonequilibrium load with generation.The harmonic wave of active filter detects the algorithm that adopts this paper to propose, and fourleg inverter adopts stagnant ring control mode, connecting system when 0.15s, and a singlephase load is connected to the B phase when 0.3s, with the dynamic perfromance of Study system.By carry out simulation study in desirable balance principal voltage, uneven principal voltage, balance distortion principal voltage and four kinds of situations of uneven distortion principal voltage, verify correctness and the validity of institute's put forward the methods.The systematic parameter that emulation is used is as shown in table 1.
The systematic parameter that table 1 emulation is used
Parameter  Value 
Power supply  ? 
Voltage  220V 
Frequency  50Hz 
Impedance  10mΩ，50μH 
Four bridgearm active filters  ? 
Dc terminal voltage  800V 
DC terminal electric capacity  1500μF 
Exchange the end filter inductance  0.1Ω，1mH 
Load  ? 
Threephase load  12Ω，20mH 
Singlephase load  15Ω，1mH，470μF 
Wave filter  ? 
Centerfrequency parameters  314rad/s 
The speed of convergence parameter  1.8（t<0.01s），0.5（t>0.01s） 
Embodiment 1
Desirable balance principal voltage situation: Fig. 2 and Fig. 3 have shown the simulation result of system in the three phase sine principal voltage situation of perfect symmetry.Provided respectively principal voltage waveform V in Fig. 2
_{abc}(a), load current waveform i
_{Labc}(b), source current waveform i
_{Sabc}(c), the fundamental waveform i of each phase source current of twodimensional linear wave filter output
_{Fabc}(d) and the current first harmonics positivesequence component waveform i of symmetrical components after calculating
_{Fpabc}(e).Fig. 3 has provided the compensation situation of current in middle wire and reactive current.Current in middle wire iln (a) before compensation, the current in middle wire isn (b) after compensation, the waveform Vsa of A phase voltage electric current before compensation, ila (c), the voltage current waveform Vsa after compensation, isa (d).Load current, source current, current in middle wire and percent harmonic distortion thereof (Total Harmonic Distortion, the THD) value of system in desirable principal voltage situation is as shown in table 2.Simulation result shows, the threephase fourwire system harmonic current detecting method based on linear filter provided by the invention has been issued to good effect in desirable principal voltage situation.
Load current, source current and THD under the desirable principal voltage of table 2
Embodiment 2
Unbalanced sinusoidal principal voltage situation: unbalanced sinusoidal principal voltage is provided by following formula:
v
_{a}＝380sinωt+38sinωt
v
_{b}＝380sin(ωt120°)+38sin(ωt+120°)
v
_{c}＝380sin(ωt+120°)+38sin(ωt120°)
The simulation result of system in unbalanced three phase sine principal voltage situation as shown in Figure 4 and Figure 5, in figure, the implication of each waveform is identical with Fig. 2 and Fig. 3.
Simulation result shows, under nonequilibrium sinusoidal voltage effect, comparatively serious imbalance has appearred in the load current of A, B, C threephase, has occurred very large harmonic current on center line.When after 0.15s access AFP, the threephase supply electric current reaches balance, the THD of B phase current is between 0.15s and 0.3s, drop to 1.14% by original 39.34%, the time greater than 0.3s after, THD drops to 1.27% by original 51.55%, simultaneously, current in middle wire also drops to respectively 0.44A and 0.48A by original 40.27A and 80.4A, sees table 3 for details.As seen, threephase fourwire system harmonic current detecting method based on linear filter provided by the invention is in nonequilibrium principal voltage situation, the harmonic component of system be can detect exactly, harmonic current and current in middle wire in threephase fourwire system effectively eliminated.
Load current, source current and THD under the nonequilibrium principal voltage of table 3
Embodiment 3
Balance distortion principal voltage situation: in order to study the system performance under balance distortion principal voltage, the threephase supply that is constructed as follows:
v
_{a}＝380sinωt+4.56sin3ωt+22.8sin(5ωt120°)+
5.7sin7ωt+3.8sin(11ωt120°)
v
_{b}＝380sin(ωt120°)+4.56sin3ωt+22.8sin5ωt+
5.7sin(7ωt120°)+3.8sin11ωt
v
_{c}＝380sin(ωt+120°)+4.56sin3ωt+
22.8sin(5ωt+120°)+5.7sin(7ωt+120°)+
3.8sin(11ωt+120°)
The simulation result of system in unbalanced three phase sine principal voltage situation as shown in Figure 6 and Figure 7, in figure, the implication of each waveform is identical with Fig. 2 and Fig. 3.
Simulation result shows, under the effect of balance distortion principal voltage, serious distortion and imbalance have appearred in load current and source current, the THD of current in middle wire is 70.81%, after 0.15s access APF, the harmonic wave in source current is suppressed well, and imbalance is compensated, current in middle wire is also reduced to respectively 0.63A and 0.59A by original 14.7A and 29.37A, has improved source current and current in middle wire characteristic.Load current, source current, current in middle wire and the THD value thereof of system in distortion principal voltage situation is as shown in table 4.As seen, the threephase fourwire system harmonic current detecting method based on linear filter provided by the invention has good performance in the situation that the distortion principal voltage can detect each phase harmonic wave well.
Load current, source current and THD under table 4 balance distortion principal voltage
Embodiment 4
Uneven distortion principal voltage situation: in order to study the system performance under nonequilibrium distortion principal voltage, the threephase supply that is constructed as follows:
v
_{a}＝380sinωt+38sinωt+4.56sin3ωt+
22.8sin(5ωt120°)+5.7sin7ωt+
3.8sin(11ωt120°)
v
_{b}＝380sin(ωt120°)+38sin(ωt+120°)+
4.56sin3ωt+22.8sin5ωt+5.7sin(7ωt120°)+
3.8sin11ωt
v
_{c}＝380sin(ωt+120°)+38sin(ωt120°)+
4.56sin3ωt+22.8sin(5ωt+120°)+
5.7sin(7ωt+120°)+3.8sin(11ωt+120°)
Simulation result such as the Fig. 8 and shown in Figure 9 in unbalanced threephase distortion principal voltage situation of system.In figure, the implication of each waveform is identical with Fig. 2 and Fig. 3.
This nonsinusoidal, unbalanced power supply simulation work state are near the working condition of practical power voltage, and simulation result shows, the algorithm of carrying still can detect harmonic wave in system well, balance source current, compensating reactive power.APF is after the 0.15s connecting system, and the THD of source current reduces greatly, and threephase imbalance is inhibited, and reactive current is compensated, and current in middle wire is limited in the 0.6A left and right.Although the THD of source current will be before several situations slightly high, but still in national approval standard scope.Load current, source current, current in middle wire and the THD value thereof of system in distortion principal voltage situation is as shown in table 5.As seen, the threephase fourwire system harmonic current detecting method based on linear filter provided by the invention has good performance in the situation that nonequilibrium distortion principal voltage also can detect each phase harmonic wave well.
Load current, source current and THD under the nonequilibrium distortion principal voltage of table 5
by in balance ideal sinusoidal principal voltage situation, uneven sinusoidal principal voltage situation, emulation in balance distortion principal voltage situation and uneven distortion principal voltage situation, result verification validity and the correctness of the threephase fourwire system harmonic current detecting method based on linear filter provided by the invention, system has good sound performance, utilize the threephase fourwire system harmonic current detecting method based on linear filter provided by the invention and four brachium pontis parallel A PF, can realize the harmonic compensation of asymmetric threephase fourwire system in the situations such as nonequilibrium distortion principal voltage, reactiveload compensation, the balance of load current and the inhibition of current in middle wire.
Compared with prior art, threephase fourwire system harmonic current detecting method based on linear filter provided by the invention, method is simple, operand is little, need not to carry out complicated rotational transform and the calculating of trigonometric function, realtime is good, and accuracy of detection is high, can detect harmonic current, reactive current, current in middle wire in threephase fourwire system under unbalanced nonlinear loads and nonideal voltage condition.
To the abovementioned explanation of the disclosed embodiments, make this area professional and technical personnel can realize or use the present invention.Multiple modification to these embodiment will be apparent concerning those skilled in the art, and General Principle as defined herein can be in the situation that do not break away from the spirit or scope of the present invention, realization in other embodiments.Therefore, the present invention will can not be restricted to these embodiment shown in this article, but will meet the widest scope consistent with principle disclosed herein and features of novelty.
Claims (1)
1. the threephase fourwire system harmonic current detecting method based on linear filter, is characterized in that, comprises the steps:
(a) utilize the design feature of threephase fourwire system, regard as mutually its each three independently singlephase, the unbalanced threephase load current i
_{a}, i
_{b}, i
_{c}, each uses a twodimensional linear wave filter to extract fundamental current; If the fundamental frequency of electrical network is f, its angular frequency is ω, and the electric power signal of wave filter input is i (t), has:
State variable is chosen as x (t)=[x
_{1}(t) x
_{2}(t)]
^{T}:
The estimated value of i (t)
Can be expressed as:
The evaluated error of signal equals:
According to minimum variance thought, the definition cost function is:
According to gradient descent method, have:
Matrix Ν regulates x (t) and converges to the speed of optimal value and the stability of algorithm, makes μ
_{1}, μ
_{2}＞0, select:
Thus, can obtain the periodic coefficient linear dynamic system:
Wherein:
X=[x
_{1}(t) x
_{2}(t)]
^{T}
Do rotational transform, order
Known the estimated value of i (t) by formula (1) and formula (3)
For:
With formula (9) substitution formula (8), have:
Make μ
_{1}=μ
_{2}=μ, η=ω obtains the twodimensional linear filtering algorithm:
Wherein: y (t)=[y
_{1}(t) y
_{2}(t)]
^{T}Be state variable; μ, η are the parameter of wave filter, all greater than zero real constant; y
_{1}(t) be wave filter output, in its expression input signal, contained angular frequency is the sinusoidal component of ω, is also the fundametal compoment of input signal, y
_{2}(t) be and y
_{1}(t) quadrature and leading homogenous frequency signal;
If starting condition y
_{1}(0)=0, y
_{2}(0)=0 is got i (t) and is input, y
_{1}(t) be output, the transport function of twodimensional linear wave filter is
Make s=j ω, have:
Its amplitude versus frequency characte and phasefrequency characteristic are respectively:
When getting μ=0.5, during η=ω=314rad/s, have:
(b) unbalanced fundamental current i
_{af}, i
_{bf}, i
_{cf}Calculate by instantaneous symmetrical components, obtain symmetrical fundamental positive sequence current component i
_{afp}, i
_{bfp}, i
_{cfp}If the unbalanced threephase load current instantaneous value is:
Wherein: i
_{a}(t), i
_{b}(t), i
_{c}(t) be respectively the threephase current instantaneous value; I
_{am}, I
_{bm}, I
_{cm}Be respectively the amplitude of threephase current;
Be respectively the initial phase of threephase current;
If with threephase current i
_{a}(t), i
_{b}(t), i
_{c}(t) corresponding rotating phasor is respectively
Have:
Known phasor by formula (20)
The imaginary part coefficient be the threephase current instantaneous value, as long as can determine its real part, just can determine these rotating phasors; If the instantaneous value of a phase current can be expressed as:
Also namely:
Therefore have:
Again because:
Can get after arrangement:
In like manner can get:
In formula (25)～(27) substitution formula (20), consist of instantaneous rotating phasor, unbalanced threephase load electric current positivesequence component value is:
Wherein: Im{} is the imaginarypart operation of asking plural number; A=e
^{J2 π/3}, a
^{2}=e
^{j2 π/3}, 1+a+a
^{2}=0;
(c) with former load current computing after, can isolate harmonic current components i
_{ah}, i
_{bh}, i
_{ch}
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