CN104092394B - Staircase waveform multi-level converter particular harmonic eliminates the method for solving of switch angle - Google Patents
Staircase waveform multi-level converter particular harmonic eliminates the method for solving of switch angle Download PDFInfo
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Abstract
The present invention relates to a kind of method for solving eliminating switch angle about staircase waveform multi-level converter particular harmonic, employ elementary symmetric polynomial equation group is carried out fall time and groebner base algorithm equation group is carried out trigonometric ratio, solving of higher polynomial equation group is converted into two unitary higher order polynomial equation and one group of monotropic unit the solving of linear function, and combine constraints and can draw all solutions of harmonic elimination equation group, by evaluating the total harmonic distortion of each group of solution, and then optimized switching angle can be drawn.Compared with the most conventional numerical algorithm and random search algorithm, it is not necessary to given initial value, and can be derived that all solutions of harmonic elimination equation group；Compared with other Algebraic Algorithm the most existing, amount of calculation is greatly reduced, and can process and more switch angle, has higher practical value.
Description
Technical field
The present invention relates to power electronic system and control method field thereof, especially with regard to inverter and multilevel
The control of device and harmonic Elimination Method, specifically, be a kind of about particular harmonic elimination (Selective Harmonic
Eliminated, SHE) switch angle method for solving.
Background technology
Particular harmonic eliminates (hereinafter referred to as SHE) modulation and is different from traditional waveform modulated PWM technology, and it is that one is passed through
Solving equation group tries to achieve the method for switch angle.Compared with SPWM Technique, SHE modulation have switching frequency low,
Switching loss is little and waveform quality high.Owing to eliminating low-order harmonic, residue harmonic wave focuses mostly in high frequency, can be significantly
Reduce the requirement to wave filter, additionally can obtain higher voltage gain, save the energy.It is illustrated in figure 1 staircase waveform 11
The output waveform of level SHE modulation, whereinα 1, α 2…α 5It is the switch angle in 1/4th cycles, VdcFor DC level
Voltage swing.According to the odd-even symmetry of function, the Fourier space of output waveform comprises only odd-times sinusoidal component, as
Shown in (formula 1).
(formula 1)
Wherein n=2k-1, k is natural number, bnFor the amplitude of each odd harmonic, computing formula is as follows:
(formula 2)
The basic thought of SHE modulation is by controlling the switch angle [alpha] in 1/4th periodic waveforms1, α2…α5So that
The amplitude of some higher hamonic wave of output voltage is zero, i.e. harmonic amplitude b as shown in (formula 2)n =0.So can obtain as
Under about the Nonlinear System of Equations of switching angle degree:
(formula 3)
Wherein 0 < α1 < α2 < … < α5 < pi/2, modulation is than m=0.25 π b1/Vdc, represent fundamental voltage amplitude b1With directly
Stream voltage VdcRatio relation, the number of equation counts 5 equal to switch, it is possible to the harmonic wave number of elimination is 4, for three phase inversion
Device, Zero-pharse harmonic will cancel out each other in on-Line Voltage, therefore without listing three times and integral multiple subharmonic in (formula 3)
Equation.(formula 3) is easy to extend to switch the situation counted as N, hereinafter referred to as SHE equation group.
About solving of SHE equation group, use numerical method (such as Newton iteration method, Homotopy etc.) or random at present more
Searching algorithm (such as genetic algorithm, Swarm Intelligence Algorithm etc.) solves, due to numerical algorithm or the local of random search algorithm
Convergence, solution procedure depends critically upon the selection of initial value, and suitable initial value can make the speed of convergence greatly speed up, and otherwise can
Convergence even dissipates very slowly, and does not still have the effective method of system at present about choosing of initial value.And SHE equation group have very much can
Can there is multiple Local Extremum, the locally optimal solution tried to achieve by numerical algorithm or random search algorithm is it cannot be guaranteed that be exactly
Global optimum, thus the performance of changer cannot guarantee that it is optimum.In recent years, have based on algebraic process (such as eliminant
The elimination, Wu Ritt's method etc.) method for solving propose, this type of method without given initial value and all solutions that SHE equation group can be given,
But the switch that can effectively process at present is counted and is up to 5, more situation is counted owing to amount of calculation is too big for switch
Final result cannot be given.Therefore, research can process more switch angle, without given initial value and all switching angles can be drawn
The method for solving of degree combination has important for the Harmonics elimination effect of raising changer further, the quality of power supply of raising electrical network
Actual application value.
Summary of the invention
The invention solves the problems that the most existing method for solving about staircase waveform multi-level converter SHE modulation switch angle
Existing three below problem: 1. the choosing of initial value.At present, initial value choose still without system, effective method, research
The means that the commonly used examination of person is gathered, it is possible to obtain some empirical equations or effective method in certain special cases, but refer to
The shortage of the property led theory is it cannot be guaranteed that these methods existing can adapt to all of situation, and limits the practicality of this technology
Change.2. the asking for of globally optimal solution.Due to numerical algorithm and the local convergence of random search algorithm itself, not only solved
Journey depends critically upon the selection of initial value, and also can only be to converge to a locally optimal solution for a given initial value, and
Actually SHE equation group often exists organizes solution more, how to find out all of locally optimal solution and then determines that globally optimal solution is to spy
The design determining harmonic elimination inverter has important value.3. Algebraic Algorithm amount of calculation is the biggest.The switch that can process at present is counted
Up to 5, final result cannot be provided for switching more situation of counting owing to amount of calculation is too big.
For reaching described purpose, staircase waveform multi-level converter particular harmonic of the present invention eliminates the method for solving of switch angle
Comprise the steps.
Step S1: utilize trigonometric function double angle formula and substitution of variableBy SHE
Equation group is converted into Polynomial equations, it is seen thatIn each multinomial about
ArgumentIt is all symmetrical.
Step S2: can be expressed as about elementary symmetric polynomial uniquely according to arbitrary symmetric polynomialThis conclusion of multinomial, willBe converted into about
Polynomial equations。
Step S3: order modulation ratioIt is a concrete numerical value, then willSubstitute into, and
CalculateAt pure lexcographical orderUnder reduction groebner base, for, whereinFor aboutUnitary high-order moment,For aboutWithBinary polynomial, and aboutIt is once.
Step S4: solve unitary higher order polynomial equation, and combine constraints, obtainThe individual reality solution meeting constraints。
Step S5: each tried to achieve in step S4 is substituted into, after substitution
For system of linear equations, obtain。
Step S6: to each group, the unitary n times multinomial side that is constructed as follows
Journey:
According to the definition of elementary symmetric polynomial, can be bySolve and be converted into above-mentioned unitary
Solving of n times polynomial equation.
Step S7: obtain in solution procedure S6Individual unitary higher order polynomial equation, obtainsGroup solves。
Step S8: check successivelyWhether meet following two condition: (1)ForIndividual inequality is real to be solved；(2) all real solutions are all located at intervalBetween.Give up be unsatisfactory for
The solution of upper two conditions, finally gives one and co-exists inGroup solves。
Step S9: to obtain in step S8Group solvesIn each group,
According toCalculate and switch angle accordingly, and will switch angle according to from little to
Big order arrangement obtains。
Step S10: evaluateGroup switch angleHarmonic elimination effect, be given
That group switch angle of harmonic elimination effect optimum is globally optimal solution.
The technical scheme that the present invention proposes is by utilizing symmetric polynomial and groebner based method to carry out SHE equation group
Simplify and trigonometric ratio, the equivalence that solves of polynary equation of higher degree group is converted into and solves two Equation of Higher Degree with One Unknowns and linear equation
Group.Being different from conventional numerical algorithm and random search algorithm, the method is without given initial value, and can obtain SHE equation group
All solutions.Relative to current existing Algebraic Algorithm, it is possible to the switching angle number of degrees amount of process is significantly enhanced.Have higher
Practicality.All solutions of SHE equation group are evaluated and then optimized switching angle can be obtained, such that it is able to the property designed
The changer that energy is optimum.
Accompanying drawing explanation
Fig. 1 is the output waveform of staircase waveform 11 level SHE modulation.
Fig. 2 is the flow chart that staircase waveform multi-level converter particular harmonic of the present invention eliminates the method for solving of switch angle.
Fig. 3 is the phase voltage waveform of staircase waveform 11 level converter output.
Fig. 4 is the Fourier analysis of the phase voltage waveform of staircase waveform 11 level converter output.
Fig. 5 is the line voltage waveform of staircase waveform 11 level converter output.
Fig. 6 is the Fourier analysis of the line voltage waveform of staircase waveform 11 level converter output.
Detailed description of the invention
Below some specific embodiments are provided with regard to the technical solution adopted in the present invention, it is noted that described
Embodiment be intended merely to facilitate the understanding of the present invention, and it is not played any restriction effect.
Specific embodiment one: combine the staircase waveform multi-level converter switching the N=3 that counts to each step in method for solving
Suddenly it is described in detail.
For the three-phase staircase waveform multi-level converter of N=3, its SHE equation group is:
(formula 4)
Wherein 0 < α1 < α2 < α3 < pi/2, modulation is generally 0 < m < 3 than m span, the value of m in practice
General the most given, might as well illustrate as a example by m=2 here.
Step S1: have according to the many double angle formulas of cosine function:
(formula 5)
(formula 6)
(formula 5) and (formula 6) is substituted into (formula 4), and makes, harmonic elimination equation group is converted into following multinomial side
Journey group:
(formula 7).
During step S2:N=3, the definition of elementary symmetric polynomial is:
(formula 8)
Theoretical according to symmetric polynomial, arbitrary symmetric polynomial can be expressed as about elementary symmetric polynomial uniquely
Multinomial.That is (formula 7) can be converted into aboutPolynomial equations.The method converted has progressively
The first term that disappears method and the method for undetermined coefficients etc., have invoked Symbolic Computation Software mathematica's in the present embodimentOrder, obtains as follows and the equation group of (formula 7) equivalence:
(formula 9).
Step S3: make m=2, is obtained by (formula 9) first equation, substitute into latter two equation of (formula 9), obtain
(formula 10)
Two multinomials to (formula 10) equation left side, calculate its reduction groebner base under pure lexcographical order, about
The computational methods of groebner base are prior art, and concrete ins and outs are referred to relevant document (such as: " computer algebra
Basis: algebraically and the ultimate principle of sign computation ", Zhang Shugong edits, Science Press, and 2005), it is not described in detail.
Can calculate with the Basis () function in Groebner workbox in call sign computation system Maple in being embodied as,
Concrete method of calling is:
with(Groebner);
G1 := Basis([h2, h3], plex(e2, e3));
Wherein h2, h3For the multinomial on the equation left side in (formula 10), obtain following Polynomial equations of equal value:
(formula 11)
Theoretical according to groebner base, (formula 11) and (formula 10) has identical solution.
Step S4: solve first equation in (formula 11).Fsolve () function can be called under Maple environment ask
Solve, show that equation only has 1 real solution, and meet, for:
。
Step S5: willSubstitute into second equation of (formula 11), try to achieveAs follows:
。
Step S6: according toIt is constructed as follows equation:
(formula 12).
Step S7: solve (formula 12), obtain:
。
Step S8: three solutions obtained in testing sequence S7:It is that three inequalities are real to solve, and is all located at intervalBetween, therefore it is the solution of equation group (formula 7).
Step S9: according toThe solution about (formula 7) tried to achieve in step S8 is converted into switching angle
Degree, and arrange according to order from small to large, finally giving switch angle is:
。
Step S10: owing to as m=2, only existing the one group of switch angle tried to achieve in step 9, so this group switchs naturally
Angle is global optimum.If there is organizing solution, the humorous of each group of solution can be evaluated with following total harmonic distortion index more
Ripple eradicating efficacy:
。
Wherein V1,V11,V13,V17,V19It is respectively fundamental voltage amplitude and the amplitude of the 11st, 13,17,19 inferior non-Zero-pharse harmonics,
Its computing formula is as shown in (formula 2).Final that minimum for selection THD group solution is as optimal solution.Have selected in the present embodiment
11st, 13,17, the summation of 19 subharmonic evaluate switch angle harmonic elimination effect, it is also possible to select more higher hamonic wave come
Calculate.
In order to the technical solution adopted in the present invention is better described, provide second embodiment with the situation of N=5, due to
During N=5, the expression formula of pilot process is huger, the most only provides result of calculation, omits concrete expression formula.Each step institute
The method used is identical with specific embodiment one.
Specific embodiment two: as N=5, three-phase staircase waveform changer needs the overtone order eliminated to be 5,7,11,13 times
Harmonic wave, therefore SHE equation group is in addition to first-harmonic equation, also needs to make 5, and the amplitude of 7,11,13 subharmonic is zero, altogether 5 equations.
Step S1: according to the many double angle formulas of cosine function and substitution of variableBy original SHE equation group
It is converted into Polynomial equations, wherein。
Step S2: willIt is converted into the multinomial about elementary symmetric polynomial
Equation group, whereinFor aboutElementary symmetric many
Item formula.
Step S3: order modulation ratio m=3.5, according to?, substitute into,
And calculateAt pure lexcographical orderUnder groebner base, equation group is converted into, wherein first equation isUnitary higher order polynomial equation,
Except argumentOutward, an argument is comprised only, and aboutIt is once.
Step S4: solve, and give up and be unsatisfactory forReality solution, final one has 3
Meet the reality solution of condition, be respectively as follows:
,,。
Step S5: willSubstitute into, try to achieveFor:；WillWithAlso substitute intoTry to achieve corresponding, finally give about's
Three groups of solutions are as follows:
Step S6: the three groups of solutions obtained according to step S5Construct respectively such as
Under unitary quintic algebra curve equation:
。
Step S7: three the unitary quintic algebra curve equations obtained in solution procedure S6 respectively, obtain aboutThree groups of solutions。
Step S8: the three groups of solutions obtained in testing sequence S7Whether meet
Following two condition: (1)It is 5 inequality real roots；(2)The most all
It is positioned at intervalBetween.The result of inspection is that the 3rd group of solution is unsatisfactory for condition, and front two groups of solutions meet condition, as follows:
。
Step S9: according toTry to achieve in step S8 two groups of solutions are converted into switch angle, and
Arranging according to order from small to large, finally giving two groups of switch angles is:
。
Step S10: to the two groups of solutions obtained in step S9, respectively according to below equation calculating THD:
。
The THD trying to achieve two groups of solutions is respectively as follows:, so second group of solution
For optimal solution.
With the two groups of switch Angle ambiguity three-phase 11 level tandem type staircase waveform multi-level converters obtained in the present embodiment
The output voltage waveforms obtained and the Fourier analysis of voltage waveform are as shown in Fig. 3, Fig. 4, Fig. 5 and Fig. 6, and direct current power source voltage is
100 volts, output voltage frequency is 50 hertz.Fig. 3 is the phase voltage waveform of output, and abscissa is time (unit: second), vertical seat
Mark represents output voltage (unit: volt), and VP1 is the phase voltage that first group of switch angle generates, and VP2 is second group of switch angle
The phase voltage generated；In Fig. 4, VP1_fft and VP2_fft is respectively as phase voltage VP1 and the Fourier analysis of VP2, abscissa
For frequency (unit: hertz), vertical coordinate is voltage (unit: volt), represents first-harmonic and the amplitude size of each harmonic.Can see
It is eliminated to the 5th, 7,11,13 subharmonic, but the 3rd, 9 inferior Zero-pharse harmonic component has still existed.Fig. 5 is output
Line voltage waveform, VL1 and VL2 be respectively first group and second group switch angle generate line voltage, in Fig. 6 VL1_fft and
VL2_fft is the Fourier analysis of line voltage VL1 and VL2 respectively, it can be seen that the 5th, 7,9,11,13,15 subharmonic
Being eliminated, minimum subharmonic is the 17th time.
It is pointed out that and staircase waveform many level SHE modulation can be extended to general many level SHE modulation, now
In the PWM waveform of output, the universal calculation equation of first-harmonic and each harmonic component amplitude is
(formula 13)
Wherein N is that switch is counted, before cosine term "" number depend on the transition state of this switching time PWM waveform,
If rising edge, then be "+" number, as trailing edge, being then "-" number.The harmonic wave eliminated as required, equally draws class
Be similar to the SHE equation group of (formula 3), difference be the most all before cosine term "+" number, but have just have negative.This kind of situation
Under, owing to when n is odd number, having:.So to being above the cosine term of "-" number,
Do substitution of variable, to being above "+" number cosine term, do substitution of variable, SHE
Equation group just can be converted into symmetric form, and the method for solving that the present invention proposes still is suitable for.
The above, the only detailed description of the invention in the present invention, but protection scope of the present invention is not limited thereto, and appoints
What is familiar with the people of this technology in the technical scope that disclosed herein, it will be appreciated that the conversion expected or replacement, all should contain
Within the scope of the comprising of the present invention, therefore, protection scope of the present invention should be as the criterion with the protection domain of claims.
Claims (1)
1. eliminate a method for solving for switch angle about staircase waveform multi-level converter particular harmonic, comprise the following steps:
Step 1: for having the changer of N number of switch angle, utilize trigonometric function double angle formula and substitution of variable to disappear specific
Humorous equation group is converted into Polynomial equations [f1,f2,…,fN], the most each fi, (i=1,2 ..., N) it is about argument
(x1,x2,…,xN) multinomial, and argument xiWith switch angle [alpha]iBetween relation be xi=cos αi(i=1,2 ..., N)；
Step 2: by [f1,f2,…,fN] be converted into about (e1,e2,…,eN) Polynomial equations [h1,h2,…,hN], wherein
ei(i=1,2 ..., N) it is about (x1,x2,…,xN) elementary symmetric polynomial；
Step 3: order modulation is concrete numerical value than m, by e1=m substitutes into [h2,h3,…,hN], and calculate [h2,h3,…,hN] pure
Lexcographical order (e2,e3,…,eNReduction groebner base under), obtains [g1,g2,…,gN-1]；
Step 4: solve unitary higher order polynomial equation g1=0, and combine constraints 0 < eN< 1, obtain b and meet constraints
Reality solution (eN)k, k=1,2 ..., b；
Step 5: by each (eN)kSubstitute into [g2,…,gN-1] solve (e2,e3,…,eN-1), try to achieve altogether b group and solve (e2,e3,…,
eN-1)k, k=1,2 ..., b；
Step 6: to each group of (e1,e2,…,eN)k, k=1,2 ..., b, it is constructed as follows equation:
xN-e1xN-1+e2xN-2-e3xN-3+…+(-1)N-1eN-1x+(-1)NeN=0
Step 7: solve b unitary n times polynomial equation obtained in the previous step, obtains b group and solves (x1,x2,…,xN)k, k=1,
2,…,b；
Step 8: inspection (x1,x2,…,xN)k, k=1,2 ..., whether b meets following two condition: (1) (x1,x2,…,xN)kFor
N number of inequality is real to be solved；(2) all real solutions are all located between interval [0,1]；Give up the solution being unsatisfactory for two above condition, final one
There are n group and solve (x1,x2,…,xN)s, s=1,2 ..., n；
Step 9: according to anticosine formula αi=cos-1(xi) calculate (x1,x2,…,xN)s, s=1,2 ..., opening corresponding to n
Close angle, and switch angle is obtained final switch angle (α according to order arrangement from small to large1,α2,…,αN)s, s=
1,2,…,n；
Step 10: calculate each group of switch angle (α1,α2,…,αN)s, s=1,2 ..., the total harmonic distortion of n, with minimum that
One group of solution is as the optimal solution switching angle.
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| CN102983768A (en) * | 2012-11-14 | 2013-03-20 | 国网智能电网研究院 | Optimization control method based on selective harmonic elimination pulse width modulation (SHEPWM) |
| CN103324864A (en) * | 2013-07-15 | 2013-09-25 | 中国矿业大学(北京) | Method for solving specific harmonic cancellation pulse width modulation inverter switching angle |
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| CN102983768A (en) * | 2012-11-14 | 2013-03-20 | 国网智能电网研究院 | Optimization control method based on selective harmonic elimination pulse width modulation (SHEPWM) |
| CN103324864A (en) * | 2013-07-15 | 2013-09-25 | 中国矿业大学(北京) | Method for solving specific harmonic cancellation pulse width modulation inverter switching angle |
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| Real Solution Number of the Nonlinear Equations in the SHEPWM Technology;Kehu Yang et al.;《International Conference on Intelligent Control and Information Processing》;20100815;第446-450页 * |
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