CN111446878B - Modeling method of three-phase voltage source type converter based on harmonic state space - Google Patents
Modeling method of three-phase voltage source type converter based on harmonic state space Download PDFInfo
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- CN111446878B CN111446878B CN202010263323.5A CN202010263323A CN111446878B CN 111446878 B CN111446878 B CN 111446878B CN 202010263323 A CN202010263323 A CN 202010263323A CN 111446878 B CN111446878 B CN 111446878B
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
- H02M7/00—Conversion of ac power input into dc power output; Conversion of dc power input into ac power output
- H02M7/42—Conversion of dc power input into ac power output without possibility of reversal
- H02M7/44—Conversion of dc power input into ac power output without possibility of reversal by static converters
- H02M7/48—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode
- H02M7/53—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal
- H02M7/537—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal using semiconductor devices only, e.g. single switched pulse inverters
- H02M7/5387—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal using semiconductor devices only, e.g. single switched pulse inverters in a bridge configuration
- H02M7/53871—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal using semiconductor devices only, e.g. single switched pulse inverters in a bridge configuration with automatic control of output voltage or current
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
- H02J3/00—Circuit arrangements for ac mains or ac distribution networks
- H02J3/38—Arrangements for parallely feeding a single network by two or more generators, converters or transformers
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
- H02M1/00—Details of apparatus for conversion
- H02M1/12—Arrangements for reducing harmonics from ac input or output
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
- H02M7/00—Conversion of ac power input into dc power output; Conversion of dc power input into ac power output
- H02M7/42—Conversion of dc power input into ac power output without possibility of reversal
- H02M7/44—Conversion of dc power input into ac power output without possibility of reversal by static converters
- H02M7/48—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode
- H02M7/53—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal
- H02M7/537—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal using semiconductor devices only, e.g. single switched pulse inverters
- H02M7/539—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal using semiconductor devices only, e.g. single switched pulse inverters with automatic control of output wave form or frequency
- H02M7/5395—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode using devices of a triode or transistor type requiring continuous application of a control signal using semiconductor devices only, e.g. single switched pulse inverters with automatic control of output wave form or frequency by pulse-width modulation
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Abstract
The invention discloses a three-phase voltage source type converter modeling method based on a harmonic state space, which comprises the following steps: 1, establishing a three-phase voltage source type converter time domain model according to a topological structure of the three-phase voltage source type converter; 2, performing Fourier transform on each state variable and input variable in the time domain equation; 3, establishing a harmonic state space model of the three-phase voltage source type converter according to the time domain model; and 4, establishing a single-phase simplified model of a harmonic state space. The method can simplify the modeling process of the harmonic state space, reduce the size of the harmonic state space model, and accurately obtain the harmonic information and the harmonic interaction mechanism of each variable of the three-phase voltage source type converter on the premise of not influencing the modeling precision, thereby providing a theoretical basis for solving the harmonic interaction problem.
Description
Technical Field
The invention relates to the field of modeling of power electronic devices, in particular to a three-phase voltage source type converter modeling method based on a harmonic state space.
Background
Under the trend of renewable energy sources such as distributed energy power generation and microgrid and the trend of new load grid connection such as energy storage and electric vehicles, the technology based on the power electronic converter, in particular the voltage source converter, is widely applied. The voltage source type converter is used as a nonlinear device, resonance can be generated when the voltage source type converter is connected to a power grid, and a large amount of harmonic waves can be injected into the power grid during operation, so that the quality of electric energy is deteriorated. Meanwhile, with the continuous increase of the number of converters, the harmonic interaction problem is increasingly complex, and a new challenge is provided for the stable operation of the system. To solve these problems, a detailed mathematical model of steady-state and transient analysis must be established to analyze the system harmonic conditions.
Researchers at home and abroad have studied on the modeling of the converter, and modeling methods can be divided into a numerical simulation method and an analytical modeling method. The numerical simulation method adopts different algorithms to carry out numerical calculation on the converter, thereby obtaining a numerical solution of certain characteristics of the converter, and the physical significance of the numerical solution is ambiguous; the analytical modeling method adopts analytical expressions to describe the characteristics of the converter, and has clear physical significance, so that most of the existing documents adopt the analytical modeling method, wherein a state space average method is the most commonly adopted method. The harmonic state space is used as a new modeling method in an analytic modeling method, can contain each harmonic of system variables for modeling, and provides great potential for solving the harmonic interaction problems such as harmonic interaction among multiple converters and the influence of switching transient voltage distortion on the converters. However, the derivation and processing of the harmonic state space model considering the switching harmonics are very complicated, and the size of the model is also very large, so that the application of the harmonic state space modeling in the power electronic system is limited.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, and provides a modeling method of a three-phase voltage source type converter based on a harmonic state space, so that the modeling process of the harmonic state space can be simplified, the size of a harmonic state space model is reduced, and the harmonic information and the harmonic interaction mechanism of each variable of the three-phase voltage source type converter can be accurately obtained on the premise of not influencing the modeling precision, thereby providing a theoretical basis for solving the harmonic interaction problem.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
the invention relates to a modeling method of a three-phase voltage source type converter based on a harmonic state space, which is characterized by comprising the following steps of:
step one, according to the topological structure of the three-phase voltage source type converter, establishing a time domain model of the three-phase voltage source type converter by using a formula (1):
in formula (1): rLA load resistor at the direct current side of the three-phase voltage source type converter; cdcA voltage stabilizing capacitor at the direct current side of the three-phase voltage source type converter; l isgThe filter inductor is an alternating current side filter inductor of the three-phase voltage source type converter; rgThe equivalent resistance is an alternating current side circuit of the three-phase voltage source type converter; i.e. igIs a matrix of the output current of the AC side of the three-phase voltage source type converter, and ig＝[iga igb igc]TWherein i isga、igb、igcThree-phase output currents of alternating current sides a, b and c of the three-phase voltage source type converter; v. ofdcThe voltage is the direct current side capacitor voltage of the three-phase voltage source type converter; v. ofsa、vsb、vscThe voltage of a, b and c three-phase power grid; s is a matrix formed by three-phase switching functions, and s ═ sa sb sc]TWherein s isa、sb、scRepresenting a switch function for controlling the on-off of the three-phase switches a, b and c; ga、gb、gcIs a, b and c three-phase AC-DC conversion function, and w intersects the DC conversion function
w＝a，b，c；swIs a w-phase switching function and has:
in formula (2): c (t) is a triangular carrier function, mw(t) is a w-phase modulated wave function and has:
mw(t)＝masin(ωt+θ)；w＝a，b，c (3)
in formula (3): m isaFor modulation index, theta is the phase angle of the modulated wave, and omega is the modulated wave function m of the w phasew(t) angular frequency, t being a time variable;
performing Fourier transform on each state variable and input variable in the time domain equation:
step 2.1, the state variable and the input variable i contained in the formula (1)ga、igb、igc、vsa、vsb、vsc、vdcIs denoted as x (t), and a fourier transform representation of any variable x (t) is obtained using equation (4):
in formula (4): omega0Is the fundamental angular frequency of any one variable x (t), and ω0＝2π/T0，T0Fundamental period, X, of any one variable X (t)kIs the kth Fourier coefficient of any one variable x (t), and
j is the imaginary unit;
and 2.2, obtaining a Fourier transform matrix representation of the formula (4) by using the formula (5):
x(t)＝E(t)X (5)
in formula (5): e (t) is an orthogonal basis matrix related to Fourier series, and
x is a matrix formed by fourier coefficients of any variable X (t), and X ═ X-h … X-1 X0 X1 … Xh]TWherein h is a self-defined finite number, XhThe h-th Fourier coefficient of any variable x (t);
step 2.3, obtaining the moment of the nth switching action of the w-phase switch of the three-phase voltage source type converter after the (i + 1) th iteration by using the formula (6)
In formula (6): when i is 0, let
a(n)For the triangular carrier function c (t) at the nth time equal to zero, kmIs the slope and modulation index m of the triangular carrier function c (t)aA ratio of (A) to (B), and
wherein m isfIs a frequency modulation index;
step 2.4, when the frequency modulation index mfWhen the number is odd, the w-phase switching function s is obtained by using the formula (7)wK-th Fourier coefficient S ofwk：
In formula (7): 2mfCalculating the number of switching moments as required;
step 2.5, the formula (7) is brought into the formula (5) to obtain a w-phase switching function s shown as a formula (8)wThe fourier transform matrix of (a):
sw＝E(t)Sw，w＝a，b，c (8)
in formula (8): swAs a function of the w-phase switchingwAnd a matrix of Fourier coefficients of, and Sw＝[S-wh … S-w1 Sw0Sw1 … Swh]TWherein S iswhAs a function of the w-phase switchingwThe h-th Fourier coefficient of (1);
step three, establishing a harmonic state space model according to the time domain model of the three-phase voltage source type converter:
equations (9) to (11) are derived from equation (5):
a(t)x(t)＝E(t)AX (10)
cx(t)＝E(t)cLX (11)
in formulae (9) to (11): a (t) is a periodic function, c is a constant,
is the derivative of X, N is the derivative matrix of the harmonic state space, and N ═ diag ([ -jh ω0 ... -jω0 0 jω0 ... jhω0]) A is a Topritz matrix formed by Fourier coefficients of a periodic function a (t), and
wherein A ishIs the h-th Fourier coefficient of the periodic function a (t), and I is an identity matrix of 2h +1 order;
formula (5) and formula (8) are taken together in formula (1) according to formulas (9) to (11), so that a harmonic state space model of the three-phase voltage source type converter shown in formula (12) is obtained:
in formula (12): i isga、Igb、IgcFor a, b and c three-phase output current iga、igb、igcRespectively, IgIs three matrixes Iga、Igb、IgcA matrix formed, and Ig＝[Iga Igb Igb]，VdcIs a DC side capacitor voltage vdcOf Fourier coefficients, Vsa、Vsb、VscIs a, b, c three-phase network voltage vsa、vsb、vscAre formed by three matrices Sa、Sb、ScA matrix formed, and S ═ Sa Sb Sc]TIn which S isa、Sb、ScAs a, b, c three-phase switching function sa、sb、scRespectively constructed of fourier coefficientsFormed matrix, Ga、Gb、GcIs a, b, c three-phase AC-DC conversion function ga、gb、gcThe Fourier coefficients of (A) are respectively formed into a matrix;
step four, establishing a single-phase simplified model of a harmonic state space:
according to the three-phase symmetric characteristic of the three-phase voltage source type converter during balanced operation, obtaining a single-phase simplified model of a harmonic state space of the three-phase voltage source type converter by using a formula (13):
in formula (13): l isaMatrix S formed by Fourier coefficients of a-phase switching functionaA simplified matrix of, and
wherein q is the maximum value of m, h-6 is more than or equal to q and less than or equal to h, La，h+m+1、Sa，h+m+Respectively representing a reduced matrix LaSum matrix SaRow h + m + 1.
Compared with the prior art, the invention has the beneficial effects that:
the method is based on the calculation of the switching time, the harmonic content of the switching function is included for modeling, the harmonic influence of the modulation process of the three-phase voltage source converter is definitely considered, and the problem that the system modeling including the switching behavior is extremely difficult due to the nonlinearity and discontinuity of a power switch is solved, so that a universal and easily-realized method is provided for obtaining the harmonic state space model of the voltage source converter;
2, each harmonic of the time domain variable is considered as an independent state variable, so that a detailed harmonic dynamic evolution process and a harmonic interaction mechanism can be obtained, the steady-state and dynamic conditions of the system can be accurately described, and a theoretical basis is provided for solving the harmonic interaction problem;
based on the three-phase symmetric characteristic of the balance system, the three-phase voltage source type converter harmonic state space model is simplified into the three-phase voltage source type converter harmonic state space single-phase model, the model size is reduced on the premise of not influencing the model precision, the problem of huge modeling size of the harmonic state space is solved, and therefore the model calculation speed is increased.
Drawings
FIG. 1 is a schematic diagram of a three-phase voltage source type grid-connected converter based on SPWM modulation in the prior art;
FIG. 2 is a schematic flow chart of a three-phase voltage source type converter harmonic state space modeling method.
Detailed Description
In this embodiment, as shown in fig. 1, the three-phase voltage source type converter adopts an SPWM modulation method, and as shown in fig. 2, a modeling method of the three-phase voltage source type converter based on a harmonic state space is performed according to the following steps:
step one, establishing a time domain model of the three-phase voltage source type converter by using a formula (1) according to a topological structure of the three-phase voltage source type converter:
in formula (1): rLA load resistor at the direct current side of the three-phase voltage source type converter; cdcA voltage stabilizing capacitor at the direct current side of the three-phase voltage source type converter; l isgThe filter inductor is an alternating current side filter inductor of the three-phase voltage source type converter; rgThe equivalent resistance is an alternating current side circuit of the three-phase voltage source type converter; i.e. igIs a matrix of the output current of the AC side of the three-phase voltage source type converter, and ig＝[iga igb igc]TWherein i isga、igb、igcThree-phase output currents of alternating current sides a, b and c of the three-phase voltage source type converter; v. ofdcThe voltage is the direct current side capacitor voltage of the three-phase voltage source type converter; v. ofsa、vsb、vscThe voltage of a, b and c three-phase power grid; s is a matrix formed by three-phase switching functions, and s ═ sa sb sc]TWherein s isa、sb、scRepresenting a switch function for controlling the on-off of the three-phase switches a, b and c; ga、gb、gcIs a, b and c three-phase AC-DC conversion function, and w intersects the DC conversion function
w＝a，b，c；swIs a w-phase switching function and has:
in formula (2): c (t) is a triangular carrier function, mw(t) is a w-phase modulated wave function and has:
mw(t)＝masin(ωt+θ)；w＝a，b，c (3)
in formula (3): m isaFor modulation index, theta is the phase angle of the modulated wave, and omega is the modulated wave function m of the w phasew(t) angular frequency, t being a time variable;
performing Fourier transform on each state variable and input variable in the time domain equation:
step 2.1, the state variable and the input variable i contained in the formula (1)ga、igb、igc、vsa、vsb、vsc、vdcIs denoted as x (t), and a fourier transform representation of any variable x (t) is obtained using equation (4):
in formula (4): omega0Is the fundamental angular frequency of any one variable x (t), and ω0＝2π/T0，T0Fundamental period, X, of any one variable X (t)kIs the kth Fourier coefficient of any one variable x (t), and
j is the imaginary unit;
and 2.2, obtaining a Fourier transform matrix representation of the formula (4) by using the formula (5):
x(t)＝E(t)X (5)
in formula (5): e (t) is an orthogonal basis matrix related to Fourier series, and
x is a matrix formed by fourier coefficients of any variable X (t), and X ═ X-h … X-1 X0 X1 … Xh]TWherein X ishThe h-th Fourier coefficient of any variable x (t), h is a self-defined finite number, h determines the number of considered harmonic times, the larger the h value is, the higher the model precision is, but the model size will be increased, the slower the calculation speed is, on the contrary, the smaller the h value is, the lower the model precision is, the modeling is convenient and simple, the calculation speed is high, and in a special case, when h is 1, the harmonic state space model is equivalent to an average model;
step 2.3, obtaining the moment of the nth switching action of the w-phase switch of the three-phase voltage source type converter after the (i + 1) th iteration by using the formula (6)
In formula (6): when i is 0, let
a(n)For the triangular carrier function c (t) at the nth time equal to zero, kmIs the slope and modulation index m of the triangular carrier function c (t)aA ratio of (A) to (B), and
wherein m isfIs a frequency modulation index;
step 2.4, when the frequency modulation index mfWhen the number is odd, the w-phase switching function s is obtained by using the formula (7)wK-th Fourier coefficient S ofwk：
In formula (7): 2mfCalculating the number of switching moments as required;
step 2.5, the formula (7) is brought into the formula (5) to obtain a w-phase switching function s shown as a formula (8)wThe fourier transform matrix of (a):
sw＝E(t)Sw，w＝a，b，c (8)
in formula (8): swAs a switching function s of the w-phasewAnd a matrix of Fourier coefficients of, and Sw＝[S-wh … S-w1Sw0 Sw1 … Swh]TWherein S iswhAs a function of the w-phase switchingwThe h-th Fourier coefficient of (1);
step three, establishing a harmonic state space model according to the time domain model of the three-phase voltage source type converter:
in order to convert the three-phase voltage source type converter time domain model into the harmonic state space model, equations (9) to (11) are obtained by derivation of equation (5):
a(t)x(t)＝E(t)AX (10)
cx(t)＝E(t)cLX (11)
in formulae (9) to (11): a (t) is a periodic function, c is a constant,
is the derivative of X, N is the derivative matrix of the harmonic state space, and N ═ diag ([ -jh ω0 ... -jω0 0 jω0 ... jhω0]) A is the Fourier coefficient of the periodic function a (t)A Topritz matrix of
Wherein A ishIs the h-th Fourier coefficient of the periodic function a (t), and I is an identity matrix of 2h +1 order;
formula (5) and formula (8) are taken together in formula (1) according to formulas (9) to (11), so that a harmonic state space model of the three-phase voltage source type converter shown in formula (12) is obtained:
in formula (12): i isga、Igb、IgcFor a, b and c three-phase output current iga、igb、igcRespectively, IgIs three matrixes Iga、Igb、IgaA matrix formed, and Ig＝[Iga Igb Igc]，VdcIs a DC side capacitor voltage vdcOf Fourier coefficients, Vsa、Vsb、VscIs a, b, c three-phase network voltage vsa、vsb、vscAre formed by three matrices Sa、Sb、ScA matrix formed, and S ═ Sa Sb Sc]TIn which S isa、Sb、ScAs a, b, c three-phase switching function sa、sb、scRespectively, G, of the Fourier coefficientsa、Gb、GcIs a, b, c three-phase AC-DC conversion function ga、gb、gcThe Fourier coefficients of (A) are respectively formed into a matrix;
step four, establishing a single-phase simplified model of a harmonic state space:
the three-phase symmetric characteristic of the three-phase voltage source type converter during balanced operation is shown as a formula (13):
in formula (13): l isgak、Igbk、IgckFor a, b and c three-phase output current iga、igb、igcK-th Fourier coefficient of (V)sak、Vsbk、VsckIs a, b, c three-phase network voltage vsa、vsb、vscK-th Fourier coefficient of (1), Sak、Sbk、SckAs a function of three-phase switching sa、sb、scThe kth fourier coefficient of (1);
therefore, the current-voltage conditions of the two phases b and c can be obtained from the equation (13) only by using the current-voltage condition of the phase a. However, the DC-side capacitor voltage v in the formula (12)dcIs a matrix V formed by fourier coefficients ofdcIn the equation of state of (2), three matrices Sa、Sb、ScThe formed matrix S and three matrices Iga、Igb、IgcThe formed matrix IgThe three phases are coupled to each other, so the coupling terms are decoupled by equation (14):
STIg＝SaIga+SbIgb+ScIgc＝3LaIga (14)
in formula (14): l isaMatrix S formed by Fourier coefficients of a-phase switching functionaA simplified matrix of, and
wherein q is the maximum value of m, h-6 is more than or equal to q and less than or equal to h, La，h+m+1、Sa，h+m+Respectively representing a reduced matrix LaSum matrix SaRow h + m + 1.
The formula (14) is introduced into the formula (12), so that a harmonic state space single-phase simplified model of the three-phase voltage source type converter shown as the formula (15) is obtained:
the solution formula (15) can obtain the three-phase voltage source type converter AC measured a-phase output current igaAnd a DC side capacitor voltage vdcThe content of each subharmonic can be obtained by using the formula (13) to measure a and b phase output current i of the three-phase voltage source type convertergb、igcThe harmonic content of (1). The three-phase output current i of a, b and c obtained by the solution is then utilized by the formula (5)ga、igb、igcRespectively, of Fourier coefficientsga、Igb、IgcAnd a DC side capacitor voltage vdcIs a matrix V formed by fourier coefficients ofdcConverted into the time domain.
Claims (1)
1. A modeling method of a three-phase voltage source type converter based on a harmonic state space is characterized by comprising the following steps:
step one, according to the topological structure of the three-phase voltage source type converter, establishing a time domain model of the three-phase voltage source type converter by using a formula (1):
in formula (1): rLA load resistor at the direct current side of the three-phase voltage source type converter; cdcA voltage stabilizing capacitor at the direct current side of the three-phase voltage source type converter; l isgThe filter inductor is an alternating current side filter inductor of the three-phase voltage source type converter; rgThe equivalent resistance is an alternating current side circuit of the three-phase voltage source type converter; i.e. igIs a matrix of the output current of the AC side of the three-phase voltage source type converter, and ig＝[igaigb igc]TWherein i isga、igb、igcThree-phase output currents of alternating current sides a, b and c of the three-phase voltage source type converter; v. ofdcThe voltage is the direct current side capacitor voltage of the three-phase voltage source type converter; v. ofsa、vsb、vscFor a, b, c three-phase networksA voltage; s is a matrix formed by three-phase switching functions, and s ═ sa sb sc]TWherein s isa、sb、scRepresenting a switch function for controlling the on-off of the three-phase switches a, b and c; ga、gb、gcIs a, b and c three-phase AC-DC conversion function, and w intersects the DC conversion function
w＝a，b，c；swIs a w-phase switching function and has:
in formula (2): c (t) is a triangular carrier function, mw(t) is a w-phase modulated wave function and has:
mw(t)＝masin(ωt+θ)；w＝a，b，c (3)
in formula (3): m isaFor modulation index, theta is the phase angle of the modulated wave, and omega is the modulated wave function m of the w phasew(t) angular frequency, t being a time variable;
performing Fourier transform on each state variable and input variable in the time domain equation:
step 2.1, the state variable and the input variable i contained in the formula (1)ga、igb、igc、vsa、vsb、vsc、vdcIs denoted as x (t), and a fourier transform representation of any variable x (t) is obtained using equation (4):
in formula (4): omega0Is the fundamental angular frequency of any one variable x (t), and ω0＝2π/T0，T0Fundamental period, X, of any one variable X (t)kIs any one variable x(t) the kth Fourier coefficient, and
j is the imaginary unit;
and 2.2, obtaining a Fourier transform matrix representation of the formula (4) by using the formula (5):
x(t)＝E(t)X (5)
in formula (5): e (t) is an orthogonal basis matrix related to Fourier series, and
x is a matrix formed by fourier coefficients of any variable X (t), and X ═ X-h … X-1 X0 X1 … Xh]TWherein h is a self-defined finite number, XhThe h-th Fourier coefficient of any variable x (t);
step 2.3, obtaining the moment of the nth switching action of the w-phase switch of the three-phase voltage source type converter after the (i + 1) th iteration by using the formula (6)
In formula (6): when i is 0, let
a(n)For the triangular carrier function c (t) at the nth time equal to zero, kmIs the slope and modulation index m of the triangular carrier function c (t)aA ratio of (A) to (B), and
wherein m isfIs a frequency modulation index;
step 2.4, when the frequency modulation index mfIf the number is odd, the w-phase switch is obtained by using the formula (7)Function swK-th Fourier coefficient S ofwk：
In formula (7): 2mfCalculating the number of switching moments as required;
step 2.5, the formula (7) is brought into the formula (5) to obtain a w-phase switching function s shown as a formula (8)wThe fourier transform matrix of (a):
sw＝E(t)Sw，w＝a，b，c (8)
in formula (8): swAs a function of the w-phase switchingwAnd a matrix of Fourier coefficients of, and sw＝[S-wh… S-w1 Sw0 Sw1… Swh]TWherein S iswhAs a function of the w-phase switchingwThe h-th Fourier coefficient of (1);
step three, establishing a harmonic state space model according to the time domain model of the three-phase voltage source type converter:
equations (9) to (11) are derived from equation (5):
a(t)x(t)＝E(t)AX (10)
cx(t)＝E(x)cIX (11)
in formulae (9) to (11): a (t) is a periodic function, c is a constant,
is the derivative of X, N is the derivative matrix of the harmonic state space, and N ═ diag ([ -jh ω0 ... -jω0 0 jω0 ... jhω0]) A is a Topritz matrix formed by Fourier coefficients of a periodic function a (t), and
wherein A ishIs the h-th Fourier coefficient of the periodic function a (t), and I is an identity matrix of 2h +1 order;
formula (5) and formula (8) are taken together in formula (1) according to formulas (9) to (11), so that a harmonic state space model of the three-phase voltage source type converter shown in formula (12) is obtained:
in formula (12): i isga、Igb、IgcFor a, b and c three-phase output current iga、igb、igcRespectively, IgIs three matrixes Iga、Igb、IgcA matrix formed, and Ig＝[Iga Igb Igc]，VdcIs a DC side capacitor voltage vdcOf Fourier coefficients, Vsa、Vsb、VscIs a, b, c three-phase network voltage vsa、vsb、vscAre formed by three matrices Sa、Sb、ScA matrix formed, and S ═ Sa Sb Sc]TIn which S isa、Sb、ScAs a three-phase switching function S of a, b and ca、Sb、ScRespectively, G, of the Fourier coefficientsa、Gb、GcIs a, b, c three-phase AC-DC conversion function ga、gb、bcThe Fourier coefficients of (A) are respectively formed into a matrix;
step four, establishing a single-phase simplified model of a harmonic state space:
according to the three-phase symmetric characteristic of the three-phase voltage source type converter during balanced operation, obtaining a single-phase simplified model of a harmonic state space of the three-phase voltage source type converter by using a formula (13):
in formula (13): l isaMatrix S formed by Fourier coefficients of a-phase switching functionaA simplified matrix of, and
wherein q is the maximum value of m, h-6 is more than or equal to q and less than or equal to h, La，h+m+1、Sa，h+m+Respectively representing a reduced matrix LaSum matrix SaRow h + m + 1.
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