WO2022042158A1 - Mmc small-signal impedance modeling method based on fourier decomposition - Google Patents

Abstract

Disclosed in the present invention is an MMC small-signal impedance modeling method based on Fourier decomposition, comprising the following steps: step 1, establishing an MMC time-domain state-space model; step 2, converting the MMC time-domain state-space model into a frequency domain by means of Fourier decomposition to obtain an MMC frequency-domain steady-state model; step 3, solving a steady-state working point of the MMC on the MMC frequency-domain steady-state model; step 4, performing perturbation linearization on the MMC at the steady-state working point to obtain a small-signal model of the MMC; step 5, injecting a voltage disturbance Δup having a frequency of ωp into the MMC, and calculating a corresponding current response Δip at the frequency ωp by means of the established small-signal model; step 6, dividing the disturbance voltage by the corresponding disturbance current to obtain the impedance of the MMC at the disturbance frequency; and step 7, changing the disturbance frequency, and repeating steps 5 and 6 to obtain an impedance curve of the MMC within a certain range. The present invention solves the problem that existing models cannot take harmonic coupling in an MMC into consideration, making the established model more accurate.

Description

MMC Small Signal Impedance Modeling Method Based on Fourier Decomposition technical field

The invention relates to the technical field of power transmission and distribution of power systems, in particular to an MMC small-signal impedance modeling method based on Fourier decomposition.

Background technique

Because the modular multilevel converter (MMC) has the advantages of modular structure, high efficiency, small size, high output waveform quality, easy installation and maintenance, etc., in the high voltage direct current transmission (High Voltage Direct Current, HVDC) ) and asynchronous grid interconnection systems have been widely used. However, in recent years, many flexible DC projects at home and abroad have experienced high-frequency harmonic resonance during commissioning or operation, and MMC-based photovoltaic, wind power and DC transmission systems have also experienced subsynchronous oscillations with a frequency of 20 to 30 Hz. , which seriously affects the safe operation of the system. Compared with traditional two-level and three-level converters, the structure of MMC is more complex, resulting in a more complex control system than traditional converters. The multi-time scale dynamic control characteristics of MMC and its relationship with the power grid The interaction between them is the main factor leading to the frequent occurrence of system oscillation accidents. Therefore, the stability analysis of the MMC-HVDC system is particularly important.

The MMC small-signal model is an important tool for analyzing the stability of MMC-HVDC systems. Because MMC has multiple frequency harmonic components in the bridge arm current and capacitor voltage during steady-state operation, and has typical time-varying nonlinear multi-frequency response characteristics, the modeling methods and linear system analysis methods of traditional power electronic converters are all different. Difficult to apply directly to MMC. At present, the small signal modeling methods for MMC can be roughly divided into two categories: time domain and frequency domain. The time-domain modeling method analyzes the small-signal stability of the system based on eigenvalues and root locus by obtaining the parameter matrix of the system. Although the time-domain modeling method can describe the internal dynamic characteristics of MMC, the modeling process is complex, and it is difficult to explain the complex harmonic coupling problem within the MMC, which has great limitations when applied to the stability analysis of AC and DC systems.

The frequency domain analysis method establishes the system frequency domain impedance model, and then uses frequency domain analysis tools such as impedance stability criterion and Bode diagram to study the small-signal stability of the AC and DC systems. The measurement verification is more applicable in the stability analysis of the actual MMC-HVDC system. At present, the methods of frequency domain modeling for MMC mainly include harmonic linearization and Fourier decomposition. The basic idea of multi-harmonic linearization is to inject a small perturbation signal at a specific frequency into the excitation of the system, and deduce the response corresponding to the perturbation frequency in the state variable, so as to obtain a linear model considering the small perturbation component; The small-signal impedance of the system can be obtained by dividing the voltage and current in the system, thereby realizing the analysis of system stability. However, when using the harmonic linearization method to model MMC, the process is complicated, which is not conducive to computer programming. The above shortcomings can be overcome when the MMC is modeled by the Fourier decomposition method. The Fourier decomposition method can obtain the small-signal impedance model of MMC by converting the time domain model to the frequency domain, and then performing small disturbance analysis at the steady-state operating point. The modeling process is relatively simple, and it is easy to program and solve, and the model accuracy is higher. . However, the traditional Fourier decomposition method adopts the derivation method for modeling, and does not consider the internal harmonic coupling of the MMC, and the model accuracy still needs to be improved.

SUMMARY OF THE INVENTION

In view of the above-mentioned deficiencies of the prior art, the present invention provides an MMC small-signal impedance modeling method based on Fourier decomposition, aiming to solve the technical problem that the existing MMC small-signal impedance modeling method is inaccurate, and can make the model It is applied to analyze the stability of the MMC-based HVDC transmission system, and then improve the operational reliability of the MMC-based HVDC transmission system.

For solving the above-mentioned technical problems, the technical scheme of the present invention is as follows:

An MMC small-signal impedance modeling method based on Fourier decomposition, the MMC includes three-phase bridge arms corresponding to three-phase alternating current one-to-one, and the three-phase bridge arms are connected in parallel; each phase bridge arm includes an upper bridge arm connected in series and the lower bridge arm, and the upper bridge arm contains N sub-modules connected in series, and the lower bridge arm contains N sub-modules connected in series, including the following steps:

Step 1, transform the single-phase branch of the MMC into an average value equivalent circuit, write the basic circuit equation of the MMC, and obtain the mathematical model of the MMC; represent the mathematical model of the MMC in the form of a state space , get the MMC time-domain state-space model

Among them, A(t), B(t) are the matrices representing the parameters of the MMC circuit, x(t) is the selected state variable,

i _c (t) is the phase circulating current,

is the sum of the capacitor voltages of the upper bridge arm,

is the sum of the capacitor voltages of the lower bridge arm, i _g (t) is the AC side current, u(t) is the selected output variable, u(t)=[U _dc ,0,0,u _g (t)] ^T , U _dc is the DC side voltage, _ug (t) is the AC side voltage;

Step 2, based on Fourier series decomposition, transform the MMC time-domain state space model into a frequency-domain equation through Fourier decomposition, and obtain the MMC frequency-domain steady-state model sX=(A-Q)X+BU;

Wherein, elements X, U, A, B correspond to x(t), u(t), A(t), B(t) in the MMC time-domain state space model, respectively, and Q represents the diagonal of the frequency information matrix;

In step 3, the steady-state operating point of the MMC is solved on the MMC frequency-domain steady-state model. When the MMC is in steady-state operation, the complex variable s in the MMC frequency-domain steady-state model tends to zero. The frequency domain steady model sX=(AQ)X+BU is inverted to obtain the steady-state operating point of the system X _ss =-(AQ) ^-1 (BU);

Step 4, the small perturbation analysis method is applied to the MMC time domain state space model to obtain the time domain state equation:

Among them, the symbol "Δ" represents a small disturbance signal, the disturbance term ΔB(t)=0, and the linearized time-domain state equation is obtained after ignoring the quadratic term in the formula:

Transform the linearized time-domain state equation into the frequency domain. When the system is running in a steady state, the complex variable s tends to 0. After arranging the steady-state operating point and the linearized time-domain state equation into a matrix form, we get MMC small signal model:

ΔAX+(A-ΔQ)ΔX+BΔU=0;

After substituting the steady-state operating point into the MMC small-signal model, for a given input signal perturbation ΔU, the perturbation response of the state variable is solved:

ΔX=-(A-ΔQ) ^-1 (ΔAX+BΔU);

Step 5, inject a voltage disturbance _Δup with a frequency of ω _p into the MMC, and calculate the corresponding current response _Δip at the frequency of ω _p by using the established MMC small signal model;

Step 6, by calculating the ratio of the disturbance voltage generated at the disturbance frequency _ωp to the corresponding disturbance current, obtain the AC side small-signal impedance of the MMC at the disturbance frequency _ωp , which is defined as:

Step 7: Change the magnitude of the disturbance frequency within a preset range according to requirements, and repeat steps 5 and 6 to obtain an impedance curve of the MMC within the preset range.

Compared with the prior art, the present invention has the following beneficial effects:

1. In the process of modeling the small-signal impedance of MMC, the Fourier decomposition method is adopted to consider the harmonic coupling problem inside the converter, so that the impedance model of the MMC is more accurate. The MMC impedance curve obtained by modeling is almost consistent with the actual MMC impedance curve, and the modeling accuracy is high.

2. Due to the high accuracy of the model established by the present invention, the model can be applied to analyze the stability of the MMC-based HVDC transmission system, thereby improving the operational reliability of the MMC-based HVDC transmission system.

Description of drawings

Fig. 1 is the topological structure schematic diagram of MMC;

Fig. 2 is the flow chart of the MMC small-signal impedance modeling method based on Fourier decomposition in the present embodiment;

Fig. 3 is the schematic diagram of the average value equivalent circuit of the MMC single-phase branch;

FIG. 4 is a schematic diagram of the comparison between the existing MMC small-signal impedance modeling method and the actual MMC impedance;

FIG. 5 is a schematic diagram of a comparison between the MMC small-signal impedance modeling method based on Fourier decomposition and the actual MMC impedance using this specific embodiment.

detailed description

In order to make the objectives, technical solutions and advantages of the present invention clearer and clearer, the content of the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention, but not to limit the present invention. In addition, it should be noted that, for the convenience of description, the drawings only show some but not all of the contents related to the present invention.

A MMC small-signal impedance modeling method based on Fourier decomposition, the adopted MMC topology structure is shown in FIG. 1 , the MMC includes three-phase bridge arms corresponding to three-phase alternating current one-to-one, and the three-phase bridge arms are connected in parallel; Each phase bridge arm includes an upper bridge arm and a lower bridge arm in series, and the upper bridge arm includes N sub-modules connected in series, and the lower bridge arm includes N sub-modules connected in series, including the following steps, and the process is shown in Figure 2:

Step 1, transform the single-phase branch of the MMC into an average value equivalent circuit, the equivalent circuit refers to the schematic diagram of FIG. 3, write the basic circuit equation of the MMC, and obtain the mathematical model of the MMC; The mathematical model of is expressed in the form of state space, and the MMC time domain state space model is obtained.

Among them, A(t), B(t) are the matrices representing the parameters of the MMC circuit, x(t) is the selected state variable,

i _c (t) is the phase circulating current,

is the sum of the capacitor voltages of the upper bridge arm,

is the sum of the capacitor voltages of the lower bridge arm, i _g (t) is the AC side current, u(t) is the selected output variable, u(t)=[U _dc ,0,0,u _g (t)] ^T , U _dc is the DC side voltage, _ug (t) is the AC side voltage;

Step 2, based on Fourier series decomposition, transform the MMC time domain state space model into a frequency domain equation through Fourier decomposition, that is, transform the MMC time domain model into the frequency domain through Fourier decomposition, and obtain the MMC frequency domain. Steady-state model sX=(AQ)X+BU;

Wherein, elements X, U, A, B correspond to x(t), u(t), A(t), B(t) in the MMC time-domain state space model, respectively, and Q represents the diagonal of the frequency information matrix;

In step 3, the steady-state operating point of the MMC is solved on the MMC frequency-domain steady-state model. When the MMC is in steady-state operation, the complex variable s in the MMC frequency-domain steady-state model tends to zero. The frequency domain steady model sX=(AQ)X+BU is inverted to obtain the steady-state operating point of the system X _ss =-(AQ) ^-1 (BU);

Step 4, the small perturbation analysis method is applied to the MMC time domain state space model to obtain the time domain state equation:

Among them, the symbol "Δ" represents a small disturbance signal. Considering that B(t) is a constant coefficient matrix in MMC, the disturbance term is 0, the disturbance term ΔB(t)=0, and the linearity is obtained after ignoring the quadratic term in the formula The time-domain state equation is:

Transform the linearized time-domain state equation into the frequency domain. When the system is running in a steady state, the complex variable s tends to 0. After arranging the steady-state operating point and the linearized time-domain state equation into a matrix form, we get MMC small signal model:

ΔAX+(A-ΔQ)ΔX+BΔU=0;

After substituting the steady-state operating point into the MMC small-signal model, for a given input signal perturbation ΔU, the perturbation response of the state variable is solved:

ΔX=-(A-ΔQ) ^-1 (ΔAX+BΔU);

Step 5, inject a voltage disturbance _Δup with a frequency of ω _p into the MMC, and calculate the corresponding current response _Δip at the frequency of ω _p by using the established MMC small signal model;

Step 6, by calculating the ratio of the disturbance voltage generated at the disturbance frequency _ωp to the corresponding disturbance current, obtain the AC side small-signal impedance of the MMC at the disturbance frequency _ωp , which is defined as:

Step 7: Change the magnitude of the disturbance frequency within a preset range according to requirements, repeat steps 5 and 6, and obtain an impedance curve of the MMC within the preset range.

Further, the matrices A(t) and B(t) in the step 1 are respectively expressed as:

Among them, R is the MMC bridge arm resistance, C=C _arm /N, C _arm is the capacitance value of the half-bridge sub-module in parallel, N is the number of bridge arm sub-modules, and L is the MMC bridge arm inductance; s _u (t) , s _l (t) are the switching functions of the upper and lower arms of the MMC, respectively, expressed as:

where m ₁ and θ ₁ are the modulation ratio and phase of the fundamental frequency modulation voltage generated by the controller, m ₂ and θ ₂ are the modulation ratio and phase of the double frequency modulation voltage generated by the second harmonic circulating current controller, and ω ₁ =2πf, f is the fundamental frequency of 50Hz.

Further, the elements X, U, A, B in the step 2 are respectively expressed as:

X=[…,X _-3 ,X _-2 ,X _-1 ,X ₀ ,X ₁ ,X ₂ ,X ₃ ,…];

U=[…,U _-3 ,U _{- 2} ,U _{- 1} , _U0 ,U1,U2, _U3 ,…];

Q=diag[…,-j3ω ₁ I,-j2ω ₁ I,-jω ₁ I,O,jω ₁ I,j2ω ₁ I,j3ω ₁ I,…];

where I is an identity matrix with the same order as the state variables, O is a zero matrix with the same order as the state variables, and the subscripts of the elements represent the harmonic orders considered.

Further, the Fourier coefficients of the h-th harmonic of x(t), u(t), A(t), and B(t) in the MMC time-domain state space model correspond to the elements X _h , U _h , A _h , B _h ,

in,

U ₀ =[U _dc ,0,0,0],U ₁ =[0,0,0,0.5U _g ],U _±h =[0](h≥2);

B _±h = [0 0 0 0] ^T (h≥1);

Among them, M1=m ₁ /2, M ₂ =m ₂ /2, m ₁ is the modulation ratio of the fundamental frequency modulation voltage generated by the controller, and m ₂ is the double frequency modulation voltage generated by the second harmonic circulating current controller. modulation ratio, C _arm is the capacitance value connected in parallel with the half-bridge sub-module, I _c ,

and I _g are the upper arm of the phase circulating current, the sum of the capacitor voltage, the sum of the capacitor voltage of the lower arm, and the frequency domain state variable corresponding to the AC side current; the subscript h is the considered harmonic order, and U _dc is the direct current side voltage value, U _g is the AC side voltage amplitude, and the superscript T represents the transposed matrix.

Further, ΔA(t), Δx(t), and Δu(t) in the time-domain state equation are matrices formed by disturbance signals, which are respectively expressed as:

ΔX=[..., X _p-3 , X _p-2 , X _p-1 , X _p , X _p+1 , X _p+2 , X _p+3 ,...] ^T ;

ΔU=[..., U _p-3 , U _p-2 , U _p-1 , U _p , U _p+1 , U _p+2 , U _p+3 ,...] ^T ;

Among them, A _p±h represents the complex Fourier coefficient of the conjugate corresponding to the matrix A at the frequency of ωp±hω ₁ ; X _p±h represents the complex Fourier of the conjugate corresponding to the matrix X at the frequency of ωp±hω ₁ Coefficient; U _p±h represents the complex Fourier coefficient of the conjugate corresponding to the U matrix at the frequency of ωp±hω ₁ .

In the prior art, in the process of small-signal impedance modeling for MMC, the MMC converter is treated as a two-level VSC converter, and the harmonic coupling problem inside the converter is not considered, so that the obtained The accuracy of the MMC impedance model is greatly reduced. The corresponding MMC AC impedance is compared with the actual MMC impedance, as shown in Figure 4. It can be seen from the figure that the MMC impedance curve modeled by the traditional method is different from the actual MMC impedance curve. larger. However, since the small signal impedance model of the present invention takes into account the internal harmonic coupling of the MMC, as shown in FIG. 5 , the MMC impedance curve and the actual MMC impedance curve are almost identical, and the model accuracy is higher.

The above-mentioned embodiments are only for illustrating the technical concept and characteristics of the present invention, and the purpose thereof is to enable those of ordinary skill in the art to understand the content of the present invention and implement them accordingly, and not to limit the protection scope of the present invention. All equivalent changes or modifications made according to the essence of the present invention shall be included within the protection scope of the present invention.

Claims (5)

1. An MMC small-signal impedance modeling method based on Fourier decomposition, the MMC includes three-phase bridge arms corresponding to three-phase alternating current one-to-one, and the three-phase bridge arms are connected in parallel; each phase bridge arm includes an upper bridge arm connected in series With the lower bridge arm, and the upper bridge arm includes N sub-modules connected in series, and the lower bridge arm includes N sub-modules connected in series, it is characterized in that, comprising the following steps:

   Step 1, transform the single-phase branch of the MMC into an average value equivalent circuit, write the basic circuit equation of the MMC, and obtain the mathematical model of the MMC; represent the mathematical model of the MMC in the form of a state space , get the MMC time-domain state-space model

   

   Among them, A(t), B(t) are the matrices representing the parameters of the MMC circuit, x(t) is the selected state variable,

   

   i _c (t) is the phase circulating current,

   

   is the sum of the capacitor voltages of the upper bridge arm,

   

   is the sum of the capacitor voltage of the lower bridge arm, i _g (t) is the AC side current, u(t) is the selected output variable, u(t)=[U _dc ,0,0,u _g (t)] ^T , U _dc is the DC side voltage, _ug (t) is the AC side voltage;

   Step 2, based on Fourier series decomposition, transform the MMC time-domain state space model into a frequency-domain equation through Fourier decomposition, and obtain the MMC frequency-domain steady-state model sX=(A-Q)X+BU;

   Wherein, elements X, U, A, B correspond to x(t), u(t), A(t), B(t) in the MMC time-domain state space model, respectively, and Q represents the diagonal of the frequency information matrix;

   In step 3, the steady-state operating point of the MMC is solved on the MMC frequency-domain steady-state model. When the MMC is in steady-state operation, the complex variable s in the MMC frequency-domain steady-state model tends to zero. The frequency domain steady model sX=(AQ)X+BU is inverted to obtain the steady-state operating point of the system X _ss =-(AQ) ^-1 (BU);

   Step 4, the small perturbation analysis method is applied to the MMC time domain state space model to obtain the time domain state equation:

   

   Among them, the symbol "Δ" represents a small disturbance signal, the disturbance term ΔB(t)=0, and the linearized time-domain state equation is obtained after ignoring the quadratic term in the formula:

   

   Transform the linearized time-domain state equation into the frequency domain. When the system is running in a steady state, the complex variable s tends to 0. After arranging the steady-state operating point and the linearized time-domain state equation into a matrix form, we get MMC small signal model:

   ΔAX+(A-ΔQ)ΔX+BΔU=0;

   After substituting the steady-state operating point into the MMC small-signal model, for a given input signal perturbation ΔU, the perturbation response of the state variable is solved:

   ΔX=-(A-ΔQ) ^-1 (ΔAX+BΔU);

   Step 5, inject a voltage disturbance _Δup with a frequency of ω _p into the MMC, and calculate the corresponding current response _Δip at the frequency of ω _p by using the established MMC small signal model;

   Step 6, by calculating the ratio of the disturbance voltage generated at the disturbance frequency _ωp to the corresponding disturbance current, obtain the AC side small-signal impedance of the MMC at the disturbance frequency _ωp , which is defined as:

   

   Step 7: Change the magnitude of the disturbance frequency within a preset range according to requirements, repeat steps 5 and 6, and obtain an impedance curve of the MMC within the preset range.
2. The MMC small-signal impedance modeling method based on Fourier decomposition as claimed in claim 1, is characterized in that, the matrix A (t) in described step 1, B (t) are respectively expressed as:

   

   

   Among them, R is the MMC bridge arm resistance, C=C _arm /N, and C _arm is the capacitance value of the half-bridge sub-modules connected in parallel, N is the number of bridge arm sub-modules, and L is the MMC bridge arm inductance; s _u (t ), s _l (t) are the switching functions of the upper and lower arms of the MMC, respectively, expressed as:

   

   where m ₁ and θ ₁ are the modulation ratio and phase of the fundamental frequency modulation voltage generated by the controller, m ₂ and θ ₂ are the modulation ratio and phase of the double frequency modulation voltage generated by the second harmonic circulating current controller, and ω ₁ =2πf, f is the fundamental frequency of 50Hz.
3. The MMC small-signal impedance modeling method based on Fourier decomposition according to claim 1, wherein the elements X, U, A, B in the step 2 are respectively expressed as:

   X=[…,X _-3 ,X _-2 ,X _-1 ,X ₀ ,X ₁ ,X ₂ ,X ₃ ,…];

   U=[…,U _-3 ,U _{- 2} ,U _{- 1} , _U0 ,U1,U2, _U3 ,…];

   

   

   Q=diag[…,-j3ω ₁ I,-j2ω ₁ I,-jω ₁ I,O,jω ₁ I,j2ω ₁ I,j3ω ₁ I,…];

   where I is an identity matrix with the same order as the state variables, O is a zero matrix with the same order as the state variables, and the subscripts of the elements represent the harmonic orders considered.
4. The MMC small-signal impedance modeling method based on Fourier decomposition according to claim 1, wherein x(t), u(t), A(t) in the MMC time-domain state space model, The Fourier coefficients of the h-th harmonic of B(t) correspond to the elements X _h , U _h , A _h , B _h , respectively,

   in,

   

   U ₀ =[U _dc ,0,0,0],U ₁ =[0,0,0,0.5U _g ],U _±h =[0](h≥2);

   

   

   

   

   

   Among them, M1=m ₁ /2, M ₂ =m ₂ /2, m ₁ is the modulation ratio of the fundamental frequency modulation voltage generated by the controller, and m ₂ is the double frequency modulation voltage generated by the second harmonic circulating current controller. modulation ratio, C _arm is the capacitance value connected in parallel with the half-bridge sub-module, I _c ,

   

   and I _g are the upper arm of the phase circulating current, the sum of the capacitor voltage, the sum of the capacitor voltage of the lower arm, and the frequency domain state variable corresponding to the AC side current; the subscript h is the considered harmonic order, and U _dc is the direct current side voltage value, U _g is the AC side voltage amplitude, and the superscript T represents the transposed matrix.
5. The MMC small-signal impedance modeling method based on Fourier decomposition according to claim 1, wherein ΔA(t), Δx(t) and Δu(t) in the time-domain state equation are disturbance signals The formed matrices are represented as:

   

   ΔX=[..., X _p-3 , X _p-2 , X _p-1 , X _p , X _p+1 , X _p+2 , X _p+3 ,...] ^T ;

   ΔU=[..., U _p-3 , U _p-2 , U _p-1 , U _p , U _p+1 , U _p+2 , U _p+3 ,...] ^T ;

   Among them, A _p±h represents the complex Fourier coefficient of the conjugate corresponding to the matrix A at the frequency of ωp±hω ₁ ; X _p±h represents the complex Fourier of the conjugate corresponding to the matrix X at the frequency of ωp±hω ₁ Coefficient; U _p±h represents the complex Fourier coefficient of the conjugate corresponding to the U matrix at the frequency of ωp±hω ₁ .

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