A modular multilevel matrix converter modeling method based on harmonic state space principle

Abstract

A modular multilevel matrix converter modeling method based on harmonic state space principle, S1: obtaining M3C grid-connected system grid topology and element parameters, establishing M3C system circuit topology nine bridge arm state space equation; S2: according to the symmetry of M3C circuit topology and the stable operation of two side system, establishing the single bridge arm state space equation of M3C; S3: establishing the state space model of M3C control system, and according to the corresponding relationship of the interface with the main circuit, establishing the state space model of M3C containing closed loop control; S4: using double fundamental frequency harmonic state space method, establishing the closed loop control M3C harmonic state space model containing each order component and small signal model. S5: carrying out two-dimensional Fourier inverse transform of each order harmonic component obtained in step S4 to time domain, and then comparing with the model built by MALAB; S6: according to the established small signal model, small disturbance analysis is carried out.

Description

Technical Field

[0001] This invention relates to the field of multilevel power electronic converter technology, and in particular to a modular multilevel matrix converter modeling method based on the harmonic state-space principle. Background Technology

[0002] Modular Multilevel Matrix Converters (M3Cs), as a new type of matrix three-phase AC-AC converter in the Modular Multilevel Converter (MMC) family, are increasingly widely used in flexible frequency division transmission due to their advantages in directly realizing high-voltage, high-power AC-AC conversion and their superior circuit cost, operating efficiency, and reliability. Furthermore, M3Cs have strong application potential in areas such as large-capacity wind turbine control, asynchronous grid interconnection, solid-state transformers, and high-power, high-torque, low-speed motor drives. However, due to the complex topology and dual-fundamental-frequency electrical coupling characteristics of M3Cs, harmonic pollution will be introduced into flexible frequency division systems, affecting the safe and stable operation of the system; harmonic interaction and stability issues will become more complex and prominent. Therefore, in order to more effectively design decoupled control systems, evaluate dynamic / steady-state performance, and suppress harmonics, high-order dynamic modeling and harmonic coupling characteristic analysis of M3C converters are particularly important.

[0003] Establishing a high-order dynamic model of the M3C converter is a prerequisite for analyzing its grid-connected stability. Current research only considers its low-order model, neglecting higher-order harmonics (third order and above) and their dynamic processes. Two-dimensional sequence component symmetric analysis solves the phase sequence coupling problem caused by the complex topology of the M3C. Preliminary research shows that the complex nonlinear coupling process within the M3C, involving submodule capacitor voltage, arm current, and the control system, has a significant impact on its external characteristics. For converters like the M3C, which are based on submodule cascade technology, the multiple couplings between submodule capacitor voltage ripple and arm current harmonics significantly affect the dynamic characteristics of modular multilevel converters. However, at present, there is no perfect solution for modeling and analyzing the high-order dynamic characteristics of the M3C. For similar research scenarios involving the modeling of the internal dynamic processes of MMC converter stations, multi-harmonic linearization or Harmonic State-Space (HSS) is a common method for MMC modeling. This involves using Fourier series theory to decompose time-varying periodic quantities containing multiple harmonics into time-invariant series sequences, establishing discrete circuit models for each frequency component, and describing the multiplicative coupling process between components through frequency domain convolution. Harmonic matrices are used to describe the relationships between harmonic components, and the established model can characterize the complex internal dynamic behavior and multi-harmonic characteristics of the MMC. However, none of these methods can be directly applied to M3C modeling. M3C exhibits dual-fundamental-frequency characteristics, where AC electrical quantities from both grids are directly coupled in the M3C circuit, forming a ripple / harmonic spectrum extending infinitely outwards from the dual fundamental frequencies. Traditional methods, however, only address single-fundamental-frequency systems and are insufficient to model and describe the cross-coupling process of components under a dual-fundamental-frequency spectrum. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings and deficiencies of the prior art and provide a modular multilevel matrix converter modeling method based on the harmonic state space principle.

[0005] The objective of this invention is achieved through the following technical solution: A modular multilevel matrix converter modeling method based on the harmonic state-space principle includes the following steps: S1. Obtain the grid topology and component parameters of the M3C grid-connected system, and establish the state-space equations of the nine bridge arms of the M3C system circuit topology; S2. Based on the symmetry of the M3C circuit topology and the stable operation of the two-sided systems, establish the state-space equation of the single bridge arm of M3C. S3. Establish the state space model of the M3C control system, and establish the M3C state space model with closed-loop control according to the interface correspondence with the main circuit. S4. Using the dual fundamental frequency harmonic state-space method, a closed-loop control M3C harmonic state-space model and a small-signal model containing components of each order are established.

[0006] S5. Perform a two-dimensional inverse Fourier transform on each harmonic component obtained in step S4 to the time domain, and then compare it with the model built by MALAB. S6. Based on the established small signal model, perform small perturbation analysis.

[0007] Furthermore, the nine-bridge-arm circuit topology model established in S1 is as follows: (1) In the formula, E S and E L These represent the three-phase power supply voltages on the mains frequency side and the frequency division side, respectively. S ∈{ u, v, w}、 L ∈{ a, b, c}; I, V, M and V C These represent the bridge arm current, bridge arm voltage, modulation signal, and submodule capacitor voltage, respectively. L , R and C These represent the bridge arm resistance, bridge arm inductance, and submodule capacitance, respectively, which are inherent parameters of the system. U xy It is a 1×9 matrix of all ones. The state-space model of the nine-arm M3C can be obtained from equation (1): (2) (3) (4) (5) (6) Furthermore, the state-space model of the single-arm M3C of S2 is as follows: (7) (8) Equation (8) is a simplified description of equation (7). In the equation, L eq The equivalent impedance represents the current in the bridge arm. Its internal impedance varies depending on the current flow path, and can be categorized into the following cases: current flowing through a power frequency system or a frequency division system and M... 3 The C-arm forms a loop, and its internal impedance is equivalent to 1 / 3 of the arm impedance, i.e. L eq = L / 3; The current flows only within the bridge arm, i.e., the internal bridge arm circulating current component, and its impedance is equivalent to the bridge arm impedance. L eq = L The current forms a loop with the bridge arm through the zero-sequence network on both sides, and its internal impedance is equivalent to infinity, i.e. L eq = ∞.

[0008] Furthermore, the state-space model of the M3C control system of S3 is as follows: M 3 The C-type control system generally follows the classic architecture of the voltage source converter, which includes three stages: voltage / power outer loop, current inner loop, and modulation. (9) (10) (11) Equation (8) represents the outer loop component, where, K P_So and K I_So These are the PI gains of the outer loop controller on the power frequency side; K P_Lo and K I_Lo These represent the PI gain of the outer loop controller on the frequency divider side; superscript ref Represents the reference value of the corresponding variable; Q S , P L and Q L These represent the reactive power output from the power frequency side, the active power output from the frequency divider side, and the reactive power, respectively. Equation (9) is the inner current loop, where, v S ref ( t ) / v L ref ( t ), i S ref ( t ) / i L ref ( t )and i S ( t ) / i L ( tThese represent the output modulation signal of the inner current loop on the power frequency / frequency division side, the input of the outer voltage loop, and the feedback current on the main circuit side, respectively. K P_Si / K P_Li and K I_Si / K I_Li These are the proportional gain and integral gain of the current inner loop control loop, respectively. K P_C The proportional gain of the circulating current suppression controller; T αβ0 This is the Clack transformation equation. (10) is the modulation stage. Based on the interface correspondence between the main circuit and the control system, the state-space equation of the M3C control system can be obtained: (12) u ctrl ( t It is divided into three parts. u ctrl1 ( t ) represents the outer loop input command value; u ctrl2 ( t () represents the power frequency / division frequency side voltage; u ctrl3 ( t This refers to the grid-side current and bridge arm harmonic circulating current. It also allows for better integration with the main circuit system (i.e., using the modulation signal output by the control system as a time-varying matrix). A cqt Time-varying parameters; X cqt As input to the control system U ctrl3 Based on the mathematical relationship between the grid-side currents and the arm currents on both sides, and using the two-dimensional sequence component symmetry method, an M3C state-space model with closed-loop control can be established: (13) Furthermore, the dual fundamental harmonic state-space method of S4 is as follows: For a time-domain function composed of two fundamental frequency components x ( t This can be represented as a two-dimensional Fourier series: (14) In the formula, ω 1 and ω 2 represents the fundamental frequency of two independent AC systems, and satisfies ω 1 / 2 = 2π / T 1 / 2; k 1.k 2 represents the corresponding order. X (k1,k2) These are the time-varying two-dimensional Fourier coefficients in complex form, i.e., the Fourier coefficients under the dual-fundamental-frequency decomposition framework. Further expansion of equation (8) yields the harmonic state-space equation containing dual-fundamental-frequency harmonic components: (15) Transforming equation (9) from the time domain to the complex frequency domain, and considering the analysis of a dual-fundamental-frequency system... n During the first harmonic, the... n First harmonic ( k ∈ Z + )Include{( k 1, k 2)│| k 1|+| k 2|= n All components of}. Therefore, the expression needs to be modified. n Step cutoff, proceed n The matrix after truncation can be expressed as: (16) (17) (18) (19) In the formula, For [2( n -| j |)+1] ×[2( n -| k - j |)+1] order The double Toeplitz matrix centered at the center; k The state variable vector X k It can be represented as: (20) (twenty one) Substituting the mentioned dual-fundamental-frequency harmonic state modeling method into the established M3C state-space model containing a control loop yields the harmonic state-space model of the M3C: (twenty two) Further linearization at the stable operating point of the harmonic state-space model of M3C yields the small-signal model of M3C: (twenty three)

[0009] Compared with the prior art, the present invention has the following advantages and beneficial effects: 1. Based on the mathematical definition and algebraic operation properties of the two-dimensional Fourier transform, this invention proposes a state-space modeling method for dual fundamental frequency harmonics, realizing the frequency decomposition of dual fundamental frequency ripples / harmonics, and using double convolution in the frequency domain to model and describe the multiplicative coupling process between frequency components. This method can effectively solve the problem of high-order modeling of dual fundamental frequency systems and provides ideas for high-order dynamic modeling of M3C and other AC-AC direct conversion topologies.

[0010] 2. The greatest advantage of this invention lies in the modularity and scalability of the modeling process of the harmonic state-space modeling method. Through HSS, higher harmonic components can be conveniently included in the M3C model. Attached Figure Description

[0011] Figure 1 This is a flowchart of an M3C frequency converter modeling method based on dual fundamental frequency harmonic state space according to the present invention; Figure 2 This is the power grid structure according to an embodiment of the present invention; Figure 3 This is a closed-loop control block diagram according to an embodiment of the present invention; Figure 4 This is a comparison of the steady-state characteristics of the dynamic analytical model and the electromagnetic transient model; Figure 5 This is a small-signal model small-perturbation analysis diagram. Detailed Implementation

[0012] To better understand the technical solution of the present invention, the embodiments provided by the present invention are described in detail below with reference to the accompanying drawings, but the implementation of the present invention is not limited thereto. Example

[0013] like Figure 1 As shown in the figure, this embodiment of the M3C frequency converter modeling method based on dual fundamental frequency harmonic state space mainly includes the following steps: S1. Obtain the grid topology and component parameters of the M3C grid-connected system, and establish the state-space equations of the nine bridge arms of the M3C system circuit topology; S2. Based on the symmetry of the M3C circuit topology and the stable operation of the two-sided systems, establish the state-space equation of the single bridge arm of M3C. S3. Establish the state space model of the M3C control system, and establish the M3C state space model with closed-loop control according to the interface correspondence with the main circuit. S4. Using the dual fundamental frequency harmonic state-space method, a closed-loop control M3C harmonic state-space model and a small-signal model containing components of each order are established.

[0014] S5. Perform a two-dimensional inverse Fourier transform on each harmonic component obtained in step S4 to the time domain, and then compare it with the model built by MALAB. S6. Based on the established small signal model, perform small perturbation analysis.

[0015] In this embodiment, a dual-fundamental-frequency harmonic state-space modeling method is proposed for M3C grid-connected systems. For example... Figure 2 In this invention, the M3C topology adopts a three-phase x three-phase nine-arm structure. uvw Indicates the three-phase port on the power frequency side. abc This indicates the three-phase port on the frequency division side. The M3C consists of 9 arms, connecting the ports on both sides sequentially, forming a 3×3 matrix topology. Each arm consists of... n It consists of a full-bridge submodule and a bridge arm reactor cascaded together. Each full-bridge submodule can output 0 and ±. v C Three levels, therefore n Each cascaded module can output - nv C to nv C ,by v C For intervals of (2) n +1) level. Bridge arms are named according to the ports they connect to, such as bridge arms. au , bu , cw Furthermore, the nine-bridge-arm circuit topology model established in S1 is as follows: (twenty four) In the formula, E S and E L These represent the three-phase power supply voltages on the mains frequency side and the frequency division side, respectively. S ∈{ u, v, w}、 L ∈{ a, b, c}; I, V, M and V C These represent the bridge arm current, bridge arm voltage, modulation signal, and submodule capacitor voltage, respectively. L , R and C These represent the bridge arm resistance, bridge arm inductance, and submodule capacitance, respectively, which are inherent parameters of the system. U xy It is a 1×9 matrix of all ones. The state-space model of the nine-arm M3C can be obtained from equation (1): (25) (26) (27) (28) (29) Furthermore, the state-space model of the single-arm M3C of S2 is as follows: (30) (31) Equation (8) is a simplified description of equation (7).

[0016] in Figure 3 For M3C's control system, such as Figure 3 As shown, this illustrates the interface correspondence with the main circuit system and the coordinate transformation relationship. M 3 The C control system generally follows the classic architecture of a voltage source converter, comprising three stages: a voltage / power outer loop, a current inner loop, and modulation. Furthermore, the state-space model of the S3 M3C control system is as follows: (32) (33) (34) Equation (8) represents the outer loop component, where, K P_So and K I_So These are the PI gains of the outer loop controller on the power frequency side; K P_Lo and K I_Lo These represent the PI gain of the outer loop controller on the frequency divider side; superscript ref Represents the reference value of the corresponding variable; Q S , P L and Q L These represent the reactive power output from the power frequency side, the active power output from the frequency divider side, and the reactive power, respectively. Equation (9) is the inner current loop, where, v S ref ( t ) / v L ref ( t ), i S ref ( t ) / i L ref ( t )and i S (t ) / i L ( t These represent the output modulation signal of the inner current loop on the power frequency / frequency division side, the input of the outer voltage loop, and the feedback current on the main circuit side, respectively. K P_Si / K P_Li and K I_Si / K I_Li These are the proportional gain and integral gain of the current inner loop control loop, respectively. K P_C The proportional gain of the circulating current suppression controller; T αβ0 This is the Clack transformation equation. (10) is the modulation stage. Based on the interface correspondence between the main circuit and the control system, the state-space equation of the M3C control system can be obtained: (35) u ctrl ( t It is divided into three parts. u ctrl1 ( t ) represents the outer loop input command value; u ctrl2 ( t () represents the power frequency / division frequency side voltage; u ctrl3 ( t This refers to the grid-side current and bridge arm harmonic circulating current. It also allows for better integration with the main circuit system (i.e., using the modulation signal output by the control system as a time-varying matrix). A cqt Time-varying parameters; X cqt As input to the control system U ctrl3 Based on the mathematical relationship between the grid-side currents and the arm currents on both sides, and using the two-dimensional sequence component symmetry method, an M3C state-space model with closed-loop control can be established: (36) Furthermore, the dual fundamental harmonic state-space method of S4 is as follows: For a time-domain function composed of two fundamental frequency components x ( t This can be represented as a two-dimensional Fourier series: (37) In the formula, ω 1 and ω 2 represents the fundamental frequency of two independent AC systems, and satisfies ω 1 / 2 = 2π / T 1 / 2; k 1. k 2 represents the corresponding order. X (k1,k2) These are the time-varying two-dimensional Fourier coefficients in complex form, i.e., the Fourier coefficients under the dual-fundamental-frequency decomposition framework. Further expansion of equation (8) yields the harmonic state-space equation containing dual-fundamental-frequency harmonic components: (38) Transforming equation (9) from the time domain to the complex frequency domain, and considering the analysis of a dual-fundamental-frequency system... n During the first harmonic, the... n First harmonic ( k ∈ Z + )Include{( k 1, k 2)│| k 1|+| k 2|= n All components of}. Therefore, the expression needs to be modified. n Step cutoff, proceed n The matrix after truncation can be expressed as: (39) (40) (41) (42) In the formula, For [2( n -| j |)+1] ×[2( n -| k - j |)+1] order The double Toeplitz matrix centered at the center; k The state variable vector X k It can be represented as: (43) (44) Substituting the mentioned dual-fundamental-frequency harmonic state modeling method into the established M3C state-space model containing a control loop yields the harmonic state-space model of the M3C: (45) Further linearization at the stable operating point of the harmonic state-space model of M3C yields the small-signal model of M3C: (46) The established M3C harmonic state-space model and small-signal model are further compared with the Matlab simulation model; for example... Figure 4 , Figure 5 As shown; Figure 4 The results show a comparison between the HSS model and the small-signal model with electromagnetic transient simulation outputs under different cutoff orders. The figures illustrate the capacitor voltage of the submodules in the established model. v Cau and bridge arm current i au The results show a high degree of agreement with the electromagnetic transient simulation output, and the agreement becomes even more accurate with increasing model order. The above analysis demonstrates that the proposed improved harmonic state-space modeling method can fully reflect the characteristics of electromagnetic transients. 3 The dynamic characteristics within C are applicable to M. 3 Modeling of the dual-fundamental-frequency electrical coupling characteristics of AC direct frequency converters of type C. To further quantitatively verify the correctness of the established harmonic transmission process, this section injects positive and negative sequence disturbance voltages with a frequency of 40Hz and an amplitude of 8% of the frequency division voltage at the stable operating point of the simulation model, harmonic state-space model and small-signal model, respectively. The dynamic characteristics of the system are shown in Figure (5). It can be seen from the figure that after introducing the disturbance voltage, the simulation waveform basically matches the established model waveform.

[0017] In summary, the curves of the M3C harmonic state-space model, small-signal model, and electromagnetic transient simulation model show a high degree of agreement, verifying the correctness of the dual-fundamental-frequency harmonic state-space modeling method. This invention is particularly applicable to grid-connected systems of frequency converters with dual-fundamental-frequency characteristics, such as M3C. Compared with existing methods, the proposed method can effectively reflect the high-order dynamic characteristics inside the M3C and establishes a high-precision high-order dynamic harmonic state-space model.

[0018] In the description of this specification, references to terms such as "an embodiment" and "example" refer to specific features, structures, materials, or characteristics incorporated in that embodiment or example, which are included in at least one embodiment or example of the present invention. The basic principles, main features, and advantages of the present invention have been shown and described above. The implementation of the present invention is not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention should be considered equivalent substitutions and are included within the scope of protection of the present invention.

Claims

1. A modular multilevel matrix converter modeling method based on the harmonic state-space principle, characterized in that... This includes the following steps: S1. Obtain the grid topology and component parameters of the M3C grid-connected system, and establish the state-space equation of the nine-bridge arm circuit topology of the M3C system. S2. Based on the symmetry of the M3C circuit topology and the stable operation of the two-sided systems, establish the state-space equation of the single bridge arm of M3C. S3. Establish the state space model of the M3C control system, and establish the M3C state space model with closed-loop control according to the interface correspondence with the main circuit. S4. Introduce two-dimensional Fourier transform into the harmonic state-space modeling method, and use the dual fundamental frequency harmonic state-space method to establish a closed-loop control M3C harmonic state-space model and small-signal model containing components of each order. S5. Perform a two-dimensional inverse Fourier transform on each harmonic component obtained in step S4 to the time domain, and then compare it with the model built in MATLAB. S6. Based on the established small signal model, perform small perturbation analysis.

2. The modular multilevel matrix converter modeling method based on the harmonic state-space principle according to claim 1, characterized in that, The state-space model of the M3C nine-bridge arm in step S1 is as follows: ， Among them, E S and E L These represent the three-phase power supply voltage matrices on the mains frequency side and the frequency division side, respectively; I, V, M, and V C These represent the nine-arm current matrix, the nine-arm voltage matrix, the nine-arm modulation signal matrix, and the nine-arm submodule capacitor-voltage matrix, respectively. v n The neutral point voltage of the system, i.e. the voltage difference between the power frequency system and the frequency division system, is defined as the average of the sum of the voltages of the nine bridge arms. N , L , R and C Representing the number of submodules respectively Bridge arm inductance, bridge arm resistance, and submodule capacitance are inherent parameters of the system; U is a 1×9 matrix of all ones; variables x and y These represent the power frequency side and the frequency divider side ports, respectively. y ∈{ u,v,w }, u,v,w Represents the three-phase port on the power frequency side. x ∈{ a,b,c }, a,b, c The bridge arm represents the three-phase port on the frequency division side; the bridge arm is named after the two ports it connects to. xy ; v xy Represents bridge arm xy The bridge arm voltage; ， in, v xy , i xy , m xy and v Cxy bridge arms xy Bridge arm voltage, bridge arm current, modulation signal, and submodule capacitor voltage; e x , e y These are the corresponding phase voltages on the frequency division side and the power frequency side, respectively. uvw Indicates the three-phase port on the power frequency side. abc This indicates the three-phase port on the frequency division side; the M3C contains 9 bridge arms, connecting the ports on both sides sequentially, forming a 3×3 matrix topology; each bridge arm consists of... n It consists of a full-bridge submodule and a bridge arm reactor cascaded together. Each full-bridge submodule can output 0 and ±. v C Three levels, therefore n Each cascaded module can output - nv C to nv C ,by v C For intervals of (2) n +1) level; the bridge arm is named through the ports on both sides to which it is connected.

3. The modular multilevel matrix converter modeling method based on the harmonic state-space principle according to claim 2, characterized in that, The state-space model of the M3C single-bridge arm in step S2 is as follows: ， in, v Cau , i au and m au Representing bridge arms au The submodule capacitor voltage, bridge arm current, and modulation signal; e u and e a Representing the power frequency side u Phase voltage and frequency divider side a Phase voltage; R and L eq These represent the resistance and equivalent inductance of the bridge arm, respectively.

4. The modular multilevel matrix converter modeling method based on the harmonic state-space principle according to claim 3, characterized in that, The equivalent inductance of a single bridge arm proposed in step S2 varies depending on the current flow path. It can be categorized into the following cases: When the current flows through a power frequency system or a frequency divider system and forms a loop with the M3C bridge arm, its internal inductance is equivalent to 1 / 3 of the bridge arm inductance. L eq = L / 3; The current flows only within the bridge arm, i.e., the internal bridge arm circulating current component, and its inductance is equivalent to the bridge arm inductance. L eq = L The current forms a loop with the bridge arm through the zero-sequence network on both sides, and its internal inductance is equivalent to infinity, i.e. L eq =∞.

5. The modular multilevel matrix converter modeling method based on the harmonic state-space principle according to claim 1, characterized in that, The basic principle of the dual fundamental harmonic state-space method in step S4 is as follows: it introduces the two-dimensional Fourier transform into the harmonic state-space modeling method. For a time-domain function composed of two fundamental frequency components x ( t This can be represented as a two-dimensional Fourier series: ， in, ω 1 and ω 2 represents the fundamental angular frequency of two independent AC systems, and satisfies ω 1 / 2 = (2π / T 1) / 2=π / T 1. T1 is the period; k 1 and k 2 represents the coupling coefficients of the two independent AC systems corresponding to the two fundamental frequency components within the frequency decomposition framework of the two fundamental frequencies, respectively. k 1|+| k 2| represents the harmonic order corresponding to the two fundamental frequency components; X (k1,k2) These are time-varying two-dimensional Fourier coefficients in complex form, i.e., Fourier coefficients under the dual-fundamental-frequency decomposition framework.

6. The modular multilevel matrix converter modeling method based on the harmonic state-space principle according to claim 5, characterized in that, The dual-fundamental harmonic state-space time-domain model in step S4: ， in, s Represents the Laplace operator; n To represent the truncation order of the state-space time-domain model of the two fundamental harmonics, i.e., the model covers ≤ n Subharmonics; A, B, C, and D are the state transition matrices composed of system topology information under the dual-fundamental-frequency decomposition framework; x, u, and y are the state variable matrix, input matrix, and output matrix under the dual-fundamental-frequency decomposition framework, respectively.

7. The modular multilevel matrix converter modeling method based on the harmonic state-space principle according to claim 6, characterized in that, Considerations in step S4 n State-space frequency domain model of a two-fundamental harmonic truncated by order: ， Among them, superscript ntr express n State-space truncation model of two fundamental harmonics, subscript df Represents a dual-fundamental-frequency harmonic state-space model; X, U, Y, and N are the state variable matrix, input matrix, output matrix, and diagonal coefficient matrix under the dual-fundamental-frequency decomposition framework, respectively; the specific mathematical forms of each matrix are as follows: ， Among them, matrix elements Representative of the power frequency side k The first-order principal state variable matrix has the following mathematical form: , where matrix elements Represents angular frequency k 1 ω S + k 2 ω L of( k 1, k 2) The state variable matrix of the subharmonics; ω S and ω L These represent the fundamental angular frequencies of the power frequency side and the frequency divider side, respectively. ， ， Among them, matrix elements This represents Toeplitz's efforts, primarily focused on the power frequency side. k Order state transition matrix, For [2( n -| j |)+1]×[2( n -| k - j |)+1] order The double Toeplitz matrix centered at point has the following mathematical form: ， Among them, matrix elements Representative with ( k 1, k 2) In a double Toeplitz matrix with order as the primary expansion, ( k The state transition matrix of the 1st and 0th harmonics.

8. The modular multilevel matrix converter modeling method based on the harmonic state-space principle according to claim 7, characterized in that, Considerations in step S4 n A small-signal state-space model of M3C dual fundamental harmonics with closed-loop control and truncated order: ， Among them, superscript S Indicates the steady-state value of the corresponding variable; subscript cqt Represents the dual fundamental harmonic state-space model of the M3C main circuit; subscript ctrl Represents a control system model; X ctrl U represents the state variables of the outer / inner loop integral elements selected under the dual-fundamental-frequency decomposition framework; ctrl This represents the input variable of the control system under the dual-fundamental-frequency decomposition framework; to connect with the M3C main circuit model, the above equations are used to transform U... ctrl Divided into three parts: U ctrl1 U represents the reference value for the submodule capacitor voltage and grid-side power. ctrl2 U represents the grid-side voltage. ctrl3 Representing the bridge arm current and submodule capacitor voltage, i.e., the state variables of the main circuit model; correspondingly, the coefficient matrix B ctrl and D ctrl According to U ctrl The partitioning method divides it into three parts; W is the input matrix corresponding to the modulation signal.

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