A method of DRU steady-state harmonic analysis

Abstract

The application discloses a DRU steady-state harmonic analysis method, and belongs to the technical field of high-voltage direct current (HVDC) power transmission. The method takes a six-pulse DRU as a basic object, constructs a bridge arm current switching function and a voltage switching function, and accurately represents AC / DC current distribution and voltage synthesis mechanism. Double switching function Fourier coefficients are solved, a Toeplitz convolution matrix is constructed to convert time domain periodic time-varying relationship into harmonic domain convolution mapping, and commutation overlap angle is solved by simultaneously solving AC / DC network equations and closing loop. The method can be directly extended to a 6N-pulse rectifier structure. The method discards traditional commutation overlap angle fixed value simplification, completely considers natural commutation, commutation overlap and multi-harmonic coupling characteristics, has clear modeling physical mechanism and high steady-state harmonic calculation precision, can effectively adapt to multi-pulse rectifier topology, and provides core theory and calculation support for harmonic analysis, parameter optimization and stable control of the DRU-HVDC system.

Description

Technical Field

[0001] This invention relates to the field of high voltage direct current transmission technology, and specifically to a DRU steady-state harmonic analysis method. Background Technology

[0002] High-voltage direct current (HVDC) transmission, as a crucial technology for large-capacity, long-distance power transmission, has broad application prospects in renewable energy grid integration and long-distance power transmission scenarios. Compared with traditional controlled converter schemes, uncontrolled diode rectifier units (DRUs) feature simpler structures, fewer power devices, and lower control system complexity, which can reduce equipment investment, maintenance difficulty, and failure risks to a certain extent, while improving the long-term reliability of the system. Therefore, DRU-HVDC has gradually become an important research direction in the field of DC transmission in recent years.

[0003] However, the simplified structure of the DRU does not reduce the complexity of its internal bridge mechanism. As a typical natural commutation device, the conduction sequence of the DRU's bridge arms is jointly determined by the instantaneous phase voltage on the AC side, the interphase commutation voltage, the leakage inductance voltage drop, and the DC side current state. When commutation overlap, AC side harmonic injection, and DC side harmonic feedback coexist, the conduction interval, output voltage formation, and phase current distribution all exhibit significant periodic time-varying and multi-frequency coupling characteristics. Therefore, the steady-state harmonic problem of the DRU is essentially a periodic time-varying nonlinear problem shaped by natural commutation, network coupling, and multi-frequency interaction.

[0004] In recent years, significant progress has been made in research both domestically and internationally regarding the operation control, impedance characteristics, fault ride-through, and harmonic propagation of DRU-HVDC systems. Although most related studies are based on offshore wind power transmission scenarios, existing literature has revealed the close relationship between the AC voltage establishment conditions of DRU systems, system stability, and harmonic coupling from the perspectives of operating mechanisms under fault conditions, DRU-side control coordination, impedance modeling, filtering, and harmonic propagation. These studies indicate that the research focus on DRU-related issues is gradually shifting from verifying system operability to elucidating the formation mechanisms of electrical behavior under specific harmonic and commutation conditions.

[0005] Against this backdrop, the core issue in DRU steady-state harmonic research is how to establish an analytical model that retains the physical essence of natural commutation while uniformly handling multiple harmonics, voltage-current coupling, and multi-pulse propagation. Remaining at the level of average value approximation or partial simulation not only makes it difficult to theoretically reveal the intrinsic relationship between conduction switching and harmonic propagation, but also hinders subsequent parameter sensitivity analysis, frequency domain mechanism explanation, and the generalization of the 6N pulsation structure. Summary of the Invention

[0006] To address the above problems, this invention proposes a DRU steady-state harmonic analysis method, which includes the following steps: S1. Determine the modeling basis and harmonic representation: Taking the six-pulse DRU rectifier bridge as the core object, set the three-phase leakage inductance to be symmetrical and the parameters to be consistent, take the electrical angle as the unified independent variable and the A-phase fundamental phase voltage as the phase reference; expand the AC side voltage and DC side voltage / current into Fourier series, select the harmonic cutoff order, and complete the harmonic domain discretization of the periodic quantity.

[0007] Specifically, the angle of power extraction To unify the independent variables and the phase reference, this invention takes the fundamental phase voltage of phase A as the sinusoidal reference. Correspondingly, the fundamental voltages of phases B and C lag and lead phase A by 120°, respectively. Subsequent construction of the switching function, derivation of Fourier coefficients, and expression of harmonic phases are all based on the above phase definition. For any 2π periodic quantity... A Fourier expansion yields (1) in, The corresponding harmonic complex coefficients are given. Therefore, the phase voltages on the AC side, the DC side voltage, and the DC side current can be expressed as shown in formula (2): (2) in, The instantaneous voltage of the m-th phase is expressed as a function of the electrical angle θ; Let the complex coefficients of the kth harmonic be the two-sided Fourier series expansion of the m-th phase voltage. The DC-side port voltage is expressed as a function of the electrical angle θ; The complex coefficients of the kth harmonic in the two-sided Fourier series expansion of the DC-side port voltage; The DC-side terminal current is expressed as a function of the electrical angle θ; The complex coefficients of the kth harmonic in the two-sided Fourier series expansion of the DC-side port current.

[0008] S2. Establish the bridge arm current switching function: Combining the natural commutation mechanism of DRU, divide the non-commutation and commutation intervals. Based on the leakage inductance voltage drop and current continuity constraints, construct a piecewise continuous current switching function to accurately depict the distribution law of DC side current to each AC phase and the current transfer process in the commutation interval.

[0009] Specifically, given the conduction state within the bridge, the AC-side phase current can be considered as the result of the DC-side current being distributed through the bridge arms. Therefore, the first... Phase bridge arm current switching function Therefore, the first The phase current can be expressed as shown in formula (3).

[0010] (3) in, The commutation overlap angle is given. This expression unifies the in-bridge conduction sequence, commutation boundary, and two-phase current transfer law into a single periodic time-varying function. Then, the mapping relationship between DC side current harmonics and AC side phase current harmonics can be expressed within the same framework.

[0011] During the non-commutation region, the rectifier bridge is in a stable conducting state, and the common anode diode is conducting at this time, denoted as . The conduction of a common cathode diode is denoted as... The diode being turned off is denoted as During the commutation interval, adjacent phases conduct together due to leakage inductance, and the DC current continuously transfers between the outgoing and incoming phases. The ternary function is transformed into a continuous piecewise function.

[0012] At this point, the establishment of the current switching function essentially boils down to two issues: first, how the start and end boundaries of the commutation interval are determined by the natural commutation point and the overlap angle; and second, how the two-phase currents within the commutation interval can complete continuous transfer under the constraint of leakage inductance.

[0013] Standard commutation process For example, if the commutation start angle is Then in the interval Satisfy the current continuity relationship: (4) Meanwhile, the voltage difference across the two-phase leakage inductance is the direct driving factor determining the current transfer rate; regarding the influence of DC-side current fluctuations, this invention further incorporates closed-loop solutions into the subsequent unified frequency domain steady-state model. Therefore, the arm current distribution within the commutation interval is not a pre-defined empirical function, but a constrained response determined by both voltage drive and current continuity conditions. This invention establishes the current switching function within the commutation segment based on this physical fact, thereby avoiding simplifying the commutation process into an instantaneous jump or a fixed-slope transition.

[0014] To establish the basic analytical structure of the current switching function, this invention first constructs the basic function form based on the standard commutation interval; during this process, the higher-order corrections to the function shape caused by the DC-side current fluctuations within the commutation segment are not explicitly expanded. Subsequently, in the unified frequency domain steady-state model, the DC-side current fluctuations are included as harmonic variables in the closed-loop solution, thereby providing a consistent description of the commutation process and the steady-state harmonic response. Based on this, the piecewise expression of the bridge arm current switching function within one power frequency cycle can be further obtained. Taking phase A as an example, its typical form is shown in formula (5).

[0015] According to formula (5), the typical waveform of the bridge arm current switching function within one power frequency cycle is plotted. In the non-commutation interval, the function maintains a piecewise constant value characteristic; while in the commutation interval, due to the simultaneous conduction of two adjacent phases, the function transitions from a constant value segment to a continuously changing segment. This waveform characteristic indicates that the current switching function is not a simple sign switching function, but a periodic time-varying function containing commutation boundary information and current transfer law.

[0016] Structurally, the six-pulse rectifier bridge operates every [time period missing]. A single commutation occurs, and the commutation intervals are mathematically similar, differing only in phase position. Therefore, after obtaining the function structure within a standard interval, the remaining intervals can be obtained through periodic extension and phase shift. Further, considering the three-phase symmetry relationship... It can be organized into a set of periodic functions that are phase-shifted to each other. This expression not only preserves the physical meaning of current distribution during commutation, but also provides a unified functional basis for subsequent Fourier expansion and harmonic domain modeling.

[0017] (5) As shown in formula (5), the A-phase current switching function is expressed in electrical angle. and commutation overlap angle The independent variable describes the proportional relationship between the instantaneous current of phase A on the AC side and the current on the DC side.

[0018] The rectifier operates in different states within one power frequency cycle: In the fully conducting range, the switching function takes a constant value of ±1, indicating that phase A bridge arm is fully conducting and the current is equal to the DC side current; in the fully turning-off range, the switching function takes 0, indicating that phase A bridge arm is fully turning off and the current is 0; in the commutation transition range, based on the assumption of DC side pulsation, it can be deduced that the switching function changes smoothly according to a cosine law, reflecting the actual physical process that the current switching between the two phase bridge arms cannot be completed instantaneously due to the presence of AC side inductance.

[0019] S3. Establish the bridge arm voltage switching function: Define the bridge arm equivalent voltage after deducting the leakage inductance voltage drop, construct a three-valued piecewise voltage switching function, standardize the conduction boundary treatment method of the commutation interval, and clearly characterize the synthesis mapping relationship of the AC side equivalent voltage to the DC side output voltage.

[0020] Specifically, the current switching function is used to characterize the distribution relationship of DC current to each phase of AC, while the voltage switching function is used to characterize the mapping relationship of the equivalent voltage of each bridge arm to the DC output voltage.

[0021] The construction process of the current switching function is as follows: In a natural commutation rectifier bridge, the voltage actually involved in forming the DC side voltage is not the original phase voltage on the AC side itself, but rather the bridge arm equivalent voltage after subtracting the corresponding leakage inductance voltage drop from the phase voltage. Therefore, the first... Phase bridge arm equivalent voltage for (6) in, For the exchange side Phase-to-phase voltage, To address the leakage inductance during bridge arm commutation, For the first Phase bridge arm current. Combining this with the current switching function established earlier, we have... (7) Therefore, the bridge arm equivalent voltage can be further expressed as (8) Based on this, the first Phase bridge arm voltage switching function Then the DC-side output voltage can be expressed as the result of the equivalent voltages of each phase bridge arm after selection and combination by the switching function, i.e. (9) Equation (9) shows that the DC-side output voltage is not a simple replication of a fixed line voltage, but rather the result of selecting and combining the equivalent voltages of each phase arm of the rectifier bridge under a given conduction state. Therefore, as long as the equivalent voltage of the bridge arm and the switching function of the bridge arm voltage are accurately established, the combined effect of the AC-side phase voltage and the leakage inductance voltage drop within the bridge on the DC-side output voltage can be expressed within the same framework.

[0022] In the non-commutation region, the rectifier bridge is in a stable conducting state. According to the definition of this invention, when the common anode diode is turned on, the corresponding voltage switching function takes the value... When the common cathode diode is turned on, the corresponding voltage switching function takes the value of The value when the diode is turned off is [value]. Therefore, within the stable conduction range, the voltage switching function exhibits a value of , or A three-valued piecewise function.

[0023] The voltage switching function describes the equivalent contribution of the bridge arm to the DC-side output voltage, rather than the continuous transfer process of the bridge arm current within the commutation interval. Therefore, within the commutation interval, this invention processes the bridge arm voltage switching function as follows: For the input diode, from the time it meets the conduction condition until it actually completes conduction, the corresponding voltage switching function is still taken as... For off-phase diodes, from the start of the turn-off preparation until complete turn-off, the corresponding voltage switching function maintains its original conduction value, i.e., the common anode arm remains unchanged. common cathode bridge arm remains This approach reflects the modeling idea that the DC-side output voltage during commutation is still mainly determined by the original conducting bridge arm. On the other hand, under the condition of finite harmonic cutoff, it also helps to reduce unnecessary jumps of the function at the commutation boundary, weaken the Gibbs oscillations in the Fourier series approximation, thereby improving the accuracy of the frequency domain expression of the voltage switching function and the calculation accuracy of the subsequent steady-state solution.

[0024] To establish the basic analytical structure of the bridge arm voltage switching function, this invention takes a six-pulse DRU rectifier bridge as an example, and takes the A-phase bridge arm as an example to give its typical piecewise expression within one power frequency cycle. According to the aforementioned definition, the piecewise expression of the A-phase bridge arm voltage switching function is as shown in formula (10): (10) According to formula (10), the typical waveform of the voltage switching function of phase A bridge arm within one power frequency cycle can be plotted. The voltage switching function maintains the three-valued piecewise characteristics as a whole. Its main difference from the current switching function is not reflected in the continuous change within the commutation interval, but in the treatment of the conduction boundary. This treatment makes the voltage switching function have a more regular time-domain structure under finite harmonic cutoff conditions, which is beneficial to the accurate calculation of its Fourier coefficients and subsequent frequency domain modeling.

[0025] Structurally, the voltage switching functions of each phase arm of the six-pulse rectifier bridge also exhibit periodicity and phase-shift symmetry in mathematical form. Therefore, after obtaining the basic expression of phase A, the voltage switching functions of the remaining phase arms can be obtained through periodic extension and phase shift. Further combining this with the three-phase symmetry... This can be organized as a set of periodic functions with phase shifts. This expression provides a unified foundation for the subsequent Fourier expansion of the bridge arm voltage switching function and the harmonic domain modeling of the DC-side output voltage.

[0026] S4. Solving for the Fourier coefficients of the dual-switching function: By utilizing the periodic symmetry and conjugate characteristics of the rectifier bridge, piecewise integration is performed on the two types of switching functions to derive the Fourier coefficients in a unified analytical form, providing a parameter basis for harmonic domain mapping.

[0027] Specifically, for any periodic function , its first The subharmonic coefficient is defined as follows: (11) in, These are harmonic orders. Due to the bridge arm current switching function... With bridge arm voltage switching function Both have been written as piecewise functions for the conduction and commutation intervals, so their Fourier coefficients can be obtained by piecewise integration based on the expressions for each interval. For the non-commutation interval, the function is usually represented as a constant segment or a sign segment, and its integration result is mainly determined by the position of the interval boundary; for the commutation interval, the current switching function corresponds to the overlap angle. The continuous transfer function corresponds to the delayed switching segmented structure with overlap angle, while the voltage switching function corresponds to the continuous transfer function. Therefore, the Fourier coefficients of both will explicitly reflect the influence of commutation overlap angle and conduction boundary on the spectrum distribution.

[0028] The Fourier coefficients of the switching function can be obtained using the inherent periodicity and symmetry of the rectifier bridge. First, the conduction pattern of the six-pulse rectifier bridge strictly repeats within the power frequency cycle; therefore, processing the reference phase switching function within one cycle yields its complete spectral representation. Second, the differences between the bridge arms are mainly phase shifts; thus, given the Fourier coefficients of the reference phase, the coefficients of the other phases can be directly generated using phase shift factors without repeated piecewise integration. Third, since the switching functions are all real-valued periodic functions, their positive and negative harmonic coefficients satisfy a conjugate symmetry relationship, further simplifying both analytical expression and numerical implementation.

[0029] Therefore, after completing the time-domain construction of the two types of switching functions, this invention does not immediately proceed to numerical solution, but first establishes a unified analytical structure for their Fourier coefficients.

[0030] Based on this, the conduction mechanism, which originally exhibits a periodic time-varying product relationship within the bridge, can be further written into a linear mapping form in the harmonic domain using convolution relationships and the Toeplitz matrix, thus laying the foundation for the subsequent establishment of the HSS–HBM joint solution framework.

[0031] S5. Construct the Toeplitz convolution matrix: Based on the Fourier coefficients of the switching function, construct a Toeplitz type convolution matrix to transform the time-varying product relationship of the internal time-domain period of the rectifier bridge into a linear convolution mapping relationship that can be solved in the harmonic domain.

[0032] Specifically, the communication side Phase current can be expressed as the product of the bridge arm current switching function and the DC side current, i.e. (12) Will , and Expanding each harmonic into a Fourier series and comparing the harmonic coefficients of the same order, we can obtain... (13) Similarly, from the relationship between the bridge arm voltage switching function and the bridge arm equivalent voltage, we have (14) Therefore, in the harmonic domain it can be written as (15) In the formula, Indicates the communication side The first phase current Subharmonic coefficient The first character representing the DC side current Subharmonic coefficient The first value representing the DC-side output voltage Subharmonic coefficient Indicates the first The first phase bridge arm equivalent voltage Subharmonic coefficient; and The first and second arms of the bridge arm current switching function and the second arm voltage switching function are respectively the first and second arms of the bridge arm current switching function. Fourth Fourier coefficients.

[0033] As can be seen from formulas (13) and (15), the first side of the AC... The first phase current The second harmonic is not caused by the DC-side current. The subharmonics are determined independently, but are composed of the various harmonics on the DC side. In the spectrum of current switching function The weighted superposition is formed under the influence of the voltage; similarly, the first voltage of the DC-side output voltage is formed by the weighted superposition of the voltages. The subharmonics are not determined solely by the same-order harmonics of the equivalent voltage of a single phase arm, but rather by the various harmonics of the equivalent voltage of each phase arm. In the voltage switching function spectrum They are formed through joint coupling under the influence of each other.

[0034] Therefore, the essence expressed by formulas (13) and (15) is that the multiplication relationship in the time domain is transformed into a convolution relationship in the harmonic domain. In other words, the switching function does not simply retain or cut off a certain harmonic, but rather recouples and maps different frequency components of the input signal into the output harmonic. It is this frequency coupling effect introduced by the switching function that enables the switching of the rectifier bridge to manifest as an explicit interaction between multiple frequencies in the harmonic domain.

[0035] To express the above convolutional relationship in a compact matrix form, this invention introduces a Toeplitz-type convolution matrix composed of the Fourier coefficients of the switching functions. Let the highest truncated harmonic order of the system be... The range of values ​​for the harmonic index is: The dimension of the corresponding harmonic column vector is For any switching function Let its Fourier coefficient be . Define the corresponding Toeplitz convolution matrix. If we denote the matrix row and column indices as... Then its elements can be defined as (16) As can be seen from equation (16), the matrix elements are only related to the difference between the row and column indices. Therefore, the matrix takes constant values ​​along each subdiagonal, exhibiting a typical Toeplitz structure. This structure precisely corresponds to the mathematical characteristic of "harmonic order shift" in convolution summation. For the bridge arm current switching function and bridge arm voltage switching function described in this invention, the above definitions respectively correspond to the... and Construct the Toeplitz product matrix.

[0036] Therefore, formula (13) can be further written as (17) in, and They represent the first on the communication side. The phase current and DC side current are stacked according to their harmonic indices to obtain a finite-dimensional column vector. Similarly, formula (15) can be written as (18) in, and Representing the DC-side output voltage and the first Harmonic column vector of the equivalent voltage of the phase bridge arm.

[0037] Furthermore, based on the equivalent voltage definition of the bridge arm mentioned above... (19) It can be seen that it satisfies the following in the harmonic domain. (20) In the formula, For the exchange side Harmonic column vectors of phase voltages Let be the harmonic domain differential operator matrix. Combining with formula (17), we can further obtain... ;(twenty one) Substituting formula (21) into formula (18), the DC-side voltage harmonic mapping relationship under the combined action of the bridge arm voltage switching function and the bridge arm current switching function can be obtained. Thus, the conduction mechanism, which originally exhibited a periodic time-varying product relationship inside the rectifier bridge, has been uniformly transformed into a linear mapping structure in the harmonic domain.

[0038] The above transformation has two implications. First, the in-bridge conduction switching is elevated to a matrix operator that can directly participate in the system's simultaneous solution, thus avoiding repeated returns to the time domain for piecewise determination during the harmonic solution stage. Second, the coupling paths between multiple frequencies are explicitly given by the convolution structure, and the harmonic components of the AC side, DC side, and the switching function itself can all be organized in a unified dimension. Therefore, the Toeplitz convolution matrix is ​​not simply a notational compression, but rather the core mathematical carrier through which HSS transforms the in-bridge time-domain conduction mechanism into a harmonic-domain solvable equation.

[0039] S6. Establish the HSS-HBM joint solution framework: Transform the periodic time-varying system into a harmonic domain matrix model through the harmonic state space (HSS), apply steady-state frequency domain balance constraints by combining the harmonic balance method (HBM), and combine the AC and DC side network equations to form a unified harmonic domain algebraic solution system.

[0040] Specifically, let's set The state vector is composed of the three-phase current harmonics on the AC side, the voltage harmonics on the DC side, the current harmonics on the DC side, and other variables to be determined within the bridge. Given an input vector consisting of AC side voltage harmonics and other external excitations, then at a given overlap angle... Under these conditions, the overall harmonic domain equations of the rectifier bridge and its AC / DC side network can be uniformly organized as follows: ;(twenty two) in, It is composed of the Toeplitz convolution matrix corresponding to the dual-switch function, the harmonic domain differential operator matrix, the constraint relationship between harmonic variables, and the AC and DC port equations. For a given overlap angle... Equation (22) can be regarded as a system of algebraic equations in the finite-dimensional harmonic domain; however, due to These are not externally given independent parameters, but rather internal variables determined by the natural commutation process. It will change with the commutation state and form a coupling relationship with the steady-state quantity of the harmonic to be determined.

[0041] From the perspective of methodological division of labor, the role of HSS is to elevate the periodic time-varying conduction relation originally defined in the time domain to a matrix operator in the harmonic domain, enabling the frequency coupling caused by the switching of bridge arms to be explicitly incorporated into the system equations. The role of HBM is to simultaneously apply steady-state equilibrium constraints to each retained harmonic component, thereby transforming the continuous-time periodic steady-state problem into a system of finite-dimensional algebraic equations that can be directly solved. In other words, HSS is responsible for establishing the mathematical structure of "how the conduction relation enters the harmonic domain," while HBM is responsible for applying the solution conditions of "what constraints each order of harmonics must simultaneously satisfy in steady state." In the framework of this invention, the two play complementary roles of modeling and solving, respectively.

[0042] Compared to purely time-domain piecewise analysis methods, this joint framework avoids the derivation and solution complexity caused by repeatedly switching time-domain intervals under conditions of high-order harmonics, multi-bridge coupling, and complex commutation. Compared to simplified models that only retain average quantities or fundamental components, it can explicitly preserve key influencing factors such as commutation overlap, multiple harmonics, and frequency coupling. Therefore, for the DRU natural commutation steady-state harmonic problem studied in this invention, the combination of HSS and HBM constitutes the core methodological foundation for unified analytical modeling and steady-state solution.

[0043] S7. Closed-loop solution for commutation overlap angle: Based on leakage inductance voltage drop, interphase commutation voltage, and current continuity constraints, a coupled equation for commutation overlap angle and AC / DC harmonic variables is established, and the joint solution of overlap angle and harmonic variables is completed iteratively.

[0044] Specifically, the angle of overlap The solution can be compressed into a single commutation boundary condition. Taking the standard commutation process A→B as an example, in Inside ;(twenty three) ;(twenty four) (25) (26) Therefore, the commutation condition satisfied by the overlap angle can be obtained as shown in formula (27).

[0045] (27) If the phase-to-phase voltage and DC current are further expanded into Fourier series, the above equation can be written as shown in formula (28).

[0046] (28) This formula directly shows that, It is no longer an externally supplied parameter determined solely by the fundamental phase-to-phase voltage and the average DC current, but rather an internal coupling quantity jointly determined by the AC-side phase-to-phase voltage harmonics, the DC-side current harmonics, and the leakage inductance parameter. Furthermore, There is an explicit closed-loop relationship between it and the HSS-HBM harmonic state, which can be summarized as shown in formula (29).

[0047] (29) Therefore, the steady-state solution essentially corresponds to and The joint solution.

[0048] S8. Extension to 6N pulsating rectifier structure: Based on the principle of phase shift and inter-bridge harmonic superposition, the six-pulsating basic model is extended to adapt to a unified steady-state harmonic calculation model for 6N pulsating multi-rectifier structure.

[0049] Specifically, as shown in S1-S7, both the arm current switching function and the arm voltage switching function in the six-pulse rectifier bridge have strict periodicity and phase shift properties. Therefore, when the... Each rectifier bridge has a phase offset relative to the reference bridge. At this time, both the current switching function and the voltage switching function of the bridge can be obtained from the reference bridge function by phase shifting, that is... (30) Furthermore, if the Fourier coefficients corresponding to the reference bridge are denoted as follows: and Based on the Fourier transform property of phase shift of a periodic function, the first... The Fourier coefficients of the rectifier bridge can be directly written as shown in formula (31). (31) This formula shows that the differences between the rectifier bridges in a multi-bridge structure are not reflected in changes to the conduction law at the analytical level within the bridge, but mainly in the different phase shift factors attached to the Fourier coefficients of each order. In other words, once the current switching function, voltage switching function, and their Fourier coefficient analytical structures of the reference bridge have been established, the corresponding expressions of the other bridges can be directly generated from the phase shift relationship, without the need to repeat the complete time-domain piecewise derivation and harmonic expansion for each rectifier bridge.

[0050] Based on this, the Toeplitz convolution matrix structure established in the aforementioned steps can also be naturally generalized. Since the construction basis of the convolution matrix is ​​precisely the Fourier coefficients of the switching function, when the Fourier coefficients of each rectifier bridge are introduced due to phase offset... After the phase shift factor in the form of the bridge, the convolution matrix of the current switching function and the convolution matrix of the voltage switching function will also change accordingly, but its Toeplitz structure itself does not change.

[0051] From a system perspective, the 6N pulsating rectifier structure can be understood as the coupled superposition of several phase-shifted six-pulsating basic units in a common AC and DC network. For AC-side variables, the phase bias between different rectifier bridges alters the phase relationship of each characteristic harmonic in the output of each bridge, causing some harmonic components to cancel each other out, while others are retained or redistributed. For DC-side common variables, the contributions of each bridge are superimposed in the same harmonic vector space and further coupled through the common DC network. This means that the harmonic suppression and spectral reconstruction phenomena commonly found in multi-pulsating structures do not require entirely new topology-specific models for explanation, but can be directly explained by the phase-shift superposition characteristics of the Fourier coefficients of the single-bridge switching function. Therefore, the engineering phenomenon of multi-pulsating rectifier structures improving harmonic performance through phase-shifting transformers can obtain a clear mathematical correspondence within the framework of this invention, rather than merely remaining at the level of empirical description or average values ​​in qualitative analysis.

[0052] Based on the HSS-HBM joint solution framework established in this invention, it can be seen that, given the phase offset of each bridge and the corresponding commutation state, the intra-bridge conduction relationship, AC side excitation, leakage inductance voltage drop, and DC side network equations of the multi-bridge system can still be uniformly written as a harmonic domain algebraic system. At this point, compared to the six-pulse basic unit, the main change in model structure is not in the form of the intra-bridge equations themselves, but in the fact that the dimensions of the system state vector and input vector will expand with the increase in the number of rectifier bridges. The inter-bridge phase offset will be explicitly entered into the overall equations through the Fourier coefficients of the switching function and the convolution matrix, while the common AC side and DC side networks are responsible for connecting the harmonic variables of each bridge.

[0053] Therefore, the dual-switch function, Fourier coefficient analytical structure, Toeplitz convolution matrix, and overlap angle calculation logic established in steps S1-7 will not become invalid due to the increase in the number of pulsations; the changes are only reflected in the expansion of the number of bridges, phase offset, and network combination method. It is precisely because of this uniformity in modeling structure that the framework established in this invention is not only applicable to the six-pulse example, but can also be naturally extended to higher pulsation rectification structures, and can serve as a unified theoretical basis for subsequent research on multi-bridge coupling, multi-pulsation harmonic cancellation mechanisms, and system-level frequency domain propagation problems.

[0054] Compared with the prior art, the beneficial effects of the present invention include: (1) The dual-switching function can clearly characterize the coupling relationship between AC and DC variables inside the DRU. The current switching function is used to describe the conduction state of the bridge arm, the phase current distribution, and the continuous current transfer within the commutation interval; the voltage switching function is used to describe the combination relationship between the equivalent phase voltage on the AC side and the output voltage on the DC side. Compared with the modeling method that only starts from the average quantity at the port, the dual-switching function can more directly retain the physical meaning of the conduction sequence and commutation process inside the rectifier bridge, providing a clear interface for subsequent harmonic domain modeling.

[0055] (2) The harmonic domain representation based on Fourier coefficients and the Toeplitz convolution matrix can effectively describe the frequency coupling relationship caused by the switching function. The periodic time-varying relationship formed by multiplying the switching function and electrical variables in the time domain can be transformed into a convolution mapping in the frequency domain; after being further represented by the Toeplitz matrix, it can be combined with the DC-side network equations, commutation constraints, and harmonic balance conditions to form a set of algebraic equations. This approach enables the DRU steady-state harmonic problem to be solved uniformly within the HSS-HBM framework.

[0056] (3) Angle of overlap These are key internal variables in the natural commutation process. This invention does not simply treat them as external constants determined by the fundamental frequency and average DC current, but rather links them to the interphase commutation voltage, bridge-side leakage inductance, and DC-side current state, and iteratively updates them during the HSS-HBM solution process. This approach allows the continuous current transfer within the commutation interval to be coupled with steady-state harmonic variables, thereby improving the model's ability to describe the natural commutation process.

[0057] (4) Simulation results verified the effectiveness of the established model under the baseline operating conditions. Time-domain results show that the model can reproduce the AC side phase current conduction distribution, continuous transfer of commutation interval, and six-pulse ripple characteristics of DC side current well. Frequency-domain results show that the average DC current and main current calculated by HSS-HBM are... The amplitude and phase of the secondary ripple are in good agreement with the EMT-FFT results. The DC component error is approximately... The amplitude error of the 6th ripple is approximately ,main The phase error of the subharmonics is less than This demonstrates that the model of this invention can accurately predict the steady-state harmonic characteristics of the DC side of a six-pulse DRU.

[0058] (5) The method of this invention has a theoretical basis for extension to multi-pulsation structures. For structures composed of multiple six-pulsation bridges combined by transformer phase shifting... The pulsating structure can be derived by introducing a phase shift relationship into the single-bridge dual-switch function model, and obtaining a unified expression for the multi-bridge structure through harmonic domain variable extension and simultaneous network equations. Therefore, The pulsation generalization does not require rebuilding a completely different intra-bridge model, but can be regarded as an extension of the six-pulsation basic unit in the bridge number dimension and phase dimension. Attached Figure Description

[0059] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0060] Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a waveform diagram of the A-phase current switching function in an embodiment of the method of the present invention; Figure 3 This is a waveform diagram of the voltage switching function of phase A in an embodiment of the method of the present invention; Figure 4 The following are transient and steady-state periodic waveforms of the DC-side current in an embodiment of the method of the present invention. Figure 5 This is a steady-state waveform diagram of the three-phase current on the AC side in an embodiment of the method of the present invention; Figure 6 This is a comparison of the phase current in the embodiment of the method of the present invention with that of ideal commutation and commutation considering leakage reactance. Figure 7 This is a comparison diagram of DC-side current HSS-HBM calculation and time-domain simulation in the embodiments of the present invention; Figure 8 This is a flowchart of the full voltage coupling iterative process of the method of the present invention. Detailed Implementation

[0061] Example 1 like Figure 1 As shown, a DRU steady-state harmonic analysis method includes the following steps: S1. Determine the modeling basis and harmonic representation: Taking the six-pulse DRU rectifier bridge as the core object, set the three-phase leakage inductance to be symmetrical and the parameters to be consistent, take the electrical angle as the unified independent variable and the A-phase fundamental phase voltage as the phase reference; expand the AC side voltage and DC side voltage / current into Fourier series, select the harmonic cutoff order, and complete the harmonic domain discretization of the periodic quantity.

[0062] S2. Establish the bridge arm current switching function: Combining the natural commutation mechanism of DRU, divide the non-commutation and commutation intervals. Based on the leakage inductance voltage drop and current continuity constraints, construct a piecewise continuous current switching function to accurately depict the distribution law of DC side current to each AC phase and the current transfer process in the commutation interval.

[0063] The current switching function is: ; According to this formula, as Figure 2As shown, a typical waveform of the bridge arm current switching function over one power frequency cycle can be plotted. It can be seen that in the non-commutation interval, the function maintains a piecewise constant value; while in the commutation interval, due to the simultaneous conduction of adjacent phases, the function transitions from a constant value segment to a continuously changing segment. This waveform characteristic indicates that the current switching function is not a simple sign-switching function, but a periodic time-varying function containing commutation boundary information and current transfer patterns.

[0064] S3. Establish the bridge arm voltage switching function: Define the bridge arm equivalent voltage after deducting the leakage inductance voltage drop, construct a three-valued piecewise voltage switching function, standardize the conduction boundary treatment method of the commutation interval, and clearly characterize the synthesis mapping relationship of the AC side equivalent voltage to the DC side output voltage.

[0065] The voltage switching function is: ; Based on this formula, a typical waveform of the voltage switching function of phase A bridge arm within one power frequency cycle can be plotted, as follows: Figure 3 As shown, the voltage switching function maintains a three-valued piecewise characteristic overall. Its main difference from the current switching function lies not in the continuous change within the commutation interval, but in the way the conduction boundary is handled.

[0066] S4. Solving for the Fourier coefficients of the dual-switching function: By utilizing the periodic symmetry and conjugate characteristics of the rectifier bridge, piecewise integration is performed on the two types of switching functions to derive the Fourier coefficients in a unified analytical form, providing a parameter basis for harmonic domain mapping.

[0067] S5. Construct the Toeplitz convolution matrix: Based on the Fourier coefficients of the switching function, construct a Toeplitz type convolution matrix to transform the time-varying product relationship of the internal time-domain period of the rectifier bridge into a linear convolution mapping relationship that can be solved in the harmonic domain.

[0068] S6. Establish the HSS-HBM joint solution framework: Transform the periodic time-varying system into a harmonic domain matrix model through the harmonic state space (HSS), apply steady-state frequency domain balance constraints by combining the harmonic balance method (HBM), and combine the AC and DC side network equations to form a unified harmonic domain algebraic solution system.

[0069] S7. Closed-loop solution for commutation overlap angle: Based on leakage inductance voltage drop, interphase commutation voltage, and current continuity constraints, a coupled equation for commutation overlap angle and AC / DC harmonic variables is established, and the joint solution of overlap angle and harmonic variables is completed iteratively.

[0070] The joint solution process is as follows: Figure 8 As shown.

[0071] S8. Extension to 6N pulsating rectifier structure: Based on the principle of phase shift and inter-bridge harmonic superposition, the six-pulsating basic model is extended to adapt to a unified steady-state harmonic calculation model for 6N pulsating multi-rectifier structure.

[0072] Example 2 Based on the method described in Example 1, this example compares simulation with theoretical results.

[0073] To ensure the persuasiveness of the comparative conclusions, the theoretical model and the time-domain simulation model must be strictly consistent in terms of parameters and reference standards, including AC fundamental and harmonic injection conditions, transformer leakage inductance, DC side parameters, inter-bridge phase offset, phase reference definition, and spectrum extraction window. Especially for the HSS-HBM model, harmonic amplitude and phase are key output quantities. If the phase reference, sampling starting point, or Fourier definition is inconsistent, even corresponding to the same physical waveform, it may manifest as a systematic phase shift in the frequency domain. Therefore, it is necessary to unify the time-domain window, frequency-domain reference, and complex coefficient definition before comparing results.

[0074] To ensure comparability between the theoretical model and the time-domain simulation results, this invention uses consistent main circuit parameters in both the EMT simulation model and the HSS-HBM theoretical calculation model. The benchmark example selects a three-phase six-pulsating diode rectifier bridge as the research object, with a symmetrical three-phase voltage source on the AC side and an inductor-resistor series load on the DC side. The main system parameters are shown in Table 1.

[0075] Table 1 System Simulation Parameters and Theoretical Calculation Parameters

[0076] Depend on Figure 4 As can be seen, the DC-side current rises rapidly in the initial stage of startup and gradually enters a stable periodic fluctuation state after about 0.12~0.15s. To avoid the impact of the startup transient process on the frequency domain analysis results, this invention selects the steady-state data after 0.30s as the object for subsequent time domain comparison and FFT analysis. Figure 4 The lower half presents a locally magnified waveform within the 0.30~0.32s interval. It can be seen that during this time period, the DC-side current fluctuates slightly periodically around the steady-state average value, exhibiting six repetitive ripple peaks and valleys within one fundamental cycle, consistent with the basic characteristics of the 6kth characteristic harmonic on the DC side of a six-pulse rectifier. Therefore, this steady-state window can be used for subsequent extraction and comparison of harmonic amplitude, phase, and model errors.

[0077] Example 3 To verify the effectiveness of the established dual-switching function HSS-HBM model in the time domain, this embodiment first compares and analyzes the three-phase AC current, the current transfer process during the commutation interval, and the steady-state waveform of the DC current. Specifically, the three-phase AC current is used to verify the ability of the bridge arm current switching function to characterize the conduction distribution relationship; the local waveform during the commutation interval is used to illustrate the influence of bridge-side leakage inductance on the continuous phase current transfer process; and the DC current waveform is used to verify the theoretical model's ability to calculate the steady-state operating point and periodic ripple. All time-domain waveforms are displayed within the steady-state interval after 0.30s.

[0078] Depend on Figure 5 It can be seen that under steady-state operating conditions, the three-phase current on the AC side exhibits a clear periodic alternating conduction characteristic, with the three phases maintaining approximately [a certain level of continuity]. The phase relationship is as follows. The currents of each phase remain relatively stable within the main conduction range, but a continuous transition occurs at the junction of adjacent phases. This indicates that the arm currents do not switch instantaneously, but rather undergo a finite-time commutation transfer under the influence of bridge-side leakage inductance. This phenomenon is consistent with the basic operating mechanism of a natural commutation rectifier bridge and also aligns with the modeling idea in step S2 where the arm current switching function is expressed as a continuous transfer within the commutation range.

[0079] From the overall waveform, the currents of phases A, B, and C sequentially enter and exit the conduction state within one power frequency cycle, with a clear conduction sequence and obvious periodic repeatability. This result shows that the established current switching function can accurately reflect the distribution process of DC-side current to each phase of AC-side current and can describe the basic phase current waveform characteristics of the six-pulse diode rectifier bridge under steady-state conditions.

[0080] To further illustrate the impact of commutation overlap on the phase current waveform, Figure 6 A comparison of phase currents under ideal commutation conditions and those considering bridge-side leakage inductance is presented. It can be seen that under ideal commutation conditions, the phase current exhibits an approximately step change at the conduction boundary; however, considering bridge-side leakage inductance, the phase current transitions from an abrupt change to a continuous transition within the commutation interval. This difference indicates that bridge-side leakage inductance delays the current transfer process between adjacent bridge arms, causing the outgoing phase current and incoming phase current to participate in DC current transmission together within a finite time, thus forming a commutation overlap interval.

[0081] Figure 7 A comparison between the calculated results of the DC-side current HSS-HBM and the time-domain simulation results is presented. It can be seen that under the reference operating condition, the DC-side current stabilizes at approximately 1717A, with a small-amplitude periodic ripple superimposed. This ripple repeats six times within one power frequency cycle, reflecting the basic pattern that the DC side of the six-pulse rectifier mainly contains the 6kth characteristic harmonic.

[0082] The average DC-side current obtained from the time-domain simulation is approximately 1717 A, while the DC component calculated by the HSS-HBM model is 1715.82 A, which are quite close. This result indicates that the model of this invention can accurately provide the steady-state operating point of the DC side under the reference operating conditions. Furthermore, the theoretical calculation results and the time-domain simulation results are consistent in terms of periodic ripple shape, repetition frequency, and overall trend, demonstrating that the harmonic domain model established based on the dual-switching function and the Toeplitz convolution matrix can effectively reflect the formation process of the DC-side current ripple.

[0083] Based on the above time-domain results, the dual-switching function HSS-HBM model established in this invention can effectively reflect the AC side phase current distribution, continuous current transfer during commutation, and the periodic ripple characteristics of the DC side current under steady-state conditions. Time-domain comparisons show that the model not only reproduces the main morphology of the rectifier bridge's steady-state waveform but also provides a reasonable description of the key physical aspects of the natural commutation process. Further analysis will involve comparing the amplitude and phase of the main harmonic components from a frequency-domain perspective to verify the model's accuracy in calculating steady-state harmonic characteristics.

[0084] Example 4 After comparing the time-domain waveforms, this embodiment further verifies the calculation accuracy of the HSS-HBM model for the main harmonic components from a frequency-domain perspective. For a naturally commutated diode rectifier, the harmonic distribution is not only determined by the six-pulse conduction sequence, but also affected by bridge-side leakage inductance, commutation overlap, and DC-side network parameters. Therefore, comparing only the time-domain waveform morphology is insufficient to fully demonstrate the model's frequency-domain predictive capability; further comparison of the amplitude and phase of the main harmonic components is also necessary.

[0085] This invention selects the DC-side current as the frequency domain comparison object to verify the model's ability to characterize the 6kth DC-side ripple component. The HSS-HBM model directly provides the complex frequency domain components of the DC-side current, while the time-domain simulation results extract the corresponding harmonic components through steady-state window FFT. The two are compared under a unified amplitude definition and phase reference, and the results are shown in Table 2.

[0086] Table 2 Comparison of Main Harmonic Components of DC Side Current

[0087] As shown in Table 2, the DC-side current is dominated by the DC component. The DC component calculated by HSS-HBM is 1716.82A, and the DC component extracted by EMT-FFT is 1716.99A, with an amplitude error of approximately 0.01%. This result indicates that the HSS-HBM model established in this invention can accurately predict the steady-state operating point of the DC side under the reference operating condition.

[0088] For the DC-side ripple components, the main harmonics are concentrated at the 6k order, including the 6th, 12th, 18th, 24th, and 30th harmonics. Among them, the 6th harmonic has the largest amplitude, with a calculated value of 6.50A from HSS-HBM and a extracted value of 6.51A from EMT-FFT, resulting in an amplitude error of only 0.21%. As the harmonic order increases, the amplitudes of the 12th, 18th, 24th, and 30th harmonics gradually decrease, consistent with the basic law that the DC-side ripple of a six-pulse rectifier attenuates with increasing frequency. This result demonstrates that the model of this invention can not only provide the DC average quantity but also accurately characterize the main 6k-order ripple components on the DC side.

[0089] From the phase results, the phase errors of the HSS-HBM calculation results and the EMT-FFT extraction results are relatively small on the main 6k harmonics. Specifically, the phase errors of the 6th, 12th, 18th, 24th, and 30th harmonics are -0.27°, -0.35°, -0.32°, -0.24°, and -0.29°, respectively. This indicates that after unifying the fundamental phase voltage of phase A as the phase reference, there is no obvious systematic phase shift between the theoretical calculation and the post-simulation processing, demonstrating the consistency of the Fourier coefficient definition, phase reference, and harmonic domain mapping relationship adopted in this invention.

[0090] It should be noted that some phase results in MATLAB FFT analysis tools are usually displayed in the range of 0° to 360°. However, to facilitate comparison with the HSS-HBM complex frequency domain calculation results, this invention uniformly converts the phase to the range of [-180°, 180°]. After this processing, the phases of the 12th, 18th, 24th, and 30th harmonics are in good agreement with the HSS-HBM results. Therefore, the phase errors in Table 2 mainly reflect the actual deviation between the model calculation and the simulation extraction, rather than the apparent deviation caused by the difference in the phase display range.

[0091] As shown in Table 2, the HSS-HBM model can accurately predict the average DC-side current and the main 6k-order ripple components. Combined with the time-domain waveforms in Section 3.2, it can be seen that the model of this invention can not only reproduce the steady-state waveform and six-pulse ripple characteristics of the DC-side current in the time domain, but also provide the amplitude and phase of the main harmonic components in the frequency domain. Therefore, the dual-switching function, Fourier coefficient analytical structure, and Toeplitz harmonic mapping relationship established in this invention can effectively describe the steady-state harmonic characteristics of the six-pulse DRU under natural commutation conditions on the DC side.

[0092] The comparison results between Examples 3 and 4 show that the HSS-HBM calculation results and the EMT simulation results differ in terms of DC-side average current and main... The amplitude of the secondary ripple and the phase of the harmonics remain well consistent. The DC component error in Table 2 is approximately... The amplitude error of the 6th ripple is approximately ,main The phase error of the subharmonics is less than This demonstrates that, under the periodic steady-state and parameter symmetry conditions set in this invention, the established harmonic domain model can accurately describe the DC-side steady-state harmonic characteristics of a six-pulse DRU. The remaining errors mainly arise from spectrum post-processing, finite harmonic truncation, commutation boundary identification, and differences in model assumptions.

[0093] First, FFT post-processing directly affects the frequency domain comparison results. If the time-domain data truncation interval is not a complete fundamental period, or if the theoretical calculation and simulation post-processing use different sampling starting points, phase references, and sine / cosine definitions, the same physical waveform may exhibit amplitude deviations or phase shifts. To reduce this type of comparison error, this invention selects... The steady-state integer-cycle data were subjected to FFT analysis, with the fundamental phase voltage of phase A used as the phase reference. Simultaneously, the MATLAB FFT analysis tool... The phase displayed in the form is uniformly converted to Range. After the above processing, the phase error in Table 2 can more directly reflect the difference between the model calculation and the simulation extraction, rather than the apparent deviation caused by the inconsistency of the display range or reference definition.

[0094] Secondly, finite harmonic cutoff affects higher-order small-amplitude components. The HSS-HBM model transforms a periodic steady-state problem into finite-dimensional harmonic algebraic equations, thus it cannot retain all high-frequency components with infinite precision. For DC components and lower-order main harmonic components... For the lower-order ripple components, their amplitudes are relatively large and their energy is concentrated, thus high accuracy can still be achieved under finite cutoff conditions. For higher-order harmonics, their amplitudes are inherently smaller, and even small absolute deviations can lead to increased relative errors. Furthermore, the current and switching functions change steeply within the commutation interval, making high-frequency components more sensitive to the cutoff order. Therefore, the errors of the main lower-order ripple components are relatively small, while the errors of the higher-order small-amplitude components increase slightly, consistent with the general characteristics of finite-order harmonic approximation models.

[0095] Furthermore, commutation boundary identification affects harmonic amplitude and phase. The natural commutation process is determined by the interphase commutation voltage, bridge-side leakage inductance, and DC-side current. Overlap angle. Even small deviations can alter the boundary positions of the switching function and further affect its Fourier coefficients. Since higher-order harmonics are more sensitive to boundary positions, even small shifts in the commutation start and stop angles can manifest as phase or amplitude errors in higher-order components. In contrast, the DC component and lower-order dominant ripple components are less sensitive to local boundary disturbances, making their calculation results more likely to align with EMT-FFT results.

[0096] Furthermore, subtle differences may still exist between the theoretical model and the EMT simulation model. The derivation of this invention is based on assumptions such as periodic steady-state, symmetrical three-phase parameters, consistent bridge-side leakage inductance, and idealized devices. In EMT simulations, factors such as the numerical solution step size, device conduction criteria, initial transient decay, sampling accuracy, and diode equivalent characteristics can all affect local waveform details. These factors typically do not change the main DC-side characteristics. The distribution pattern of subharmonics is observed, but slight differences may be introduced near higher-order small-amplitude components or commutation boundaries. Therefore, this invention focuses more on DC average quantities and main... The model focuses on the consistency of the amplitude and phase of the secondary ripple, rather than using the relative error of higher-order small components as the sole criterion for judging the effectiveness of the model.

[0097] In summary, the HSS-HBM model of this invention is suitable for harmonic analysis of six-pulse DRUs with symmetric parameters, periodic steady state, consistent bridge-side leakage inductance, and sufficient harmonic cutoff order.

[0098] The embodiments described above are merely illustrative of implementation methods of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of this patent should be determined by the appended claims.

Claims

1. A DRU steady-state harmonic analysis method, characterized in that, Includes the following steps: S1: Determine the modeling basis and harmonic representation, unify the phase reference and expand the AC and DC side electrical quantities into Fourier series, and select the harmonic truncation order; S2: Establish the bridge arm current switching function, divide the conduction and commutation intervals by combining the natural commutation mechanism, and construct a piecewise continuous function to characterize the distribution law of DC side current to AC phase current; S3: Establish the bridge arm voltage switching function, construct a piecewise function based on the bridge arm equivalent voltage, and characterize the synthesis mapping relationship of the AC side equivalent voltage to the DC side output voltage; S4: Solve for the Fourier coefficients of the double-switching function. Utilize the periodicity and symmetry of the rectifier bridge to obtain the unified analytical Fourier coefficients of the current and voltage switching functions through piecewise integration. S5: Construct a Toeplitz convolution matrix, build a matrix based on the Fourier coefficients of the dual-switch function, and transform the time-varying product relationship of the rectifier bridge in the time domain into a harmonic domain convolution mapping relationship; S6: Build a joint HSS-HBM solution framework, establish a frequency domain matrix model through the harmonic state space, apply steady-state balance constraints by combining the harmonic balance method, and form a unified algebraic solution system by combining the AC and DC side network equations. S7: Closed-loop solution of commutation overlap angle. Based on the commutation constraint conditions, establish the coupled equation of overlap angle and AC / DC harmonic variables, and complete the iterative solution. S8: Extended to the 6N pulsating rectifier structure, the six-pulsating model is extended into a multi-pulsating general model by phase shifting and inter-bridge harmonic superposition.

2. The DRU steady-state harmonic analysis method according to claim 1, characterized in that, In step S3, the equivalent voltage of the bridge arm is the electrical quantity after deducting the voltage drop of the bridge arm commutation leakage inductance from the AC side phase voltage; within the commutation interval, the voltage switching function is 0 before the input diode is turned on, and the original conduction value is maintained before the output diode is turned off.

3. The DRU steady-state harmonic analysis method of claim 2, wherein, In step S3, the expression for the bridge arm voltage switching function is: ； in, Represents electrical angle, Represents the commutation overlap angle. This represents the voltage switching function for phase A.

4. The DRU steady-state harmonic analysis method according to claim 1, characterized in that, In step S5, the Toeplitz convolution matrix is ​​constructed from the Fourier coefficients of the bridge arm current switching function and the bridge arm voltage switching function, respectively, and the matrix elements are only related to the difference between the row and column indices.

5. The DRU steady-state harmonic analysis method according to claim 4, characterized in that, In step S5, the Toeplitz convolution matrix transforms the product relationship between the switching function and the electrical quantity in the time domain into the coupling mapping relationship of different orders of harmonics in the harmonic domain.

6. The DRU steady-state harmonic analysis method according to claim 1, characterized in that, In step S6, the harmonic state space transforms the DRU periodic time-varying system into a harmonic domain steady-state matrix model, and the harmonic balance method synchronously applies frequency domain steady-state balance conditions to each harmonic component.

7. The DRU steady-state harmonic analysis method according to claim 6, characterized in that, In step S6, the unified algebraic solution system integrates the dual-switch function convolution matrix, harmonic domain differential operator, and AC / DC port constraint equations.

8. The DRU steady-state harmonic analysis method according to claim 1, characterized in that, In step S2, the bridge arm current switching function is a continuous transition function within the commutation interval, satisfying the continuous current transfer law under leakage inductance constraint.

9. The DRU steady-state harmonic analysis method according to claim 1, characterized in that, In step S7, the commutation overlap angle is determined by the AC side phase-to-phase harmonic voltage, the DC side harmonic current, and the bridge arm leakage inductance, forming a closed-loop iterative relationship with the harmonic variables.

10. The DRU steady-state harmonic analysis method according to claim 1, characterized in that, In step S8, the model of the 6N pulsating rectifier structure is extended by generating the switching functions and Fourier coefficients of each sub-bridge through phase shifting of the reference bridge, and then completing the simultaneous equations through harmonic superposition.

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