A hybrid reduced-order control method for large-scale dynamic systems

Abstract

The application discloses a hybrid reduced-order control method for large-scale dynamic systems and relates to the technical field of system reduced-order control.The application builds a multi-time-scale state space model, quantitatively divides fast and slow subsystems based on a preset time scale threshold value, accurately screens dominant state variables by using eigenvalue dominant factors and participation factor analysis, performs energy-optimal reduction on the slow subsystem by adopting balance truncation and singular perturbation principles, and introduces a closed-loop deviation evaluation and automatic optimization mechanism, thereby solving the problems that the division of fast and slow subsystems is subjective in the prior art, dominant modes are easily lost, transient accuracy is poor, decoupling and reduction links are mutually fragmented and depend on artificial trial and error, and realizing automation, high precision and strong universality of the reduction process, so that the model order can be reduced while the dominant modes and transient characteristics of the system are completely retained, and an efficient and reliable technical scheme is provided for analysis, simulation and control design of large-scale dynamic systems.

Description

Technical Field

[0001] This invention relates to the field of system order reduction control technology, specifically a hybrid order reduction control method for large-scale dynamic systems. Background Technology

[0002] Modern power systems, with their expanded scale, complex structure, and high proportion of renewable energy grid integration, exhibit strong coupling characteristics across multiple time scales, encompassing both electromechanical and electromagnetic transients. Using full-order detailed models for small-signal stability analysis, controller design, and real-time simulation presents challenges such as high order, large computational load, and unclear mechanisms, necessitating efficient decoupling and order reduction methods. Existing technologies suffer from several shortcomings: First, the division of fast and slow subsystems relies on empirical thresholds, lacks quantitative evaluation, easily loses dominant modes, and suffers from insufficient decoupling accuracy and physical clarity. Second, traditional order reduction algorithms have poor adaptability; direct order reduction across the entire system is computationally complex and cannot specifically preserve key fast and slow dynamics, resulting in significant loss of accuracy in transient simulation and stability analysis. Third, the decoupling and order reduction stages are fragmented, the process relies on manual trial and error, lacks automation and versatility, and is difficult to promote in engineering applications.

[0003] Therefore, it is necessary to propose a hybrid order reduction method for large-scale power systems that maintains transient accuracy. This method simplifies the model while fully preserving the dominant modes and transient characteristics of the system, providing an efficient, universal, and automated technical solution for the analysis, simulation, and control design of complex power systems. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a hybrid order reduction control method for large-scale dynamic systems. This method solves the problems of existing order reduction methods for large-scale power systems with multiple time scales, which suffer from subjective loss of dominant modes in the division of fast and slow subsystems, poor transient accuracy in order reduction, unsystematic processes, and an inability to balance order reduction efficiency with core dynamic accuracy.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a hybrid order reduction control method for large-scale dynamic systems, comprising the following steps: constructing a state-space model with multiple time scales.

[0006] Based on a preset time scale threshold, the state-space model is decoupled into fast subsystems and slow subsystems with corresponding electromagnetic transient characteristics.

[0007] The internal state variables of the tachy subsystem are decoupled based on linear coordinate transformation to obtain the decoupled tachy subsystem.

[0008] Calculate the eigenvalue dominance factor of the decoupled fast subsystem, screen the dominant state variables, and merge the non-dominant state variables into the slow subsystem to obtain the refined fast subsystem and the refined slow subsystem.

[0009] The refined slow subsystem is reduced in order using the balanced truncation method and the singular perturbation principle to obtain a reduced-order slow subsystem model. This reduced-order slow subsystem model is then integrated with the refined fast subsystem model to obtain the initial reduced-order model.

[0010] Time-domain and frequency-domain simulations were performed on the state-space model and the initial reduced-order model to calculate the H∞ norm deviation and damping ratio deviation, as well as the mode division deviation and state truncation deviation.

[0011] If the H∞ norm deviation and damping ratio deviation meet the preset conditions, the output of the initial reduced-order model is the final reduced-order model. Otherwise, the process of selecting the dominant state variables and refining the slow subsystem reduction process are optimized based on the mode splitting deviation and state truncation deviation.

[0012] Compared with existing technologies, this invention has the following advantages: By constructing a multi-timescale state-space model, quantizing and dividing fast and slow subsystems based on preset timescale thresholds, accurately screening dominant state variables using eigenvalue dominant factors and participating factor analysis, employing the principle of balance truncation and singular perturbation to perform energy-optimal order reduction of the slow subsystem, and introducing a closed-loop deviation evaluation and automatic optimization mechanism, this invention solves the problems in existing technologies such as subjective loss of dominant modes in the division of fast and slow subsystems, poor transient accuracy in order reduction, and the separation of decoupling and order reduction links and reliance on manual trial and error. It achieves automation, high precision, and strong versatility in the order reduction process, and can completely retain the dominant modes and transient characteristics of the system while reducing the model order, providing an efficient and reliable technical solution for the analysis, simulation, and control design of large-scale dynamic systems. Attached Figure Description

[0013] Figure 1 This is a flowchart of a hybrid order reduction control method for large-scale dynamic systems according to the present invention.

[0014] Figure 2 This is a flowchart illustrating the refined fast subsystem and refined slow subsystem obtained in the hybrid order reduction control method for large-scale dynamic systems of the present invention.

[0015] Figure 3 This is a flowchart illustrating the initial reduced-order model obtained in a hybrid reduced-order control method for large-scale dynamic systems according to the present invention. Detailed Implementation

[0016] The present invention will now be described in detail with reference to the accompanying drawings and embodiments. Please refer to the accompanying drawings. Figure 1 The present invention provides a technical solution: a hybrid order reduction control method for large-scale dynamic systems, comprising the following steps: S1, constructing a state-space model of small signals with multiple time scales that can simultaneously reflect electromagnetic transient and electromechanical transient response characteristics.

[0017] The general form of the state-space model, without considering the output variables, is as follows.

[0018] (1).

[0019] in, For state vectors, For the input vector, The state matrix, The input matrix is ​​denoted as .

[0020] The general form of the state-space model provides a unified mathematical foundation for subsequent decoupling of fast and slow subsystems based on time-scale thresholds, avoiding the subjectivity and blindness of traditional methods that rely on empirical division. At the same time, the state-space form facilitates mathematical operations such as linear transformations and eigenvalue analysis, enabling quantitative characterization of the system's dynamic characteristics. This lays the model foundation for accurately selecting the dominant mode and automating the decoupling and order reduction processes.

[0021] S2. Based on a preset time scale threshold, the state-space model is decoupled into a fast subsystem and a slow subsystem with corresponding electromagnetic transient characteristics. For example, the preset time scale threshold is 100ms.

[0022] Considering the order-of-magnitude difference in response speed between electromagnetic transient processes and electromechanical transient processes in large-scale, multi-timescale dynamic systems (the former is typically on the order of milliseconds, while the latter is on the order of hundreds of milliseconds or more), if these two types of dynamics are not distinguished and the entire system is uniformly reduced in order, the coupling between fast and slow dynamics will make it difficult for the reduced-order model to simultaneously maintain the key characteristics of both. Therefore, a quantifiable time scale threshold needs to be introduced as an objective classification benchmark to categorize the original system into fast subsystems and slow subsystems based on variable response time. Simultaneously, to avoid information loss between subsystems due to simple separation, a bidirectional coupling relationship must be explicitly established by setting the output of the fast subsystem equal to the input of the slow subsystem, and the output of the slow subsystem equal to the input of the fast subsystem. The specific process is as follows: S201, Based on the preset time scale threshold, classify the state variables, input variables, and output variables in the state-space model according to their dynamic response time scale. Specifically: S202, Variables with dynamic response times less than or equal to the preset time scale threshold are classified as the fast dynamic part corresponding to electromagnetic transient characteristics, and variables with dynamic response times greater than the preset time scale threshold are classified as the slow dynamic part corresponding to electromechanical transient characteristics.

[0023] S203. Construct state-space models for the fast and slow subsystems based on the fast and slow dynamic components, respectively. The state vector, input vector, and output vector of the fast subsystem's state-space model are composed of the state variables, input variables, and output variables of the fast dynamic component, respectively. The state vector, input vector, and output vector of the slow subsystem's state-space model are composed of the state variables, input variables, and output variables of the slow dynamic component, respectively.

[0024] S204. Set the output of the fast subsystem to be equal to the input of the slow subsystem, and the output of the slow subsystem to be equal to the input of the fast subsystem, and establish the coupling relationship between the fast subsystem and the slow subsystem.

[0025] Specifically, the spatial models of the decoupled fast subsystem and slow subsystem are as follows.

[0026] (2).

[0027] (3).

[0028] in, , , , , and These are the state vector, input vector, and output vector of the fast subsystem, and the state vector, input vector, and output vector of the slow subsystem, respectively. , , , , and Let be the state matrix, input matrix, and output matrix of the fast subsystem, and be the state moments, input matrix, and output matrix of the slow subsystem, respectively.

[0029] Since the fast subsystem and the slow subsystem originate from the same overall system, a dynamic coupling relationship can be established between them by setting the output of the fast subsystem to be equal to the input of the slow subsystem, and the output of the slow subsystem to be equal to the input of the fast subsystem.

[0030] This embodiment achieves objective quantitative division of fast and slow subsystems by setting a preset time scale threshold, replacing the subjective threshold setting based on experience in traditional methods. This effectively reduces the risk of losing the dominant mode due to improper division. At the same time, through the coupling relationship design where the outputs of fast and slow subsystems are inputs to each other, it ensures that the decomposed model is mathematically equivalent to the original system, avoiding the breakage of dynamic information caused by artificial decoupling. This provides a standardized model structure that is both separated and coupled for subsequent operations such as dominant factor screening and balance truncation, thereby improving the versatility and engineering scalability of the order reduction process.

[0031] S3. Based on linear coordinate transformation, the internal state variables of the tachy subsystem are decoupled to obtain the decoupled tachy subsystem.

[0032] After decoupling the fast and slow subsystems, although the electromagnetic transient (fast) and electromechanical transient (slow) dynamics are macroscopically separated, the fast subsystem still contains multiple coupled state variables that collectively affect its output. Due to these couplings, a change in any eigenvalue indirectly affects multiple state variables through the off-diagonal elements of the state matrix, making it impossible to directly determine the independent contribution of each eigenvalue to the fast subsystem's output. If dominant mode selection is performed directly on this basis, the coupling effect will mask the true dominant dynamics, easily leading to the misclassification and discarding of critical modes that should be retained as non-dominant, thus causing a decrease in the transient accuracy of the subsequent reduced-order model.

[0033] Therefore, it is necessary to first decouple the internal state variables of the tachy subsystem, that is, to diagonalize its state matrix through linear coordinate transformation, so that each new state variable after transformation is only related to one eigenvalue, thereby completely separating the originally entangled dynamic modes. The specific process is as follows: S301, diagonalize the state matrix of the tachy subsystem and construct the eigenvalue diagonal matrix calculation formula.

[0034] S302. Substitute the eigenvalue diagonal matrix calculation formula into the state space model of the fast subsystem to obtain the decoupled fast subsystem.

[0035] It should be noted that during the decoupling transformation of the tachy subsystem, the tachy subsystem... Let the size be represented as n×1. The size is denoted as k×1. The size is denoted as p×1, where n is the number of state variables of the fast subsystem, k is the number of input variables of the fast subsystem, and p is the number of output variables of the fast subsystem.

[0036] To clearly analyze the independent dynamic characteristics of each state variable in the tachy subsystem, a decoupled coordinate transformation is performed on the tachy subsystem. The eigenvalue diagonal matrix calculation formula for the decoupled transformation can be expressed as follows:

[0037] (4).

[0038] in, The state matrix of the fast subsystem , where P is the eigenvalue diagonal matrix, and P is the invertible eigenvector matrix (non-singular linear transformation matrix) of the fast subsystem. Let P be the inverse matrix. , ... State matrix of fast subsystem The eigenvalues ​​represent the inherent characteristics of each independent dynamic mode of the fast subsystem, and L, M and O are placeholders.

[0039] Substituting equation (4) into equation (2) yields the decoupled tachy subsystem, which can be expressed as follows.

[0040] (5).

[0041] in, , and To decouple the state vector, input vector, and output vector in the coordinate system, , and To decouple the input and output matrices in the coordinate system, it can be represented as follows.

[0042] (6).

[0043] (7).

[0044] in, — Scalar elements within the matrix — It is also a scalar element within the matrix.

[0045] After the above transformation, the state matrix of the fast subsystem Transform into a diagonal matrix, state vector The dynamic characteristics are only related to their corresponding characteristic values, thus achieving dynamic decoupling between state variables.

[0046] This embodiment achieves complete dynamic decoupling of internal state variables by performing linear coordinate transformation based on eigenvalue diagonalization on the tachy subsystem. This allows the independent influence of each eigenvalue on the system output to be quantified individually, avoiding misjudgment or omission of the dominant mode caused by the coupling of state variables. It also improves the ability of the reduced-order model to retain transient dynamic characteristics and ensures the accuracy of the reduced-order model.

[0047] S4. Calculate the eigenvalue dominance factor of the decoupled fast subsystem, screen the dominant state variables, and merge the non-dominant state variables into the slow subsystem to obtain the refined fast subsystem and the refined slow subsystem.

[0048] Dynamic decoupling of the internal state variables of the fast subsystem was achieved through linear coordinate transformation, allowing each eigenvalue to independently correspond to a dynamic mode. However, it is still impossible to determine which modes have a dominant influence on the system output. Since the number of state variables in a fast subsystem is usually large, retaining all fast dynamic state variables would still result in a significant computational burden for subsequent order reduction. Simply selecting the dominant mode based on the real part of the eigenvalue (i.e., the rate of decay) would ignore the coupling strength between the mode and the input and output.

[0049] Therefore, a quantitative index is needed that comprehensively considers eigenvalues, input matrices, and output matrices to evaluate the true contribution of each eigenvalue to the output of the fast subsystem, thereby objectively screening out the truly dominant dynamic modes. Based on this, participation factor analysis can be used to map the dominant modes onto specific state variables, accurately identifying which state variables should be retained in the fast subsystem and which can be safely merged into the slow subsystem. For example... Figure 2 As shown, the specific process is as follows: S401, perform a Laplace transform on the decoupled fast subsystem and then perform an inverse Laplace transform to obtain the time-domain expression of the output variable.

[0050] S402. Determine the influence weight of the eigenvalues ​​on each output vector based on the time-domain expression, and output the dominance factor.

[0051] S403. Sum the dominance factors of the same eigenvalue to obtain the dominance of the eigenvalue on all output vectors.

[0052] S404. If the degree of dominance of a certain feature value is greater than a set threshold, then the corresponding feature value is the dominant feature value, and the corresponding dynamic mode is the dominant mode of the fast subsystem.

[0053] S405. Perform participation factor analysis on the dominant eigenvalues. If the participation factor is greater than the preset threshold, it will be identified as the dominant state variable.

[0054] It should be noted that after performing a Laplace transform on the decoupled fast subsystem, the following expression can be obtained.

[0055] (8).

[0056] Where s is the Laplace operator, For the increment of the i-th state variable in the decoupled coordinate system in the complex frequency domain, Let q be the increment of the fast subsystem output variable in the complex frequency domain. For the decoupled input matrix elements, For the decoupled output matrix elements, Let be the complex frequency domain expression of the j-th input variable.

[0057] Furthermore, there is.

[0058] (9).

[0059] The time-domain expression of the output variable obtained by performing an inverse Laplace transform on (9) is as follows.

[0060] (10).

[0061] Where t is the time variable, This represents the increment of the q-th output variable in the time domain.

[0062] For a stable system, its eigenvalues They are all located in the left half of the complex plane, therefore If it is a decreasing function, then...

[0063] (11).

[0064] Therefore, the time-varying term in equation (10) This is a decay function, whose long-term response is dominated by the dynamic process before steady state. The coefficient term... The i-th eigenvalue is determined For the q-th output The influence weight of dynamic components can be more intuitively expressed as follows.

[0065] (12).

[0066] To quantify this degree of dominance, we define an output dominance factor. .

[0067] (13).

[0068] To comprehensively evaluate the combined impact of eigenvalues ​​on the output of the entire fast subsystem, the dominance of eigenvalues ​​is defined. , which represents the summation of the eigenvalues ​​over the dominance factors of all output components.

[0069] (14).

[0070] The dominant factors corresponding to all eigenvalues ​​of the fast subsystem are calculated and sorted. A threshold is set; for example, eigenvalues ​​with dominant factors greater than this threshold are identified as dominant eigenvalues, and their corresponding dynamic modes are the dominant modes of the fast subsystem. Participation factor analysis is performed on the identified dominant eigenvalues, and state variables with participation factors greater than a preset threshold, such as 0.1, are defined as dominant state variables. Furthermore, all non-dominant state variables in the fast subsystem are separated from the original fast subsystem model and merged into the slow subsystem. Thus, a refined fast and slow subsystem model with a set of state variables that are physically and dynamically completely decoupled and hierarchically distinct is obtained.

[0071] The refined state-space model of the system is represented as follows.

[0072] (15).

[0073] in, and These are the state vectors of the refined fast subsystem and the slow subsystem, respectively; , , , Divide the state matrix A of the system to be studied into blocks; and The input matrix B is divided into blocks for the system to be studied.

[0074] Furthermore, the process of conducting participation factor analysis on the identified dominant eigenvalues ​​is as follows: First, for each eigenvalue determined to be dominant, the left and right eigenvectors associated with that eigenvalue are calculated. The left eigenvector reflects the observability of the eigenvalue for each state variable, while the right eigenvector reflects the excitability of the eigenvalue for each state variable. Then, the components corresponding to the same state variable in the left and right eigenvectors are multiplied together to obtain the participation factor of the eigenvalue with respect to the state variable. A larger participation factor indicates a greater role for the eigenvalue in the dynamic response of the state variable.

[0075] When calculating the left and right eigenvectors, the eigenvalue decomposition is usually performed directly on the state matrix of the fast subsystem using a numerical linear algebra library (such as the eig function in MATLAB or numpy.linalg.eig in Python). Each column of the resulting right eigenvector matrix is ​​the right eigenvector of the corresponding eigenvalue, while the left eigenvector is obtained by solving for the eigenvector of the transpose of the state matrix, or directly from the left eigenvector matrix in the eigenvalue decomposition (if the function returns). Then, the eigenvectors are paired and normalized according to the same eigenvalue so that the dot product of the left and right eigenvectors is 1, which can then be used to calculate the participation factor of each eigenvalue with each state variable.

[0076] This embodiment defines an eigenvalue dominance factor to extract the influence weight of eigenvalues ​​on the output from the time-domain analytical expression, achieving quantitative and automated screening of dominant modes. This replaces the traditional method's reliance on experience or subjective judgment based solely on eigenvalue magnitude, avoiding misjudgment or omission of dominant modes. Simultaneously, by separating non-dominant state variables from the fast subsystem and merging them into the slow subsystem, the size of the fast subsystem is significantly reduced. Although some non-dominant fast dynamic variables are added to the slow subsystem, their impact on the system output is negligible and does not compromise the overall accuracy of the slow subsystem.

[0077] S5. Based on the balanced truncation method and the singular perturbation principle, the refined slow subsystem is reduced in order to obtain the reduced slow subsystem model. The reduced slow subsystem model is then integrated with the refined fast subsystem to obtain the initial reduced-order model.

[0078] In S4, refined fast and slow subsystems have been obtained. The size of the fast subsystem has been significantly reduced (only the dominant state variables are retained), but the slow subsystem may still have a high order. Directly reducing the order of the slow subsystem arbitrarily may disrupt the coupling relationship between it and the fast subsystem, resulting in distortion of the transient response of the overall model after order reduction.

[0079] Considering that the slow subsystem is controllable and observable, and that the contributions of its various state variables to the system energy differ greatly—some state variables contribute significantly to energy and must be retained, while others contribute very little and can be safely truncated—a method for order reduction is needed that can quantitatively truncate based on energy contribution while maintaining the coupling relationship between the fast and slow subsystems.

[0080] Since the balanced truncation method can quantify the energy contribution of each state by sorting it according to Hankel singular values, and the singular perturbation principle allows states with minimal energy to be approximated as zero without introducing significant errors, combining the two can maximize the order reduction efficiency while ensuring accuracy. Figure 3 As shown, the specific process is as follows: S501, based on a non-singular transformation matrix, the state vector of the refined slow subsystem after transformation is made to have the controllable Gramian matrix and the observable Gramian matrix equal and diagonal matrix in the equilibrium coordinate system.

[0081] S502. Substitute the state vector in the equilibrium coordinate system into the refined slow subsystem to obtain the refined slow subsystem in the equilibrium coordinate system.

[0082] S503. Arrange the state vectors of the refined slow subsystem in the equilibrium coordinate system in descending order according to the corresponding Hankel singular values, select the first r states, and make the cumulative energy contribution reach the preset threshold to obtain the state vector expression.

[0083] S504. Substitute the state vector expression into the refined slow subsystem in the equilibrium coordinate system to obtain the transformed refined slow subsystem.

[0084] S505. Based on the singular perturbation principle, the truncated reduced-order state is identified as 0. After approximating the transformed refined slow subsystem, the truncated reduced-order state is re-substituted into the transformed refined slow subsystem to eliminate the truncated reduced-order state and obtain the preliminary reduced-order refined slow subsystem model.

[0085] S506. Let the state to be retained corresponding to the larger Hankel singular value in the refined slow subsystem model after the initial order reduction be equal to the state vector of the refined slow subsystem after order reduction, and obtain the slow subsystem model after order reduction.

[0086] S507. Substitute the output vector expression of the reduced-order slow subsystem model into the refined fast subsystem model.

[0087] S508. By combining the state vector expressions of the reduced-order slow subsystem model with the refined fast subsystem model, the initial reduced-order model is obtained.

[0088] It should be noted that after the fine partitioning is completed, the state space model of the system has been expressed as Equation (15). For the convenience of subsequent order reduction, we rewrite it as the following coupling form to clarify the input-output relationship between the fast and slow subsystems.

[0089] (16).

[0090] (17).

[0091] Find a non-singular transformation matrix This ensures that the transformed refined slow subsystem achieves equal and diagonal controllable Gramian matrices to observable Gramian matrices in the equilibrium coordinate system. This is achieved by introducing an equilibrium coordinate transformation.

[0092] (18).

[0093] in, Let be the state vector of the refined slow subsystem in the equilibrium coordinate system. Substituting equation (18) into equation (17) yields the state space of the refined slow subsystem in the equilibrium coordinate system, which can be expressed as .

[0094] (19).

[0095] in, , , These are the state matrix, input matrix, and output matrix of the refined slow subsystem in the equilibrium coordinate system, respectively. This is the output vector of the refined slow subsystem in the equilibrium coordinate system. Sort the states in descending order of their corresponding Hankel singular values, select the first r states, and ensure their cumulative energy contribution reaches a preset threshold, such as 99%. It can be represented as.

[0096] (20).

[0097] in, The states to be retained corresponding to larger Hankel singular values; Let be the state to be truncated and reduced in order. Equation (20) is the state vector expression.

[0098] Then equation (19) can be transformed into the following form.

[0099] (twenty one).

[0100] in, , , and for The elements in and for Key elements and for The elements in.

[0101] because The corresponding Hankel singular value is extremely small, and its impact on the system dynamics is negligible. According to the singular perturbation principle, it can be approximated as... Then, according to equation (21), we have...

[0102] (twenty two).

[0103] Substituting equation (22) into equation (21) will eliminate the error. This yields a refined model of the slow subsystem after order reduction. And let... Let the state vector of the refined slow subsystem be the state vector after the order reduction. Then the model of the slow subsystem after the order reduction can be represented as follows.

[0104] (twenty three).

[0105] in, and These are the state vector and output vector of the refined slow subsystem model after order reduction, respectively. , , and These are the state matrix, input matrix, output matrix, and transmission matrix of the refined slow subsystem model after order reduction.

[0106] The output relationship of the refined slow subsystem after order reduction, i.e., equation (23) Substituting the expression into the refined fast subsystem, i.e., equation (16), and eliminating the expression... This process recouples the fast and slow subsystems, resulting in the final initial reduced-order model.

[0107] (twenty four).

[0108] in, and They represent correspond and The block matrix. The simultaneous equation (23) The expression and equation (24) are combined and rearranged to obtain the reduced-order model of the whole system that maintains transient accuracy.

[0109] (25).

[0110] in, and These are the total state matrix and total input matrix of the reduced-order model of the entire system, respectively. and They represent correspond and The block matrix.

[0111] This embodiment employs a balanced truncation method to quantitatively reduce the order of a refined slow subsystem based on energy contribution. It also utilizes the singular perturbation principle to safely truncate states with minimal energy, avoiding the subjectivity and blindness of traditional order reduction methods that rely on experience to select retained states. This enhances the reduced model's ability to preserve the system's dominant dynamics. Simultaneously, the reduced slow subsystem is recoupled and integrated with the refined fast subsystem, ensuring consistency between the reduced model and the original system in input-output characteristics. This achieves full automation and high precision in the order reduction process, providing a reliable simplified model for real-time simulation and controller design of large-scale power systems.

[0112] S6. Perform time-domain and frequency-domain simulations on the state-space model and the initial reduced-order model, calculate the H∞ norm deviation and damping ratio deviation, and simultaneously calculate the mode division deviation and state truncation deviation.

[0113] After constructing the initial reduced-order model, how can we objectively determine whether it meets engineering accuracy requirements, and if not, how should we guide subsequent optimization and adjustments? Since the reduction process involves multiple stages, the decisions at each stage affect the accuracy of the final model. Relying solely on a single metric for evaluation makes it impossible to distinguish which specific stage the error originates from, thus hindering targeted corrections.

[0114] Therefore, a multi-dimensional quantitative evaluation system needs to be designed to decompose the error into different sources: damping ratio deviation and H∞ norm deviation evaluate the overall dynamic approximation degree between the reduced model and the original model from the time domain and frequency domain, respectively; mode splitting deviation evaluates the rationality of the selection of dominant state variables in step S4; and state truncation deviation evaluates the degree of energy loss in the balanced truncation reduction in step S5. The specific process is as follows: S601, apply the same unit step input vector or unit impulse input vector to the state space model and the initial reduced model respectively, record the output vector values ​​at each discrete time point, and extract the damping ratio corresponding to the dominant eigenvalues ​​of the state space model and the initial reduced model according to the output vector values, and take the absolute difference between the two damping ratios as the damping ratio deviation.

[0115] S602. Calculate the transfer function matrices for the state-space model and the initial reduced-order model, respectively. The H∞ norm of the difference between the two transfer function matrices is taken as the H∞ norm deviation. The transfer function matrix describes the frequency domain mapping relationship between the system input and output; each element corresponds to the frequency response characteristic from one input to one output.

[0116] S603. Obtain the set of fast subsystem state variables after preliminary division according to the preset time scale threshold, and at the same time count the number of non-dominant state variables merged into the slow subsystem. Use the ratio of the number of non-dominant state variables to the total number of fast subsystem state variables as the mode division deviation.

[0117] S604. Obtain the Hankel singular values ​​corresponding to the truncated reduced state when reducing the order of the refined slow subsystem in the balanced truncation method, as well as all Hankel singular values. The ratio of the sum of the truncated Hankel singular values ​​to the sum of all Hankel singular values ​​is used as the state truncation bias.

[0118] The damping ratio is an indicator reflecting the rate of oscillation decay in a system, and is equal to the absolute value of the real part of the eigenvalue divided by the modulus of the eigenvalue. The damping ratio deviation is used to measure how closely the reduced-order model approximates the original model in terms of dynamic stability characteristics.

[0119] The H∞ norm is a metric in the frequency domain that measures the maximum gain of a transfer function matrix. It is defined as the supremum of the maximum singular value of the matrix across all frequencies, representing the maximum amplification factor of the system for the worst-case input. The H∞ norm deviation is used to measure the accuracy with which a reduced-order model approximates the input-output characteristics of the original model across the entire frequency range.

[0120] Modal partitioning bias reflects how many state variables that originally belonged to the fast subsystem were removed from the fast subsystem because they were deemed to have a negligible impact on the system output during the process of refining the fast and slow subsystem partitioning. It is used to assess the degree of conservatism in the partitioning.

[0121] The state truncation bias reflects the proportion of energy discarded during the order reduction process relative to the total energy, and is used to quantify the approximation error introduced by the order reduction.

[0122] This embodiment achieves a quantitative and objective evaluation of the dynamic accuracy and frequency domain characteristics of the reduced-order model by calculating the damping ratio deviation and H∞ norm deviation. This replaces the traditional method of relying on subjective judgment by manually observing waveforms, effectively avoiding misjudgments caused by inconsistent evaluation standards. At the same time, by calculating the mode division deviation and state truncation deviation, the overall error is decomposed into specific reduction stages. This allows it to clearly identify whether the main state variable selection is too strict (or too loose) or the equilibrium truncation energy is insufficient when the reduced-order model does not meet the accuracy requirements.

[0123] S7. If the H∞ norm deviation and damping ratio deviation meet the preset conditions, the output of the initial reduced-order model is the final reduced-order model. Otherwise, the dominant state variable selection process and the refined slow subsystem reduction process are optimized based on the mode splitting deviation and state truncation deviation.

[0124] When the overall accuracy does not meet the requirements, it is essential to determine which part of the process the error originates from in order to make effective corrections. Blindly adjusting all parameters simultaneously is not only inefficient but may also lead to interference between parameters, making convergence difficult.

[0125] The preset conditions are that the H∞ norm deviation is less than the preset H∞ norm deviation threshold and the damping ratio deviation within the same frequency range is less than the set damping ratio deviation threshold.

[0126] The process of optimizing the dominant state variable screening process and the process of refining the slow subsystem order reduction process is as follows: S701, if the mode division deviation is greater than 0, then in the process of the dominant state variable screening process, the set threshold used to determine the dominant feature value is reduced.

[0127] S702. If the state truncation deviation is greater than 0, then during the process of refining the slow subsystem order reduction, the preset threshold for the cumulative energy contribution used to select the retained state is increased.

[0128] This embodiment achieves closed-loop automatic optimization of the reduced-order model by decomposing the overall accuracy failure into response strategies for modal partitioning deviation and state truncation deviation. When the H∞ norm deviation or damping ratio deviation exceeds the limit, it can automatically determine whether the dominant state variable screening is too strict or the equilibrium truncation energy retention is insufficient, and accordingly reduce the judgment threshold of the dominant eigenvalue or increase the preset threshold of the cumulative energy contribution, thereby improving the model accuracy through the most direct parameter adjustment. This makes the hybrid reduced-order control method adaptive, repeatable, and efficient, improving the feasibility and robustness of engineering application.

[0129] The above embodiments can be implemented, in whole or in part, by software, hardware, firmware, or any other combination thereof. When implemented using software, the above embodiments can be implemented, in whole or in part, in the form of a computer program product.

[0130] Those skilled in the art will recognize that the modules and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0131] In addition, the functional modules in the various embodiments of this application can be integrated into one processing module, or each module can exist physically separately, or two or more modules can be integrated into one module.

[0132] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

[0133] Finally, the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A hybrid order reduction control method for large-scale dynamic systems, characterized in that, Includes the following steps: Construct a state-space model with multiple time scales; Based on a preset time scale threshold, the state-space model is decoupled into a fast subsystem and a slow subsystem with corresponding electromagnetic transient characteristics. The internal state variables of the tachy subsystem are decoupled based on linear coordinate transformation to obtain the decoupled tachy subsystem. Calculate the eigenvalue dominance factor of the decoupled fast subsystem, screen the dominant state variables, and merge the non-dominant state variables into the slow subsystem to obtain the refined fast subsystem and the refined slow subsystem. The refined slow subsystem is reduced in order based on the balanced truncation method and the singular perturbation principle to obtain the reduced slow subsystem model; the reduced slow subsystem model is then integrated with the refined fast subsystem to obtain the initial reduced model. Time-domain and frequency-domain simulations were performed on the state-space model and the initial reduced-order model to calculate the H∞ norm deviation and damping ratio deviation, as well as the mode division deviation and state truncation deviation. If the H∞ norm deviation and damping ratio deviation meet the preset conditions, the output of the initial reduced-order model is the final reduced-order model; otherwise, the dominant state variable selection process and the refined slow subsystem reduction process are optimized based on the mode division deviation and state truncation deviation.

2. The hybrid order reduction control method for large-scale dynamic systems according to claim 1, characterized in that, The general form of the state-space model, without considering the output variables, is as follows: ,in, For state vectors, For the input vector, The state matrix, The input matrix is ​​denoted as .

3. The hybrid order reduction control method for large-scale dynamic systems according to claim 1, characterized in that, The process of decoupling the state-space model into a fast subsystem and a slow subsystem with corresponding electromagnetic transient characteristics is as follows: Based on a preset time scale threshold, the state variables, input variables, and output variables in the state-space model are categorized according to their dynamic response time scale, where: Variables with dynamic response times less than or equal to a preset time scale threshold are classified as the fast dynamic part of the corresponding electromagnetic transient characteristics, and variables with dynamic response times greater than the preset time scale threshold are classified as the slow dynamic part of the corresponding electromechanical transient characteristics. State-space models for the fast and slow subsystems are constructed based on the fast and slow dynamic components, respectively, where: The state vector, input vector, and output vector of the state-space model of the fast subsystem are composed of the state variables, input variables, and output variables of the fast dynamic part, respectively; the state vector, input vector, and output vector of the state-space model of the slow subsystem are composed of the state variables, input variables, and output variables of the slow dynamic part, respectively. The fast subsystem's output is set to equal the slow subsystem's input, and vice versa, thus establishing the coupling relationship between the fast and slow subsystems.

4. The hybrid order reduction control method for large-scale dynamic systems according to claim 1, characterized in that, The process of obtaining the decoupled fast subsystem is as follows: Diagonalize the state matrix of the fast subsystem to construct the eigenvalue diagonal matrix calculation formula; Substituting the eigenvalue diagonal matrix calculation formula into the state-space model of the fast subsystem yields the decoupled fast subsystem.

5. The hybrid order reduction control method for large-scale dynamic systems according to claim 1, characterized in that, The process of calculating the eigenvalue dominance factor of the decoupled fast subsystem and selecting the dominant state variable is as follows: The time-domain expression of the output variable is obtained by performing a Laplace transform and then an inverse Laplace transform on the decoupled fast subsystem. The influence weights of eigenvalues ​​on each output vector are determined based on the time-domain expression, and the dominance factor is output. Summing the dominance factors for the same eigenvalue yields the degree of dominance of the eigenvalue over all output vectors; If the degree of dominance of a certain feature value is greater than a set threshold, then the corresponding feature value is the dominant feature value, and the corresponding dynamic mode is the dominant mode of the fast subsystem. Participation factor analysis is performed on the dominant eigenvalues. If the participating factor is greater than a preset threshold, it will be identified as the dominant state variable.

6. The hybrid order reduction control method for large-scale dynamic systems according to claim 1, characterized in that, The process of reducing the order of the refined slow subsystem based on the balanced truncation method and the singular perturbation principle, to obtain the reduced-order slow subsystem model, is as follows: Based on a non-singular transformation matrix, the state vector of the refined slow subsystem after the transformation is made so that the controllable Gramian matrix and the observable Gramian matrix are equal and diagonal matrices in the equilibrium coordinate system. Substituting the state vector in the equilibrium coordinate system into the refined slow subsystem yields the refined slow subsystem in the equilibrium coordinate system. Arrange the state vectors of the refined slow subsystem in the equilibrium coordinate system in descending order of their corresponding Hankel singular values, select the first r states, and make the cumulative energy contribution reach a preset threshold to obtain the state vector expression. Substituting the state vector expression into the refined slow subsystem in the equilibrium coordinate system yields the transformed refined slow subsystem. Based on the singular perturbation principle, the truncated reduced-order state is identified as 0. After approximating the transformed refined slow subsystem, the truncated reduced-order state is re-substituted into the transformed refined slow subsystem to eliminate the truncated reduced-order state and obtain the preliminary reduced-order refined slow subsystem model. Let the state to be retained corresponding to the larger Hankel singular value in the refined slow subsystem model after the initial order reduction be equal to the state vector of the refined slow subsystem after order reduction, and then we get the slow subsystem model after order reduction.

7. The hybrid order reduction control method for large-scale dynamic systems according to claim 1, characterized in that, The process of integrating the reduced-order slow subsystem model with the refined fast subsystem model to obtain the initial reduced-order model is as follows: Substitute the output vector expression of the reduced-order slow subsystem model into the refined fast subsystem model; By combining the state vector expressions of the reduced-order slow subsystem model with the refined fast subsystem model, the initial reduced-order model is obtained.

8. The hybrid order reduction control method for large-scale dynamic systems according to claim 1, characterized in that, The process of calculating the H∞ norm deviation and damping ratio deviation, and simultaneously calculating the mode division deviation and state cutoff deviation, is as follows: Calculate the state matrices of the state-space model and the initial reduced-order model respectively, and solve for their eigenvalues; The dominant damping ratios of the two are calculated based on the eigenvalues, and the absolute difference between the two dominant damping ratios is taken as the damping ratio deviation. Calculate the transfer function matrices of the state-space model and the initial reduced-order model respectively, and use the H∞ norm of the difference between the two transfer function matrices as the H∞ norm deviation; Obtain the set of fast subsystem state variables after preliminary division based on a preset time scale threshold, and simultaneously count the number of non-dominant state variables merged into the slow subsystem. The ratio of the number of non-dominant state variables to the total number of fast subsystem state variables is used as the mode division bias. Obtain the Hankel singular values ​​corresponding to the truncated reduced state when reducing the order of the refined slow subsystem in the balanced truncation method, as well as all Hankel singular values. The ratio of the sum of the truncated Hankel singular values ​​to the sum of all Hankel singular values ​​is used as the state truncation bias.

9. The hybrid order reduction control method for large-scale dynamic systems according to claim 1, characterized in that, The preset conditions are that the H∞ norm deviation is less than the preset H∞ norm deviation threshold and the damping ratio deviation within the same frequency range is less than the set damping ratio deviation threshold.

10. A hybrid order reduction control method for large-scale dynamic systems according to claim 1, characterized in that, The process of optimizing the selection of dominant state variables and the refinement of slow subsystem order reduction based on modality partitioning bias and state truncation bias is as follows: If the modal division deviation is greater than 0, then during the process of selecting the dominant state variable, the threshold used to determine the dominant feature value is reduced. If the state truncation deviation is greater than 0, then during the refinement of the slow subsystem order reduction process, the preset threshold for the cumulative energy contribution used to select the retained state is increased.

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