A method for estimating the stable domain of a dynamic system based on active learning gaussian process regression

Abstract

The application discloses a power system stability domain estimation method based on active learning Gaussian process regression, comprising the following steps: constructing a state space model of an electric power system to be evaluated; constructing a bounded inverse Lyapunov function; establishing a non-analytic Lyapunov function estimation model based on Gaussian process regression; performing Gaussian process active learning sampling, sampling the region with the maximum uncertainty in the current estimated stability domain according to the predicted mean and the predicted variance, updating the training sample and the Gaussian process regression model, and taking the uncertain parameters as input features to jointly participate in the training of the Gaussian process regression model; considering the model parameter uncertainty and estimating the minimum stability domain; and outputting the minimum stability domain result. The application can provide the bounded inverse Lyapunov function value, the safe stability margin and the minimum stability domain information considering the parameter uncertainty, improve the generalization ability of the Gaussian process regression model, and solve the influence of the model parameter uncertainty in the electric power system on the stability domain.

Description

Technical Field

[0001] This invention relates to the field of dynamical system stability domain estimation technology, and specifically to a dynamical system stability domain estimation method based on active learning Gaussian process regression. Background Technology

[0002] The large-scale grid connection of renewable energy sources, such as wind and solar power, has led to a decrease in grid inertia, weakened frequency and voltage regulation capabilities, and stability issues such as wideband oscillations due to their randomness, volatility, and weak support characteristics. This has resulted in a continuous increase in the computational burden and complexity of power system transient stability analysis. Stability domain estimation methods, by characterizing the set of initial states that converge to a stable equilibrium point after a disturbance, determine the transient stability of the operating point and provide stability margin information, making them an important method in power system transient stability analysis.

[0003] Current methods for estimating the stability region of power systems mainly include:

[0004] (1) Stability domain estimation methods based on time-domain simulation or numerical integration. Among them, the time-domain simulation method requires point-by-point simulation calculation of a large number of initial states, operating modes and fault scenarios. As the number of state variables of power systems containing new energy increases and the operating conditions become more complex, the amount of calculation and complexity increase significantly. This type of method can usually only give the stability judgment of a single sample point. The stability domain boundary needs to be obtained indirectly by relying on a large number of sampling results, and the calculation efficiency is low.

[0005] (2) Direct methods based on Lyapunov functions or energy functions. Direct methods require constructing analytical forms of Lyapunov functions, energy functions, or corresponding stability criteria. Power systems containing new energy sources are characterized by high dimensionality, strong nonlinearity, multi-time-scale coupling, and complex power electronic control links. It is difficult to construct analytical functions. In practical applications, the system model needs to be simplified or approximated, which leads to a certain degree of conservatism in the stability domain estimation results.

[0006] (3) Data-driven stability domain estimation method based on machine learning. The estimation effect of machine learning depends on the quantity, distribution and quality of training samples. Actual operating data is limited by the operating principle of power system with safety and stability as the primary goal. The number of unstable samples under specific topologies is extremely small. Although Monte Carlo simulation can generate a large number of samples, its input space is highly random, which not only increases the training cost, but may also cause overfitting problems due to the uneven quality of samples.

[0007] In addition, the control parameters of new energy power generation equipment are difficult to obtain accurately in actual power grid operation. The models provided by manufacturers are black boxes, and the impact of uncertain parameters on the stability domain is difficult to be effectively assessed. Summary of the Invention

[0008] To overcome the shortcomings and deficiencies of existing technologies, this invention provides a dynamic system stability domain estimation method based on active learning Gaussian process regression. This method enables the construction of a non-analytical Lyapunov function and efficient estimation of the stability domain or minimum stability domain in complex, nonlinear, high-dimensional power systems with uncertainties in control parameters of renewable energy equipment. The obtained stability domain estimation results can be used to determine whether the operating point in the state space can eventually converge to the equilibrium point. Furthermore, it can provide bounded inverse Lyapunov function values, safety margins, and minimum stability domain information considering parameter uncertainties. This solves the problems of strong conservatism, difficulty in handling non-polynomial differential-algebraic equation systems, low sampling efficiency in large-scale dynamic systems under random sampling conditions, and poor generalization ability of Gaussian process regression models in dynamic systems with renewable energy components. It also addresses the technical problem of the impact of model parameter uncertainties on the stability domain in renewable energy power systems. By using uncertain parameters as input features in the training of the Gaussian process regression model and comprehensively evaluating the stability of the same operating point under various parameter values, the minimum stability domain of the dynamic system can be estimated.

[0009] To achieve the above objectives, the present invention adopts the following technical solution:

[0010] This invention provides a method for estimating the stability region of a dynamical system based on active learning Gaussian process regression, comprising the following steps:

[0011] Construct a state-space model of the power system to be evaluated;

[0012] Construct a bounded inverse Lyapunov function and set the function values ​​of sample points outside the stability region to finite values;

[0013] The bounded inverse Lyapunov function is used as the unknown function to be learned and modeled as a Gaussian process. The observed values ​​are calculated based on the trajectory of the sampling points and the Gaussian process regression model is trained.

[0014] The posterior prediction results of the bounded inverse Lyapunov function are calculated using the trained Gaussian process regression model to obtain the prediction mean and prediction variance.

[0015] Active learning sampling of Gaussian process is performed to construct an initial sample set. Based on the predicted mean and predicted variance, the region with the greatest uncertainty in the currently estimated stability region is sampled to update the training samples and Gaussian process regression model, and the stability region is estimated.

[0016] Uncertain parameters are used as input features to participate in the training of the Gaussian process regression model. During prediction, the Latin hypercube sampling method is used to evaluate the stability of different parameter combinations at the same running point and output the minimum stability region result.

[0017] As a preferred technical solution, a state-space model of the power system to be evaluated is constructed, specifically including:

[0018] Construct a nonlinear state-space model or differential-algebraic equation model that reflects the transient process of the power system to be evaluated;

[0019] The sampling space, sampling time, initial sample size, total sample size, stability determination condition parameters, stability domain estimation probability parameters, and function level set parameters of the established nonlinear dynamic system are determined. Uncertain control parameters of the power generation equipment are denoted as uncertain parameters.

[0020] As a preferred technical solution, a bounded inverse Lyapunov function is constructed, expressed as:

[0021] ;

[0022] in, This represents a bounded inverse Lyapunov function. Represents the state variables of a dynamic system. Indicates time, Let be the trajectory function of the dynamical system in the characteristic space. for Class function, It is the stability region of the equilibrium point of the dynamic system, ||·|| denotes the vector norm, and exp(·) denotes the exponential function;

[0023] When the state point of the dynamic system is within the stability region, the function value is obtained by the exponential transformation of the state trajectory integral;

[0024] When the state point of the dynamical system is outside the stability region, the function value is 1. The function satisfies the Lyapunov function condition within the stability region, and for any 0... <c<1:

[0025] ;

[0026] Where c represents the level set parameter of the bounded inverse Lyapunov function.

[0027] As a preferred technical solution, The class function sets a scaling factor. function satisfy:

[0028] ;

[0029] in, Let be any positive real number. Describes the space of two-order continuously differentiable functions. To represent the positive real exponent parameter, we introduce a scaling factor, and obtain:

[0030] ;

[0031] Where s represents the scaling factor.

[0032] As a preferred technical solution, the bounded inverse Lyapunov function is used as the unknown function to be learned and modeled as a Gaussian process, specifically expressed as:

[0033] ;

[0034] in, It is a mean function; Let covariance function be used. This represents a bounded inverse Lyapunov function;

[0035] The Gaussian process regression model is expressed as:

[0036] ;

[0037] in, This represents the bounded inverse Lyapunov function approximating the observations. This represents Gaussian white noise, and , This represents the noise variance.

[0038] As a preferred technical solution, the radial basis covariance function is used as the covariance function of the Gaussian process regression model, expressed as:

[0039] ;

[0040] in, Indicates the signal variance. Represents the length scale matrix;

[0041] Estimate model hyperparameters by maximizing the log-likelihood function:

[0042] ;

[0043] in, Let y represent the probability density function, y represent the training label vector, and K be the probability density function. The covariance matrix formed Implicit in K, I represents the identity matrix, and n is the number of training samples.

[0044] As a preferred technical solution, an initial sample set is constructed, and samples are taken from the region with the greatest uncertainty within the currently estimated stable region based on the predicted mean and predicted variance. The training samples and the Gaussian process regression model are then updated, specifically including:

[0045] Select initial sample points, perform time-domain trajectory simulation for each initial sample point and calculate the bounded inverse Lyapunov function value to construct the initial stability region and the initial training sample set;

[0046] Using the current training sample set to train a Gaussian process regression model, the next candidate sample point is searched, which is represented as:

[0047] ;

[0048] in, Let X represent the candidate sample points selected during the (i+1)th active learning iteration, and let X represent the sampling space. and Let represent the posterior prediction mean and posterior prediction variance of the Gaussian process regression model at the state point after the i-th training iteration, respectively; argmax represents the state point where the objective function is maximized; and st represents the constraints. This represents the inverse cumulative distribution function of the standard Gaussian distribution;

[0049] Perform trajectory integration on the candidate sample point. If the candidate sample point satisfies the stability criterion, calculate its bounded inverse Lyapunov function value; otherwise, set its function value to 1.

[0050] Add the candidate sample points and their function values ​​to the training sample set, and retrain the Gaussian process regression model until the total number of samples is reached.

[0051] As a preferred technical solution, stability region estimation is performed, specifically including:

[0052] For a trained Gaussian process regression model, given the level set parameters c and the probability parameter δ, calculate the predicted mean for the test state point. Prediction variance and the corresponding stability region estimation set , represented as:

[0053] ;

[0054] ;

[0055] in, and Let represent the mean and variance of the Gaussian posterior prediction distribution followed by the state points to be tested, respectively, and D represent the set of candidate state points. This represents the set of stability region estimates.

[0056] As a preferred technical solution, uncertain parameters are used as input features to participate in the training of the Gaussian process regression model. During prediction, the Latin hypercube sampling method is used to evaluate the stability of different parameter combinations at the same running point, specifically including:

[0057] The state variables and uncertain parameters are used together as inputs to the Gaussian process regression model, and the corresponding training labels are calculated.

[0058] For the same operating point, the Latin hypercube sampling method is used to generate multiple sets of uncertain parameter combinations, and it is determined whether the stability region discrimination condition is met under each parameter combination.

[0059] If all sampled parameter values ​​converge to the equilibrium point, then the system is considered to be within its minimum stability region.

[0060] As a preferred technical solution, the determination of whether the stability region discrimination condition is met under each parameter combination is specifically expressed as follows:

[0061] ;

[0062] in, t represents the estimated minimum stability region of the system. n Let ξ represent the simulation termination time, ξ represent the conditional parameters for stability determination, and u represent a set of uncertain parameter values. This indicates that when the uncertain parameter takes the value u, the dynamic system operates from the initial state to t. n The state trajectory function value, This represents the set of uncertain parameters extracted using the Latin hypercube sampling method.

[0063] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0064] (1) This invention proposes a non-analytical Lyapunov function construction method and a stability domain estimation method by combining the bounded inverse Lyapunov function and the Gaussian process regression model. This method can be effectively applied to power systems whose dynamic models are characterized by non-polynomial differential-algebraic equations.

[0065] (2) The bounded inverse Lyapunov function proposed in this invention supplements the points outside the stable domain on the basis of the traditional inverse Lyapunov function. Unstable sample points can also be assigned finite function values, which solves the problem that the traditional inverse Lyapunov function cannot utilize unstable sample information.

[0066] (3) The present invention distributes the bounded inverse Lyapunov function values ​​of all stable state points in the state space evenly to the interval between 0 and 1 by using scaling coefficients, thereby improving the ability of the Gaussian process regression model to approximate the function values ​​inside the stable domain and the generalization performance.

[0067] (4) The present invention introduces a Gaussian process active learning framework, which can estimate the stable domain results with higher accuracy with a smaller number of samples.

[0068] (5) The present invention sets a probability margin based on the prediction mean and prediction variance of Gaussian process, retains a safety margin during prediction, avoids misjudging unstable operating points as stable operating points, and can maintain a 100% recall rate in the example.

[0069] (6) The present invention can effectively consider the impact of uncertain parameters in the dynamic system on the stability region. By taking the uncertain parameters as input features and combining them with Latin hypercube sampling to evaluate the stability of the same operating point under various parameter values, the minimum stability region of the system is obtained. Attached Figure Description

[0070] Figure 1 This is a schematic diagram of the overall process of the dynamic system stability domain estimation method based on active learning Gaussian process regression of the present invention;

[0071] Figure 2 This is a schematic diagram illustrating the specific process of the dynamic system stability domain estimation method based on active learning Gaussian process regression of the present invention.

[0072] Figure 3 This is a wiring diagram of the IEEE-39 node system of the present invention;

[0073] Figure 4 This is a schematic diagram of the projection of the stability region estimation results of the present invention into the state space of each generator;

[0074] Figure 5 This is a heatmap of the bounded inverse Lyapunov function value of the present invention;

[0075] Figure 6 This invention considers the minimum stability region estimation result under model parameter uncertainty. Detailed Implementation

[0076] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0077] Example

[0078] like Figure 1 , Figure 2 As shown, this embodiment provides a method for estimating the stability domain of a dynamical system based on active learning Gaussian process regression, including the following steps:

[0079] S1. Establish a state-space model of the power system containing new energy sources to be evaluated: determine the state variables, equilibrium point, sampling space, and control parameters of the new energy power generation equipment that may have uncertainties in the target system. Specific steps include:

[0080] S11. Based on the dynamic characteristics of the power system containing new energy sources to be evaluated, establish a nonlinear state-space model or differential-algebraic equation model that can reflect its transient process.

[0081] S12. Determine the sampling space X and sampling time of the established nonlinear dynamic system. Initial sample size The total sample size N, the conditional parameter ξ for stability determination, and the probability parameter for stability region estimation. The function level set parameter c, and the control parameters of the new energy power generation equipment that may have uncertainties are denoted as uncertain parameters u, and u satisfies the set of maximum and minimum value constraints. Let this be the range of parameter values;

[0082] S2. Constructing a bounded inverse Lyapunov function: Based on the inverse Lyapunov theorem, the traditional inverse Lyapunov function is bounded, and the numerical distribution characteristics of large-scale nonlinear dynamical systems are improved by scaling coefficients. Specifically, this includes:

[0083] S21. For a nonlinear dynamical system, if its equilibrium point is asymptotically stable and has a corresponding stability region, then the bounded inverse Lyapunov function can be expressed as:

[0084] ;

[0085] In the formula, This represents a bounded inverse Lyapunov function; Represents the system's state variables. Indicates time, Let be the trajectory function of the dynamical system in the characteristic space. for Class function, Let ||·|| be the stability region of the equilibrium point, ||·|| denotes the vector norm, and exp(·) denotes the exponential function. When the state point is within the stability region, the function value is obtained by the exponential transformation of the integral of the state trajectory; when the state point is outside the stability region, the function value is 1. The function satisfies the Lyapunov function condition within the stability region, and for any 0... <c<1:

[0086] ;

[0087] In the formula, c represents the level set parameter of the bounded inverse Lyapunov function.

[0088] S22, Selection Class function When, a scaling factor s is set for it, where, function satisfy:

[0089] ;

[0090] In the formula, a is any positive real number. Describes the space of two-order continuously differentiable functions. To represent the positive real exponent parameter, and to improve the problem of function values ​​concentrating around 1, a scaling factor s is introduced, resulting in:

[0091] ;

[0092] In subsequent applications, the scaling factor s is taken as the optimal value obtained from the test.

[0093] S3. Establish a nonanalytic Lyapunov function estimation model based on Gaussian process regression: Model the unknown bounded inverse Lyapunov function as a Gaussian process, calculate the observed values ​​based on the trajectory of the sampling points, and train the Gaussian process regression model to obtain the predicted mean and predicted variance. Specifically, this includes:

[0094] S31, the bounded inverse Lyapunov function As an unknown function to be learned, it is modeled as a Gaussian process, that is:

[0095] ;

[0096] In the formula, It is a mean function; Here, the covariance function is... All are prior distribution functions;

[0097] S32. Considering numerical observation errors, establish a Gaussian process regression model, namely:

[0098] ;

[0099] In the formula, This represents the approximate observation of the bounded inverse Lyapunov function obtained through numerical methods. This represents Gaussian white noise, and , Indicates the noise variance;

[0100] S33. The radial basis covariance function is used as the covariance function of the Gaussian process regression model to describe the correlation between different state points, as follows:

[0101] ;

[0102] In the formula, Indicates the signal variance; Representing the length scale matrix, these parameters (using...) (represented by) and noise standard deviation Together, they constitute the hyperparameters of the Gaussian process regression model;

[0103] Estimate model hyperparameters by maximizing the log-likelihood function:

[0104] ;

[0105] In the formula, Let y represent the probability density function, y represent the training label vector, and K be the probability density function. The covariance matrix formed Implicit in K, I represents the identity matrix, and n is the number of training samples;

[0106] S34. For the test input, the posterior prediction result of the bounded inverse Lyapunov function is calculated using the trained Gaussian process regression model to obtain the prediction mean and prediction variance. The prediction mean is used to characterize the estimation result of the function value corresponding to the test state point, and the prediction variance is used to characterize the uncertainty of the estimation result.

[0107] S4. Perform Gaussian process active learning sampling: Construct an initial sample set from the vicinity of the equilibrium point. During training, based on the mean and uncertainty information output by the Gaussian process regression model, prioritize sampling in the region with the greatest uncertainty within the currently estimated stability region to update the training samples and model, and perform stability region estimation. Specifically, this includes:

[0108] S41. Select initial sample points in the initial sampling space near the equilibrium point, perform time-domain trajectory simulation on each initial sample point and calculate the bounded inverse Lyapunov function value to form an initial stable region and an initial training sample set around the equilibrium point.

[0109] S42. Begin a complete exploration of the stable region boundary. Train a Gaussian process regression model using the current training sample set, and search for the next candidate sample point through the following optimization problem:

[0110] ;

[0111] In the formula, Let X represent the candidate sample points selected during the (i+1)th active learning iteration, and let X represent the sampling space. and Let represent the posterior prediction mean and posterior prediction variance of the Gaussian process regression model at the state point after the i-th training iteration, respectively; argmax represents the state point where the objective function is maximized; and st represents the constraints. This represents the inverse cumulative distribution function of the standard Gaussian distribution;

[0112] S43. Perform trajectory integration on the candidate sample point. If the candidate sample point satisfies the stability discrimination condition, calculate its bounded inverse Lyapunov function value; otherwise, set its function value to 1.

[0113] S44. Add the candidate sample points and their function values ​​to the training sample set, and retrain the Gaussian process regression model. When the number of samples is less than the total number of samples, repeat steps S43 to S45 until the total number of samples is reached.

[0114] S45. For the trained Gaussian process regression model, given the level set parameters c and probability parameters δ; for the state point to be tested, calculate the predicted mean. Prediction variance and the corresponding stability region estimation set ;

[0115] ;

[0116] ;

[0117] In the formula, and Let represent the mean and variance of the Gaussian posterior prediction distribution followed by the state points to be tested, respectively, and D represent the set of candidate state points. This represents the set of stability region estimates.

[0118] S5. Considering model parameter uncertainty and estimating the minimum stability region: Uncertain parameters are also included as input features in the training of the Gaussian process regression model. During prediction, the Latin hypercube sampling method is used to evaluate the stability of different parameter combinations at the same running point. If all sampled parameter values ​​converge to the equilibrium point, then the point is considered to be within the minimum stability region of the system. Specifically, this includes:

[0119] S51. Use the state variable and the uncertain parameter u as inputs to the Gaussian process regression model, and calculate the corresponding training labels;

[0120] S52. For the same state point, use the Latin hypercube sampling method to generate multiple sets of uncertain parameter combinations, and determine whether the state point satisfies the stability domain discrimination condition under each parameter combination.

[0121] ;

[0122] In the formula, t represents the estimated minimum stability region of the system. n Let ξ represent the simulation termination time, ξ represent the conditional parameters for stability determination, and u represent a set of uncertain parameter values. This indicates that when the uncertain parameter takes the value u, the dynamic system operates from the initial state to t. n The state trajectory function value, This represents the set of uncertain parameters extracted using the Latin hypercube sampling method;

[0123] S53. Output the minimum stability region result and compare it with the ordinary stability region estimation result without considering parameter uncertainty.

[0124] This embodiment is used to verify the applicability of the method of the present invention in transient stability domain estimation of high-dimensional differential-algebraic power systems at IEEE-39 nodes. The results are presented using accuracy rates. and recall rate To evaluate model performance, accuracy measures the model's classification accuracy across all test samples, while recall measures the model's classification accuracy on unstable test samples.

[0125] To further verify the applicability of the method of the present invention in analyzing the transient stability problem of power systems, this embodiment... Figure 3 The system was tested using the IEEE-39 node system shown. This system comprises 10 generators, 39 nodes, and 46 transmission lines. In power flow calculations, node 31 was designated as the slack node, with its node voltage magnitude and phase angle fixed at 0.982 and 0, respectively. In transient stability calculations, the synchronous generators on this node were designated as slack machines, and their dynamic state variables were not considered. After eliminating load nodes and intermediate nodes from the node admittance matrix using Gaussian elimination, an 18th-order dynamic model of the system was constructed based on the synchronous machine rotor motion equations and electromagnetic power calculation formulas. The model contains 18 differential equations and 18 algebraic equations, which are difficult to calculate using most stability domain estimation methods based on Lyapunov functions. Some system parameters and equilibrium points are shown in Table 1; other relevant parameters adopt the standard IEEE-39 node system parameters.

[0126] Table 1. Some parameters of the IEEE-39 Node system

[0127]

[0128] First, experiments were conducted without considering the uncertainty of the model parameters. The covariance function of the Gaussian process regression model still adopted the radial basis covariance function; the scaling factor was used in the stability region estimation. Function level set parameters and probability parameters (To avoid symbols and angles) A conflict occurs, and the probability parameter is... Change to use (Represented); During sampling, the ode23t function built into MATLAB is used to calculate the system's differential-algebraic equations; the sampling space is and The initial sampling space is one-tenth of its size; the initial number of samples... and total sample size The sampling times were set to 50 and 1500 respectively; and sampling step size The values ​​are 100 seconds and 0.01 seconds respectively. The projections of the final stability region estimation results into the state spaces corresponding to each generator are as follows: Figure 4 As shown, the model training time was 67246.93 seconds. It should be noted that the actual stability region in the figure was obtained by performing a stability scan on the grid points in the state space corresponding to each generator, rather than directly calculating the actual stability region of the entire system and then projecting it.

[0129] To further illustrate the stability domain estimation accuracy of the method of this invention, a trained Gaussian process regression model was applied to a test set for testing. The test set in this embodiment consists of two parts: one part is randomly generated in different test spaces using the rand function, divided into 4 groups, each with 10,000 test samples; the other part is generated by setting a random three-phase short-circuit fault in any transmission line and performing transient stability calculations using DSP simulation software, totaling 3,000 test samples. The composition of the test set and the test results are shown in Table 2. The average test time for the first part was 116.61 seconds, and the average test time for the second part was 17.19 seconds.

[0130] Table 2. Composition and test results of the test set for the IEEE-39 Node system without considering model parameter uncertainties.

[0131]

[0132] Through observation Figure 4 As can be seen from Table 2, the results obtained when estimating the stability region of a power system using the method of this invention exhibit strong conservatism. The reasons may include: firstly, the sample size is relatively small, and the model is insufficient to estimate a more accurate stability region; secondly, points near the actual stability region boundary have similar bounded inverse Lyapunov function values, such as... Figure 5 As shown, the model may prematurely determine that it has reached the actual stability domain boundary during the active learning process and stop further sampling and correction of these regions. However, from the perspective of engineering safety, the stability domain projection estimated by the method of this invention is located inside the corresponding actual stability domain projection, and the recall rate of the model in the test set reaches 100%, which can meet the safety and stability requirements of the power system.

[0133] To further verify the effectiveness of the method proposed in this embodiment in handling uncertain parameters in the dynamic model of a power system, this embodiment assumes that the damping coefficients of all synchronous generators participating in the transient calculation are... All are uncertain parameters, and their uncertainty range is set within the actual value. Within the scope, the total number of samples The value is set to 2000, and the remaining settings are the same as those described above when the influence of uncertain parameters is not considered. Using the Latin hypercube sampling method, 1000 different sets of uncertain parameter values ​​are extracted, and multiple stability assessments are performed on the same operating point in the state space to estimate the minimum stability region of the system. Its projection into the state space corresponding to each generator is as follows: Figure 6 As shown in Table 2. Furthermore, the model was applied to the same test set as in Table 2, and the results are shown in Table 3:

[0134] Table 3 Test results considering model parameter uncertainties

[0135]

[0136] It is evident that, after considering the impact of uncertain parameters in the system on the stability region, the minimum stability region estimation result lies within the aforementioned stability region estimation result without considering model parameter uncertainties, consistent with the general rule that the smaller the damping coefficient of the synchronous generator, the worse the overall transient stability of the system. Furthermore, the model achieves an accuracy of over 72% and a recall of 100% on the test set, without misclassifying unstable sample points, meeting the power system safety and stability requirements, thus proving the effectiveness of the method in the IEEE-39 Node system.

[0137] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A method for estimating the stability region of a dynamical system based on active learning Gaussian process regression, characterized in that, Includes the following steps: Construct a state-space model of the power system to be evaluated; Construct a bounded inverse Lyapunov function and set the function values ​​of sample points outside the stability region to finite values; The bounded inverse Lyapunov function is used as the unknown function to be learned and modeled as a Gaussian process. The observed values ​​are calculated based on the trajectory of the sampling points and the Gaussian process regression model is trained. The posterior prediction results of the bounded inverse Lyapunov function are calculated using the trained Gaussian process regression model to obtain the prediction mean and prediction variance. Active learning sampling of Gaussian process is performed to construct an initial sample set. Based on the predicted mean and predicted variance, the region with the greatest uncertainty in the currently estimated stability region is sampled to update the training samples and Gaussian process regression model, and the stability region is estimated. Uncertain parameters are used as input features to participate in the training of the Gaussian process regression model. During prediction, the Latin hypercube sampling method is used to evaluate the stability of different parameter combinations at the same running point and output the minimum stability region result.

2. The method for estimating the stability region of a dynamical system based on active learning Gaussian process regression according to claim 1, characterized in that, Constructing a state-space model of the power system to be evaluated includes: Construct a nonlinear state-space model or differential-algebraic equation model that reflects the transient process of the power system to be evaluated; The sampling space, sampling time, initial sample size, total sample size, stability determination condition parameters, stability domain estimation probability parameters, and function level set parameters of the established nonlinear dynamic system are determined. Uncertain control parameters of the power generation equipment are denoted as uncertain parameters.

3. The method for estimating the stability region of a dynamical system based on active learning Gaussian process regression according to claim 1, characterized in that, Construct a bounded inverse Lyapunov function, expressed as: ； in, This represents a bounded inverse Lyapunov function. Represents the state variables of a dynamic system. Indicates time, Let be the trajectory function of the dynamical system in the characteristic space. for Class function, It is the stability region of the equilibrium point of the dynamic system, ||·|| denotes the vector norm, and exp(·) denotes the exponential function; When the state point of the dynamic system is within the stability region, the function value is obtained by the exponential transformation of the state trajectory integral; When the state point of the dynamical system is outside the stability region, the function value is 1. The function satisfies the Lyapunov function condition within the stability region, and for any 0... <c<1: ； Where c represents the level set parameter of the bounded inverse Lyapunov function.

4. The method for estimating the stability region of a dynamical system based on active learning Gaussian process regression according to claim 3, characterized in that, Give The class function sets a scaling factor. function satisfy: ； in, Let be any positive real number. Describes the space of two-order continuously differentiable functions. To represent the positive real exponent parameter, we introduce a scaling factor, and obtain: ； Where s represents the scaling factor.

5. The method for estimating the stability region of a dynamical system based on active learning Gaussian process regression according to claim 1, characterized in that, The bounded inverse Lyapunov function is taken as the unknown function to be learned and modeled as a Gaussian process, specifically represented as: ； in, It is a mean function; Let covariance function be used. This represents a bounded inverse Lyapunov function; The Gaussian process regression model is expressed as: ； in, This represents the bounded inverse Lyapunov function approximating the observations. This represents Gaussian white noise, and , This represents the noise variance.

6. The method for estimating the stability region of a dynamical system based on active learning Gaussian process regression according to claim 5, characterized in that, The radial basis covariance function is used as the covariance function of the Gaussian process regression model, expressed as: ； in, Indicates the signal variance. Represents the length scale matrix; Estimate model hyperparameters by maximizing the log-likelihood function: ； in, Let y represent the probability density function, y represent the training label vector, and K be the probability density function. The covariance matrix formed Implicit in K, I represents the identity matrix, and n is the number of training samples.

7. The method for estimating the stability region of a dynamical system based on active learning Gaussian process regression according to claim 1, characterized in that, Construct an initial sample set, sampling the region with the greatest uncertainty within the currently estimated stability region based on the predicted mean and predicted variance, and update the training samples and the Gaussian process regression model, specifically including: Select initial sample points, perform time-domain trajectory simulation for each initial sample point and calculate the bounded inverse Lyapunov function value to construct the initial stability region and the initial training sample set; Using the current training sample set to train a Gaussian process regression model, the next candidate sample point is searched, which is represented as: ； in, Let X represent the candidate sample points selected during the (i+1)th active learning iteration, and let X represent the sampling space. and Let represent the posterior prediction mean and posterior prediction variance of the Gaussian process regression model at the state point after the i-th training iteration, respectively; argmax represents the state point where the objective function is maximized; and st represents the constraints. This represents the inverse cumulative distribution function of the standard Gaussian distribution; Perform trajectory integration on the candidate sample point. If the candidate sample point satisfies the stability criterion, calculate its bounded inverse Lyapunov function value; otherwise, set its function value to 1. Add the candidate sample points and their function values ​​to the training sample set, and retrain the Gaussian process regression model until the total number of samples is reached.

8. The method for estimating the stability region of a dynamical system based on active learning Gaussian process regression according to claim 7, characterized in that, The stability region estimation includes: For a trained Gaussian process regression model, given the level set parameters c and the probability parameter δ, calculate the predicted mean for the test state point. Prediction variance and the corresponding stability region estimation set , represented as: ； ； in, and Let represent the mean and variance of the Gaussian posterior prediction distribution followed by the state points to be tested, respectively, and D represent the set of candidate state points. This represents the set of stability region estimates.

9. The method for estimating the stability region of a dynamical system based on active learning Gaussian process regression according to claim 1, characterized in that, Uncertain parameters are used as input features in the training of the Gaussian process regression model. During prediction, the stability of different parameter combinations at the same running point is evaluated using the Latin hypercube sampling method. Specifically, this includes: The state variables and uncertain parameters are used together as inputs to the Gaussian process regression model, and the corresponding training labels are calculated. For the same operating point, the Latin hypercube sampling method is used to generate multiple sets of uncertain parameter combinations, and it is determined whether the stability region discrimination condition is met under each parameter combination. If all sampled parameter values ​​converge to the equilibrium point, then the system is considered to be within its minimum stability region.

10. The method for estimating the stability region of a dynamical system based on active learning Gaussian process regression according to claim 9, characterized in that, The determination of whether the stability region criterion is satisfied under various parameter combinations is specifically expressed as follows: ； in, t represents the estimated minimum stability region of the system. n Let ξ represent the simulation termination time, ξ represent the conditional parameters for stability determination, and u represent a set of uncertain parameter values. This indicates that when the uncertain parameter takes the value u, the dynamic system operates from the initial state to t. n The state trajectory function value, This represents the set of uncertain parameters extracted using the Latin hypercube sampling method.

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