Transient voltage instability identification method based on Koopman operator

Abstract

The invention discloses a transient voltage instability identification method based on a Koopman operator, and belongs to the field of power system operation and maintenance. The method comprises the following steps: firstly, on the basis of a Koopman operator theory, extracting a leading Koopman operator in voltage response after large disturbance through a Hankel matrix combined with an extended dynamic mode decomposition (Hankel-EDMD) method; secondly, a dominant Koopman operator is adopted to carry out continuation on a voltage response dominant component, and an enough voltage dominant mode time sequence is obtained; on the basis, a transient voltage stability discrimination model of a Koopman operator coupled with a Lyapunov exponent is provided, due to the dimension expansion effect of a Hankel matrix on measured data, the timeliness of the method is good, and therefore the method has high practical application value.

Description

Technical Field

[0001] The invention belongs to the technical field of power system operation and maintenance, and in particular relates to a transient voltage instability identification method based on a Koopman operator. Background Art

[0002] Transient voltage instability is one of the important factors threatening the stability of the power system. The dynamic reactive power reserve and support capacity of the power grid with a high proportion of renewable energy have dropped sharply. The control models and operating characteristics of the grid-connected renewable energy are diverse, which leads to rapid fluctuations in the reactive voltage of the system after the fault, making the voltage stability problem more prominent. In recent years, with the development of wide-area measurement technology, it has provided an important technical means for the analysis of power voltage stability. The Lyapunov energy function method based on measurement data has gradually attracted widespread attention from experts and scholars due to its low model dependence. Based on chaos theory, this method relies on the strong coupling relationship between the large-disturbance transient voltage collapse of the power system and the chaotic characteristics of the nonlinear dynamic system, and uses the maximum Lyapunov exponent value jump to achieve stability judgment. At present, the research on transient voltage stability judgment based on the maximum Lyapunov index is still in progress, and the research work on improving its timeliness, measurement error interference, and measurement window selection still needs to be further promoted.

[0003] At present, the method of transient voltage stability judgment based on the maximum Lyapunov exponent requires more observation data and a longer analysis time window for calculation, which takes a long time to calculate. It is not suitable for instability identification after power system faults and is difficult to adapt to modern power systems with changeable operating modes.

[0004] Therefore, a new technical solution is urgently needed in the prior art to solve this problem. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide a transient voltage instability identification method based on Koopman operator to solve the technical problem that the existing transient voltage stability judgment method has poor timeliness and is difficult to adapt to the modern power system with large-scale new energy grid connection.

[0006] The transient voltage instability identification method based on Koopman operator includes the following steps, and the following steps are performed in sequence:

[0007] Step 1: Collect the voltage response data online and pre-process the collected data. Determine whether the system has a fault according to the set conditions. If no fault occurs, continue to collect data. If a fault occurs, measure and obtain the voltage response data of each sampling point after the fault, and construct the t1 to t2 at sampling point 1 to sampling point m based on the voltage response data. k The continuous time series data matrix of the moment;

[0008] Step 2: Use the constructed continuous time series data matrix to take the time series from t1 to t k-1 The sampling value vector at the time corresponds to a Hankel matrix with p rows and q columns; then the same sampling point t2 to t k The sampling value vector at the time corresponds to a Hankel matrix with p rows and q columns; the proportional factor is obtained through two Hankel matrices; and the data matrix of the voltage response data is reconstructed through the proportional factor and the Hankel matrix at different sampling points;

[0009] Step 3: Based on the Koopman operator theory, the observation function is used to map the reconstructed data matrix to the observation space;

[0010] Step 4: Obtain the Koopman operator by solving the least squares optimization problem;

[0011] Step 5: Perform eigendecomposition on the Koopman operator to obtain its eigenvalue, right eigenvector, and left singular vector;

[0012] Step 6: Calculate and obtain the amplitude of each Koopman operator, and establish a time domain prediction model of the voltage dominant mode through the amplitude of the Koopman operator;

[0013] Step 7. A transient voltage stability discrimination model based on the Koopman operator is established through the time domain prediction model of the voltage-dominant mode. The Lyapunov index is defined in the transient voltage stability discrimination model based on the Koopman operator, and the calculation result of the Lyapunov index is used as the final result of the model. If the result of the Lyapunov index is greater than 0, it indicates that transient voltage instability occurs in the system at this time; if the result of the Lyapunov index is not greater than 0, it indicates that the fault occurring in the system at this time is not transient voltage instability, and return to step 1 to wait for the next detection.

[0014] In step 1, the voltage response data is obtained through time domain simulation or vector measurement unit PMU, and the voltage response data is expressed as:

[0015] x k+1 =f(x k );

[0016] In the formula, x k is the sampling value of the system voltage response at the kth sampling moment, and x k =x(kΔt); Δt is the sampling interval.

[0017] In step 1, the time from t1 to t k The continuous time series data matrix of the moment is expressed as follows:

[0018]

[0019] In the formula, From t1 to t k The numerical matrix composed of the sampling values ​​at the moment; x(c m ,t k ) indicates t k Sample data at sampling point m at the moment; x l Indicates the tth l A vector consisting of the data values ​​of all sampling points at the time instant.

[0020] In step 2, the interval from t1 to t k-1 The sampling value vector at the time corresponds to the Hankel matrix with p rows and q columns as follows:

[0021]

[0022] Where m represents the sampling point;

[0023] The same sampling point t2 to t k The sampling value vector at the time corresponds to the Hankel matrix with p rows and q columns as follows:

[0024]

[0025] Define the scale factor as follows:

[0026]

[0027] In the formula, is the last column of the Hankel matrix formed at sampling point m, is the first column of the Hankel matrix formed at sampling point m, and ||·|| is the Euclidean norm of the matrix.

[0028] The data matrix of the voltage response data reconstructed in step 2 is expressed as follows:

[0029]

[0030] In the formula, X' and Y' are The voltage response matrix after the Hankel matrix expansion at different sampling times. In step 3, the reconstructed data matrix is ​​mapped to the observation space and expressed as follows:

[0031] X lift =φ(X')

[0032] Y lift =φ(Y');

[0033] In the formula, X lift and Y liftare the data matrices that map X' and Y' to the observation space, and φ is the observation function;

[0034] According to Koopman operator theory, we have:

[0035]

[0036] In the formula, is the Koopman operator.

[0037] The Koopman operator obtained in step 4 is expressed as follows:

[0038]

[0039] In the formula, Represents the matrix X lift The Moore-Penrose pseudoinverse.

[0040] The amplitude of the Koopman operator in step 5 is expressed as follows:

[0041]

[0042] In the formula, X1 is a matrix The first column of data is the initial condition; φ j is the jth Koopman operator, φ j =Uw j ;w j is the right eigenvector after eigendecomposition of the Koopman operator, and U is the left singular vector after eigendecomposition of the Koopman operator.

[0043] The time domain prediction model of the voltage dominant mode in step 6 is shown as follows:

[0044] Δd j (t)=(λ j ) t-1 B j ;

[0045] In the formula, Δd j (t) is the dominant modal response after extension; λ j is the eigenvalue after eigendecomposition of the Koopman operator.

[0046] The transient voltage stability discrimination model based on the Koopman operator in step 7 is as follows:

[0047] In the formula, Λ(kΔt) is the Lyapunov exponent, N is the set time interval parameter; Δd j ((k+i)Δt),Δd j((k+i-1)Δt),Δd j (iΔt), Δd j ((i-1))Δt) are the dominant modal responses after extension times (k+i)Δt, (k+i-1)Δt, iΔt, and (i-1))Δt, respectively.

[0048] Through the above design scheme, the present invention can bring the following beneficial effects:

[0049] Firstly, based on the Koopman operator theory, the present invention extracts the dominant Koopman operator in the voltage response after a large disturbance by combining the Hankel matrix with the extended dynamic mode decomposition (Hankel-EDMD) method; secondly, the dominant Koopman operator is used to extend the dominant component of the voltage response to obtain sufficient voltage dominant modal time series; on this basis, a transient voltage stability discrimination model of the Koopman operator coupled with the Lyapunov index is proposed. Due to the dimensional expansion effect of the Hankel matrix on the measured data, the timeliness of the method is better, and thus it has a higher practical application value.

[0050] The present invention realizes transient voltage stability discrimination based on the short-time window measurement data obtained by the PMU (Phasor Measurement Unit) after the fault. This method is based on the system PMU measurement data, avoids the complex calculation of system modeling, improves the practical application efficiency of the discrimination method, and provides effective data support for the discrimination of system transient voltage instability with rapid development and serious consequences, thus having high practical application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] The present invention is further described below with reference to the accompanying drawings and specific embodiments:

[0052] Figure 1 It is a flowchart of the transient voltage instability identification method based on Koopman operator of the present invention;

[0053] Figure 2 A Nordic system wiring diagram in an embodiment of the transient voltage instability identification method based on the Koopman operator of the present invention;

[0054] Figure 3 This is a diagram showing the comparison results of mode energy calculation in an embodiment of the transient voltage instability identification method based on the Koopman operator of the present invention;

[0055] Figure 4 It is a comparison diagram of the dominant mode extension of the transient voltage instability identification method based on the Koopman operator of the present invention.

[0056] Figure 5 This is a comparison diagram of voltage instability identification results in an embodiment of a transient voltage instability identification method based on the Koopman operator of the present invention. DETAILED DESCRIPTION

[0057] Transient voltage instability identification method based on Koopman operator, such as Figure 1 As shown, the process includes the following steps, and the following steps are performed in sequence:

[0058] Step 1: First, collect the voltage response data online and perform preprocessing such as data filling, data cleaning and normalization on the collected data. Determine whether the system has a fault according to the set conditions. If no fault occurs, continue to collect data. If a fault occurs, obtain the voltage response data of each sampling point after the fault through time domain simulation or vector measurement unit (Phasor Measurement Unit, PMU). The voltage response data is expressed as:

[0059] x k+1 =f(x k );

[0060] In the formula, x k is the sampling value of the system voltage response at time k and x k =x(kΔt); Δt is the sampling interval.

[0061] From sampling point 1 to sampling point m, t1 to t k The continuous time series data matrix of the moment is expressed as follows:

[0062]

[0063] In the formula, From t1 to t k The numerical matrix composed of the sampling values ​​at the moment; x(c m ,t k ) indicates t k Sample data at sampling point m at the moment; x l Indicates the tth l A vector consisting of the data values ​​of all sampling points at the time instant.

[0064] Step 2: Use the sampling point t1 to t k-1 The sampling value vector at the time corresponds to the Hankel matrix with p rows and q columns as follows:

[0065]

[0066] Where m represents the sampling point;

[0067] The same sampling point t2 to t kThe sampling value vector at the time corresponds to the Hankel matrix with p rows and q columns as follows:

[0068]

[0069] Define the scale factor as follows:

[0070]

[0071] In the formula, is the last column of the Hankel matrix formed at sampling point m, is the first column of the Hankel matrix formed at sampling point m, and ||·|| is the Euclidean norm of the matrix.

[0072] The data matrix for reconstructing the voltage response data is represented as follows:

[0073]

[0074] In the formula, X' and Y' are Voltage response matrix after Hankel matrix expansion at different sampling times.

[0075] Step 3: Use the observation function to map the data matrices X' and Y' reconstructed in step 2 to the observation space:

[0076] X lift =φ(X')

[0077] Y lift =φ(Y');

[0078] In the formula, X lift and Y lift are the data matrices that map X' and Y' to the observation space, and φ is the observation function;

[0079] According to Koopman operator theory, we have:

[0080]

[0081] In the formula, is the Koopman operator.

[0082] Step 4: Obtain the Koopman operator by solving the least squares optimization problem:

[0083]

[0084] In the formula, Represents the matrix X lift The Moore-Penrose pseudoinverse.

[0085] Step 5: Perform eigendecomposition on the Koopman operator and obtain its eigenvalue as λ j , the right eigenvector w j , left singular vector U, then the j-th Koopman operator can be expressed as:

[0086] φ j =Uw j ;

[0087] Step 6: Calculate the amplitude of each Koopman operator. The amplitude of the Koopman operator is:

[0088]

[0089] Where X1 is the response measurement matrix The first column of data is the initial conditions.

[0090] The Koopman operator amplitude can characterize the influence of each operator on the time domain signal. Therefore, the dominant mode that can reflect the dynamic characteristics of the system can be determined according to the Koopman operator amplitude, and the transient voltage dynamic process can be approximately characterized. The time domain prediction model for the voltage dominant mode is established as shown in the following formula:

[0091] Δd j (t)=(λ j ) t-1 B j ;

[0092] In the formula, Δd j (t) is the dominant modal response after extension.

[0093] Step 7: Establish a transient voltage stability discrimination model based on Koopman operator:

[0094]

[0095] In the formula, Λ(kΔt) is the Lyapunov exponent, N is the set time interval parameter; Δd j ((k+i)Δt),Δd j ((k+i-1)Δt),Δd j (iΔt), Δd j ((i-1))Δt) are the dominant modal responses after extension times (k+i)Δt, (k+i-1)Δt, iΔt, and (i-1))Δt, respectively.

[0096] If the Lyapunov index is greater than 0, it indicates that the system transient voltage is unstable; if the Lyapunov index result is not greater than 0, it indicates that the fault occurring in the system at this time is not transient voltage instability, and other methods are used to judge other faults. This method returns to step 1 and waits for the next detection.

[0097] The present invention is further described in detail below in conjunction with embodiments:

[0098] like Figure 2 The Nordic system shown in the figure simulates the following operating conditions to verify the effectiveness of the transient voltage stability judgment method of the invention:

[0099] Working condition a: When the load bus 1 is connected to 100% induction motor load, a three-phase short circuit fault occurs in the system line 1041-1043 near the bus 1043 at 1 second. After the fault lasts for 100ms, the line is tripped and the fault is cleared;

[0100] Working condition b: When the load bus 1 is connected to 100% induction motor load, a three-phase short circuit fault occurs in the system line 1041-1043 near the bus 1043 at 1 second. After the fault lasts for 200ms, the line is tripped and the fault is cleared;

[0101] First, the voltage amplitude and phase angle measurement data of each load bus after the system fault are collected to extract the Koopman operator. In the simulation verification of this section, the analysis window length is set to 0.5s. Therefore, the voltage and phase angle measurement data 0.5s after the fault are used as the input of the Hankel matrix combined with the extended dynamic mode decomposition (Hankel Extended Dynamic Mode Decomposition, Hankel-EDMD) algorithm, and the Koopman operator of the system and the Koopman mode corresponding to each Koopman operator are extracted under two working conditions respectively, and the energy size of each mode is calculated as follows Figure 3 shown.

[0102] The time domain extension of mode 8 in working condition a and mode 7 in working condition b is performed. In order to ensure the accuracy of the subsequent Lyapunov index calculation, the time for the mode time domain extension in this section is 4s. The extension results are as follows: Figure 4 .

[0103] Since the traditional Lyapunov exponent MLE needs to continuously collect measurement data after a system failure to realize index calculation, under the two working conditions, according to the traditional Lyapunov exponent MLE calculation method, the voltage response measurement data of each busbar in the system is continuously collected to realize transient voltage stability judgment. The transient voltage stability judgment comparison results composed of the Lyapunov exponents of the invention method and the traditional method are shown in Figure 2. Figure 5 shown.

[0104] In working condition a, that is, in the scenario of stable operation of transient voltage of the system, the calculation result of Lyapunov exponent of the proposed method is always less than 0, which is below the stability boundary. Therefore, it can be inferred that when a large disturbance fault is detected in the system, the 0.5sPMU measurement data is used as the input of the proposed method, that is, the Koopman dominant mode signal extended in the time domain can be used to quickly determine that the transient voltage is stable at this time. However, the Lyapunov exponent MLE result calculated by the traditional method is above the stability boundary at the initial moment. If the transient voltage of the system is judged to be unstable at this time, a misjudgment will occur, resulting in subsequent control actions and wasting control resources. If the voltage is not judged to be unstable at this time, the system continues to be monitored and the measurement signal is collected. In the subsequent calculation process, the transient voltage stability of the system can also be accurately judged. In summary, with the traditional calculation method, the operator cannot give the evaluation result of system stability through shorter measurement data, and needs to continuously collect signals to provide more accurate stability judgment. Therefore, if the traditional method wants to obtain accurate judgment results, it will affect the timeliness of stability judgment.

[0105] according to Figure 5 From the comparison of the instability identification time of the two methods shown, it can be seen that the method proposed in the present invention takes a shorter time and is more suitable for the identification of transient voltage instability in the power system.

Claims

1. The transient voltage instability identification method based on Koopman operator is characterized by: The process includes the following steps, which are performed in sequence: Step 1: Collect the voltage response data online and pre-process the collected data. Determine whether the system has a fault according to the set conditions. If no fault occurs, continue to collect data. If a fault occurs, measure and obtain the voltage response data of each sampling point after the fault, and construct the t1 to t2 at sampling point 1 to sampling point m based on the voltage response data. k The continuous time series data matrix of the moment; Step 2: Use the constructed continuous time series data matrix to take the time series from t1 to t k-1 The sampling value vector at the time corresponds to a Hankel matrix with p rows and q columns; then the same sampling point t2 to t k The sampling value vector at the time corresponds to a Hankel matrix with p rows and q columns; the proportional factor is obtained through two Hankel matrices; and the data matrix of the voltage response data is reconstructed through the proportional factor and the Hankel matrix at different sampling points; Step 3: Based on the Koopman operator theory, the observation function is used to map the reconstructed data matrix to the observation space; Step 4: Obtain the Koopman operator by solving the least squares optimization problem; Step 5: Perform eigendecomposition on the Koopman operator to obtain its eigenvalue, right eigenvector, and left singular vector; Step 6: Calculate and obtain the amplitude of each Koopman operator, and establish a time domain prediction model of the voltage dominant mode through the amplitude of the Koopman operator; Step 7. A transient voltage stability discrimination model based on the Koopman operator is established through the time domain prediction model of the voltage-dominant mode. The Lyapunov index is defined in the transient voltage stability discrimination model based on the Koopman operator, and the calculation result of the Lyapunov index is used as the final result of the model. If the result of the Lyapunov index is greater than 0, it indicates that transient voltage instability occurs in the system at this time; if the result of the Lyapunov index is not greater than 0, it indicates that the fault occurring in the system at this time is not transient voltage instability, and return to step 1 to wait for the next detection.

2. The transient voltage instability identification method based on Koopman operator according to claim 1 is characterized in that: In step 1, the voltage response data is obtained through time domain simulation or vector measurement unit PMU, and the voltage response data is expressed as: x k+1 =f(x k ); In the formula, x k is the sampling value of the system voltage response at the kth sampling moment, and x k =x(kΔt); Δt is the sampling interval.

3. The transient voltage instability identification method based on Koopman operator according to claim 1 is characterized in that: In step 1, the time from t1 to t k The continuous time series data matrix of the moment is expressed as follows: In the formula, From t1 to t k The numerical matrix composed of the sampling values ​​at the moment; x(c m ,t k ) indicates t k Sample data at sampling point m at the moment; x l Indicates the tth l A vector consisting of the data values ​​of all sampling points at the time instant.

4. The transient voltage instability identification method based on Koopman operator according to claim 1 is characterized in that: In step 2, the interval from t1 to t k-1 The sampling value vector at the time corresponds to the Hankel matrix with p rows and q columns as follows: Where m represents the sampling point; The same sampling point t2 to t k The sampling value vector at the time corresponds to the Hankel matrix with p rows and q columns as follows: Define the scale factor as follows: In the formula, is the last column of the Hankel matrix formed at sampling point m, is the first column of the Hankel matrix formed at sampling point m, and ||·|| is the Euclidean norm of the matrix.

5. The transient voltage instability identification method based on Koopman operator according to claim 4 is characterized in that: The data matrix of the voltage response data reconstructed in step 2 is expressed as follows: In the formula, X' and Y' are Voltage response matrix after Hankel matrix expansion at different sampling times.

6. The transient voltage instability identification method based on Koopman operator according to claim 1 is characterized in that: In step 3, the reconstructed data matrix is ​​mapped to the observation space as follows: X lift =φ(X') Ylift = φ(Y'); Where, X lift and Y lift are the data matrices that map X' and Y' to the observation space, and φ is the observation function; According to Koopman operator theory, we have: In the formula, is the Koopman operator.

7. The transient voltage instability identification method based on Koopman operator according to claim 1 is characterized in that: The Koopman operator obtained in step 4 is expressed as follows: In the formula, Represents the matrix X lift The Moore-Penrose pseudoinverse.

8. The transient voltage instability identification method based on Koopman operator according to claim 1 is characterized in that: The amplitude of the Koopman operator in step 5 is expressed as follows: In the formula, X1 is a matrix The first column of data is the initial condition; φ j is the jth Koopman operator, φ j =Uw j ;w j is the right eigenvector after eigendecomposition of the Koopman operator, and U is the left singular vector after eigendecomposition of the Koopman operator.

9. The transient voltage instability identification method based on Koopman operator according to claim 1 is characterized in that: The time domain prediction model of the voltage dominant mode in step 6 is shown as follows: Δd j (t)=(λ j ) t-1 B j ; In the formula, Δd j (t) is the dominant modal response after extension; λ j is the eigenvalue after eigendecomposition of the Koopman operator.

10. The transient voltage instability identification method based on Koopman operator according to claim 9 is characterized in that: The transient voltage stability discrimination model based on the Koopman operator in step 7 is as follows: In the formula, Λ(kΔt) is the Lyapunov exponent, N is the set time interval parameter; Δd j ((k+i)Δt),Δd j ((k+i-1)Δt),Δd j (iΔt), Δd j ((i-1))Δt) are the dominant modal responses after extension times (k+i)Δt, (k+i-1)Δt, iΔt, and (i-1))Δt, respectively.

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