A Power System Forward Uncertainty Quantification Method Based on Inverse Uncertainty Quantification

Abstract

This invention discloses a method for forward uncertainty quantification of power systems based on inverse uncertainty quantification. First, an uncertainty model of transient parameters is established using inverse uncertainty quantification. Then, parameter samples conforming to this model are obtained using the MCMC method. Next, the number of samples required for inverse uncertainty quantification is unified with the number required for forward uncertainty quantification. Finally, forward uncertainty quantification is performed on the system output. This invention improves the accuracy of the new power system uncertainty quantification results by constructing a reasonable transient parameter uncertainty model through inverse uncertainty quantification and applying it to the uncertainty quantification of power system output.

Description

Technical Field

[0001] This invention belongs to the technical field of dynamic and uncertainty quantification of power systems, and specifically relates to a method for forward uncertainty quantification of power systems based on inverse uncertainty quantification. Background Technology

[0002] New power systems are transforming into systems with strong uncertainty, which stems from factors such as model parameters and simulation algorithms. Uncertainty in model parameters affects the system's dynamic behavior and, consequently, system decision-making. Therefore, it is necessary to quantify this uncertainty, and obtaining accurate uncertainty quantification results requires establishing a correct uncertainty model.

[0003] For some model parameters, such as generator output and load, their uncertainty models can be fitted using historical data. However, for transient parameters of the power system, such as inertia and damping, the lack of historical data makes it impossible to accurately characterize their uncertainty models. Most existing methods rely on model assumptions, which leads to inaccurate uncertainty quantification results. Therefore, improving the accuracy of power system uncertainty quantification results has become a technical problem and challenge that those skilled in the art wish to overcome. Summary of the Invention

[0004] This invention addresses the difficulty of obtaining accurate power system uncertainty quantification results using existing technologies. It provides a forward uncertainty quantification method for power systems based on inverse uncertainty quantification. First, inverse uncertainty quantification is used to establish an uncertainty model for transient parameters. Then, the MCMC method is used to obtain parameter samples that conform to this model. Next, the number of samples required for inverse uncertainty quantification and forward uncertainty quantification are unified. Finally, forward uncertainty quantification is performed on the system output. This invention improves the accuracy of new power system uncertainty quantification results by constructing a reasonable transient parameter uncertainty model through inverse uncertainty quantification and applying it to the uncertainty quantification of power system output results.

[0005] To achieve the above objectives, the technical solution adopted by this invention is: a method for forward uncertainty quantification of power systems based on inverse uncertainty quantification, comprising the following steps:

[0006] S1. Model Establishment: An uncertainty model of transient parameters is established using inverse uncertainty quantification.

[0007] S2. Obtaining parameter samples: Use the MCMC method to obtain parameter samples that conform to the model in step S1;

[0008] S3. Unified Sample Number: Achieve a unified sample number for both reverse uncertainty quantification and forward uncertainty quantification;

[0009] S4. Positive uncertainty quantification: Perform positive uncertainty quantification on the power system output.

[0010] As an improvement to the present invention, the establishment of the uncertainty model in step S1 specifically includes the following steps:

[0011] S11: Establishing the framework for Bayesian inference:

[0012] d = h(z) + e

[0013] Where d is a measurement vector of dimension m; z is a vector of dimension N, whose components are transient parameters to be quantized; h(·) is a function vector mapping z to d; and e is a measurement error vector, including active and reactive errors, whose joint probability density distribution is:

[0014]

[0015] S12: The expression for the posterior probability density function is:

[0016] π post (z|d)∝π like (d|z)π prior (z)

[0017] Where, π prior (z) is the prior probability distribution of z, π like (d|z) is the likelihood function, π post (z|d) is the posterior probability distribution of z;

[0018] S13: Select simulation time from t0 to t f π in logarithmic form post (z|d) is:

[0019]

[0020] in, and The results are the likelihood, measurement, and simulation calculations at time t, respectively.

[0021] S14: Calculate the maximum a posteriori value of z, expressed as:

[0022]

[0023] As another improvement of the present invention, the likelihood function in step S12 is specifically as follows:

[0024]

[0025] Where, d i This refers to the i-th component of the measurement vector d;

[0026] Assuming that the components of z are independent, the formula for calculating the prior probability distribution function is:

[0027]

[0028] As another improvement of the present invention, the logarithmic form of π in step S13 post (z|d) is formed, including the following steps:

[0029] S131: To avoid the curse of dimensionality in subsequent MCMC sampling, a decentralized model is adopted. It is assumed that the voltage and phase of the generator node are obtained directly from the PMU device, and the algebraic variable V... di V qi ,I di ,I qi The calculation expression is:

[0030] V di =V i sin(δ i -θ vi )

[0031] V qi =V i cos(δ i -θ vi )

[0032]

[0033] Among them, V di V qi These are the direct-axis and quadrature-axis components of the voltage at the i-th node, respectively; I di ,I qi These are the direct-axis and quadrature-axis components of the current at the i-th node, respectively; V i ,θ vi These are the magnitude and phase angle of the voltage at the i-th generator node, respectively; E di ′,E qi ′ are the direct-axis and quadrature-axis subtransient electromotive forces of the i-th generator, respectively; X di ′,X qi ' and δ are the direct-axis and quadrature-axis subtransient reactances of the i-th generator, respectively; i It is the power angle of the i-th generator;

[0034] S132: Calculate the active and reactive power outputs of the i-th generator, respectively:

[0035] P ei =V di I di +V qi I qi +ePi

[0036] Q ei =-V di I qi +V qi I di +e Qi

[0037] Among them, P ei Q ei These are the active and reactive power outputs of the i-th generator, respectively; e Pi ,e Qi These are the errors related to active and reactive power, respectively.

[0038] S133: Based on a decentralized model, logarithmic form of π post (z|d) can be rewritten as:

[0039]

[0040] As another improvement of the present invention, the parameter sample acquisition in step S2 specifically includes the following steps:

[0041] S21: For k = 0, ..., N inv -1, from the assumed proposal distribution q(z) k Generate a new sample z. * , where N inv The number of samples generated that conform to the inverse uncertainty quantification result;

[0042] S22: Calculate log(π) post (z * |d));

[0043] S23: Calculate the reception rate α(z) k ,z * The expression is:

[0044]

[0045] S24: Randomly generate a number u from a uniform distribution in the interval [0,1];

[0046] S25: If u < α(z) k ,z * ), then z k+1 =z * Otherwise z k+1 =z k ;

[0047] S26: Repeat steps S21-S25 until k = N inv -1, thus obtaining the obedience to π. post(z|d) of N inv One parameter sample.

[0048] As another improvement of the present invention, step S3, which unifies the number of samples, specifically includes the following steps:

[0049] S31: Obtain the marginal cumulative density function of z using kernel density estimation:

[0050]

[0051] Among them, P i For z i Cumulative density function; K i For z i kernel function; B i For bandwidth; It is z i The j-th sample; N inv The number of samples generated that conform to the inverse uncertainty quantification result;

[0052] S32: Generate samples for positive uncertainty quantification using Gaussian Cupula, with the sample size set to N. for The samples obtained from MCMC sampling in step S2 are converted into samples under a Gaussian distribution:

[0053] z i ′=φ -1 (z i )

[0054] Among them, z i ' represents a sample under a Gaussian distribution; φ is the cumulative density function of the standard normal distribution;

[0055] Perform Cholesky decomposition on the correlation matrix of z′:

[0056] D = L·L T

[0057] Where D is the correlation matrix;

[0058] Generate N for A sample M conforming to a Gaussian distribution with correlation matrix D:

[0059] M = L·R

[0060] Where R is randomly generated from N for It consists of N standard normally distributed samples;

[0061] We obtain samples that conform to their respective marginal distributions:

[0062] M i ′=P i-1 (φ(M i ))

[0063] Among them, M i ′ is to obey z i The number of marginally distributed samples is N. for .

[0064] As a further improvement of the present invention, step S4 specifically includes the following steps:

[0065] S41: Convert the transient parameter z in the power system transient model using the corresponding N for One sample replacement;

[0066] S42: Modeling the uncertainties of generator output and load in a power system using stochastic differential equations:

[0067] S43: Perform dynamic simulation of the power system and quantify the uncertainty of the output results.

[0068] Compared with existing technologies, the technical advantages and effects of this invention are as follows: Given the high uncertainty of modern power systems, this invention proposes a method for forward uncertainty quantification of power systems based on inverse uncertainty quantification. This method constructs a reasonable transient parameter uncertainty model through inverse uncertainty quantification and applies it to the uncertainty quantification of the output results, thereby improving the accuracy of power system uncertainty quantification results. Simultaneously, to avoid the curse of dimensionality problem faced by MCMC sampling, this invention performs inverse uncertainty quantification based on a decentralized model. Furthermore, considering that the number of samples required for inverse uncertainty quantification is generally much larger than the number required for forward uncertainty quantification, this invention combines kernel density estimation and Gaussian Copula to achieve a unified sample number. Attached Figure Description

[0069] Figure 1 This is a flowchart of the steps of the power system forward uncertainty quantification method based on inverse uncertainty quantification of the present invention;

[0070] Figure 2 This is a schematic diagram of the New England 39 busbar 10 generator system to which the method of this invention is applied;

[0071] Figure 3 This is a schematic diagram of the inverse uncertainty quantification result in Test Example 1 of this invention;

[0072] Figure 4 This is a schematic diagram showing the comparison results of the mean values ​​obtained by positive uncertainty quantification based on the method and model assumption method of this invention in Test Example 1 of this invention;

[0073] Figure 5This is a schematic diagram showing the comparison results of the standard deviation obtained by positive uncertainty quantification based on the method and model assumption method of this invention in Test Example 1 of this invention;

[0074] Figure 6 This is a schematic diagram showing the comparison of the mean values ​​obtained by positive uncertainty quantification based on the method of this invention and the improved model assumption method in Test Example 2 of this invention;

[0075] Figure 7 This is a schematic diagram showing the comparison of standard deviations obtained from positive uncertainty quantification based on the method of this invention and the improved model assumption method in Test Example 2 of this invention. Detailed Implementation

[0076] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0077] Example 1

[0078] This invention addresses the importance of uncertainty models for uncertainty quantification recognized by the power system industry. Considering the problem that transient parameters cannot be accurately modeled due to a lack of historical data, this invention proposes a forward uncertainty quantification method for power systems based on inverse uncertainty quantification, aiming to improve the accuracy of uncertainty quantification results for new power systems.

[0079] The method in this embodiment is applied to Figure 2 The power system shown, Figure 2 This is a schematic diagram of a classic New England 39-bus 10-generator system. The method for quantifying forward uncertainty in power systems, based on inverse uncertainty quantification, is as follows: Figure 1 As shown, it includes the following steps:

[0080] Step S1: Establish an uncertainty model for transient parameters using inverse uncertainty quantization;

[0081] S11. Establish the framework of Bayesian inference, expressed as:

[0082] d = h(z) + e

[0083] Where d is a measurement vector of dimension m, whose data comes from the PMU device; z is a vector of dimension N, whose components are transient parameters to be quantized; h(·) is a function vector that maps z to d; e is a measurement error vector, which mainly consists of active and reactive errors. Typically, it is assumed that their components are independent, therefore their joint probability density distribution can be expressed as:

[0084]

[0085] S12. The expression for the posterior probability density function is obtained as follows:

[0086] π post (z|d)∝π like (d|z)π prior (z)

[0087] Where, π prior (z) is the prior probability distribution of z, π like (d|z) is the likelihood function, π post (z|d) is the posterior probability distribution of z;

[0088] The formula for calculating the likelihood function is:

[0089]

[0090] Where, d i This refers to the i-th component of the measurement vector d;

[0091] In this embodiment, it is assumed that the components of z are independent of each other, and the formula for calculating the prior probability distribution function is:

[0092]

[0093] S13. Select simulation time from t0 to t f π in logarithmic form post (z|d) can be further expressed as:

[0094]

[0095] in, and The figures at time t are the likelihood, measurement, and simulation results, respectively.

[0096] The logarithmic form of π formed in step S13 post In (z|d), a decentralized model is adopted to avoid the curse of dimensionality in subsequent MCMC sampling; it is assumed that the voltage and phase of the generator node are obtained directly from the PMU device, and the algebraic variable V di V qi ,I di ,I qi The calculation expression is:

[0097] V di =V i sin(δ i -θ vi )

[0098] V qi =V i cos(δ i -θvi )

[0099]

[0100] Among them, V di V qi These are the direct-axis and quadrature-axis components of the voltage at the i-th node, respectively; I di ,I qi These are the direct-axis and quadrature-axis components of the current at the i-th node, respectively; V i ,θ vi These are the magnitude and phase angle of the voltage at the i-th generator node, respectively; E di ′,E qi ′ are the direct-axis and quadrature-axis subtransient electromotive forces of the i-th generator, respectively; X di ′,X qi ' and δ are the direct-axis and quadrature-axis subtransient reactances of the i-th generator, respectively; i It is the power angle of the i-th generator;

[0101] The active and reactive power outputs of the i-th generator are calculated using the following expression:

[0102] P ei =V di I di +V qi I qi +e Pi

[0103] Q ei =-V di I qi +V qi I di +e Qi

[0104] Among them, P ei Q ei These are the active and reactive power outputs of the i-th generator, respectively; e Pi ,e Qi These are the errors related to active and reactive power, respectively.

[0105] Based on a decentralized model, the logarithmic form of π post (z|d) can be further rewritten as:

[0106]

[0107] The maximum a posteriori value of S14 and z can be expressed as:

[0108]

[0109] Step S2: Use the MCMC method to obtain parameter samples that conform to the model;

[0110] S21. For k = 0, ..., N inv -1, from the assumed proposal distribution q(z) k Generate a new sample z. * ;

[0111] S22. Calculate log(π) post (z * |d));

[0112] S23. Calculate the receiver efficiency α(z) k ,z * The expression is:

[0113]

[0114] S24. Randomly generate a number u from a uniform distribution in the interval [0,1].

[0115] S25. If u < α(z) k ,z * ), then z k+1 =z * Otherwise z k+1 =z k ;

[0116] S26: Repeat steps S21, S22, S23, S24, and S25 sequentially until k = N. inv -1, and the MCMC method can be used to obtain the π-compliant result. post (z|d) of N inv One parameter sample.

[0117] Step S3: Unify the number of samples required for reverse uncertainty quantization with the number of samples required for forward uncertainty quantization;

[0118] The marginal cumulative density function of z obtained by kernel density estimation is:

[0119]

[0120] Among them, P i For z i Cumulative density function; K i For z i kernel function; B i For bandwidth; It is z i The j-th sample; N inv The number of samples generated that conform to the inverse uncertainty quantification result;

[0121] Samples for positive uncertainty quantification are generated using Gaussian Cupu la, with the sample size set to N. forThe sample obtained from MCMC sampling is converted to a Gaussian distribution, expressed as follows:

[0122] z i ′=φ -1 (z i )

[0123] Among them, z i ' represents a sample under a Gaussian distribution; φ is the cumulative density function of the standard normal distribution;

[0124] The Cholesky decomposition of the correlation matrix of z′ is expressed as follows:

[0125] D = L·L T

[0126] Where D is the correlation matrix;

[0127] Generate N for A sample M conforming to a Gaussian distribution with correlation matrix D:

[0128] M = L·R

[0129] Where R is randomly generated from N for It consists of N standard normally distributed samples;

[0130] Further, samples that conform to their respective marginal distributions are obtained:

[0131]

[0132] Among them, M i ′ is to obey z i The number of marginally distributed samples is N. for .

[0133] Step S4: Perform positive uncertainty quantification on the system output;

[0134] S41. Replace the transient parameter z in the power system transient model with the corresponding N... for One sample replacement;

[0135] S42. Model the uncertainty of generator output and load in a power system using stochastic differential equations;

[0136] S43. Perform dynamic simulation of the power system to quantify the uncertainty of the system output results.

[0137] Test case

[0138] Test Example 1

[0139] based on Figure 2 For the generator system shown, we select the inertia H and excitation parameter K of the motor connected to system node 31.A ,K E ,K F As the research object, it is assumed that the prior probabilities of these parameters follow a Gaussian distribution, where the mean is the value provided by the manufacturer and the standard deviation is 10% of the mean. The proposal distribution q follows a multidimensional Gaussian distribution, with the mean being the previously accepted value and the standard deviation being 0.03% of that value. The components of the measurement error vector e follow a Gaussian distribution, with a mean of 0 and a standard deviation of 0.01. It is assumed that the external disturbance occurs after 0.05 s, after which the transmission line between nodes 15 and 16 is removed. It is assumed that the number of samples sampled by MCMC is 100,000.

[0140] First, the parameters H and K are obtained through inverse uncertainty quantization. A ,K E ,K F The uncertainty model is then used, and parameter samples are obtained using the MCMC method to characterize the uncertainty model. Figure 3 The uncertainty model for each parameter is presented. As can be seen from the figure, H, K A ,K E ,K F It does not follow some common distributions, such as K. E Two peaks appeared in the probability distribution plot. Furthermore, the relationships between the parameters can also be derived from the plot, for example, K... A and K E There is a positive correlation between them, K F and K A ,K E There is a negative correlation between H and K. A ,K E ,K F All parameters are relatively independent. This fully demonstrates that inverse uncertainty quantization can obtain the accurate probability distribution of each parameter and the correlation between parameters, which cannot be obtained by the commonly used parameter uncertainty model assumption method. This also explains the rationality of using inverse uncertainty quantization to obtain the transient parameter uncertainty model in this invention.

[0141] This test case uses the Monte Carlo method to perform positive uncertainty quantification of the active power output of the motor connected to node 31, with 5000 Monte Carlo simulations performed. The simulation depicts the transient process of disconnecting line 15-16 after 2 seconds, with a total simulation duration of 20 seconds. This test case compares the simulation with the widely used model assumption method in current transient parameter uncertainty quantification research. This method assumes that the uncertainty model of transient parameters is characterized by some common distributions. In this test case, it is assumed that the parameters follow two distributions: a Gaussian distribution with a mean given by the manufacturer and a standard deviation of 10% of the mean; and a uniform distribution with a mean given by the manufacturer and an interval length of 60% of the mean.

[0142] Based on the parameter uncertainty models obtained using the method in this case and the model assumption method respectively, the Monte Carlo method was used to obtain the positive uncertainty quantification results of the active power output of the motor connected to node 31. When analyzing the results, the mean and standard deviation were selected as the research objects. Figure 4 , Figure 5 The results comparing the mean and standard deviation obtained using the method described in this case and the model assumption method are presented respectively. From Figure 4 It can be seen that the mean obtained using the method proposed in this case is basically the same as the mean obtained using the Gaussian and uniform assumptions in the first 6 seconds, but the mean results differ significantly as time progresses. From Figure 5 It can be seen that the standard deviation of the results obtained using the method of this invention differs significantly from that obtained using the model assumption method after 2 seconds. This reflects that different assumed models will result in different final uncertainty quantification results, confirming the importance of uncertainty modeling and thus proving the rationality of the method proposed in this case.

[0143] Test Example 2

[0144] To ensure the comprehensiveness of the results, this test case improves the transient parameter uncertainty model assumed in the model assumption method of Test Case 1. The mean of the Gaussian and uniform distributions is replaced with the maximum a posteriori value obtained through inverse uncertainty quantification, while all other parameters remain unchanged. This test case performs uncertainty quantification on the active power output of the motor connected to node 31, and selects its mean and standard deviation as the research objects when analyzing the results. Figure 6 , Figure 7 The results comparing the mean and standard deviation obtained using this method and the improved model assumption method are presented respectively. From Figure 6 It can be seen that the difference between the mean obtained using the proposed method and the mean obtained using the improved model assumption method is significant compared to... Figure 4 Significantly reduced, but Figure 7 The results show that the standard deviations remain significantly different. These simulation results confirm that even if the mean of the assumed model is improved, its confidence interval remains uncertain, making it impossible to accurately characterize the uncertainty model of transient parameters and thus obtain correct quantification results of transient parameter uncertainty. This demonstrates the importance of transient parameter uncertainty modeling and further proves the rationality of the method in this invention.

[0145] In summary, this invention proposes a forward uncertainty quantification method for power systems based on inverse uncertainty quantification. It utilizes inverse uncertainty quantification to obtain a reasonable transient parameter uncertainty model and quantifies the system output results to improve the accuracy of uncertainty quantification results for new power systems, ensure the rationality of system decisions, and provide assurance for the accuracy of uncertainty quantification results for new power systems.

[0146] It should be noted that the above content merely illustrates the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. For those skilled in the art, various improvements and modifications can be made without departing from the principle of the present invention, and all such improvements and modifications fall within the scope of protection of the claims of the present invention.

Claims

1. A method for quantifying forward uncertainty in power systems based on inverse uncertainty quantification, characterized in that, Includes the following steps: S1. Model Establishment: An uncertainty model of transient parameters is established using inverse uncertainty quantification. S11: Establishing the framework for Bayesian inference: ； in, It is a measurement vector of dimension m; It is a vector of dimension N, whose components are transient parameters that need to be quantized; It is Mapped to The function vector; This is the measurement error vector, which includes active and reactive power errors, and its joint probability density distribution is: ； S12: The expression for the posterior probability density function is: ； in, yes The prior probability distribution, It is the likelihood function. yes The posterior probability distribution; S13: Select simulation time arrive logarithmic form for: ； in, , and They are respectively Likelihood at any given time, measured and simulated results; S14: Calculation The maximum a posteriori value is expressed as: ； S2. Obtaining parameter samples: Use the MCMC method to obtain parameter samples that conform to the model in step S1; S3. Unified Sample Number: Achieve a unified sample number for both reverse uncertainty quantification and forward uncertainty quantification; S31: Obtained using kernel density estimation Marginal cumulative density function: ； in, for The cumulative density function; for Kernel function; For bandwidth; yes The j-th sample; The number of samples generated that conform to the inverse uncertainty quantification result; S32: Generate samples for positive uncertainty quantification using Gaussian Cupula, with the sample size set to... The samples obtained from MCMC sampling in step S2 are converted into samples under a Gaussian distribution: ； in, The samples are under a Gaussian distribution; The cumulative density function of the standard normal distribution; right Perform Cholesky decomposition on the correlation matrix: ； in, This is the correlation matrix; generate A correlation matrix Samples under Gaussian distribution : ； in, Randomly generated It consists of standard normally distributed samples, with dimensions of . ; We obtain samples that conform to their respective marginal distributions: ； in, To obey The number of marginally distributed samples is ; S4. Positive Uncertainty Quantification: Performing positive uncertainty quantification on the power system output; S41: Transient parameters in the power system transient model Use the corresponding One sample replacement; S42: Modeling the uncertainties of generator output and load in a power system using stochastic differential equations: S43: Perform dynamic simulation of the power system and quantify the uncertainty of the output results.

2. The method for forward uncertainty quantification of power systems based on inverse uncertainty quantification as described in claim 1, characterized in that: The likelihood function in step S12 is specifically: ； in, For measurement vector The One component; Assumption If the components are independent of each other, then the formula for calculating the prior probability distribution function is: 。 3. The method for forward uncertainty quantification of power systems based on inverse uncertainty quantification as described in claim 1, characterized in that: In step S13, the logarithmic form The formation process includes the following steps: S131: A decentralized model is adopted, assuming that the voltage and phase of the generator node are obtained directly from the PMU device, and algebraic variables... The calculation expression is: ； in, These are the direct-axis and quadrature-axis components of the voltage at the i-th node, respectively. These are the direct-axis and quadrature-axis components of the current at the i-th node, respectively. These are the magnitude and phase angle of the voltage at the i-th generator node, respectively. These are the direct-axis and quadrature-axis subtransient electromotive forces of the i-th generator, respectively. These are the direct-axis and quadrature-axis subtransient reactances of the i-th generator, respectively; It is the power angle of the i-th generator; S132: Calculate the active and reactive power outputs of the i-th generator, respectively: ； in, These are the active and reactive power outputs of the i-th generator, respectively. These are the errors related to active and reactive power, respectively. S133: In a decentralized model, logarithmic form Rewritten as: 。 4. The power system forward uncertainty quantification method based on inverse uncertainty quantification as described in claim 2 or 3, characterized in that: The parameter sample acquisition in step S2 specifically includes the following steps: S21: Regarding From the assumed proposal distribution Generate new samples ,in, The number of samples generated that conform to the inverse uncertainty quantification result; S22: Calculation ; S23: Calculate the reception rate The expression is: ； S24: Randomly generate a number from a uniform distribution in the interval [0,1]. ; S25: If ,but ;otherwise ; S26: Repeat steps S21-S25 until... Obtain obedience of One parameter sample.

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