Robust state estimation method for power systems based on MES and data fusion

Abstract

The present invention discloses a robust state estimation method for a power system based on MES and data fusion. The method first uses the robust Mahalanobis distance to detect system outliers and assigns appropriate weights to selected PMU buffer measurements. An MES estimator then uses these weights to filter out non-Gaussian PMU measurement noise to produce a set of state estimation results. Simultaneously, the MES estimator processes received SCADA measurements containing unknown measurement noise to produce another set of state estimation results. The two sets of estimation results are then fused based on data fusion theory, obtaining a final optimal estimation result. The present invention has the advantage of embedding a zero injection expression in the MES estimation model, eliminating the need for explicit expression of equality constraints, eliminating the need for Lagrange multiplier processing, and thus improving algorithm efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of power system estimation methods, and more particularly to the technical field of power system robust state estimation methods. Background Art

[0002] State Estimation (SE) technology can be used to improve the state perception and defense capabilities of smart grid monitoring against bad data. The accurate state information obtained is crucial in various advanced applications of power systems, such as power system planning, reliability analysis, and optimal load distribution. A reliable data foundation is the cornerstone of other advanced applications.

[0003] The goal of state estimation (SE) is to obtain the best estimate of the current system node voltage magnitude and phase angle, given a set of redundant measurements and accurate network topology and parameters. Therefore, the performance of SE is highly dependent on the accuracy of the measurements and the estimation model established under the assumed conditions. Reasonable assumptions are crucial for model building.

[0004] In state estimation (SE), measurements are subject to instrument errors, even severe errors due to instrument failures, pulsed communication noise, and other factors. Network parameters also vary over time due to environmental and temperature fluctuations, introducing uncertainty into the assumed SE model. Furthermore, without sufficient field information or calibration, zero injection information at the tie node may be erroneous. Therefore, measurement noise in real power systems is unknown, highly uncertain, and not the traditional Gaussian noise.

[0005] However, in existing research on power systems, measurement noise is often assumed to be Gaussian distributed. However, actual measurement statistics show that measurement noise often deviates from the assumed Gaussian model, resulting in outliers. In this case, the performance of existing SE methods that rely on the assumption that measurement noise is Gaussian can be significantly degraded.

[0006] With the widespread deployment of Phasor Measurement Units (PMUs), PMUs provide an increasing number of synchronized voltage and current phasor measurements with timestamps. These PMUs offer higher accuracy than traditional Supervisory Control and Data Acquisition (SCADA) measurements, and are therefore often used to improve SE accuracy and enhance the ability to detect and identify bad data. However, the sampling frequencies of measurements from SCADA systems and PMUs differ significantly, leading to time skew issues. Therefore, these measurements must be aligned to the same timeframe.

[0007] To address this issue, PMU measurements can be buffered, and hypothesis testing can be used to select the optimal buffer length. The sample mean and covariance of these buffered PMU measurements are then combined with SCADA measurements for SE. PMU measurement errors have traditionally been assumed to follow a Gaussian distribution. However, recent statistical studies have shown that PMU measurement noise follows a heavy-tailed distribution, rather than a Gaussian distribution. Based on this, the paper "Robust Hybrid State Estimation Framework with Unknown Measurement Noise Statistics" in the IEEE Transactions on Industrial Informatics, Volume 2, improves traditional static SE in the presence of non-Gaussian noise. However, this paper does not consider the time offset between SCADA and PMU measurements. Summary of the Invention

[0008] The purpose of the present invention is to address the deficiencies of the existing technology and to provide a robust state estimation method for power systems based on MES and data fusion, which can improve the monitoring state perception capability and deal with unknown non-Gauss noise and time offset problems between SCADA and PMU measurements.

[0009] The technical solutions adopted by the present invention to solve the above technical problems are as follows:

[0010] A robust state estimation method for power systems based on MES and data fusion first uses the robust Mahalanobis distance to detect system outliers and assign appropriate weights to selected PMU buffer measurements. Then, the MES estimator uses these weights to filter out non-Gaussian PMU measurement noise to produce one set of state estimation results. Simultaneously, the MES estimator processes received SCADA measurements containing unknown measurement noise to produce another set of state estimation results. For these two sets of estimation results, the estimation results from two independent MES estimators are fused based on data fusion theory to obtain the final optimal estimation result.

[0011] 1) Measurement equation

[0012] According to the Wiener approximation theorem, non-Gaussian distributions p(x) can be well approximated by known Gaussian distributions. Therefore, the following model can be used to simulate non-Gaussian distribution errors:

[0013]

[0014] Where: a i represents the weight, and N A Indicates the number of Gauss elements; and ∑ i represents the mean and variance,

[0015] Based on whether the known measurement error obeys the Gauss distribution, the above formula can be further expressed as:

[0016] G(e)=(1-ε)Φ(e)+εK(e) (2)

[0017] Where: Φ(e) is the majority distribution of measurement noise, which is usually modeled as a Gauss distribution; K(e) is the unknown distribution, which is considered to be a heavy-tailed density, such as the Laplace density or Gauss density with large variance; 0≤ε≤0.5, the contamination coefficient, is used to adjust the contribution of non-Gauss components. For example, for small ε, the model indicates that most of the error follows the Gauss distribution while maintaining a small part of the non-Gauss error.

[0018] The measurement equation under non-Gauss noise can be expressed as follows:

[0019] z=h(x)+G(e) (3)

[0020] Where: z represents the measurement; h(x) represents the nonlinear measurement function; G(e) represents the measurement error;

[0021] 2) Calculation of PMU optimal buffer length

[0022] Based on Mahalanobis distance, the optimal buffer length of multiple PMU measurement points is determined. For all PMU measurement points, take them as a whole.

[0023] Assume that PMU samples n per second r =30 samples / s, SCADA sampling interval N t =5s, in one interval, the number of PMU sampling N=N t *n r = 150 samples, use Z to represent the sampling matrix, and divide it into n subset =N t = 5 subsets: Z = [Y1, Y2, Y3, Y4, Y5], calculated based on the following steps:

[0024] (1) Calculate the median of the Z matrix:

[0025] Y'=median(Z)=[y'1,y'2,y'3,y'4,y'5] (4)

[0026] (2) Perform change point detection on Y5. If a change point exists, the last PMU data is used; otherwise, Y5 is included in the PMU buffer set;

[0027] (3) Based on Mahalanobis distance P i , for η1=[y'5,y'4], η2=[y'4,y'3], …, η4=[y'2,y'1], detect system changes:

[0028]

[0029] If P in η1 i All less than Then it contains Y4, otherwise the algorithm stops; and continues to check η2=[y'4,y'3],...,η4=[y'2,y'1] until the end,

[0030] (4) Calculate the average value of the measurement within the PMU buffer set and variance

[0031]

[0032]

[0033] Where: α represents the number of columns in Z', w i is the weight of the i-th column of Z',

[0034] Based on the above steps, the optimal buffer length of multiple PMU measurement points can be obtained and the average value can be calculated. and variance After that, it can be matched with SCADA data in the same section;

[0035] 3) Robust SE method based on MES and data fusion

[0036] 3.1) MES-based SCADA robust SE

[0037] Based on the MES estimation method, the MES estimation model is established using SCADA measurements. Its expression is as follows:

[0038]

[0039] In the formula: x represents the state quantity; h i (x) represents the measurement function of the i-th SCADA; w i represents the i-th measurement weight; z i represents the i-th measurement; σ represents the Parzen window width; c(x) = 0 represents the zero injection node constraint,

[0040] The zero injection constraint in polar coordinates can be expressed as:

[0041]

[0042] Where: Ψ represents x B with x N The transformation relationship; B represents the zero injection power node; N represents the non-zero injection node,

[0043] Γ is:

[0044]

[0045] Φ is:

[0046]

[0047] Φ -1 for:

[0048]

[0049] Substituting this constraint into the MES estimation model, we get:

[0050]

[0051] make:

[0052]

[0053]

[0054] Using Newton's method to solve, we get the modified equation:

[0055]

[0056] Where: H is the m×n dimensional Jacobian matrix; F and Q are:

[0057]

[0058]

[0059] The status quantity is corrected to:

[0060]

[0061]

[0062]

[0063] The calculation process of the MES estimator is as follows:

[0064] ⑴ Initialize parameters, k = 0, set x k,N Initial value;

[0065] ⑵ By x k,N Calculate x according to formula (20) k,B ;

[0066] ⑶ Calculate matrices H, F and Q, solve equation (16), and get and update calculate

[0067] (4) If max|Δx k |<ε, the state estimation converges and the calculation ends, otherwise go to step ⑶;

[0068] After iterative convergence, the state error variance matrix of the SCADA-based MES estimation is:

[0069] The above embedding of zero injection expression in the MES estimation model makes it unnecessary to express the equality constraint explicitly. By eliminating the equality constraint, there is no need to use the Lagrange multiplier method, thus improving the efficiency of the algorithm.

[0070] 3.2) MES-based PMU robust SE

[0071] Similar to SCADA robust SE, PMU measurements are used alone to perform robust estimation based on the MES model, but PMU measurements are voltage and current phasors, and their measurement functions are linear functions:

[0072]

[0073] Where: is the mean value of the measurement in the PMU buffer, which is used as the pseudo measurement and calculated by formula (7); A is the relationship matrix between PMU and state quantity, is a constant coefficient matrix, that is, the Jacobian matrix of PMU measurement and state quantity; ε is the PMU measurement error, and its variance is Calculated by formula (6),

[0074] Based on the MES model, we can get:

[0075]

[0076] Also use Newton's method to solve and get the modified equation:

[0077]

[0078] The state quantity correction is the same as formula (19)-(21),

[0079] After iterative convergence, the state error variance matrix of the PMU-based MES estimation is:

[0080]

[0081] 3.3) State Fusion Based on Data Fusion Theory

[0082] Drawing on data fusion theory, SCADA and PMU measurements are considered as different data sources, and their measurements are respectively S and zP The SCADA-based and PMU-based SEs are called SSE and PSE respectively. S and z P Treatment can produce two independent sets of SE results and The corresponding error variance matrix is given by ∑ s and ∑ p express,

[0083] based on and ∑ s and ∑ p , the best fusion result is obtained through data fusion:

[0084]

[0085] From the above formula, we can see that x s with x p The denominator of the SE weight is the same, and the numerator is ∑ p and ∑ s , which means that the size of the weight is inversely proportional to the variance of the estimation result, that is, the weight of the estimation result with higher accuracy is correspondingly larger, and the weight of the estimation result with lower accuracy is correspondingly smaller. The above estimation method is called data fusion SE method, which combines data fusion with the MES estimation model to obtain the final optimal estimation result.

[0086] The beneficial effects that can be achieved by the present invention using the above technical solutions are:

[0087] To address the issues of unknown non-Gaussian noise and time offset between SCADA and PMU measurements, this method proposes a robust SE method for power systems based on MES and data fusion, based on data fusion theory and the MES estimation model. This method has the following characteristics:

[0088] (1) Embedding zero injection expressions in the MES estimation model eliminates the need for explicit equality constraints, eliminates equality constraints, and eliminates the need to use the Lagrange multiplier method, thereby improving algorithm efficiency.

[0089] (2) The PMU buffer measurements are selected based on the robust Mahalanobis distance and assigned weights, and the SE method based on MES is further used to filter out non-Gauss measurement noise and outliers;

[0090] (3) Based on MES and data fusion theory, SE is performed on PMU and SCADA measurements with unknown measurement noise. Based on the two sets of estimation results, the estimation results from two independent MES estimators are fused by using data fusion theory to obtain the final optimal SE.

[0091] A full simulation was carried out based on the IEEE-14 and IEEE-30 node standard systems. The obtained results demonstrated the effectiveness and robustness of the proposed method, which has obvious advantages over existing methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0092] Figure 1 This is a calculation flow chart of the MES estimation model in the present invention;

[0093] Figure 2 This is the data fusion SE calculation flow chart of the present invention;

[0094] Figure 3 This is the IEEE-14 node standard system topology diagram in the present invention;

[0095] Figure 4 This is the IEEE-30 node standard system topology diagram in the present invention;

[0096] Figure 5 This is an algorithm relationship diagram of the present method and the HWLS and TWLS methods in the present invention;

[0097] Figure 6 is the absolute error of the voltage modulus of each method under Laplace measurement noise in the present invention;

[0098] Figure 7 is the absolute error of the voltage phase angle of each method under Laplace measurement noise in the present invention;

[0099] Figure 8 is the absolute error of the voltage modulus under different cases in the present invention;

[0100] Figure 9 is the absolute error of the voltage phase angle under different cases in the present invention. DETAILED DESCRIPTION

[0101] Depend on Figures 1 to 9 As shown in the figure, a robust state estimation method for power systems based on a maximum exponential square (MES) method and data fusion is proposed. First, the robust Mahalanobis distance is used to detect system outliers and assign appropriate weights to selected PMU buffer measurements. Then, the MES estimator uses these weights to filter out non-Gaussian PMU measurement noise to produce a set of state estimation results. At the same time, the MES estimator processes the received SCADA measurements containing unknown measurement noise to produce another set of state estimation results. For these two sets of estimation results, the estimation results from the two independent MES estimators are fused based on data fusion theory to obtain the final optimal estimation result.

[0102] 1) Measurement equation

[0103] According to the Wiener approximation theorem, non-Gaussian distributions p(x) can be well approximated by known Gaussian distributions. Therefore, the following model can be used to simulate non-Gaussian distribution errors:

[0104]

[0105] Where: a i represents the weight, and N A Indicates the number of Gauss elements; and ∑ i represents the mean and variance,

[0106] Based on whether the known measurement error obeys the Gauss distribution, the above formula can be further expressed as:

[0107] G(e)=(1-ε)Φ(e)+εK(e) (2)

[0108] Where: Φ(e) is the majority distribution of measurement noise, which is usually modeled as a Gauss distribution; K(e) is the unknown distribution, which is considered to be a heavy-tailed density, such as the Laplace density or Gauss density with large variance; 0≤ε≤0.5, the contamination coefficient, is used to adjust the contribution of non-Gauss components. For example, for small ε, the model indicates that most of the error follows the Gauss distribution while maintaining a small part of the non-Gauss error.

[0109] The measurement equation under non-Gauss noise can be expressed as follows:

[0110] z=h(x)+G(e) (3)

[0111] Where: z represents the measurement; h(x) represents the nonlinear measurement function; G(e) represents the measurement error;

[0112] 2) Calculation of PMU optimal buffer length

[0113] Based on Mahalanobis distance, the optimal buffer length of multiple PMU measurement points is determined. For all PMU measurement points, take them as a whole.

[0114] Assume that PMU samples n per second r =30 samples / s, SCADA sampling interval N t =5s, in one interval, the number of PMU sampling N=N t *n r = 150 samples, use Z to represent the sampling matrix, and divide it into n subset =N t = 5 subsets: Z = [Y1, Y2, Y3, Y4, Y5], calculated based on the following steps:

[0115] (1) Calculate the median of the Z matrix:

[0116] Y'=median(Z)=[y'1,y'2,y'3,y'4,y'5] (4)

[0117] (2) Perform change point detection on Y5. If a change point exists, the last PMU data is used; otherwise, Y5 is included in the PMU buffer set;

[0118] (3) Based on Mahalanobis distance P i , for η1=[y'5,y'4], η2=[y'4,y'3], …, η4=[y'2,y'1], detect system changes:

[0119]

[0120] If P in η1 i All less than Then it contains Y4, otherwise the algorithm stops; and continues to check η2=[y'4,y'3],…,η4=[y'2,y'1] until the end.

[0121] (4) Calculate the average value of the measurement within the PMU buffer set and variance

[0122]

[0123]

[0124] Where: α represents the number of columns in Z', w i is the weight of the i-th column of Z',

[0125] Based on the above steps, the optimal buffer length of multiple PMU measurement points can be obtained and the average value can be calculated. and variance After that, it can be matched with SCADA data in the same section;

[0126] 3) Robust SE method based on MES and data fusion

[0127] 3.1) MES-based SCADA robust SE

[0128] Based on the MES estimation method, the MES estimation model is established using SCADA measurements. Its expression is as follows:

[0129]

[0130] In the formula: x represents the state quantity; h i(x) represents the measurement function of the i-th SCADA; w i represents the i-th measurement weight; z i represents the i-th measurement; σ represents the Parzen window width; c(x) = 0 represents the zero injection node constraint,

[0131] The zero injection constraint in polar coordinates can be expressed as:

[0132]

[0133] Where: Ψ represents x B with x N The transformation relationship; B represents the zero injection power node; N represents the non-zero injection node,

[0134] Γ is:

[0135]

[0136] Φ is:

[0137]

[0138] Φ -1 for:

[0139]

[0140] Substituting this constraint into the MES estimation model, we get:

[0141]

[0142] make:

[0143]

[0144]

[0145] Using Newton's method to solve, we get the modified equation:

[0146]

[0147] Where: H is the m×n dimensional Jacobian matrix; F and Q are:

[0148]

[0149]

[0150] The status quantity is corrected to:

[0151]

[0152]

[0153]

[0154] The estimated iterative calculation process based on MES is as follows Figure 1 As shown,

[0155] The calculation process of the MES estimator is as follows:

[0156] ⑴ Initialize parameters, k = 0, set x k,N Initial value;

[0157] ⑵ By x k,N Calculate x according to formula (20) k,B ;

[0158] ⑶ Calculate matrices H, F and Q, solve equation (16), and get and update calculate

[0159] (4) If max|Δx k |<ε, the state estimation converges and the calculation ends, otherwise go to step ⑶;

[0160] After iterative convergence, the state error variance matrix of the SCADA-based MES estimation is:

[0161] The above embedding of zero injection expression in the MES estimation model makes it unnecessary to express the equality constraint explicitly. By eliminating the equality constraint, there is no need to use the Lagrange multiplier method, thus improving the efficiency of the algorithm.

[0162] 3.2) MES-based PMU robust SE

[0163] Similar to SCADA robust SE, PMU measurements are used alone to perform robust estimation based on the MES model, but PMU measurements are voltage and current phasors, and their measurement functions are linear functions:

[0164]

[0165] Where: is the mean value of the measurement in the PMU buffer, which is used as the pseudo measurement and calculated by formula (7); A is the relationship matrix between PMU and state quantity, is a constant coefficient matrix, that is, the Jacobian matrix of PMU measurement and state quantity; ε is the PMU measurement error, and its variance is Calculated by formula (6),

[0166] Based on the MES model, we can get:

[0167]

[0168] Also use Newton's method to solve and get the modified equation:

[0169]

[0170] The state quantity correction is the same as formula (19)-(21),

[0171] After iterative convergence, the state error variance matrix of the PMU-based MES estimation is:

[0172]

[0173] 3.3) State Fusion Based on Data Fusion Theory

[0174] Drawing on data fusion theory, SCADA and PMU measurements are considered as different data sources, and their measurements are respectively S and z P The SCADA-based and PMU-based SEs are called SSE and PSE respectively. S and z P Treatment can produce two independent sets of SE results and The corresponding error variance matrix is given by ∑ s and ∑ p express,

[0175] based on and ∑ s and ∑ p , the best fusion result is obtained through data fusion:

[0176]

[0177] From the above formula, we can see that x s with x p The denominator of the SE weight is the same, and the numerator is ∑ p and ∑ s , which means that the size of the weight is inversely proportional to the variance of the estimation result, that is, the weight of the estimation result with higher accuracy is correspondingly larger, and the weight of the estimation result with lower accuracy is correspondingly smaller. The above estimation method is called data fusion SE method, which combines data fusion with MES estimation model to obtain the final optimal estimation result. The fusion SE calculation logic diagram is shown as follows: Figure 2 shown.

[0178] 4) Example simulation and analysis

[0179] 4.1) Basic data and simulation conditions

[0180] In order to verify the performance of the proposed robust state estimation method for power systems based on MES and data fusion, simulations were performed based on IEEE-14 node and IEEE-30 node standard systems. The topological structures are shown in the following diagrams: Figure 3 IEEE-14 node standard system topology diagram and Figure 4 The IEEE-30 node standard system topology is shown in the figure.

[0181] For the IEEE-14 node standard system, three PMUs were deployed at nodes 4, 7, and 14, and 31 SCADA measurements were considered, including 16 branch powers, 14 injection powers, and one node voltage amplitude. For the IEEE-30 node standard system, 38 pairs of power measurements were considered, including 15 pairs of active and reactive power injection and 23 pairs of branch active and reactive power measurements. Active and reactive power injection at nodes 11, 12, 24, 27, and 30, as well as active and reactive power on branches 24-23, 25-26, and 30-27, were key measurements. Six PMUs were deployed at nodes 8, 9, 12, 24, 25, and 26.

[0182] Two methods are used to compare with the method proposed in this solution:

[0183] 1) Hybrid data WLS estimator (i.e. HWLS) combines buffered PMU measurements with SCADA measurements for SE.

[0184] 2) Two-stage WLS estimator (i.e., TWLS), this method first performs traditional SE based on SCADA measurements, and its SE estimation results are further combined with PMU measurements to perform the second-stage linear SE.

[0185] Using the mean absolute error (MAE) of the voltage modulus and angle estimation values as the overall performance evaluation index, the algorithm relationship between the proposed method and the HWLS and TWLS methods is as follows: Figure 5 shown.

[0186] 4.2) Simulation results and analysis

[0187] (1) Under Gaussian noise

[0188] First, consider superimposing known Gaussian noise onto the SCADA and PMU measurements to better understand the optimal performance of each method.

[0189] To this end, the noise of each SCADA and PMU measurement is assumed to have zero mean and variance 2×10 -4 and 2×10 -6The mean absolute errors of the voltage modulus and angle estimated by each method are shown in Table 1 and Table 2, respectively.

[0190] As shown in Tables 1 and 2, under Gauss measurement noise, for the IEEE-14 node standard system, the MAE of the voltage modulus and phase angle of HWLS are 5.01×10 -4 and 3.94×10 -2 The MAE of the voltage modulus and phase angle of TWLS are 4.52×10 -4 and 3.55×10 -2 The MAE of the voltage modulus and phase angle of the proposed method are 3.74×10 -4 and 3.43×10 -2 For the IEEE-30 bus standard system, the MAE of the voltage modulus and phase angle of HWLS are 3.77×10 -4 and 2.71×10 -2 The MAE of the voltage modulus and phase angle of TWLS are 2.99×10 -4 and 9.96×10 -3 The MAE of the voltage modulus and phase angle of the proposed method are 2.53×10 -4 and 6.72×10 -3 .

[0191] Therefore, the proposed method and TWLS are better and outperform the HWLS method. This is because both the proposed method and TWLS fully consider the buffered PMU measurement, avoiding the impact of the SCADA with lower measurement accuracy on the PMU with higher measurement accuracy in the mixed measurement, thereby improving the estimation accuracy.

[0192] Table 1 Mean absolute error of voltage modulus under Gaussian noise (pu)

[0193]

[0194] Table 2 Mean absolute error of voltage phase angle under Gaussian noise (°)

[0195]

[0196] To further evaluate the performance of various methods under another Gauss noise level, it is assumed that the variance of each SCADA and PMU measurement is increased to 6×10 -3 and 6×10 -5 , verification is carried out based on the IEEE-30 node standard system, and the results are shown in Table 3.

[0197] Table 3 Mean absolute error of voltage of each method under Gaussian noise

[0198]

[0199] As shown in Table 3, the MAE of the estimated voltage modulus for HWLS, TWLS and the proposed method are 1.57×10 -3 , 1.35×10 -3 , 1.21×10 -3 , and the MAE of the estimated voltage phase angle is 4.32×10 -2 , 3.33×10 -2 and 1.94×10 -2 As can be seen, the estimation errors of all methods increase slightly with the increase of noise level, because at a given measurement redundancy level, if the noise level increases, the statistical efficiency of the estimation will decrease. However, the proposed method still outperforms the other two HWLS and TWLS methods.

[0200] (2) Non-Gaussian noise

[0201] In practical applications, the noise statistics of SCADA and PMU measurements are usually unknown and deviate from the Gauss assumption due to the aging process of voltage transformers and current transformers, changes in ambient temperature, communication channel noise, etc. To evaluate the performance of various methods for unknown statistical noise, simulations are performed using non-Gauss noise models, namely Gauss mixture models and heavy-tailed Laplace noise. Specifically, for SCADA measurements, a Gauss mixture model with two Gauss components (non-Gaussian after mixing) is assumed, which consists of zero mean and variance 2×10 -4 and 2×10 -3 Indicates that the weights are 0.95 and 0.05 respectively. For PMU measurement, the variance is assumed to be 2×10 -6 and 2×10 -4 , with weights 0.9 and 0.1. In the second case, with zero mean and scale parameter 10 -3 The Laplace distribution is used for noise in SCADA and PMU measurements.

[0202] Table 4 Mean absolute error of voltage modulus under non-Gaussian noise (pu)

[0203]

[0204] Table 5 Mean absolute error of voltage phase angle under non-Gaussian noise (°)

[0205]

[0206] Tables 4 and 5 respectively show the MAE results of the voltage magnitude and angle estimated by each method in the presence of non-Gauss measurement noise simulated by the Gauss mixture model. Under the Gauss mixture model noise, for the IEEE-14 bus standard system, the MAE of the voltage magnitude and phase angle of HWLS are 9.41×10 -3 and 0.51, the MAE of the voltage modulus and phase angle of TWLS are 8.37×10 -3 and 0.39, the MAE of voltage modulus and phase angle of the proposed method are 6.75×10 -3 and 0.18; for the IEEE-30 bus standard system, the MAE of the voltage modulus and phase angle of HWLS are 4.91×10 -3 and 0.34, the MAE of the voltage modulus and phase angle of TWLS are 1.95×10 -3 and 0.27, the MAE of voltage modulus and phase angle of the proposed method are 1.42×10 -3 and 0.13.

[0207] Compared to the Gauss noise cases in Tables 1 and 2, the estimation errors of all three methods increase. However, HWLS and TWLS experience a larger increase in estimation error because they rely heavily on the Gaussian assumption for SE. In contrast, the robust estimator based on MES and data fusion proposed in this paper is more robust and produces more reasonable SE results.

[0208] Figure 6 and Figure 7 Figure 2 shows the absolute error values of HWLS, TWLS, and the proposed method under Laplace measurement noise. The results in the figure are consistent with those shown in Tables 4 and 5. Therefore, it can be concluded that the proposed robust power system state estimation method based on MES and data fusion does not require Gaussian measurement noise and is more robust in dealing with non-Gaussian noise.

[0209] (3) Robustness to bad data

[0210] In order to verify the robustness of the robust estimator proposed in this method to various types of bad data, simulations are performed based on the IEEE-30 node standard system in the presence of bad data on bad leverage points, bad key measurements, and multiple interacting measurements.

[0211] Set up the following four simulation schemes respectively:

[0212] Solution 1: When there is a single bad data, change the active power injected into P8 of node 8 to 0.1 pu;

[0213] Solution ②: Bad leverage measurement, P 19 and P 19-20 Changed to 0.1pu;

[0214] Solution ③: Bad key measurement, P 11 Changed to 0.2pu;

[0215] Solution ④: For multiple interacting bad data, change P2 and Q2 to their opposite numbers.

[0216] Figure 8 and Figure 9 The simulation results for different types of bad data under the four schemes are shown. Clearly, because the robustness of the proposed robust estimator based on MES and data fusion is provided by the robust Mahalanobis distance and MES estimator, the impact of various types of bad data is bounded, resulting in very small estimation biases. The maximum bias in voltage modulus estimation is 0.00094 pu, and the maximum bias in voltage phase angle estimation is 0.0075°, which is negligible in practical applications.

Claims

1. A robust state estimation method for power systems based on MES and data fusion, characterized by: First, the robust Mahalanobis distance is used to detect system outliers and assign appropriate weights to the selected PMU buffer measurements. Then, the MES estimator uses these weights to filter out non-Gaussian PMU measurement noise and generate a set of state estimation results. At the same time, the MES estimator processes the received SCADA measurements containing unknown measurement noise to generate another set of state estimation results. For these two sets of estimation results, the estimation results from the two independent MES estimators are fused based on data fusion theory to obtain the final optimal estimation result. 1) Measurement Equation According to the Wiener approximation theorem, non-Gaussian distributions p(x) can be well approximated by known Gaussian distributions. Therefore, the following model can be used to simulate non-Gaussian distribution errors: Where: a i represents the weight, and N A Indicates the number of Gauss elements; and ∑ i represents the mean and variance, Based on whether the known measurement error obeys the Gauss distribution, the above formula can be further expressed as: G(e)=(1-ε)Φ(e)+εK(e) (2) Where: Φ(e) is the majority distribution of measurement noise, which is usually modeled as a Gauss distribution; K(e) is the unknown distribution, which is considered to be a heavy-tailed density, such as the Laplace density or Gauss density with large variance; 0≤ε≤0.5, the contamination coefficient, is used to adjust the contribution of non-Gauss components. For example, for small ε, the model indicates that most of the error follows the Gauss distribution while maintaining a small part of the non-Gauss error. The measurement equation under non-Gauss noise can be expressed as follows: z=h(x)+G(e) (3) Where: z represents the measurement; h(x) represents the nonlinear measurement function; G(e) represents the measurement error; 2) Calculation of PMU optimal buffer length Based on Mahalanobis distance, the optimal buffer length of multiple PMU measurement points is determined. For all PMU measurement points, take them as a whole. Assume that PMU samples n per second r =30 samples / s, SCADA sampling interval N t =5s, in one interval, the number of PMU sampling N=N t *n r = 150 samples, use Z to represent the sampling matrix, and divide it into n subset =N t = 5 subsets: Z = [Y1, Y2, Y3, Y4, Y5], calculated based on the following steps: (1) Calculate the median of the Z matrix: Y'=median(Z)=[y'1,y'2,y'3,y'4,y'5] (4) (2) Perform change point detection on Y5. If a change point exists, the last PMU data is used; otherwise, Y5 is included in the PMU buffer set; (3) Based on Mahalanobis distance P i , for η1=[y'5,y'4], η2=[y'4,y'3], …, η4=[y'2,y'1], detect system changes: If P in η1 i All less than Then it contains Y4, otherwise the algorithm stops; and continues to check η2=[y'4,y'3],…,η4=[y'2,y'1] until the end. (4) Calculate the average value of the measurement within the PMU buffer set and variance Where: α represents the number of columns in Z', w i is the weight of the i-th column of Z', Based on the above steps, the optimal buffer length of multiple PMU measurement points can be obtained and the average value can be calculated. and variance After that, it can be matched with SCADA data in the same section; 3) Robust SE method based on MES and data fusion 3.1) MES-based SCADA robust SE Based on the MES estimation method, the MES estimation model is established using SCADA measurements. Its expression is as follows: In the formula: x represents the state quantity; h i (x) represents the measurement function of the i-th SCADA; w i represents the i-th measurement weight; z i represents the i-th measurement; σ represents the Parzen window width; c(x) = 0 represents the zero injection node constraint, The zero injection constraint in polar coordinates can be expressed as: Where: Ψ represents x B with x N The transformation relationship; B represents the zero injection power node; N represents the non-zero injection node, Γ is: Φ is: Φ -1 for: Substituting this constraint into the MES estimation model, we get: make: Using Newton's method to solve, we get the modified equation: Where: H is the m×n dimensional Jacobian matrix; F and Q are: The state quantity is corrected to: The calculation process of the MES estimator is as follows: ⑴ Initialize parameters, k = 0, set x k,N Initial value; ⑵ By x k,N Calculate x according to formula (20) k,B ; ⑶ Calculate matrices H, F and Q, solve equation (16), and get and update calculate (4) If max|Δx k |<ε, the state estimation converges and the calculation ends, otherwise go to step ⑶; After iterative convergence, the state error variance matrix of the SCADA-based MES estimation is: The above embedding of zero injection expression in the MES estimation model makes it unnecessary to express the equality constraint explicitly. By eliminating the equality constraint, there is no need to use the Lagrange multiplier method, thus improving the efficiency of the algorithm. 3.2) MES-based PMU robust SE Similar to SCADA robust SE, PMU measurements are used alone to perform robust estimation based on the MES model, but PMU measurements are voltage and current phasors, and their measurement functions are linear functions: Where: is the mean value of the measurement in the PMU buffer, which is used as the pseudo measurement and calculated by formula (7); A is the relationship matrix between PMU and state quantity, is a constant coefficient matrix, that is, the Jacobian matrix of PMU measurement and state quantity; ε is the PMU measurement error, and its variance is Calculated by formula (6), Based on the MES model, we can get: Also use Newton's method to solve and get the modified equation: The state quantity correction is the same as formula (19)-(21), After iterative convergence, the state error variance matrix of the PMU-based MES estimation is: 3.3) State Fusion Based on Data Fusion Theory Drawing on data fusion theory, SCADA and PMU measurements are considered as different data sources, and their measurements are respectively S and z P The SCADA-based and PMU-based SEs are called SSE and PSE respectively. S and z P Treatment can produce two independent sets of SE results and The corresponding error variance matrix is given by ∑ s and ∑ p express, based on and ∑ s and ∑ p , the best fusion result is obtained through data fusion: From the above formula, we can see that x s with x p The denominator of the SE weight is the same, and the numerator is ∑ p and ∑ s , which means that the size of the weight is inversely proportional to the variance of the estimation result, that is, the weight of the estimation result with higher accuracy is correspondingly larger, and the weight of the estimation result with lower accuracy is correspondingly smaller. The above estimation method is called data fusion SE method, which combines data fusion with the MES estimation model to obtain the final optimal estimation result.

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