A distribution network state estimation method based on matrix filling

Abstract

The present invention discloses a distribution network state estimation method based on matrix filling, which relates to the field of distribution network. The regular term of the penalty function method is used to process the flow constraint, which weakens the influence of the distribution network line parameter deviation on the state estimation accuracy. The weight matrix for the matrix filling state estimation problem is designed to balance the bias of different error levels of different types of measurements on the matrix filling results, thereby improving the state estimation accuracy. The proposed APALM algorithm introduces an extrapolation amount to improve the PALM algorithm, accelerates the convergence speed and approaches the global optimal solution. When used to solve the distribution network state estimation problem based on matrix filling, the state estimation accuracy is improved and the solution time is shortened. The present invention realizes high-precision distribution network state estimation that only relies on the SCADA real-time measurement data of some nodes under the condition of only SCADA configuration and incomplete measurement; and realizes high-precision distribution network state estimation that is higher than the pseudo-measurement frequency.

Description

Technical Field

[0001] The invention relates to the field of distribution networks, and in particular to a distribution network state estimation method based on matrix filling. Background Art

[0002] Solving state estimation with the weighted least square (WLS) method requires redundant measurements, that is, the number of effective measurement equations is greater than the number of state variables. However, in most distribution networks, due to economic considerations, only a few nodes are equipped with a Supervisory Control And Data Acquisition (SCADA) system, resulting in partially observable or low observable measurement conditions, which makes the WLS method unable to solve the state estimation problem in partially observable distribution networks. To solve this problem, the node load forecast results obtained by smart meter data are used to supplement the pseudo-measurement results. However, the introduction of pseudo-measurements with large errors will reduce the accuracy of distribution network state estimation. The low sampling frequency of smart meters limits the interval of pseudo-measurements and the frequency of distribution network state estimation, which makes it difficult to capture the rapid system state changes caused by the integration of high penetration renewable energy. In addition, the lines in the distribution network are short, so the aging of the wires may cause non-negligible deviations between the actual line parameters and the recorded values, which will also reduce the performance of distribution network state estimation. Summary of the invention

[0003] The purpose of the present invention is to provide a distribution network state estimation method based on matrix filling to solve the problems raised in the above background technology.

[0004] To achieve the above object, the present invention provides the following technical solutions:

[0005] A distribution network state estimation method based on matrix filling includes the following steps:

[0006] Step 1: Construct a distribution network state estimation model based on matrix filling;

[0007] 1.1 Constructing the system variable matrix

[0008] Let M denote the system variable matrix containing the true values ​​to be reconstructed, Ω be the binary observation matrix describing the locations of the known items, Ω has the same size as M and is defined as follows: if M ij Known, Ω ij =1; otherwise 0, then, let P Ω (M) represents the spatiotemporal measurement matrix for which measurement data can be obtained, and its unknown items are usually set to 0. Obviously, P Ω (M) = M⊙Ω, where ⊙ represents the Hadamard product;

[0009] In the distribution network, the measured values ​​of the balancing nodes are known, and the voltage phase angle is fixed to 0°. Then, the voltage phasor, voltage amplitude, and real / imaginary parts of active / reactive power injection at the unbalanced (PQ) node are used to form M. For a part of the system variable matrix at a certain time t, M (t) , the same variables of different nodes are arranged according to the original value, and different variables of the same node are arranged by column. We will (t) Arranged as:

[0010]

[0011] In the formula and are the voltage phasor vector and node power injection vector of the unbalanced node, Represents a node index set, represents the number of nodes, considering the time series t=1,…,T to expand the time scale of the system variable matrix, M consists of multiple M (t) Vertically stacked:

[0012]

[0013] P Ω (M) is called the spatiotemporal measurement indicator matrix, which is filled with the corresponding measurement data, while the entries of unmeasured variables are zero. According to the composition of the system variable matrix, the obtained measurement data are filled into the spatiotemporal measurement matrix P Ω (M);

[0014] 1.2 Establishing a linear power flow model

[0015] First, the node admittance matrix Y is divided into block form (3). Then, according to the complex node injection power equation (4) and Ohm's law (5), formula (6) is derived:

[0016]

[0017]

[0018]

[0019] i=Y L1 v1+Y LL v (6)

[0020] in is the current injection vector of the unbalanced node, v1 is the voltage phasor of the balanced node, and then, by eliminating i in formula (6), the fixed point equation (7) of the node voltage phasor is obtained:

[0021]

[0022]

[0023] Where w is the zero-load voltage, our goal is to establish a linear model of the voltage-power relationship, which is in the form of (9):

[0024]

[0025] Where |·| represents the phasor amplitude. Substituting (7) into (9), we get the coefficient matrix B and vector b:

[0026]

[0027] b=w (11)

[0028] The fixed point equation (12) provides an iterative process where the updated equation for the voltage phasor is

[0029] v (k) =G s (v (k-1) ) (12)

[0030] Iteration (12) can be regarded as an extension of the traditional Z-bus method. Given an initial point v (0) , fixed node injection power s and iteration number k, the updated node voltage phasor is given by the function G in (13) s It is determined by formula (7):

[0031]

[0032] When (13) iteratively updates the node voltage phasor estimate When k iterations converge, the coefficient matrices B and C are also fixed values:

[0033]

[0034] Then, if the actual node power injection amount is close to the above fixed value s, the node voltage phasor estimation value Calculated by the model, this shows that there is an approximately linear relationship between the node voltage phasor and the amount of power injected;

[0035] In addition, the node voltage phasor equation is equivalently divided into three sub-equations, including the real part, imaginary part and amplitude:

[0036]

[0037]

[0038]

[0039] in

[0040]

[0041]

[0042]

[0043]

[0044]

[0045]

[0046] The power flow constraint is added to the matrix filling model to ensure that the restoration term of M complies with the operation regulations. When the power injection of all nodes is determined, the voltage phasor is determined by the power flow. Then, the linear power flow model is used to reduce the computational complexity of the optimization model.

[0047] 1.3 Establishing a complete matrix filling model for distribution network state estimation

[0048] In order to estimate the distribution network state based on the matrix filling model, the linear power flow constraints obtained by formulas (15)-(17) are added to the nuclear norm minimization problem. Considering a time series t=1,…,T, formula (13) is iterated to obtain the coefficient matrix components at each time t and supplement the three constraints (24)-(26):

[0049]

[0050]

[0051]

[0052] The real part, imaginary part, amplitude of the node voltage phasor and the node active and reactive power injection in the above constraints are all parts of the estimated value X of the system variable matrix M. Therefore, according to the arrangement of the estimated value X, the components of the estimated value X are replaced according to their corresponding positions, such as formulas (27)-(31):

[0053]

[0054]

[0055]

[0056]

[0057]

[0058] in is the basis vector, and the subscript n represents the position of the only non-zero element. In addition, the basic matrix filling model for distribution network state estimation is based on the optimization problems (32)-(36):

[0059]

[0060] st X⊙Ω=P Ω (M) (33)

[0061] For t=1,…,T:

[0062]

[0063]

[0064]

[0065] in

[0066]

[0067] In formula (37), and Correspondingly, it represents the set of nodes with SCADA and pseudo measurements. For distribution network state estimation, P Ω (M) includes the real-time voltage amplitude from SCADA, the active / reactive node power injection measurement values ​​and the active / reactive node power injection pseudo measurement values ​​from the smart meter. Then, the binary observation matrix Ω defined by formula (37) represents the SCADA and pseudo measurement in X, M and P Ω (M) position;

[0068] The optimization problem (32)-(36) is processed by noise-resistant balancing, that is, the linear constraints (33)-(36) are put into the objective function (32) as regularization terms, the constraint (33) in matrix form is processed by the F-norm, and the constraint (34)-(36) in vector form is processed by the 2-norm, as shown in the modified optimization problem (38):

[0069]

[0070] Where μ and ν are penalty coefficients;

[0071] For distribution network state estimation, different types of measurement data have different magnitudes, which may lead to unbalanced estimation accuracy of different measurement variables. Inspired by the WLS method, the weight matrix W is added to the second part of the optimization problem (38) as a heuristic improvement of (39) to deal with this problem:

[0072]

[0073] in

[0074]

[0075] In (40), σ V , σ PQ and σ pseudo They represent the voltage amplitude, the active / reactive node injection power measured by SCADA, and the standard deviation of the active / reactive node injection power measured by smart meters. Ω (M), the introduction of standard deviation distinguishes the weights of various measurements, P Ω (M) ij The inverse of the absolute error is converted to X and P Ω (M) ij The relative error between them, then, based on the penalty function method, (39) constitutes an unconstrained optimization problem, and the distribution network state estimation is solved by the matrix filling model;

[0076] Step 2: Solve the distribution network state estimation model based on matrix filling, including the following steps;

[0077] (1) Obtain P Ω (M),Ω,W,μ,ν and time series length T, set the maximum number of iterations N max_iter , step length and tolerance δ tol , initialize the number of iterations k = 0, coefficient β0 = 1;

[0078] (2) Through U SVD ΣV SVD =SVD(P Ω (M)) Calculate P Ω The matrix singular value decomposition (SVD) result of (M) is used to initialize U -1 =U0=U SVD Σ 1 / 2 ,V -1 =V0=Σ 1 / 2 V SVD ;

[0079] (3) Calculation

[0080] (4) Calculate U k ′=U k +γ k (U k -U k-1 );

[0081] (5) Calculation in:

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088]

[0089] (6) Calculation

[0090] (7) Calculation

[0091] (8) Calculate V k ′=V k +γ k (V k -V k-1 );

[0092] (9) Calculation in:

[0093]

[0094]

[0095]

[0096]

[0097] (10) Calculation

[0098] (11) Calculation

[0099] (12) Calculate X k+1 =U k+1 V k+1 ;

[0100] (13) Update k→k+1;

[0101] (14) Calculate V k ′=V k +γ k (V k -V k-1 );

[0102] (15) If ||X k+1 -Xk || F <δ tol Or k = N max_iter , go to step (16); otherwise, go to step (3);

[0103] (16) Output U=U k ,V=V k ,X=X k ;

[0104] (17) Obtain state estimation results, including:

[0105] Estimated node voltage amplitude:

[0106] Node voltage phase angle estimation:

[0107] Compared with the prior art, the present invention has the following beneficial effects:

[0108] 1) The regularization term of the penalty function method is used to deal with the power flow constraints, which weakens the influence of the distribution network line parameter deviation on the state estimation accuracy;

[0109] 2) A weight matrix for the matrix filling state estimation problem is designed to balance the bias of different error levels of different types of measurements on the matrix filling results, thereby improving the state estimation accuracy;

[0110] 3) The proposed APALM (Accelerated Proximal Alternating Linearized Minimization) algorithm introduces extrapolation to improve the PALM algorithm, accelerates the convergence speed and approaches the global optimal solution. When used to solve the distribution network state estimation problem based on matrix filling, it improves the state estimation accuracy and shortens the solution time;

[0111] 4) High-precision distribution network state estimation is achieved by relying only on SCADA real-time measurement data of some nodes under the condition of only SCADA configuration and incomplete measurement;

[0112] 5) High-precision distribution network state estimation is achieved at a frequency higher than the pseudo-measurement frequency. BRIEF DESCRIPTION OF THE DRAWINGS

[0113] Figure 1 is a flow chart of the method of the present invention;

[0114] FIG2( a ) is a schematic diagram showing an example of measurement distribution in the process of continuous power distribution network state estimation according to the method of the present invention;

[0115] FIG. 2( b ) shows the relationship between M and P in the method of the present invention. Ω(M) Detailed information and corresponding simulation setup diagram;

[0116] Figure 3 The topology of the IEEE 33-node distribution network and the schematic diagram of the SCADA installation are shown;

[0117] Figure 4 It is a schematic diagram of the performance comparison of the method of the present invention in the voltage amplitude state estimation of IEEE 33-node distribution network, including pseudo measurement and 50% ratio SCADA;

[0118] Figure 5 It is a schematic diagram of the performance comparison of the method of the present invention in the voltage phase angle state estimation of IEEE 33-node distribution network, including pseudo measurement and 50% ratio SCADA;

[0119] Figure 6 It is a schematic diagram of the performance comparison of the method of the present invention in the voltage amplitude state estimation of IEEE 33-node distribution network, including 50% SCADA;

[0120] Figure 7 The figure is a schematic diagram showing the performance comparison of the method of the present invention in voltage phase angle state estimation of IEEE 33-node distribution network, including 50% SCADA. DETAILED DESCRIPTION

[0121] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0122] See also Figures 1 to 3 In an embodiment of the present invention, a distribution network state estimation method based on matrix filling is characterized by comprising the following steps:

[0123] Step 1: Construct a distribution network state estimation model based on matrix filling;

[0124] 1.1 Constructing the system variable matrix

[0125] Let M denote the system variable matrix containing the true values ​​to be reconstructed, Ω be the binary observation matrix describing the locations of the known items, Ω has the same size as M and is defined as follows: if M ij Known, Ω ij =1; otherwise 0, then, let P Ω (M) represents the spatiotemporal measurement matrix for which measurement data can be obtained, and its unknown items are usually set to 0. Obviously, P Ω(M) = M⊙Ω, where ⊙ represents the Hadamard product;

[0126] In the distribution network, the measured values ​​of the balancing nodes are known, and the voltage phase angle is fixed to 0°. Then, the voltage phasor, voltage amplitude, and real / imaginary parts of active / reactive power injection at the unbalanced (PQ) node are used to form M. For a part of the system variable matrix at a certain time t, M (t) , the same variables of different nodes are arranged according to the original value, and different variables of the same node are arranged by column. We will (t) Arranged as:

[0127]

[0128] In the formula and are the voltage phasor vector and node power injection vector of the unbalanced node, Represents a node index set, represents the number of nodes, considering the time series t=1,…,T to expand the time scale of the system variable matrix, M consists of multiple M (t) Vertically stacked:

[0129]

[0130] P Ω (M) is called the spatiotemporal measurement indicator matrix, which is filled with the corresponding measurement data, while the entries of unmeasured variables are zero. According to the composition of the system variable matrix, the obtained measurement data are filled into the spatiotemporal measurement matrix P Ω (M);

[0131] 1.2 Establishing a linear power flow model

[0132] First, the node admittance matrix Y is divided into block form (3). Then, according to the complex node injection power equation (4) and Ohm's law (5), formula (6) is derived:

[0133]

[0134]

[0135]

[0136] i=Y L1 v1+Y LL v (6)

[0137] in is the current injection vector of the unbalanced node, v1 is the voltage phasor of the balanced node, and then, by eliminating i in formula (6), the fixed point equation (7) of the node voltage phasor is obtained:

[0138]

[0139]

[0140] Where w is the zero-load voltage, our goal is to establish a linear model of the voltage-power relationship, which is in the form of (9):

[0141]

[0142] Where |·| represents the phasor amplitude. Substituting (7) into (9), we get the coefficient matrix B and vector b:

[0143]

[0144] b=w (11)

[0145] The fixed point equation (7) provides an iterative process where the updated equation for the voltage phasor is

[0146] v (k) =G s (v (k-1) ) (12)

[0147] Iteration (12) can be regarded as an extension of the traditional Z-bus method. Given an initial point v (0) , fixed node injection power s and iteration number k, the updated node voltage phasor is given by the function G in (13) s It is determined by formula (7):

[0148]

[0149] When (13) iteratively updates the node voltage phasor estimate When k iterations converge, the coefficient matrices B and C are also fixed values:

[0150]

[0151] Then, if the actual node power injection amount is close to the above fixed value s, the node voltage phasor estimation value Calculated by model (9), this shows that there is an approximately linear relationship between the node voltage phasor and the power injection amount;

[0152] In addition, the node voltage phasor equation is equivalently divided into three sub-equations, including the real part, imaginary part and amplitude:

[0153]

[0154]

[0155]

[0156] in

[0157]

[0158]

[0159]

[0160]

[0161]

[0162]

[0163] The power flow constraint is added to the matrix filling model to ensure that the restoration term of M complies with the operation regulations. When the power injection of all nodes is determined, the voltage phasor is determined by the power flow. Then, the linear power flow model is used to reduce the computational complexity of the optimization model.

[0164] 1.3 Establishing a complete matrix filling model for distribution network state estimation

[0165] In order to estimate the distribution network state based on the matrix filling model, the linear power flow constraints obtained by formulas (15)-(17) are added to the nuclear norm minimization problem. Considering a time series t=1,…,T, formula (13) is iterated to obtain the coefficient matrix components at each time t and supplement the three constraints (24)-(26):

[0166]

[0167]

[0168]

[0169] The real part, imaginary part, amplitude of the node voltage phasor and the node active and reactive power injection in the above constraints are all parts of the estimated value X of the system variable matrix M. Therefore, according to the arrangement of the estimated value X, the components of the estimated value X are replaced according to their corresponding positions, such as formulas (27)-(31):

[0170]

[0171]

[0172]

[0173]

[0174]

[0175] in is the basis vector, and the subscript n represents the position of the only non-zero element. In addition, the basic matrix filling model for distribution network state estimation is based on the optimization problems (32)-(36):

[0176]

[0177] st X⊙Ω=P Ω (M) (33)

[0178] For t=1,…,T:

[0179]

[0180]

[0181]

[0182] in

[0183]

[0184] In formula (37), and Correspondingly, it represents the set of nodes with SCADA and pseudo measurements. For distribution network state estimation, P Ω (M) includes the real-time voltage amplitude from SCADA, the active / reactive node power injection measurement values ​​and the active / reactive node power injection pseudo measurement values ​​from the smart meter. Then, the binary observation matrix Ω defined by formula (37) represents the SCADA and pseudo measurement in X, M and P Ω (M) position;

[0185] The optimization problem (32)-(36) is processed by noise-resistant balancing, that is, the linear constraints (33)-(36) are put into the objective function (32) as regularization terms, the constraint (33) in matrix form is processed by the F-norm, and the constraint (34)-(36) in vector form is processed by the 2-norm, as shown in the modified optimization problem (38):

[0186]

[0187] Where μ and ν are penalty coefficients;

[0188] For distribution network state estimation, different types of measurement data have different magnitudes, which may lead to unbalanced estimation accuracy of different measurement variables. Inspired by the WLS method, the weight matrix W is added to the second part of the optimization problem (38) as a heuristic improvement of (39) to deal with this problem:

[0189]

[0190] in

[0191]

[0192] In (40), σ V , σ PQ and σ pseudo They represent the voltage amplitude, the active / reactive node injection power measured by SCADA, and the standard deviation of the active / reactive node injection power measured by smart meters. Ω (M), the introduction of standard deviation distinguishes the weights of various measurements, P Ω (M) ij The inverse of the absolute error is converted to X and P Ω (M) ij The relative error between them, then, based on the penalty function method, (39) constitutes an unconstrained optimization problem, and the distribution network state estimation is solved by the matrix filling model;

[0193] Step 2: Solve the distribution network state estimation model based on matrix filling, including the following steps;

[0194] (1) Obtain P Ω (M),Ω,W,μ,ν and time series length T, set the maximum number of iterations N max_iter , step length and tolerance δ tol , initialize the number of iterations k = 0, coefficient β0 = 1;

[0195] (2) Through U SVD ΣV SVD =SVD(P Ω (M)) Calculate P Ω The matrix singular value decomposition (SVD) result of (M) is used to initialize U -1 =U0=U SVD Σ 1 / 2 ,V -1 =V0=Σ 1 / 2 V SVD ;

[0196] (3) Calculation

[0197] (4) Calculate U k ′=U k +γ k (U k -U k-1 );

[0198] (5) Calculation in:

[0199]

[0200]

[0201]

[0202]

[0203]

[0204]

[0205]

[0206] (6) Calculation

[0207] (7) Calculation

[0208] (8) Calculate V k ′=V k +γ k (V k -V k-1 );

[0209] (9) Calculation in:

[0210]

[0211]

[0212]

[0213]

[0214] (10) Calculation

[0215] (11) Calculation

[0216] (12) Calculate X k+1 =U k+1 V k+1 ;

[0217] (13) Update k→k+1;

[0218] (14) Calculate V k ′=V k +γ k (V k -V k-1 );

[0219] (15) If ||X k+1 -X k || F <δ tol Or k = N max_iter , go to step (16); otherwise, go to step (3);

[0220] (16) Output U=U k ,V=V k ,X=X k ;

[0221] (17) Obtain state estimation results, including:

[0222] Estimated node voltage amplitude: Node voltage phase angle estimation: As a preferred embodiment of the present invention, the following algorithm is adopted:

[0223]

[0224] The above algorithm uses low-rank matrix decomposition to solve the problem that it is time-consuming to repeatedly perform matrix singular value decomposition (SVD) to calculate the nuclear norm of X in (39). According to (41), the nuclear norm of X is calculated by half of the sum of the F-norms of U and V:

[0225]

[0226] By replacing the nuclear norm of X, the distribution network state estimation problem (39) is transformed into a low-rank matrix factorization (LRMF) problem (42), which is solved by an alternating minimization (Alternating Minimization) or stochastic gradient descent (SGD) algorithm. The objective function of the LRMF problem (42) becomes non-convex and can be extended to a larger matrix by adjusting the parameter r:

[0227]

[0228] The non-convex LRMF problem is divided into two parts according to the form of (43);

[0229]

[0230] Where h(U,V) is a non-convex function, f(U) and g(V) are convex functions;

[0231] The Proximal Alternating Linear Minimization (PALM) algorithm is used to solve (42). PALM transforms h(U, V k ) is linearized into (44):

[0232]

[0233] Among them, U h(U k ,V k ) represents h(U,V k ) in U k The partial guide at represents the step size, substituting (44) into (45), the approximate algorithm obtains the analytical solution (46) for U k+1 Calculation and update of

[0234]

[0235] Likewise, V k+1 Calculate and update through the above linearization algorithm and approximation algorithm;

[0236] The following formula (47) and formula (48) demonstrate the update of U and V in the Accelerated Proximal Alternating Linear Minimization (APALM) algorithm based on extrapolation:

[0237] U k ′=U k +γ k (U k -U k-1 ) (47)

[0238] V k ′=V k +γ k (V k -V k-1 ) (48)

[0239] The coefficient γ k Set to a fixed value within (0,1), when γ k = 0, APALM is simplified to PALM, γ k Dynamically update through formulas (49)-(50), β0=1;

[0240]

[0241]

[0242] The low-rank decomposition-matrix completion problem (42) for distribution network state estimation is decomposed from (51)-(53) into the form of (43);

[0243]

[0244]

[0245]

[0246] Then, U k+1 and V k+1 The approximate functions of are (54)-(55);

[0247]

[0248]

[0249] The convergence criterion of Algorithm 1 is optional. k+1 and X k The F-norm of the difference between the two values ​​is reduced to a relatively small tolerance δ tol When , we consider APALM to converge;

[0250] ||X k+1 -X k || F <δ tol (56)

[0251] The partial derivative components in Algorithm 1 are listed in (57)-(64).

[0252]

[0253]

[0254]

[0255]

[0256]

[0257]

[0258]

[0259]

[0260] in

[0261]

[0262]

[0263]

[0264] Since APALM k The iteration of will cause the estimated value of the node injection power s to change. The movement of the fixed point of equation (7) will cause B (t) and C (t) Therefore, B (t) and C (t) It should be updated within a certain iteration interval to ensure the accuracy of the linear power flow model while reducing the computational burden.

[0265] As a preferred embodiment of the present invention, the simulation is set as follows:

[0266] IEEE 33 and 118-bus distribution networks are selected to demonstrate the performance of the proposed distribution network state estimation model based on APALM matrix filling. The accuracy of distribution network state estimation is evaluated by the mean absolute error (MAE) of voltage magnitude and the mean absolute percentage error (MAPE) of voltage phase angle. For ease of comparison, a unified simulation setting is used for all case studies, including:

[0267] 1) SCADA measures node voltage amplitude and active / reactive power injection at a sampling interval of 5 seconds, while smart meters measure active / reactive power injection at a sampling interval of 15 minutes. Therefore, the pseudo measurement interval is determined to be 15 minutes.

[0268] 2) The distribution network state estimation is performed continuously at 3-minute intervals. The time series length T of the distribution network state estimation based on matrix filling is 5, so at the corresponding distribution network state estimation moment, there is one set of pseudo measurements and 5 sets of SCADA measurements in the 12-minute backward time window.

[0269] 3) Only some nodes in the tested distribution network have SCADA installed. The installation rate represents the percentage of nodes with SCADA installed to the total number of nodes, which determines the observability of the system.

[0270] 4) Assume that the noise conforms to Gaussian normal distribution. The standard deviation of line impedance is 10%, the standard deviation of SCADA voltage amplitude measurement is 1%, the standard deviation of injected power measurement is 2%, and the standard deviation of injected power pseudo measurement is 20%.

[0271] The 30-minute timeline in Figure 2(a) shows an example of measurement distribution during continuous distribution network state estimation. Figure 2(b) shows the timeline of M and P. Ω (M) and the corresponding simulation settings. ΩIn each sub-matrix of (M), the real and imaginary parts of the node voltage phasors corresponding to the first two rows are zero because they are not measured. The last three rows corresponding to the node voltage amplitude, active and reactive power injection are filled with SCADA measurement data, which may also come from pseudo-measurement data in the smart meter sampling time sub-matrix.

[0272] The test run scenario is randomly selected and determined by the injected power of a real smart meter dataset of a city distribution network. All case analyses are run on the MATLAB R2022b platform, with an Intel i7-9700@3GHz CPU and 48GB memory.

[0273] (1) Fixed SCADA installation ratio

[0274] We first evaluate the performance of the proposed accelerated matrix filling-in based distribution network state estimation method against a similar approach with a fixed SCADA installation ratio of 50%. Figure 3 The topology and SCADA installation of the IEEE 33-node distribution network are shown, where nodes with SCADA installed are randomly selected except for the balancing nodes.

[0275] 1) IEEE 33-node distribution network

[0276] We first consider the condition that smart meters are installed on all nodes and pseudo measurements can be made. Table 1 lists the statistical results of distribution network state estimation using various methods.

[0277] Table 1 Performance comparison on IEEE 33-node distribution network state estimation, including pseudo-measurement and 50% SCADA

[0278]

[0279] Note: WLS: Weighted Least Squares

[0280] This method (MCSE_W_APALM): The APALM algorithm solves the distribution network state estimation problem based on matrix filling involving weighted matrices;

[0281] MCSE_APALM: APALM algorithm solves the distribution network state estimation problem based on matrix filling without weighted matrix;

[0282] MCSE_W_AM: Alternating minimization algorithm for solving distribution network state estimation problems based on matrix completion involving weighted matrices;

[0283] MCSE_AM: Alternating minimization algorithm for solving distribution network state estimation problems based on matrix completion without weighted matrices;

[0284] The traditional WLS method has stable, fast and accurate performance in distribution network state estimation, and provides redundant measurements through pseudo measurements. Due to the accurate nonlinear power flow model and low computational burden, WLS achieves a voltage amplitude estimation error of 0.05% in only 0.182 seconds, which is better than all matrix filling-based methods. However, the line impedance deviation deteriorates the voltage phase angle estimation error of WLS to 0.0222 degrees. Then, the matrix filling-based distribution network state estimation method using APALM to solve the weight matrix achieves voltage amplitude and phase angle estimation errors of 0.09% and 0.0138 degrees in 1.7577 seconds. Compared with WLS, the proposed method has higher accuracy in voltage phase angle estimation, while its estimation error in voltage amplitude is limited to a very small level. The excellent performance of the proposed method in distribution network state estimation is mainly due to the different treatment of various error measurements by the weight matrix. This can be demonstrated by comparing the results of establishing the distribution network state estimation model based on matrix filling with and without the proposed weight matrix (this method is compared with MCSE_APALM, or MCSE_W_AM is compared with MCSE_AM). When comparing different solution algorithms (this method and MCSE_W_AM algorithm), the computing time consumed by using APALM to solve the distribution network state estimation based on matrix filling is usually much less than the computing time consumed by using AM to solve the distribution network state estimation based on matrix filling. Therefore, the faster the solution speed of APALM, the more iterations it has in the same time, and the more optimal solutions it has, which also helps to improve the accuracy of distribution network state estimation. It can also be found that the introduction of the proposed weight matrix accelerates the convergence speed of the distribution network state estimation problem based on matrix filling, especially when solving it with APALM.

[0285] Figure 4 and Figure 5 The voltage amplitude and phase angle estimates and their true values ​​obtained by different methods for distribution network state estimation are compared. In general, the deviations between the true and estimated values ​​of all methods are relatively small at the beginning of the feeder line. As the feeder line extends, these deviations gradually increase and reach a maximum value at the end of the feeder line. This is mainly due to the cumulative errors caused by measurement noise, line impedance deviation and power flow model residuals. The voltage amplitude and phase angle curves estimated by WLS and the method proposed in this paper are very close to the true values, which is excellent in the statistical results of distribution network state estimation errors. However, compared with WLS, the method proposed in this paper has more reasonable error control at the end of the feeder and has better performance in voltage phase angle estimation.

[0286] Then, we consider the case where pseudo measurements are unavailable and 50% of the measurement data in SCADA makes the distribution system partially observable. Table 2 lists the statistical results of distribution network state estimation for various methods.

[0287] Table 2 Performance comparison on IEEE 33-node distribution network state estimation, including 50% SCADA

[0288]

[0289]

[0290] Due to the lack of redundant measurements, the traditional WLS method is not feasible for the partially observable distribution network state estimation problem. However, since the matrix filling model has no requirements on the available data ratio, the distribution network state estimation method based on matrix filling can still solve the problem when pseudo measurements are unavailable and SCADA measurements are few. The distribution network state estimation method based on matrix filling using APALM to solve the weight matrix achieved 0.18% and 0.0581 degrees of voltage amplitude and phase angle estimation errors in 21.6471 seconds. The test results show that the proposed method shows relatively fast and accurate performance in partially observable distribution network state estimation, and outperforms other similar methods in voltage amplitude / phase angle estimation and calculation time. The results of the proposed method and MCSE_APALM and MCSE_W_AM also verify the effectiveness of differentiating the measurement of various errors through the weight matrix and introducing APALM in the distribution network state estimation based on matrix filling. It should be noted that the distribution network state estimation based on matrix filling without APALM to solve the weight matrix will fall into the wrong convergence direction, thus showing poor distribution network state estimation performance.

[0291] Figure 6 and Figure 7 The performance of different methods in partially observable distribution network state estimation is compared. Similar to the case of redundant measurements, at the end of the feeder, the deviation between the true value and the estimated value gradually increases due to the accumulated error. The voltage amplitude and phase angle curves estimated by the proposed method are close to the true value with acceptable errors. This shows that the proposed method reduces the impact of measurement and line impedance noise and overcomes the obstacle of lack of measurement.

[0292] (2) Different SCADA installation ratios

[0293] The state estimation effect of this method was tested under different SCADA installation ratios, and the conclusions were basically consistent with those under the condition of fixed SCADA installation ratio.

[0294] Table 3 Performance comparison on IEEE 33-node distribution network state estimation, including pseudo measurements

[0295]

[0296] Table 4 Performance comparison on IEEE 33-node distribution network state estimation without pseudo measurements

[0297]

[0298] In summary, the present invention:

[0299] 1) The regularization term of the penalty function method is used to deal with the power flow constraints, which weakens the influence of the distribution network line parameter deviation on the state estimation accuracy;

[0300] 2) A weight matrix for the matrix filling state estimation problem is designed to balance the bias of different error levels of different types of measurements on the matrix filling results, thereby improving the state estimation accuracy;

[0301] 3) The proposed APALM algorithm introduces extrapolation to improve the PALM algorithm, accelerates the convergence speed and approaches the global optimal solution. When used to solve the distribution network state estimation problem based on matrix filling, it improves the state estimation accuracy and shortens the solution time;

[0302] 4) High-precision distribution network state estimation is achieved by relying only on SCADA real-time measurement data of some nodes under the condition of only SCADA configuration and incomplete measurement;

[0303] 5) High-precision distribution network state estimation is achieved at a frequency higher than the pseudo-measurement frequency.

[0304] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

[0305] In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims

1. A distribution network state estimation method based on matrix filling, characterized in that: The steps include: Step 1: Construct a distribution network state estimation model based on matrix filling; 1.1 Constructing the system variable matrix Let M denote the system variable matrix containing the true values ​​to be reconstructed, Ω be the binary observation matrix describing the locations of the known items, Ω has the same size as M and is defined as follows: if M ij Known, Ω ij =1; otherwise 0, then, let P Ω (M) represents the spatiotemporal measurement matrix for which measurement data can be obtained, and its unknown items are usually set to 0. Obviously, P Ω (M) = M⊙Ω, where ⊙ represents the Hadamard Product; In the distribution network, the measured values ​​of the balancing nodes are known, and the voltage phase angle is fixed to 0°. Then, the voltage phasor, voltage amplitude, and real / imaginary parts of active / reactive power injection at the unbalanced (PQ) node are used to form M. For a part of the system variable matrix at a certain time t, M (t) , the same variables of different nodes are arranged according to the original value, and different variables of the same node are arranged in columns. (t) Arranged as: In the formula and are the voltage phasor vector and node power injection vector of the unbalanced node, Represents a node index set, represents the number of nodes, considering the time series t=1,…,T to expand the time scale of the system variable matrix, M consists of multiple M (t) Vertically stacked: P Ω (M) is called the spatiotemporal measurement indicator matrix, which is filled with the corresponding measurement data, while the entries of unmeasured variables are zero. According to the composition of the system variable matrix, the obtained measurement data are filled into the spatiotemporal measurement matrix P Ω (M); 1.2 Establishing a linear power flow model First, the node admittance matrix Y is divided into block form (3). Then, according to the complex node injection power equation (4) and Ohm's law (5), formula (6) is derived: i=Y L1 v1+Y LL (6) in is the current injection vector of the unbalanced node, v1 is the voltage phasor of the balanced node, and then, by eliminating i in formula (6), the fixed point equation (7) of the node voltage phasor is obtained: Where w is the zero-load voltage. Our goal is to establish a linear model of the voltage-power relationship, which is in the form of (9): Where |·| represents the phasor amplitude. Substituting (7) into (9), we get the coefficient matrix B and vector b: b=w (11) The fixed point equation (7) provides an iterative process where the updated equation for the voltage phasor is v (k) =G s (v (k-1) ) (12) Iteration (12) can be regarded as an extension of the traditional Z-bus method. Given an initial point v (0) , fixed node injection power s and iteration number k, the updated node voltage phasor is given by the function G in (13) s It is determined by formula (12): When (13) iteratively updates the node voltage phasor estimate When k iterations converge, the coefficient matrices B and C are also fixed values: Then, if the actual node power injection amount is close to the above fixed value s, the node voltage phasor estimation value Calculated by model (9), this shows that there is an approximately linear relationship between the node voltage phasor and the power injection amount; In addition, the node voltage phasor equation is equivalently divided into three sub-equations, including the real part, imaginary part and amplitude: in The power flow constraint is added to the matrix filling model to ensure that the restoration term of M complies with the operation regulations. When the power injection of all nodes is determined, the voltage phasor is determined by the power flow. Then, the linear power flow model is used to reduce the computational complexity of the optimization model. 1.3 Establishing a complete matrix filling model for distribution network state estimation In order to estimate the distribution network state based on the matrix filling model, the linear power flow constraints obtained by formulas (15)-(17) are added to the nuclear norm minimization problem. Considering a time series t=1,…,T, formula (13) is iterated to obtain the coefficient matrix components at each time t and supplement the three constraints (24)-(26): The real part, imaginary part, amplitude of the node voltage phasor and the node active and reactive power injection in the above constraints are all parts of the estimated value X of the system variable matrix M. Therefore, according to the arrangement of the estimated value X, the components of the estimated value X are replaced according to their corresponding positions, such as formulas (27)-(31): in is the basis vector, and the subscript n represents the position of the only non-zero element. In addition, the basic matrix filling model for distribution network state estimation is based on the optimization problems (32)-(36): s.t.X⊙Ω=P Ω (M) (33) For t=1,…,T: in In formula (37), and Correspondingly, it represents the set of nodes with SCADA (Supervisory Control And Data Acquisition) and pseudo-measurements. For distribution network state estimation, P Ω (M) includes the real-time voltage amplitude from SCADA, the active / reactive node power injection measurement values ​​and the active / reactive node power injection pseudo measurement values ​​from the smart meter. Then, the binary observation matrix Ω defined by formula (37) represents the SCADA and pseudo measurement in X, M and P Ω (M) position; The optimization problem (32)-(36) is processed by noise-resistant balancing, that is, the linear constraints (33)-(36) are put into the objective function (32) as regularization terms, the constraint (33) in matrix form is processed by the F-norm, and the constraint (34)-(36) in vector form is processed by the 2-norm, as shown in the modified optimization problem (38): Where μ and ν are penalty coefficients; For distribution network state estimation, different types of measurement data have different magnitudes, which may lead to unbalanced estimation accuracy of different measurement variables. Inspired by the weighted least squares (WLS) method, the weight matrix W is added to the second part of the optimization problem (38) as a heuristic improvement of (39) to deal with this problem: in In (40), σ V , σ PQ and σ pseudo They represent the voltage amplitude, the active / reactive node injection power measured by SCADA, and the standard deviation of the active / reactive node injection power measured by smart meters. Ω (M), the introduction of standard deviation distinguishes the weights of various measurements, P Ω (M) ij The inverse of the absolute error is converted to X and P Ω (M) ij The relative error between them, then, based on the penalty function method, (39) constitutes an unconstrained optimization problem, and the distribution network state estimation is solved by the matrix filling model; Step 2: Solve the distribution network state estimation model based on matrix filling, including the following steps; (1) Obtain P Ω (M),Ω,W,μ,ν and time series length T, set the maximum number of iterations N max_iter , step length and tolerance δ tol , initialize the number of iterations k = 0, coefficient β0 = 1; (2) Through U SVD ΣV SVD =SVD(P Ω (M)) Calculate P Ω The matrix singular value decomposition (SVD) result of (M) is used to initialize U -1 =U0=U SVD Σ 1 / 2 ,V -1 =V0=Σ 1 / 2 V SVD ; (3) Calculation (4) Accounting U′ k = U k +γ k (U k -U k-1 ) ; (5) Calculation in: (6) Calculation (7) Calculation (8) Calculate V k ′=V k +γ k (V k -V k-1 ); (9) Calculation in: (10) Calculation (11) Calculation (12) Calculate X k+1 =U k+1 V k+1 ; (13) Update k→k+1; (14) Calculate V k ′=V k +γ k (V k -V k-1 ); (15) If ||X k+1 -X k || F <δ tol Or k = N max_iter , go to step (16); otherwise, go to step (3); (16) Output U=U k ,V=V k ,X=X k ; (17) Obtain state estimation results, including: node voltage amplitude estimation value: Node voltage phase angle estimation:

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