CN102801162A - Two-stage linear weighted least-square power system state estimation method - Google Patents
Two-stage linear weighted least-square power system state estimation method Download PDFInfo
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Abstract
The invention provides a two-stage linear weighted least-square power system state estimation method which is characterized by comprising the following steps of: forming a network model, and calculating a node admittance matrix and a branch-node relationship matrix; transforming a measurement vector and a state vector; forming an accurate linear measurement equation; performing the first stage of linear weighted least-square estimation to obtain the estimated value of the transformed state vector; performing inverse transformation, and performing the second stage of linear weighted least-square estimation to obtain the estimated values of the voltage amplitudes and phase angles of all nodes; and identifying bad data. According to the two-stage linear weighted least-square power system state estimation method provided by the invention, more scientific state estimation result can be obtained, the calculation efficiency is higher, and the engineering application prospect is good.
Description
Technical Field
The invention relates to the field of power system dispatching automation, in particular to a two-stage linear weighted least square power system state estimation method.
Background
The state estimation of the power system is the basis and core of an energy management system, and is used for filtering real-time information provided by a data acquisition and monitoring System (SCADA), so that estimated values of state variables (voltage amplitude and phase angle) of the whole network are obtained, and further estimated values of branch power, node injection power and the like can be obtained.
Since foreign scholars put forward the first model of state estimation in 1970, foreign scholars and engineers have conducted a lot of intensive research and practice on state estimation, and state estimators are deployed almost in every scheduling center all over the world, and the basic position of state estimation in safe operation of a power grid is widely accepted.
The most widely used state estimation method so far is the nonlinear Weighted Least Squares (WLS) method, which is simple in model and convenient to calculate, and can obtain unbiased estimated values of state variables in the absence of bad data. In order to improve the calculation efficiency of the nonlinear WLS, a state estimation method of a fast decoupling method is also provided. Since the non-linear WLS is not robust, a maximum normalized residual error (LNR) verification method based on the non-linear WLS, an estimation identification method (EI) of bad data, and the like have been proposed to identify the bad data.
Meanwhile, various robustness state estimation methods are also continuously proposed. The robustness state estimation can automatically inhibit the influence of bad data in the estimation process, so that a correct state variable estimation value is obtained. Of the various robust state estimation methods, the application of M-estimation is most common. M-estimation generally includes weighted least absolute value estimation (WLAV), non-quadratic criterion state estimation (such as QL estimation and QC estimation), etc., and state estimation with a yield of maximum targets, which has just been proposed in recent years, and state estimation with exponential objective functions also belong to the category of M-estimation. The problem faced by robust state estimation: 1) global optimization difficulties; 2) computational efficiency is lower than non-linear WLS. The above reasons affect the application of robust state estimation in practice.
In summary, the existing various state estimation methods are all a nonlinear optimization problem without exception, and generally a newton method or an interior point method is used for solving. From a mathematical point of view, such a non-linear optimization model and its corresponding solution method have several inevitable drawbacks: 1) the method is easy to fall into a local optimal point, namely a global optimal solution is difficult to obtain, and particularly some robust state estimation is difficult to obtain; 2) multiple iterations are required to obtain an acceptable solution, the nonlinear optimization problem sometimes has difficulty in convergence and even does not converge, and the solution is time-consuming; 3) the jacobian matrix and the residual sensitivity matrix are not constant matrices, so that a cyclic operation of deleting bad data and re-running state estimation needs to be performed multiple times when bad data identification is performed by LNR, and the non-linear state estimation needs to be re-run when multiple measurement errors are estimated if bad data are identified by EI method. The academic community has conducted some research on the above three problems, for example, foreign scholars have proposed using a trust domain method to enhance the convergence of state estimation, but so far, the above three problems have not been solved well, and the root cause is caused by the non-linear characteristic of the conventional state estimation model. If an accurate and linear state estimation model can be established, the three problems can be thoroughly solved, so that a more scientific state estimation result is theoretically ensured, and further practicability of state estimation is promoted.
Disclosure of Invention
The present invention aims to solve the above technical problems completely or at least to provide a useful commercial choice. Therefore, an object of the present invention is to provide a two-stage linear weighted least-squares power system state estimation method (TLWLS) with more scientific results and higher computational efficiency.
The two-stage linear weighted least square power system state estimation method provided by the embodiment of the invention comprises the following steps: A. forming a network model, and calculating a node admittance matrix and a branch-node association matrix; B. transforming the measurement vector and the state vector; C. forming a precise linear measurement equation; D. performing linear weighted least square estimation of a first stage to obtain an estimated value of the transformed state vector; e, performing inverse transformation and linear weighted least square estimation of a second stage to obtain the voltage amplitude values and phase angle estimation values of all nodes; and F, identifying bad data.
In one embodiment of the present invention, the step a includes: all lines and transformers in the network are equivalent to pi-type branch ij, and y is recordeds＝1/(rij+jxij)＝gs+jbsSeries susceptance, r, for a pi-branch ijij+jxijSeries impedance value of pi-type branch ij, bcIs the grounding susceptance of the pi-type branch ij, wherein if the pi-type branch ij is a transformer branch, b iscK is 0 and k is the transformation ratio of an ideal transformer, if the pi-type branch ij is a common line, k is 1, and a plurality of parallel branches are equivalent to one branch; in the equivalent circuit, let gij＝gs/k，bij＝bs/k，gsi＝(1-k)gs/k2，bsi＝(1-k)bs/k2+bc/2，gsj＝(k-1)gs/k，bsj＝(k-1)bs/k+b c2; calculating a node admittance matrix Y = G + jB, wherein G and B are a real part and an imaginary part of the node admittance matrix respectively; and calculating a branch-node association matrix as A = { a = { (a)ij(1. ltoreq. i.ltoreq.b, 1. ltoreq. j.ltoreq.N-1), wherein the respective elements are defined as:
in one embodiment of the present invention, the step B includes: transforming state vectors into <math>
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</math> Wherein N is the total number of all nodes in the network, b is the number of all branches in the network, l is the branch numberiAnd ljThe numbers of the nodes at the two ends of the branch l,
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are respectively node liAnd ljThe magnitude of the voltage of (a) is,
andare respectively node liAnd ljThe phase angle of (a) is,
is the phase angle difference between the two phases,representing the contribution of all b branches to the state vector X,
also represents the contribution of all b branches to the state vector X, X ∈ RN+2bIs a state vector; and transforming the measurement vector into y ∈ RmThe method comprises the steps of squaring node voltage amplitude, enabling branch circuits, injecting active power, injecting idle power and squaring branch circuit current amplitude, wherein m is the total number of measured quantities, and when the square of the node voltage amplitude is expressed by a transformed state vector X, the square of the node voltage amplitude is
viFor the voltage at node i, the branch from node i to node j has active powerThe branch from node i to node j is reactive <math>
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</math> Gij+jBijThe square of the branch current amplitude isWherein, IijThe amplitude of the current for the pi branch ij, a ═ gsi+gij)2+(bsi+bij)2，
D=-gsibij+bsigij。
In one embodiment of the present invention, the step C includes: let J ∈ Rm×(N+2b)Is a Jacobian matrix, in which the node voltage amplitude is measuredHas a square corresponding Jacobian matrix element of <math>
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The Jacobian matrix element corresponding to the square of the branch current amplitude measurement is
And obtaining an accurate linearized measurement equation according to the converted measurement vector and state vector obtained in the step B: y = JX + τ, where τ ∈ RmFor measuring the error vector, J ∈ Rm×(N+2b)Is a constant jacobian matrix.
In one embodiment of the present invention, the step D includes: structure of the device <math>
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G is JTWJ is an information matrix, and an estimated value X of the transformed state vector X is obtained by solving-1JTWy。
In one embodiment of the present invention, the step E comprises: using the state vector X obtained in the step D according to a formula <math>
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</math> Performing inverse transformation to obtain the amplitudes of all node voltages and the estimated values theta of the phase angle differences at the two ends of all branches2b∈R2bI.e. by <math>
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</math> And using the estimated value theta of the phase angle difference of the two ends of all the branches2bConstruction Min J (theta) ═ theta2b-A2θ)TWθ(θ2b-A2θ) linear weighted least squares problem, where WθIs a weight matrix, takes the value of an identity matrix, A2＝[AT AT]TA is the branch-node incidence matrix obtained in the step A, and theta is belonged to RN-1For the phase angles of all nodes except the reference node, wherein the optimal solution should satisfy the condition
Can obtain the productSubstituting A into A2And solving to obtain theta ═ AT A)-1ATθbWherein
in one embodiment of the present invention, the step F includes: computing regularization residual values for the ith measurement
Wherein r is [ I-J [ J ]TWJ]-1JTW]τ＝Sτ，S＝I-J(JTWJ)-1JTW is a residual sensitivity matrix which is a constant matrix r-N (0, omega), wherein omega is SR; and if the regularization residual error value of a certain measurement is larger than a preset value, removing the measurement, and operating the step D again until all the measured regularization residual errors are smaller than the preset value.
The two-stage linear weighted least squares power system state estimation method (TLWLS) of the present invention has the advantages of at least: the TLWLS model is essentially a quadratic programming problem, and a unique global optimal solution can be obtained by mathematically ensuring; TLWLS only needs to solve a linear equation set without iteration, has no convergence problem, and has estimation efficiency far higher than that of a conventional nonlinear weighted least square state estimation method (WLS); the jacobian matrix and the residual sensitivity matrix of the TLWLS are constant matrixes, which provides convenience for identifying bad data and can improve the identification efficiency of the bad data. In conclusion, the two-stage linear weighted least square state estimation method provided by the invention can obtain a more scientific state estimation result, has higher calculation efficiency and has good engineering application prospect.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a flow chart of a two-stage linear weighted least squares power system state estimation method of the present invention;
FIG. 2 is a schematic diagram of a π branch circuit;
FIG. 3 is a schematic diagram of an equivalent circuit for a pi branch circuit; and
FIG. 4 is a single line diagram and measurement configuration diagram for a three-node system.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention.
As shown in fig. 1, the method for accurately linearizing the state estimation measurement equation of the power system of the present invention includes the following steps:
and S101, forming a network model, and calculating a node admittance matrix and a branch-node association matrix.
Specifically, by equating a three-winding transformer in the network to three two-winding transformers, all lines and transformers in the network can be represented by a unified pi-type branch, as shown in fig. 2. In FIG. 2, ys＝1/(rij+jxij)＝gs+jbsIs the series susceptance of branch ij; r isij+jxijIs a series impedance value; bcIs the grounding susceptance of the branch. Wherein, if the pi-type branch ij is a transformer branch,then b isc=0 and k is the transformation ratio of an ideal transformer; if the pi-type branch ij is a common line, k is 1, and a plurality of branches connected in parallel are equivalent to one branch. It should be noted that a plurality of branches connected in parallel is equivalent to one branch.
The equivalent circuit of the pi-branch of fig. 2 is shown in fig. 3. In FIG. 3, gij=gs/k；bij＝bs/k；gsi=(1-k)gs/k2；bsi=(1-k)bs/k2+bc/2；gsj=(k-1)gs/k；bsj=(k-1)bs/k+bc/2. Then, a node admittance matrix Y = G + jB is formed, G and B being respectively a real part and an imaginary part of the node admittance matrix, and a branch-node association matrix a = { a =isformedij(1. ltoreq. i.ltoreq.b, 1. ltoreq. j.ltoreq.N-1), the elements of which are defined as:
step S102, the measurement vector and the state vector are transformed.
First, the state vector is transformed into <math>
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Wherein N is the total number of all nodes in the network; b is the number of all branches in the network (a plurality of branches connected in parallel is equivalent to one branch); l is the branch number, liAnd ljThe numbers of the nodes at the two ends of the branch l,
and
are respectively node liAnd ljThe magnitude of the voltage of (a) is,
andare respectively node liAnd ljThe phase angle of (a) is,
is the phase angle difference;
representing the contribution of all b branches to the state vector X,also represents the contribution of all b branches to the state vector X; x is formed by RN+2bIs a state vector.
Next, the measurement vector is transformed into y ∈ RmThe measurement types include: the square of the node voltage amplitude, the branch active, the branch reactive, the injection active and the injection reactive, and the square of the branch current amplitude; m is the total number of measurements. The measurement of the quantity can be expressed in terms of a state vector X.
1) Square of voltage amplitude measurement
Wherein v isiIs the voltage at node i.
2) Branch active and reactive power measurement from node i to node j
Wherein, PijAnd QijThe branch flowing from the node i to the node j is active and reactive respectively.
3) Measurement of injected active and injected reactive of node i
Wherein, PiAnd QiActive and reactive injection, G, respectively, for node iij+jBijAdmittance of corresponding elements in a matrix for a node
4) Square of branch current amplitude measurement
Wherein, IijIs the current amplitude of branch ij; a ═ g (g)si+gij)2+(bsi+bij)2；
D=-gsibij+bsigij。
Step S103, forming a precise linear measurement equation.
According to the transformed measurement vector and the new state vector, a new accurate linearized measurement equation can be obtained as follows:
y=JX+τ （7）
wherein y ∈ RmThe converted measurement vector comprises the square of the node voltage amplitude, the branch active, the branch reactive, the active injection and the reactive injection, and the square of the branch current amplitude; x is formed by RN+2bIs a transformed state vector; τ to N (0, R) is a measurement error vector, wherein
Wherein
Is tauiThe variance of (a); j is an element of Rm×(N+2b)The expression of the elements of each part of the constant jacobian matrix is shown below.
1) Jacobian matrix elements corresponding to the squares of voltage amplitude measurements
For the square of the voltage amplitude measurement, the corresponding Jacobian matrix element is
2) Jacobian matrix element corresponding to branch power measurement
For branch power measurement, the corresponding Jacobian matrix element is
3) Jacobian matrix element corresponding to injection power measurement
For implant power measurements, the corresponding Jacobian matrix element is
4) Jacobian matrix element corresponding to square of branch current amplitude measurement
For the square of the branch current amplitude measurement, the corresponding Jacobian matrix element is
It should be noted that the above equation (7) is an accurate linear measurement equation. Based on the measurement equation, any existing state estimation method can be adopted to solve and obtain the transformed state vector X, and after X is obtained, the estimated values of all branch power, node injection power, branch current amplitude and the like can be further obtained, so that the accurate sensing of the whole network state is obtained.
Step S104, linear weighted least square estimation of the first stage is carried out to obtain an estimated value of the state vector X after transformation.
Specifically, the following linear weighted least squares problem is solved to obtain an estimate of the state vector X.
Wherein W = R-1Is a weight matrix. In order to obtain the optimum value, the optimum condition must be satisfied:
namely, it is
GX=JTWy （9）
Wherein G = JTWJ is an information matrix. The information matrix G is subjected to orthogonal decomposition, and the estimated value of the state vector X can be obtained by solving, namely X is G-1JTWy。
And step S105, performing inverse transformation, and performing linear weighted least square estimation of the second stage to obtain the voltage amplitude and phase angle estimation values of all nodes.
According to the definition of the transformed state vector X, the voltage amplitudes of all nodes and the estimated values of the phase angle differences of the two ends of all branches can be obtained by inverse transformation, namely
In equation (10), the symbol of arccos should be consistent with the symbol of arcsin.
To this end, estimates of the voltage amplitudes of all nodes have been obtained, but the phase angles of all nodes are not known, and it is clear that the phase angle difference across all branches can be used for estimation.
Order to <math>
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</math> Obviously, θ can be obtained by the formula (10)2b. Using the following equation2bThe phase angles of all nodes except the reference node are estimated.
Let theta be [ theta ]2，…,θN]TRepresenting the phase angles of all nodes (except the reference node), then θ2bThe relation with theta is
θ2b=A2θ （11）
Wherein A is2＝[AT AT]T，A2Is a matrix of 2b × (N-1).
Because of theta2bIs derived from the state estimation result of the first stage, so theta can be obtained2bConsider a measurement containing noise. And theta is to be estimated, the equation (11) can be rewritten as the following measurement equation for theta
θ2b=A2θ+τ （12）
A linear weighted least squares problem can be constructed that estimates theta as follows
Min J(θ)=(θ2b-A2θ)TWθ(θ2b-A2θ) （13）
Wherein, WθFor the weight matrix, without loss of generality, W may be takenθIs an identity matrix. When the formula (13) obtains the optimum value, it must satisfy
Substituting A into A2Is obtained by
ATAθ＝ATθb （15）
Wherein, <math>
<mrow>
<msub>
<mi>θ</mi>
<mi>b</mi>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>[</mo>
<mo>[</mo>
<mi>arcsin</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>/</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>arccos</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>/</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>]</mo>
<mo>/</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>≤</mo>
<mi>l</mi>
<mo>≤</mo>
<mi>b</mi>
<mo>)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
</math> is a b-dimensional vector, and can be obviously represented by theta2bTo obtain thetab。
The estimated value of θ obtained from equation (15) is θ ═ aT A)-1ATθb。
To this end, estimates of the voltage amplitude and phase angle for all nodes have been obtained.
Step S106, identifying bad data.
In the present invention, the identification of the bad data only needs to be performed in the first stage of the two-stage linear weighted least square state estimation.
The residual of the first stage linear weighted least squares state estimation is
r=y-JX （16）
Wherein, X is the result of the first-stage state estimation and is obtained by solving the formula (9).
The relationship between the residual error and the measurement error can be derived from the equations (9) and (16) as
r＝[I-J[JTWJ]-1JTW]τ＝Sτ （17）
Wherein, S ═ I-J (J)TWJ)-1JTW is the residual sensitivity matrix. Since the jacobian matrix J is a constant matrix, the residual sensitivity matrix S is also a constant matrix. This means that the residual sensitivity matrix is not affected by bad data.
It is easy to prove that r-N (0, Ω), where Ω ═ SR, we can get normalized residual (normalized residual) rNIs composed of
Wherein, | riIs the residual riAbsolute value of (a).
If it is not
The kth measurement is considered bad data and can be removed from the measurement set and the first stage linear weighted least squares state estimation is re-run. And repeating the steps until all the regularization residual errors are less than 3, and finishing the identification of all the bad data. Since the jacobian matrix is a constant matrix, it is clear that the poor data identification efficiency based on the linear weighted least squares state estimation is very high.
For better understanding of those skilled in the art, the applicant takes a three-node system as an example to illustrate the method of the present invention. The single line diagram and the measurement configuration diagram of the three-node system are shown in fig. 4, and the network parameters are shown in table 1. And selecting the node 1 as a reference node.
TABLE 1 three-node System network parameters
Table 1 The network data of the 3-bus system
(1) Estimation results of non-linear WLS at normal load and TLWLS proposed by the present invention the measurements at normal load are shown in table 2.
TABLE 2 three-node systematic volume measurement
Table 2 Measurements of the 3-bus system
If the conventional non-linear WLS is adopted, the convergence accuracy is 10-6Then, after 4 iterations of nonlinear WLS convergence, the obtained estimated values of voltage amplitude and phase angle of all nodes are shown in table 3.
TABLE 3 estimation of non-linear WLS
Table 3 The Estimated State Vector by Nonlinear WLS
The following estimation is performed using the TLWLS proposed by the present invention.
Transforming state vectors into <math>
<mrow>
<mi>X</mi>
<mo>=</mo>
<msup>
<mrow>
<mo>[</mo>
<msubsup>
<mi>v</mi>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>v</mi>
<mn>2</mn>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>v</mi>
<mn>3</mn>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<msub>
<mi>v</mi>
<mn>1</mn>
</msub>
<msub>
<mi>v</mi>
<mn>2</mn>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mn>12</mn>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mn>1</mn>
</msub>
<msub>
<mi>v</mi>
<mn>3</mn>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mn>13</mn>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mn>2</mn>
</msub>
<msub>
<mi>v</mi>
<mn>3</mn>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mn>23</mn>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mn>1</mn>
</msub>
<msub>
<mi>v</mi>
<mn>2</mn>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mn>12</mn>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mn>1</mn>
</msub>
<msub>
<mi>v</mi>
<mn>3</mn>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mn>13</mn>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mn>2</mn>
</msub>
<msub>
<mi>v</mi>
<mn>3</mn>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mn>23</mn>
</msub>
<mo>]</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>,</mo>
</mrow>
</math> Transforming the measurement vector into The measurement equation that yields accurate linearization is y-JX + τ. Wherein the constant jacobian can be obtained from the formula given in S103. By solving the estimation value of the state vector X after transformation in equation (9), further, the estimation values of the voltage amplitudes of all nodes and the phase angle differences between both ends of all branches can be obtained from equation (10), as shown in table 4. In the context of table 4, the results are,and
are each Xi、viAnd theta2biAn estimate of (d).
TABLE 4 estimation results of the first stage of two-stage linear weighted least squares state estimation
Table 4 The estimation value by the first stage linear WLS of the TLWLS
Based on the wiring diagram of the network, a branch-node association matrix (node 1 is a reference node) of the network is obtained
By solving equation (15), the estimated values of the phase angles of all nodes can be obtained as θ = [ ]2,θ3]T=[-0.0213,-0.0479]T. Thus, the estimated values of the voltage amplitude and phase angle for all nodes obtained from the two-stage linear weighted least squares state estimation proposed by the present invention are shown in table 5.
TABLE 5 estimation results of two-stage linear weighted least squares state estimation
Table 5 The Estimated State Vector by TLWLS
Comparing table 3 and table 5, it can be seen that the state variable estimates obtained by the two-stage linear weighted least squares state estimation (TLWLS) proposed by the present invention match well with the state variable estimates obtained by the conventional non-linear WLS under normal loading. However, TLWLS does not require iteration, whereas conventional non-linear WLS requires 4 iterations to converge, in this example, TLWLS is more than 3 times more computationally efficient than non-linear WLS.
(2) Examples of trapping non-linear WLS into locally optimal solution
An example of a non-linear WLS falling into a locally optimal solution is given below, still taking the 3-node system as an example. As the system load continues to increase, another set of quantity measurements may be obtained, as shown in table 6. Obviously, a voltage collapse occurs at this time.
TABLE 6 measurement of System volume of three nodes under heavy load
Table 6 Measurements of the 3-bus system at hevy load
When this example was estimated using nonlinear WLS, the objective function value of the nonlinear WLS at the time of convergence was 0.1088 after 35 convergence. For this example, the true values of the state variables, the estimated values for nonlinear WLS, and the estimated values for two-stage linear weighted least squares state estimation (TLWLS) are shown in table 7.
TABLE 7 truth values of state variables under heavy load, non-linear WLS and estimated results of TLWLS
Table 7 The True value,Estimate Value by WLS and TLWLS at hevy load
As can be seen from table 7, under heavy loading, especially in voltage collapse, the conventional non-linear WLS falls into a locally optimal solution, whose estimate appears to have negative voltage magnitude and large phase angle value. It can be seen that under heavy load, especially in voltage collapse, the estimation result of the conventional non-linear WLS is not reliable, but the power system requires the state estimation to have the best possible performance. This sharp contradiction is caused by the inherent characteristics of conventional non-linear WLS when heavy loads, especially voltage breakdown, occur.
As can also be seen from table 7, even when the voltage collapses, the estimation result of the two-stage linear weighted least squares state estimation method (TLWLS) proposed by the present invention is very accurate, i.e., the TLWLS proposed by the present invention can obtain a globally optimal solution at any time without iteration. Therefore, the TLWLS provided by the invention can obtain a more scientific state estimation result and has higher calculation efficiency.
(3) Identifying bad data
Measurement of quantity P at regular load shown in Table 212Iterate 20% of the noise, P12Becomes bad data. Obtaining P from formula (18)12And (3) removing the regularized residual errors from the measurement set when the regularized residual errors are larger than 3, re-operating the linear weighted least square state estimation of the first stage, and finishing the identification of the bad data when all the regularized residual errors are smaller than 3. Therefore, the regularization residual error inspection method can effectively identify the bad data of the TLWLS in the first stage, and the identification efficiency is found to be more than 3 times of that of the traditional regularization residual error inspection method based on the nonlinear WLS.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an example," "a specific example," or "some examples," etc., mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made in the above embodiments by those of ordinary skill in the art without departing from the principle and spirit of the present invention.
Claims (7)
1. A two-stage linear weighted least square power system state estimation method is characterized by comprising the following steps:
A. forming a network model, and calculating a node admittance matrix and a branch-node association matrix;
B. transforming the measurement vector and the state vector;
C. forming a precise linear measurement equation;
D. performing linear weighted least square estimation of a first stage to obtain an estimated value of the transformed state vector;
E. performing inverse transformation and linear weighted least square estimation of a second stage to obtain the voltage amplitude values and phase angle estimation values of all nodes; and
F. and identifying bad data.
2. The two-stage linear weighted least squares power system state estimation method of claim 1, said step a comprising:
all lines and transformers in the network are equivalent to pi-type branch ij, and y is recordeds＝1/(rij+jxij)＝gs+jbsSeries susceptance, r, for a pi-branch ijij+jxijSeries impedance value of pi-type branch ij, bcIs the grounding susceptance of the pi-type branch ij, wherein if the pi-type branch ij is a transformer branch, b iscK is 0 and k is the transformation ratio of an ideal transformer, if the pi-type branch ij is a common line, k is 1, and a plurality of parallel branches are equivalent to one branch;
in the equivalent circuit, let gij＝gs/k，bij＝bs/k，gsi＝(1-k)gs/k2，bsi＝(1-k)bs/k2+bc/2，gsj=(k-1)gs/k，bsj=(k-1)bs/k+bc/2；
Calculating a node admittance matrix Y = G + jB, wherein G and B are a real part and an imaginary part of the node admittance matrix respectively; and calculating a branch-node association matrix a = { a = { (a) }ij(1. ltoreq. i.ltoreq.b, 1. ltoreq. j.ltoreq.N-1), wherein the respective elements are defined as:
3. the two-stage linear weighted least squares power system state estimation method of claim 2, said step B comprising:
transforming state vectors into <math>
<mrow>
<mi>X</mi>
<mo>=</mo>
<msup>
<mrow>
<mo>[</mo>
<msubsup>
<mi>v</mi>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<msubsup>
<mi>v</mi>
<mn>2</mn>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<msubsup>
<mi>v</mi>
<mi>N</mi>
<mn>2</mn>
</msubsup>
<mo>,</mo>
<msub>
<msub>
<mi>v</mi>
<mi>l</mi>
</msub>
<mi>i</mi>
</msub>
<msub>
<msub>
<mi>v</mi>
<mi>l</mi>
</msub>
<mi>j</mi>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>≤</mo>
<mi>l</mi>
<mo>≤</mo>
<mi>b</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
<msub>
<msub>
<mi>v</mi>
<mi>l</mi>
</msub>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>≤</mo>
<mi>l</mi>
<mo>≤</mo>
<mi>b</mi>
<mo>)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>,</mo>
</mrow>
</math> Wherein N is the total number of all nodes in the network, b is the number of all branches in the network, l is the branch numberiAnd ljThe numbers of the nodes at the two ends of the branch l,
and
are respectively node liAnd ljThe magnitude of the voltage of (a) is,
and
are respectively node liAnd ljThe phase angle of (a) is,
is the phase angle difference between the two phases,
representing the contribution of all b branches to the state vector X,
also represents the contribution of all b branches to the state vector X, X ∈ RN+2bIs a state vector; and
transforming the measurement vector into y ∈ RmThe method comprises the steps of squaring node voltage amplitude, enabling branch circuits, injecting active power, injecting idle power and squaring branch circuit current amplitude, wherein m is the total number of measured quantities, and when the square of the node voltage amplitude is expressed by a transformed state vector X, the square of the node voltage amplitude isviFor the voltage at node i, the branch from node i to node j has active power <math>
<mrow>
<msub>
<mi>P</mi>
<mi>ij</mi>
</msub>
<mo>=</mo>
<msubsup>
<mi>v</mi>
<mi>i</mi>
<mn>2</mn>
</msubsup>
<mrow>
<mo>(</mo>
<msub>
<mi>g</mi>
<mi>si</mi>
</msub>
<mo>+</mo>
<msub>
<mi>g</mi>
<mi>ij</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<msub>
<mi>g</mi>
<mi>ij</mi>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
<mo>-</mo>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>ij</mi>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> The branch from node i to node j is reactive
The injection active of node i is
Injected reactive of node i is
Gij+jBijThe square of the branch current amplitude is <math>
<mrow>
<msubsup>
<mi>I</mi>
<mi>ij</mi>
<mn>2</mn>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>Av</mi>
<mi>i</mi>
<mn>2</mn>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>Bv</mi>
<mi>j</mi>
<mn>2</mn>
</msubsup>
<mo>-</mo>
<msub>
<mrow>
<mn>2</mn>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>C</mi>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
<mo>-</mo>
<mi>D</mi>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
</math> Wherein, IijThe amplitude of the current for the pi branch ij, a ═ gsi+gij)2+(bsi+bij)2， D=-gsibij+bsigij。
4. The two-stage linear weighted least squares power system state estimation method of claim 3, said step C comprising:
let J ∈ Rm×(N+2b)Is a Jacobian matrix, wherein the Jacobian matrix element corresponding to the square of the node voltage amplitude measurement is <math>
<mrow>
<mfrac>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
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<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>j</mi>
<mn>2</mn>
</msubsup>
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<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
</math> The Jacobian matrix element corresponding to the branch power measurement is <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>P</mi>
</mrow>
<mi>ij</mi>
</msub>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<msub>
<mi>g</mi>
<mi>si</mi>
</msub>
<mo>+</mo>
<msub>
<mi>g</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>P</mi>
</mrow>
<mi>ij</mi>
</msub>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>j</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>P</mi>
</mrow>
<mi>ij</mi>
</msub>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>g</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>P</mi>
</mrow>
<mi>ij</mi>
</msub>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>b</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
</mrow>
<mi>ij</mi>
</msub>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>b</mi>
<mi>si</mi>
</msub>
<mo>+</mo>
<msub>
<mi>b</mi>
<mi>ij</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
</mrow>
<mi>ij</mi>
</msub>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>j</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
</mrow>
<mi>ij</mi>
</msub>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>b</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
</mrow>
<mi>ij</mi>
</msub>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>g</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> The Jacobian matrix element corresponding to the injection power measurement is <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>P</mi>
</mrow>
<mi>i</mi>
</msub>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<msub>
<mi>G</mi>
<mi>ii</mi>
</msub>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>P</mi>
</mrow>
<mi>i</mi>
</msub>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>j</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>P</mi>
</mrow>
<mi>i</mi>
</msub>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>G</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>P</mi>
</mrow>
<mi>i</mi>
</msub>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>B</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
</mrow>
<mi>i</mi>
</msub>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>B</mi>
<mi>ii</mi>
</msub>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
</mrow>
<mi>i</mi>
</msub>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>j</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
</mrow>
<mi>i</mi>
</msub>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>B</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msub>
<mrow>
<mo>∂</mo>
<mi>Q</mi>
</mrow>
<mi>i</mi>
</msub>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>G</mi>
<mi>ij</mi>
</msub>
<mo>,</mo>
</mrow>
</math> The Jacobian matrix element corresponding to the square of the branch current amplitude measurement is <math>
<mrow>
<mfrac>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>I</mi>
</mrow>
<mi>ij</mi>
<mn>2</mn>
</msubsup>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<mi>A</mi>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>I</mi>
</mrow>
<mi>ij</mi>
<mn>2</mn>
</msubsup>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>j</mi>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>=</mo>
<mi>B</mi>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>I</mi>
</mrow>
<mi>ij</mi>
<mn>2</mn>
</msubsup>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mn>2</mn>
<mi>C</mi>
<mo>,</mo>
</mrow>
</math> <math>
<mrow>
<mfrac>
<msubsup>
<mrow>
<mo>∂</mo>
<mi>I</mi>
</mrow>
<mi>ij</mi>
<mn>2</mn>
</msubsup>
<mrow>
<msub>
<mrow>
<mo>∂</mo>
<mi>v</mi>
</mrow>
<mi>i</mi>
</msub>
<msub>
<mi>v</mi>
<mi>j</mi>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mi>ij</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mi>D</mi>
<mo>;</mo>
</mrow>
</math> And
and B, obtaining an accurate linearized measurement equation according to the converted measurement vector and the state vector obtained in the step B: y = JX + τ, where τ ∈ RmFor measuring the error vector, J ∈ Rm×(N+2b)Is a constant jacobian matrix.
5. The two-stage linear weighted least squares power system state estimation method of claim 4, said step D comprising:
structure of the device <math>
<mrow>
<mi>MinJ</mi>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mi>Σ</mi>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>m</mi>
</msubsup>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>i</mi>
</msub>
<mo>-</mo>
<msub>
<mi>J</mi>
<mi>i</mi>
</msub>
<mi>X</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>/</mo>
<msub>
<mi>R</mi>
<mi>ii</mi>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>-</mo>
<mi>JX</mi>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mi>W</mi>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>-</mo>
<mi>JX</mi>
<mo>)</mo>
</mrow>
</mrow>
</math> Wherein W ═ R-1Is a weight matrix in which the optimal solution should satisfy the condition
G is JTWJ is an information matrix, and an estimated value X of the transformed state vector X is obtained by solving-1JTWy。
6. The two-stage linear weighted least squares power system state estimation method of claim 5, said step E comprising:
using the state vector X obtained in said step DAccording to the formula <math>
<mfenced open='{' close=''>
<mtable>
<mtr>
<mtd>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<msqrt>
<msubsup>
<mi>v</mi>
<mi>i</mi>
<mn>2</mn>
</msubsup>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>≤</mo>
<mi>i</mi>
<mo>≤</mo>
<mi>N</mi>
<mo>)</mo>
</mrow>
</msqrt>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>=</mo>
<mi>arcsin</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>/</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>≤</mo>
<mi>l</mi>
<mo>≤</mo>
<mi>b</mi>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>=</mo>
<mi>arccos</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>/</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>≤</mo>
<mi>l</mi>
<mo>≤</mo>
<mi>b</mi>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</math> Performing inverse transformation to obtain the amplitudes of all node voltages and the estimated values theta of the phase angle differences at the two ends of all branches2b∈R2bI.e. by <math>
<mrow>
<msub>
<mi>θ</mi>
<mrow>
<mn>2</mn>
<mi>b</mi>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mrow>
<mo>[</mo>
<mi>arcsin</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mi>sin</mi>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>/</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>arccos</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mi>cos</mi>
<msub>
<mi>θ</mi>
<mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>/</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</msub>
<msub>
<mi>v</mi>
<msub>
<mi>l</mi>
<mi>j</mi>
</msub>
</msub>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>≤</mo>
<mi>l</mi>
<mo>≤</mo>
<mi>b</mi>
<mo>)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mi>T</mi>
</msup>
<mo>;</mo>
</mrow>
</math> And
utilizing the estimated value theta of the phase angle difference of the two ends of all the branches2bConstruction Min J (theta) ═ theta2b-A2θ)TWθ(θ2b-A2θ) linear weighted least squares problem, where WθIs a weight matrix, takes the value of an identity matrix, A2＝[AT AT]TA is the branch-node incidence matrix obtained in the step A, and theta is belonged to RN-1For the phase angles of all nodes except the reference node, wherein the optimal solution should satisfy the condition
Can obtain the product
Substituting A into A2And solving to obtain theta ═ ATA)-1ATθbWherein
7. the two-stage linear weighted least squares power system state estimation method of claim 6, said step F comprising:
computing regularization residual values for the ith measurement
Wherein r is [ I-J [ J ]TWJ]-1JTW]τ＝Sτ，S＝I-J(JTWJ)-1JTW is a residual sensitivity matrix which is a constant matrix r-N (0, omega), wherein omega is SR; and
and if the regularization residual error value of a certain measurement is larger than a preset value, removing the measurement, and operating the step D again until all the measured regularization residual errors are smaller than the preset value.
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