CN102801162A - Two-stage linear weighted least-square power system state estimation method - Google Patents

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

A Two-Stage Linear Weighted Least Squares State Estimation Method for Power Systems

technical field

The invention relates to the field of power system scheduling automation, in particular to a two-stage linear weighted least squares power system state estimation method.

Background technique

Power system state estimation is the basis and core of the energy management system. Its function is to filter the real-time information provided by the data acquisition and monitoring system (SCADA), so as to obtain the estimated value of the state variables (voltage amplitude and phase angle) of the whole network. Furthermore, estimated values such as branch power and node injection power can be obtained.

Since foreign scholars proposed the first model of state estimation in 1970, scholars and engineers at home and abroad have conducted a large number of in-depth research and practice on state estimation. Now almost every dispatch center in the world has deployed state estimators. The basic position of state estimation in the safe operation of power grid has been widely recognized.

So far, the most widely used state estimation method is nonlinear weighted least squares (WLS), which has a simple model and is easy to calculate, and can obtain unbiased estimates of state variables when there is no bad data. In order to improve the computational efficiency of nonlinear WLS, people proposed a fast decoupling method for state estimation. Because non-linear WLS is not robust, people have proposed the largest normalized residual (LNR) test method based on non-linear WLS and the estimation identification method (EI) of bad data to identify bad data.

At the same time, various robust state estimation methods have been continuously proposed. Robust state estimation can automatically suppress the influence of bad data during the estimation process, so as to obtain correct state variable estimates. Among various robust state estimation methods, M-estimation is the most widely used. M-estimation generally includes weighted least absolute value estimation (WLAV), non-quadratic criterion state estimation (such as QL estimation and QC estimation, etc.), and the state estimation with the goal of maximizing the pass rate that has just been proposed in recent years. The state estimation of the type objective function also belongs to the category of M-estimation. Problems faced by robust state estimation: 1) Difficulty in global optimization; 2) Computational efficiency is lower than that of nonlinear WLS. The above reasons affect the practical application of robust state estimation.

Taken together, the existing state estimation methods are all nonlinear optimization problems without exception, which are generally solved by Newton's method or interior point method. From a mathematical point of view, this nonlinear optimization model and its corresponding solution method have several unavoidable disadvantages: 1) It is easy to fall into the local optimum, that is, it is difficult to obtain the global optimal solution, especially some robust states Estimate; 2) In order to obtain an acceptable solution, multiple iterations are required. This nonlinear optimization problem sometimes has difficulty in converging, or even does not converge, and the solution is time-consuming; 3) The Jacobian matrix and the residual sensitivity matrix are not constant matrices , so when using LNR to identify bad data, it is necessary to delete bad data and re-run the cyclic operation of state estimation multiple times, and if the EI method is used to identify bad data and estimate multiple measurement errors, it is also necessary to re-run the Linear state estimation. The academic community has conducted some research on the above three issues. For example, foreign scholars have proposed to use the trust region method to enhance the convergence of state estimation, but so far, the above three issues have not been well resolved. The fundamental reason is that It is caused by the nonlinear characteristics of traditional state estimation models. If an accurate linearized state estimation model can be established, the above three problems can be completely solved, thereby theoretically ensuring more scientific state estimation results and promoting the further practical application of state estimation.

Contents of the invention

The present invention aims to completely solve the above technical problems or at least provide a useful commercial option. Therefore, an object of the present invention is to propose a two-stage linear weighted least squares power system state estimation method (two-stage linear WLS, TLWLS) with more scientific results and higher computational efficiency.

The two-stage linear weighted least squares power system state estimation method according to an embodiment of the present invention includes the following steps: A. forming a network model, calculating a node admittance matrix and a branch-node correlation matrix; B. measuring vectors and states Transform the vector; C. Form an accurate linearized measurement equation; D. Perform the linear weighted least squares estimation of the first stage to obtain the estimated value of the transformed state vector; E perform inverse transformation and perform the second stage Linear weighted least squares estimation to obtain estimates of voltage magnitudes and phase angles for all nodes; and F. Perform bad data identification.

In one embodiment of the present invention, the step A includes: equating all lines and transformers in the network to π-type branches ij, denoting y _s =1/(r _ij +jx _ij )=g _s +jb _s is the series susceptance of the π-type branch ij, r _ij + jx _ij is the series impedance value of the π-type branch ij, b _c is the grounding susceptance of the π-type branch ij, where, if the π-type branch ij is Transformer branch, then b _c =0 and k is the transformation ratio of the ideal transformer, if the π-type branch ij is an ordinary line, then k=1, multiple parallel branches are equivalent to one branch; after equivalent In the circuit, record g _ij = g _s /k, b _ij = b _s /k, g _si = (1-k)g _s /k ² , b _si = (1-k)b _s /k ² +b _c /2, g _sj = (k-1)g _s /k, b _sj = (k-1)b _s /k+b _c /2; calculate node admittance matrix Y=G+jB, G and B respectively is the real part and imaginary part of the node admittance matrix; and the calculation branch-node incidence matrix is A={a _ij }(1≤i≤b,1≤j≤N-1), where each element is defined as:

和

分别是节点l_i和l_j的电压幅值，

和分别是节点l_i和l_j的相角，

为相角差，代表所有的b条支路对状态矢量X的贡献，

也代表所有的b条支路对状态矢量X的贡献，X∈R^N+2b为状态矢量；以及将量测矢量变换为y∈R^m，包括节点电压幅值的平方、支路有功、支路无功、注入有功、注入无功，支路电流幅值的平方，其中，m为量测量的总个数，当用变换后的状态矢量X表示时，节点电压幅值的平方为

v_i为节点i的电压，从节点i到节点j的支路有功为从节点i到节点j的支路无功为 $Q_{\mathrm{ij}}=-v_i^2(b_{\mathrm{si}}+b_{\mathrm{ij}})+v_i{v}_j{b}_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}-v_i{v}_j{g}_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}},$ 节点i的注入有功为 $P_i=v_i\underset{j\∈N_i}{\mathrm{\Σ}}{v}_j(G_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}),$ 节点i的注入无功为 $Q_i=v_i\underset{j\∈N_i}{\mathrm{\Σ}}{v}_j(G_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-B_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}),$ G_ij+jB_ij为节点导纳矩阵中的对应元素，支路电流幅值的平方为其中，I_ij为π型支路ij的电流幅值，A＝(g_si+g_ij)²+(b_si+b_ij)²，

D=-g_sib_ij+b_sig_ij。In one embodiment of the present invention, the step B includes: transforming the state vector into $x={[{\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_1^2,{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_2^2,...,{\mathrm{<figure-callout\; id="2"\; label="v\; N"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_N^{},{v_l}_i{v_l}_j\cos {\mathrm{\&theta;}}_{l_i{l}_j}(1\&le;l\&le;b),{v_l}_i{v}_{l_j}\sin {\mathrm{\&theta;}}_{l_i{l}_j}(1\&le;l\&le;b)]}^T,$ Among them, 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 number, l _i and l _j are the node numbers at both ends of the branch l,

and

are the voltage amplitudes of nodes l _i and l _j respectively,

and are the phase angles of nodes l _i and l _j respectively,

is the phase angle difference, Represents the contribution of all b branches to the state vector X,

It also represents the contribution of all b branches to the state vector X, X∈R ^N+2b is the state vector; and transforms the measurement vector into y∈R ^m , including the square of the node voltage amplitude, branch active power, branch reactive power, injected active power, injected reactive power, the square of the branch current amplitude, where m is the total number of quantity measurements, when represented by the transformed state vector X, the square of the node voltage amplitude is

v _i is the voltage of node i, and the branch active power from node i to node j is The branch reactive power from node i to node j is $Q_{\mathrm{ij}}=-v_i^2(b_{\mathrm{the\; si}}+b_{\mathrm{ij}})+v_i{v}_j{b}_{\mathrm{ij}}\cos {\mathrm{\&theta;}}_{\mathrm{ij}}-v_i{v}_j{g}_{\mathrm{ij}}\sin {\mathrm{\&theta;}}_{\mathrm{ij}},$ The injected active work of node i is $P_i=v_i\underset{j\&Element;N_i}{\mathrm{\&Sigma;}}{v}_j(G_{\mathrm{ij}}\cos {\mathrm{\&theta;}}_{\mathrm{ij}}+B_{\mathrm{ij}}\sin {\mathrm{\&theta;}}_{\mathrm{ij}}),$ The injected reactive power of node i is $Q_i=v_i\underset{j\&Element;N_i}{\mathrm{\&Sigma;}}{v}_j(G_{\mathrm{ij}}\sin {\mathrm{\&theta;}}_{\mathrm{ij}}-B_{\mathrm{ij}}\cos {\mathrm{\&theta;}}_{\mathrm{ij}}),$ G _ij + jB _ij is the corresponding element in the node admittance matrix, and the square of the branch current amplitude is Among them, I _ij is the current amplitude of π-type branch ij, A=(g _si +g _ij ) ² +(b _si +b _ij ) ² ,

D=-g _si b _ij +b _si g _ij .

支路电流幅值量测的平方对应的雅可比矩阵元素为

以及根据步骤B得到的变换后的量测矢量和状态矢量，得到精确线性化的量测方程为：y=JX+τ，其中，τ∈R^m为量测误差矢量，J∈R^m×(N+2b)为常数雅可比矩阵。In one embodiment of the present invention, the step C includes: setting ^{J∈Rm×(N+2b)} as the Jacobian matrix, wherein the Jacobian matrix element corresponding to the square of the node voltage amplitude measurement is $\frac{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">i</figure-callout>}}^2}{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">i</figure-callout>}}^2}=1,$ $\frac{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">i</figure-callout>}}^2}{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}=0,$ $\frac{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">i</figure-callout>}}^2}{{\&PartialD;v}_i{v}_j\cos {\mathrm{\&theta;}}_{\mathrm{ij}}}=0,$ $\frac{{\&PartialD;v}_i^2}{{\&PartialD;v}_i{v}_j\sin {\mathrm{\&theta;}}_{\mathrm{ij}}}=0,$ The Jacobian matrix elements corresponding to branch power measurement are $\frac{{\&PartialD;P}_{\mathrm{ij}}}{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">i</figure-callout>}}^2}=g_{\mathrm{the\; si}}+g_{\mathrm{ij}},$ $\frac{{\&PartialD;P}_{\mathrm{ij}}}{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}=0,$ $\frac{{\&PartialD;P}_{\mathrm{ij}}}{{\&PartialD;v}_i{v}_j\cos {\mathrm{\&theta;}}_{\mathrm{ij}}}=-g_{\mathrm{ij}},$ $\frac{{\&PartialD;P}_{\mathrm{ij}}}{{\&PartialD;v}_i{v}_j\sin {\mathrm{\&theta;}}_{\mathrm{ij}}}=-b_{\mathrm{ij}},$ $\frac{{\&PartialD;Q}_{\mathrm{ij}}}{{\&PartialD;v}_i^2}=-(b_{\mathrm{the\; si}}+b_{\mathrm{ij}}),$ $\frac{{\&PartialD;Q}_{\mathrm{ij}}}{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}=0,$ $\frac{{\&PartialD;Q}_{\mathrm{ij}}}{{\&PartialD;v}_i{v}_j\cos {\mathrm{\&theta;}}_{\mathrm{ij}}}=b_{\mathrm{ij}},$ $\frac{{\&PartialD;Q}_{\mathrm{ij}}}{{\&PartialD;v}_i{v}_j\sin {\mathrm{\&theta;}}_{\mathrm{ij}}}=-g_{\mathrm{ij}},$ The Jacobian matrix elements corresponding to the injection power measurement are $\frac{{\&PartialD;P}_i}{{\&PartialD;v}_i^2}=G_i,$ $\frac{{\&PartialD;P}_i}{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}=0,$ $\frac{{\&PartialD;P}_i}{{\&PartialD;v}_i{v}_j\cos {\mathrm{\&theta;}}_{\mathrm{ij}}}=G_{\mathrm{ij}},$ $\frac{{\&PartialD;P}_i}{{\&PartialD;v}_i{v}_j\sin {\mathrm{\&theta;}}_{\mathrm{ij}}}=B_{\mathrm{ij}},$ $\frac{{\&PartialD;Q}_i}{{\&PartialD;v}_i^2}=-B_i,$ $\frac{{\&PartialD;Q}_i}{{\&PartialD;v}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}=0,$ $\frac{{\&PartialD;Q}_i}{{\&PartialD;v}_i{v}_j\cos {\mathrm{\&theta;}}_{\mathrm{ij}}}=-B_{\mathrm{ij}},$

The Jacobian matrix element corresponding to the square of branch current amplitude measurement is

And according to the transformed measurement vector and state vector obtained in step B, the accurate linearized measurement equation is: y=JX+τ, where τ∈R ^m is the measurement error vector, J∈R ^{m×( N+2b)} is a constant Jacobian matrix.

记G＝J^TWJ为信息矩阵，求解得到变换的状态矢量X的估计值X＝G^-1J^TWy。In one embodiment of the present invention, said step D includes: constructing $\mathrm{MinJ}\left(x\right)={\mathrm{\&Sigma;}}_{i=1}^m{({\mathrm{the\; y}}_i-J_{i}x)}^2/R_i={(\mathrm{the\; y}-\mathrm{JX})}^{T}W(\mathrm{the\; y}-\mathrm{JX})$ The linear weighted least squares problem of , where W=R ^-1 is the weight matrix, and the optimal solution should satisfy the condition

Record G=J ^T WJ as the information matrix, and obtain the estimated value X=G ^-1 J ^T Wy of the transformed state vector X by solving.

可得将A代入A₂，并求解可得θ＝(A^T A)^-1A^Tθ_b，其中，In one embodiment of the present invention, the step E includes: using the state vector X obtained in the step D, according to the formula $\left\{\begin{array}{c}{v}_i=\sqrt{v_i^2(1\&le;i\&le;N)}\\ {\mathrm{\&theta;}}_{l_i{l}_j}=\arcsin (v_{l_i}{v}_{l_j}\sin {\mathrm{\&theta;}}_{l_i{l}_j}/\left(v_{l_i}{v}_{l_j}\right))(1\&le;l\&le;b)\\ {\mathrm{\&theta;}}_{l_i{l}_j}=\arccos (v_{l_i}{v}_{l_j}\cos {\mathrm{\&theta;}}_{l_i{l}_j}/\left(v_{l_i}{v}_{l_j}\right))(1\&le;l\&le;b)\end{array}\right.$ Perform inverse transformation to obtain the magnitude of all node voltages and the estimated value of the phase angle difference at both ends of all branches θ _2b ∈ R ^2b , namely ${\mathrm{\&theta;}}_{2b}={[\arcsin (v_{l_i}{v}_{l_j}\sin {\mathrm{\&theta;}}_{l_i{l}_j}/\left(v_{l_i}{v}_{l_j}\right)),\arccos (v_{l_i}{v}_{l_j}\cos {\mathrm{\&theta;}}_{l_i{l}_j}/\left(v_{l_i}{v}_{l_j}\right))(1\&le;l\&le;b)]}^T;$ And use the estimated value θ _2b of the phase angle difference at both ends of all branches to construct the linear weighted least squares problem of Min J(θ)=(θ _2b -A ₂ θ) ^T W _θ (θ _2b -A ₂ θ) , where W _θ is the weight matrix, the value is the identity matrix, A ₂ =[A ^T A ^T ] ^T , A is the branch-node correlation matrix obtained in step A, θ∈R ^N-1 is the The phase angles of all nodes, where the optimal solution should satisfy the condition

Available Substitute A into A ₂ and solve to get θ=(A ^T A) ^-1 A ^T θ _b , where,

${\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; [</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; arcsin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; arccos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; .</span>$

其中，r＝[I-J[J^TWJ]^-1J^TW]τ＝Sτ，S＝I-J(J^TWJ)^-1J^TW为残差灵敏度矩阵，是常数矩阵，r~N(0，Ω)，其中Ω＝SR；以及若某个量测的正则化残差值大于预定值，则去掉该量测，重新运行所述D步骤，直至所有量测的正则化残差都小于所述预定值。In one embodiment of the present invention, the step F includes: calculating the regularized residual value of the i-th measurement

Among them, r=[IJ[J ^T WJ] ^-1 J ^T W]τ=Sτ, S=IJ(J ^T WJ) ^-1 J ^T W is the residual sensitivity matrix, which is a constant matrix, r~N(0, Ω), wherein Ω=SR; and if the regularized residual value of a certain measurement is greater than a predetermined value, then remove the measurement and re-run the D step until the regularized residuals of all measurements are less than the predetermined value.

The advantages of the two-stage linear weighted least squares power system state estimation method (TLWLS) of the present invention at least include: the TLWLS model is a quadratic programming problem in essence, and the unique global optimal solution can be guaranteed mathematically; TLWLS only needs Solving linear equations does not require iteration, there is no convergence problem, and the estimation efficiency is much higher than that of the conventional nonlinear weighted least squares state estimation method (WLS); the Jacobian matrix and residual sensitivity matrix of TLWLS are constant matrices, which gives The identification of bad data provides convenience and can improve the identification efficiency of bad data. In summary, the two-stage linear weighted least squares state estimation method proposed by the present invention can obtain more scientific state estimation results, and has higher calculation efficiency, and has a good engineering application prospect.

Additional aspects and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.

Description of drawings

The above and/or additional aspects and advantages of the present invention will become apparent and comprehensible from the description of the embodiments in conjunction with the following drawings, wherein:

Fig. 1 is the flow chart of two-stage linear weighted least squares power system state estimation method of the present invention;

Fig. 2 is the schematic diagram of π-type branch;

Fig. 3 is the schematic diagram of π-type branch equivalent circuit; And

Figure 4 is a single-line diagram and measurement configuration diagram of a three-node system.

Detailed ways

Embodiments of the present invention are described in detail below, examples of which are shown in the drawings, wherein the same or similar reference numerals designate the same or similar elements or elements having the same or similar functions throughout. The embodiments described below by referring to the figures are exemplary and are intended to explain the present invention and should not be construed as limiting the present invention.

As shown in Figure 1, the accurate linearization method of the power system state estimation measurement equation of the present invention comprises the following steps:

In step S101, a network model is formed, and a node admittance matrix and a branch-node correlation matrix are calculated.

Specifically, if the three-winding transformer in the network is equivalent to three two-winding transformers, then all the lines and transformers in the network can be represented by a unified π-shaped branch, as shown in Figure 2. In Fig. 2, y _s =1/(r _ij +jx _ij )=g _s +jb _s is the series susceptance of the branch ij; r _ij +jx _ij is the series impedance value; b _c is the ground susceptance of the branch . Among them, if the π-type branch ij is a transformer branch, then b _c =0 and k is the transformation ratio of the ideal transformer; if the π-type branch ij is an ordinary line, then k=1, and multiple branches connected in parallel are equivalent for a branch. It should be noted that multiple branches connected in parallel are equivalent to one branch.

The equivalent circuit of the π-type branch in Figure 2 is shown in Figure 3. In Fig. 3, g _ij =g _s /k; b _ij =b _s /k; g _si =(1-k)g _s /k ² ; b _si =(1-k)b _s /k ² +b _c /2; g _sj = (k-1)g _s /k; b _sj = (k-1)b _s /k+b _c /2. Then form the node admittance matrix Y=G+jB, G and B are the real part and imaginary part of the node admittance matrix respectively, and form the branch-node correlation matrix as A={a _ij }(1≤i≤b, 1≤j≤N-1), each element is defined as:

Step S102, transforming the measurement vector and the state vector.

First, transform the state vector into $x={[{\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_1^2,{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_2^2,...,{\mathrm{<figure-callout\; id="2"\; label="v\; N"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_N^{},{v_l}_i{v_l}_j\cos {\mathrm{\&theta;}}_{l_i{l}_j}(1\&le;l\&le;b),{v_l}_i{v}_{l_j}\sin {\mathrm{\&theta;}}_{l_i{l}_j}(1\&le;l\&le;b)]}^T.$

和

分别是节点l_i和l_j的电压幅值，

和分别是节点l_i和l_j的相角，

为相角差；

代表所有的b条支路对状态矢量X的贡献，也代表所有的b条支路对状态矢量X的贡献；X∈R^N+2b为状态矢量。Among them, N is the total number of all nodes in the network; b is the number of all branches in the network (multiple branches connected in parallel are equivalent to one branch); l is the branch number, l _i and l _j are the branches The node numbers at both ends of l,

and

are the voltage amplitudes of nodes l _i and l _j respectively,

and are the phase angles of nodes l _i and l _j respectively,

is the phase angle difference;

Represents the contribution of all b branches to the state vector X, It also represents the contribution of all b branches to the state vector X; X∈R ^N+2b is the state vector.

Secondly, transform the measurement vector into y∈R ^m , including the measurement types: square of node voltage amplitude, branch active power, branch reactive power, injected active power, injected reactive power, square of branch current amplitude ; m is the total number of measurements. Then the quantity measurement can be expressed by the state vector X.

1) The square of the voltage amplitude measurement

${\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Among them, v _i is the voltage of node i.

2) Branch active and reactive power measurement from node i to node j

${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; the\; si</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; the\; si</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

Among them, P _ij and Q _ij are the branch active power and reactive power flowing from node i to node j respectively.

3) Measurement of injected active power and injected reactive power of node i

${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Among them, P _i and Q _i are the injected active power and injected reactive power of node i respectively, and G _ij + jB _ij are the corresponding elements in the node admittance matrix

4) The square of the branch current amplitude measurement

${\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="Av\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">Av</figure-callout></span><figure-callout\; id="2"\; label="Av\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Bv</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; C</span>}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; D.</span>}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

D=-g_sib_ij+b_sig_ij。Wherein, I _ij is the current amplitude of branch ij; A=(g _si +g _ij ) ² +(b _si +b _ij ) ² ;

D=-g _si b _ij +b _si g _ij .

Step S103, forming an accurate linearized measurement equation.

According to the above transformed measurement vector and new state vector, the new accurate linearized measurement equation can be obtained as:

y=JX+τ （7）

其中

为τ_i的方差；J∈R^m×(N+2b)为常数雅可比矩阵，其各个部分的元素的表达式如下所示。Among them, y∈R ^m is the transformed measurement vector, including the square of node voltage amplitude, branch active power, branch reactive power, injected active power, injected reactive power, and the square of branch current amplitude; X∈R ^{N +2b} is the transformed state vector; τ~N(0,R) is the measurement error vector, where

in

is the variance of τ _i ; J∈R ^m×(N+2b) is a constant Jacobian matrix, and the expressions of the elements of each part are as follows.

1) Jacobian matrix elements corresponding to the square of the voltage amplitude measurement

For the square of the voltage amplitude measurement, the corresponding Jacobian matrix element is

$\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">i</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">j</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; .</span>$

2) Jacobian matrix elements corresponding to branch power measurement

For branch power measurement, the corresponding Jacobian matrix elements are

$\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">i</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; the\; si</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">j</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; ,</span>$

$\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">i</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; the\; si</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">j</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; g</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; .</span>$

3) Jacobian matrix elements corresponding to injection power measurement

For the injection power measurement, the corresponding Jacobian matrix elements are

$\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">i</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">j</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; ,</span>$

$\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">i</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">j</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; Q</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; G</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; .</span>$

4) Jacobian matrix elements corresponding to the square of branch current amplitude measurement

For the square of branch current amplitude measurement, the corresponding Jacobian matrix element is

$\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">i</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout></span><figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">j</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; ,</span>$ $\frac{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; .</span>$

It should be noted that the above formula (7) is an accurate linearized measurement equation. Based on this measurement equation, any existing state estimation method can be used to solve the transformed state vector X. After obtaining X, all branch powers, node injection powers, and branch current amplitudes can be further obtained. In this way, an accurate perception of the state of the entire network can be obtained.

Step S104, performing the first-stage linear weighted least squares estimation to obtain the estimated value of the transformed state vector X.

Specifically, the following linear weighted least squares problem is solved to obtain the estimated value of the state vector X.

$\mathrm{<span\; class="notranslate">\; MinJ</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; m</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; JX</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; JX</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

即Where W=R ^-1 is the weight matrix. In order to obtain the optimal value, the optimal condition must be satisfied:

Right now

GX=J ^T Wy (9)

Among them, G=J ^T WJ is the information matrix. The estimated value of the state vector X can be obtained by performing orthogonal decomposition on the information matrix G, that is, X=G ^-1 J ^T Wy.

Step S105, perform inverse transformation, and perform linear weighted least squares estimation in the second stage to obtain estimated values of voltage amplitudes and phase angles of all nodes.

According to the definition of the transformed state vector X, the inverse transformation can be used to obtain the estimated values of the voltage amplitudes of all nodes and the phase angle differences at both ends of all branches, namely

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\sqrt{{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; arcsin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}\mathrm{<span\; class="notranslate">\; sin</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; arccos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}\mathrm{<span\; class="notranslate">\; cos</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

In formula (10), the sign of arccos should be consistent with that of arcsin.

So far, the estimated values of the voltage amplitudes of all nodes have been obtained, but the phase angles of all nodes are still unknown. Obviously, the phase angle differences at both ends of all branches can be used to estimate.

make ${\mathrm{\&theta;}}_{2b}={\left[\arcsin (v_{l_i}{v}_{l_j}\sin {\mathrm{\&theta;}}_{l_i{l}_j}/\left(v_{l_i}{v}_{l_j}\right))(1\&le;l\&le;b),\arccos (v_{l_i}{v}_{l_j}\cos {\mathrm{\&theta;}}_{l_i{l}_j}/\left(v_{l_i}{v}_{l_j}\right))(1\&le;l\&le;b)\right]}^T,$ Obviously, θ _2b can be obtained by formula (10). The following uses θ _2b to estimate the phase angles of all nodes except the reference node.

Let θ=[θ ₂ ,…,θ _N ] ^T represent the phase angles of all nodes (except the reference node), then the relationship between θ _2b and θ is

θ _2b = A ₂ θ (11)

Wherein, A ₂ =[ ^AT A ^T ] ^T , and A ₂ is a 2b×(N-1) matrix.

Since θ _2b is derived from the state estimation result of the first stage, θ _2b can be regarded as a noisy quantity measurement. And θ is the value to be estimated, then formula (11) can be rewritten as the following measurement equation about θ

θ _2b =A ₂ θ+τ (12)

Then the following linear weighted least squares problem can be constructed to estimate θ

Min J(θ)=(θ _2b -A ₂ θ) ^T W _θ (θ _2b -A ₂ θ) (13)

Among them, W _θ is the weight matrix, without loss of generality, W _θ can be taken as the identity matrix. When formula (13) obtains the optimal value, it must satisfy

${\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="T\; A"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">T</figure-callout></span><figure-callout\; id="2"\; label="T\; A"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">\; </figure-callout>}}{\mathrm{<span\; class="notranslate">\; </span>}}_{}{\mathrm{<span\; class="notranslate">A</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 14</span>}<span\; class="notranslate">\; )</span>$

Substituting A into A ₂ , we get

A ^T Aθ = A ^T θ _b (15)

in, ${\mathrm{\&theta;}}_b={\left[\right[\arcsin (v_{l_i}{v}_{l_j}\sin {\mathrm{\&theta;}}_{l_i{l}_j}/\left(v_{l_i}{v}_{l_j}\right))+\arccos (v_{l_i}{v}_{l_j}\cos {\mathrm{\&theta;}}_{l_i{l}_j}/\left(v_{l_i}{v}_{l_j}\right))]/2(1\&le;l\&le;b)]}^T$ is a b-dimensional vector, obviously θ _b can be obtained from θ _2b .

The estimated value of θ can be obtained from formula (15) θ=(A ^T A) ^-1 A ^T θ _b .

So far, the estimated values of the voltage amplitudes and phase angles of all nodes have been obtained.

Step S106, identifying bad data.

In the present invention, it is only necessary to identify bad data in the first stage of the two-stage linear weighted least squares state estimation.

The residuals of the first-stage linear weighted least squares state estimation are

r=y-JX （16）

Among them, X is the result of state estimation in the first stage, which is obtained by solving equation (9).

From equations (9) and (16), it can be deduced that the relationship between residual error and measurement error is

r=[IJ[J ^T WJ] ^-1 J ^T W]τ=Sτ (17)

Wherein, S=IJ(J ^T WJ) ^-1 J ^T W 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, then the normalized residual (normalized residual) r ^N can be obtained as

${{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}^{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; |</span>}{\sqrt{{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$

where | _ri | is the absolute value of the residual _ri .

则第k个量测即被认为是不良数据，可将其从量测集中去掉，重新运行第一阶段的线性加权最小二乘状态估计。重复以上步骤，直到所有的正则化残差都小于3，即可完成对所有不良数据的辨识。由于雅可比矩阵是常数矩阵，显然基于线性加权最小二乘状态估计的不良数据辨识效率是非常高的。if

Then the kth measurement is considered bad data, it can be removed from the measurement set, and the linear weighted least squares state estimation of the first stage is re-run. Repeat the above steps until all the regularized residuals are less than 3, then the identification of all bad data can be completed. Since the Jacobian matrix is a constant matrix, it is obvious that the identification efficiency of bad data based on linear weighted least squares state estimation is very high.

In order to make those skilled in the art understand better, the applicant takes a three-node system as an example to illustrate the method of the present invention. The single-line diagram and measurement configuration diagram of the three-node system are shown in Figure 4, and its network parameters are shown in Table 1. Select node 1 as the 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 and TLWLS proposed by the present invention under normal load. The quantity measurement under normal load is shown in Table 2.

Table 2 Three-node system quantity measurement

Table 2 Measurements of the 3-bus system

If the traditional nonlinear WLS is used, when the convergence accuracy is 10 ^-6 , after 4 iterations of nonlinear WLS convergence, the estimated values of the voltage amplitudes and phase angles of all nodes are shown in Table 3.

Table 3 Estimation results of nonlinear WLS

Table 3 The Estimated State Vector by Nonlinear WLS

In the following, the TLWLS proposed by the present invention is used for estimation.

分别是X_i、v_i和θ_2bi的估计值。Transform the state vector into $x={[{\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_1^2,{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_2^2,{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_3^2,v_1{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\cos {\mathrm{\&theta;}}_{12},v_1{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_3\cos {\mathrm{\&theta;}}_{13},v_2{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_3\cos {\mathrm{\&theta;}}_{\mathrm{twenty\; three}},v_1{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_2\sin {\mathrm{\&theta;}}_{12},v_1{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_3\sin {\mathrm{\&theta;}}_{13},v_2{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_3\sin {\mathrm{\&theta;}}_{\mathrm{twenty\; three}}]}^T,$ Transform the measurement vector into $\mathrm{the\; y}={[{\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_1^2,{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">v</figure-callout>}}_2^2,Q_{12},Q_{\mathrm{twenty\; one}},Q_{13},Q_{31},{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="HDA00002049112900012.png,HDA00002049112900022.png"\; state="\{\{state\}\}">Q</figure-callout>}}_2,P_{12},P_{\mathrm{twenty\; one}},P_{13},P_{31},P_2]}^T,$ Then an accurate linearized measurement equation can be obtained as y=JX+τ. The constant Jacobian can be obtained by the formula given in S103. By solving the estimated value of the transformed state vector X in Equation (9), the estimated value of all node voltage amplitudes and phase angle differences at both ends of all branches can be obtained from Equation (10), as shown in Table 4. In Table 4, and

are the estimated values of Xi _, vi and θ _{2bi ,} respectively.

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

According to the wiring diagram of the network, the branch-node association matrix of the network (node 1 is the reference node) can be obtained as

$\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{cc}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]$

By solving formula (15), the estimated value of the phase angle of all nodes can be obtained as θ=[θ ₂ ,θ ₃ ] ^T =[-0.0213,-0.0479] ^T . Therefore, the estimated values of the voltage amplitudes and phase angles of all nodes obtained by 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 found that the estimated value of the state variable obtained by the two-stage linear weighted least squares state estimation (TLWLS) proposed by the present invention and the estimated value of the state variable obtained by the traditional nonlinear WLS It fit very well. However, TLWLS does not need iterations, while traditional nonlinear WLS needs 4 iterations to converge. In this example, the calculation efficiency of TLWLS is more than three times that of nonlinear WLS.

(2) Calculation example of nonlinear WLS trapped in local optimal solution

Still taking the 3-node system as an example, an example of a nonlinear WLS trapped in a local optimal solution is given below. When the system load continues to increase, another set of quantity measurements can be obtained, as shown in Table 6. Apparently, a voltage collapse occurred at this point.

Table 6 Measurement of three-node system volume under heavy load

Table 6 Measurements of the 3-bus system at heavy load

When using nonlinear WLS to estimate this example, after 35 times of convergence, the value of the objective function of nonlinear WLS is 0.1088. For this example, the true values of the state variables, the estimated values of the nonlinear WLS, and the estimated values of the two-stage linear weighted least squares state estimation (TLWLS) are shown in Table 7.

Table 7 The true value of the state variable, the estimation results of nonlinear WLS and TLWLS under heavy load

Table 7 The True value, Estimate Value by WLS and TLWLS at heavy load

It can be seen from Table 7 that under heavy load, especially when the voltage collapses, the traditional nonlinear WLS falls into a local optimal solution, and its estimated value has a negative voltage amplitude and a large phase angle value. It can be seen that the estimation results of traditional nonlinear WLS are unreliable under heavy load, especially during voltage collapse, but at this time the power system requires state estimation to have as good a performance as possible. Under heavy load, especially voltage collapse, the sharp contradiction is caused by the inherent characteristics of traditional nonlinear WLS.

It can also be seen from Table 7 that 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, that is, the TLWLS proposed by the present invention can obtain the global optimal solution without iteration. Therefore, the TLWLS proposed by the present invention can obtain more scientific state estimation results, and the calculation efficiency is higher.

(3) Identify bad data

The 20% noise is iterated on the quantity measurement P ₁₂ at regular load shown in Table 2, so that P ₁₂ becomes bad data. The regularized residual of _P12 can be obtained from formula (18), and it is found that it is greater than 3, so it is removed from the measurement set, and the linear weighted least squares state estimation of the first stage is re-run, and all the regularized residuals are found to be If it is less than 3, bad data identification ends. It can be seen that the regularized residual test method can effectively identify bad data in the first stage of TLWLS, and it is found that its identification efficiency is more than three times that of the traditional regularized residual test method based on nonlinear WLS.

In the description of this specification, descriptions referring to the terms "one embodiment", "some embodiments", "example", "specific examples", or "some examples" mean that specific features described in connection with the embodiment or example , structure, material or characteristic is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and cannot be construed as limitations to the present invention. Variations, modifications, substitutions, and modifications to the above-described embodiments are possible within the scope 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 recorded_s＝1/(r_ij+jx_ij)＝g_s+jb_sSeries susceptance, r, for a pi-branch ij_ij+jx_ijSeries impedance value of pi-type branch ij, b_cIs the grounding susceptance of the pi-type branch ij, wherein if the pi-type branch ij is a transformer branch, b is_cK 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 g_ij＝g_s/k，b_ij＝b_s/k，g_si＝(1-k)g_s/k²，b_si＝(1-k)b_s/k²+b_c/2，g_sj=(k-1)g_s/k，b_sj=(k-1)b_s/k+b_c/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>&theta;</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>&le;</mo> <mi>l</mi> <mo>&le;</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>&theta;</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>&le;</mo> <mi>l</mi> <mo>&le;</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 number_iAnd l_jThe numbers of the nodes at the two ends of the branch l,

and

are respectively node l_iAnd l_jThe magnitude of the voltage of (a) is,

and

are respectively node l_iAnd l_jThe 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 ∈ R^N+2bIs a state vector; and

transforming the measurement vector into y ∈ R^mThe 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 isv_iFor 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>&theta;</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>&theta;</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

G_ij+jB_ijThe 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>&theta;</mi> <mi>ij</mi> </msub> <mo>-</mo> <mi>D</mi> <mi>sin</mi> <msub> <mi>&theta;</mi> <mi>ij</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math> Wherein, I_ijThe amplitude of the current for the pi branch ij, a ═ g_si+g_ij)²+(b_si+b_ij)²， $B=g_{\mathrm{ij}}^2+b_{\mathrm{ij}}^2,$ $C=g_{\mathrm{ij}}^2+b_{\mathrm{ij}}^2+g_{\mathrm{si}}{g}_{\mathrm{ij}}+b_{\mathrm{si}}{b}_{\mathrm{ij}},$ D=-g_sib_ij+b_sig_ij。

4. The two-stage linear weighted least squares power system state estimation method of claim 3, said step C comprising:

let J ∈ R^m×(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>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> <mn>2</mn> </msubsup> <msubsup> <mrow> <mo>&PartialD;</mo> <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>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> <mn>2</mn> </msubsup> <msubsup> <mrow> <mo>&PartialD;</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> <msubsup> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> <mn>2</mn> </msubsup> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>cos</mi> <msub> <mi>&theta;</mi> <mi>ij</mi> </msub> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </math> <math> <mrow> <mfrac> <msubsup> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> <mn>2</mn> </msubsup> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>sin</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>P</mi> </mrow> <mi>ij</mi> </msub> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>P</mi> </mrow> <mi>ij</mi> </msub> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>P</mi> </mrow> <mi>ij</mi> </msub> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>cos</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>P</mi> </mrow> <mi>ij</mi> </msub> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>sin</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>Q</mi> </mrow> <mi>ij</mi> </msub> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>Q</mi> </mrow> <mi>ij</mi> </msub> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>Q</mi> </mrow> <mi>ij</mi> </msub> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>cos</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>Q</mi> </mrow> <mi>ij</mi> </msub> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>sin</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>P</mi> </mrow> <mi>i</mi> </msub> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>P</mi> </mrow> <mi>i</mi> </msub> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>P</mi> </mrow> <mi>i</mi> </msub> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>cos</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>P</mi> </mrow> <mi>i</mi> </msub> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>sin</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>Q</mi> </mrow> <mi>i</mi> </msub> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>Q</mi> </mrow> <mi>i</mi> </msub> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>Q</mi> </mrow> <mi>i</mi> </msub> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>cos</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>Q</mi> </mrow> <mi>i</mi> </msub> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>sin</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>I</mi> </mrow> <mi>ij</mi> <mn>2</mn> </msubsup> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>I</mi> </mrow> <mi>ij</mi> <mn>2</mn> </msubsup> <msubsup> <mrow> <mo>&PartialD;</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>&PartialD;</mo> <mi>I</mi> </mrow> <mi>ij</mi> <mn>2</mn> </msubsup> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>cos</mi> <msub> <mi>&theta;</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>&PartialD;</mo> <mi>I</mi> </mrow> <mi>ij</mi> <mn>2</mn> </msubsup> <mrow> <msub> <mrow> <mo>&PartialD;</mo> <mi>v</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>v</mi> <mi>j</mi> </msub> <mi>sin</mi> <msub> <mi>&theta;</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 ∈ R^mFor measuring the error vector, J ∈ R^m×(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>&Sigma;</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 J^TWJ is an information matrix, and an estimated value X of the transformed state vector X is obtained by solving^-1J^TWy。

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>&le;</mo> <mi>i</mi> <mo>&le;</mo> <mi>N</mi> <mo>)</mo> </mrow> </msqrt> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&theta;</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>&theta;</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>&le;</mo> <mi>l</mi> <mo>&le;</mo> <mi>b</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&theta;</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>&theta;</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>&le;</mo> <mi>l</mi> <mo>&le;</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 branches_2b∈R^2bI.e. by <math> <mrow> <msub> <mi>&theta;</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>&theta;</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>&theta;</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>&le;</mo> <mi>l</mi> <mo>&le;</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 branches_2bConstruction Min J (theta) ═ theta_2b-A₂θ)^TW_θ(θ_2b-A₂θ) linear weighted least squares problem, where W_θIs a weight matrix, takes the value of an identity matrix, A₂＝[A^T A^T]^TA is the branch-node incidence matrix obtained in the step A, and theta is belonged to R^N-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 ═ A^TA)^-1A^Tθ_bWherein

<math> <mrow> <msub> <mi>&theta;</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>&theta;</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>&theta;</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>&le;</mo> <mi>l</mi> <mo>&le;</mo> <mi>b</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>.</mo> </mrow> </math>

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]^-1J^TW]τ＝Sτ，S＝I-J(J^TWJ)^-1J^TW 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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