CN102738794B - Grid topology identification method based on seidel-type recursion bayesian estimation - Google Patents

Abstract

The invention discloses a Seidel-type recursion bayesian method and application thereof to the state estimation. According to the invention, the recursive estimation calculation of posterior probabilities of different modes is implemented by extracting characteristic quantities of different modes; and in the recursive calculation, the Seidel-type recursion bayesian method utilizes the posterior probabilities just obtained in the recursion to replace prior probabilities to implement the calculation of the posterior probabilities. The method can be used for rapidly and accurately carrying out mode identification and has strong robustness for noise. A power grid topology error identification method or a transformation tapping point position estimation method on the basis of the Seidel-type recursion bayesian method can be used for rapidly and accurately obtaining a correct power grid operation mode or a correct transformer tapping point position, so that the qualified rate of the state estimation can be greatly improved and the practicality of an integral energy management system is further promoted.

Description

Based on the power network topology misidentification method of your formula recursion Bayesian Estimation of Saden

Technical field

The present invention relates to a kind of Saden that formula recursion bayes method and the application in Power system state estimation thereof, belong to power system analysis and control field.

Background technology

Along with the development of intelligent grid, more and more higher to reliability and the required precision of EMS (Energy Management System, EMS) analysis decision, the further practical focus that becomes people's concern of EMS.Power system state estimation (State Estimation, SE) is basis and the core of EMS, for the advanced applied software in EMS system provides real time data source.The factor that affects state estimation qualification rate comprises bad data, Topology Error and parameter error.At present, research and the application to bad data identification of academia and engineering circles is relatively many, and the research to topology error identification and parameter Estimation and application are less.And the accuracy of the correctness of topological structure and parameter (particularly load tap changer position) is the key factor of Guarantee Status computed reliability and precision.Therefore, how to ensure that the correctness of topological structure and the accuracy of network parameter (particularly load tap changer position) become the prerequisite of the EMS application software reliability services such as state estimation, stability analysis, Security Checking.

Existing topology error identification method mainly comprises residual error method, the augmented state estimation technique, regular method, minimum information loss method, innovation graph approach, transfer trend method etc.What residual error method obtained state estimation exceedes the standardized residual of threshold value owing to relevant topology mistake, but bad data also can cause standardized residual to be crossed the border, and in the time that Topology Error and bad data coexist, the identification capability of residual error method is limited.The augmented state estimation technique is using the voltage amplitude value difference at the trend of suspicious branch road, branch road two ends and phase angle difference as augmented state variable, estimate with voltage magnitude and the phase angle of other computing nodes simultaneously, but it is high that the part that this method needs measures redundancy, and easily produce numerical value instability problem.Rule method is carried out identification Topology Error by setting up a set of logic rules, can process some simple Topology Errors, but in the time of the mode of connection more complicated of electrical network, the regular formulation that is adapted to various operational modes is cumbersome.Topology error identification problem is converted into a mixed integer programming problem by minimum information loss method, provides new thinking from information-theoretical angle for topology error identification, but the modeling comparison complexity of this method, and the difficulty of application is larger.Innovation graph approach can Fast Identification Topology Error and the simultaneous situation of bad data, but identification while simultaneously occurring for branch road Topology Error and multiple bad data also has difficulties.Can effectively pick out branch road Topology Error and the simultaneous situation of multiple bad data although shift trend method, the precondition of this method is that the result of a upper time section state estimation is right-on, supposes too harsh.In order to improve precision and the qualification rate of state estimation, studying practical topology error identification method becomes the task of top priority.

The accuracy of parameter is also the key factor of Guarantee Status computed reliability and precision.In all parameters, because transformer voltage ratio occurs with quadratic term in the calculating of reactive power flow, its error is the most remarkable on the impact of state estimation result, therefore, accurately estimates that load tap changer position is the important prerequisite that realizes reliable state estimation.Up to now, the method for estimation of load tap changer position is mainly comprised to the augmented state estimation technique, residual error method, the tap position tracking estimation technique and perturbation method etc.The augmented state estimation technique, using transformer voltage ratio as augmented state variable, is estimated jointly with node voltage amplitude and phase angle.The method is simple, but it is higher to require to measure redundancy, and is prone to numerical value instability problem.Residual error method utilizes the sensitivity relation between residual error and ratio error to estimate the tram of tap, but is easily subject to the impact that residual contamination and residual error are flooded.Tap is followed the tracks of the estimation technique and is regarded over time tap joint position as a Markov Chain, utilize markovian state transition probability to carry out tap joint position estimation, this method has certain robustness to measurement noise, but the precision that tap position is estimated depends on state transition probability, and currently also do not have reliable method to determine rational state transition probability, test shows that this method only can obtain with little probability the estimation of tap joint position as a rule, and the confidence level of its result still can not be satisfactory.Perturbation ratio juris is suspicious transformer all to be carried out one time under all tap joint position to state estimation, and the corresponding tap joint position of least residual is considered to correct tap joint position.Due to the existence of measurement noise and be subject to residual contamination and impact that residual error is flooded, when tap joint position error hour, perturbation method may obtain wrong estimation.There is scholar's research in recent years and utilized PMU data to carry out the method for estimation of tap joint position, but needed transformer branch road two ends all will have PMU to measure.For this reason, measure and layout less in the situation that at current PMU, the method for estimation of research based on RTU data, to obtain accurately and rapidly correct load tap changer position, has important practical significance.

Integrate consideration, topology error identification is in possible power network topology pattern, to find unique correct topological mode; And load tap changer location estimation is to select a correct position from multiple definite tap joint position.From the angle of statistical learning, topology error identification and load tap changer location estimation all belong to the category of pattern recognition.So-called pattern recognition is according to the feature of research object or attribute, by constructing certain system, uses certain analytical method to judge the classification of sample, and system should make the result of sample classification identification meet as much as possible the fact.At present, pattern recognition theory and technology successfully apply to multiple fields such as industry, agricultural, biology, scientific research, and this field is also in continuous expansion.A very important link in pattern recognition is feature extraction and feature selecting, its role is to extract and select to lie in fixing, essence and important feature or attribute in sample data, thereby produce the pattern that can represent a certain special object, get final product on this basis learning of structure system, and complete Classification and Identification.For different objects and different objects, can select different mode identification methods.In these methods, Bayesian Estimation (grader) combines prior information and sample information, can fine sample be identified, because of but a kind of outstanding mode identification method has obtained increasingly extensive application.

In the time using Bayesian Estimation to carry out topology error identification and load tap changer location estimation, choose suitable feature or attribute most important.The characteristic feature that is different from another kind of topological structure of electric due to a kind of topological structure of electric is under different patterns, to obtain state estimation residual error difference, so can be using the state estimation residual error under different topological structure of electric as using Bayesian Estimation to carry out the characteristic quantity of topology error identification; Equally, the difference of load tap changer position and the characteristic feature of another tap joint position are that the state estimation residual error obtaining under different tap joint position is different, therefore can carry out using the state estimation residual error under different tap joint position as utilization Bayesian Estimation the characteristic quantity of load tap changer location estimation.In Bayesian Estimation, the preferred features amount when likelihood function being formed by residual error becomes power network topology misidentification and load tap changer location estimation.In order to increase ornamental, in the time carrying out power network topology misidentification or load tap changer location estimation by Bayesian Estimation, can introduce pseudo-measurement and virtual measurement, they form measurement vector together with actual measurements.Move multiple state estimation by measuring different topological structure of electric or the load tap changer positions of vector, can obtain the residual error under different topological structure of electric or different load tap changers position, and then can obtain corresponding likelihood function numerical value.Owing to inevitably containing noise in metric data, therefore likelihood function numerical value corresponding to topological structure of electric or correct load tap changer position can not become to reflect topological structure of electric and correct tap joint position strictly according to the facts, and the result of single Bayesian Estimation is always not reliable.

In order to eliminate the impact of measurement noise on estimated result, can adopt recursion Bayesian Estimation.In each step of recursion Bayesian Estimation, all need under different topological structure of electric or load tap changer position, to move multiple state estimation according to measurement vector and obtain the residual error under different topological structure of electric or load tap changer position, and then obtain likelihood function numerical value; In different recursion steps, substantial amount is measured constant, and pseudo-measurement and all differences of virtual measurement.In recursion, the posterior probability of the different topological structure of electric that last recursion is obtained or different load tap changers position, as the prior probability of this recursion, can realize the recursion of posterior probability and calculate.Suppose according to Bayes, initial distribution does not affect final posterior probability values, therefore can all topological structure of electric in the time of recursion first or the posterior probability of load tap changer position be set to be uniformly distributed, calculate through recursion, correct topological structure of electric or the corresponding posterior probability in load tap changer position can progressively level off to 1, and the posterior probability of other topological structure of electric or load tap changer position can progressively level off to 0.Thus, can obtain correct topological structure of electric or load tap changer position.

Based on above thought, foreign scholar has proposed the power network topology misidentification method based on recursion Bayesian Estimation, domestic scholars has proposed the load tap changer location estimation method based on recursion Bayesian Estimation, can estimate correct topological structure of electric or correct load tap changer position.But in these methods, ought in recursion last time, be all on posterior probability that once recursion obtains different topological structure of electric or load tap changer position as prior probability, and the posterior probability having obtained in recursion last time is not used in this recursion, they are until be just used in recursion next time, theory analysis and test all show, the recursion Bayesian Estimation of adopting is in this way calculated, and its efficiency and robustness all can not be satisfactory.Adopt more rational recursion Bayesian Estimation method, with quick, accurate, to estimate correct topological structure of electric or load tap changer tram robust, for the precision and the qualification rate that improve state estimation, thereby promote the practical of whole EMS system to have great importance.

Summary of the invention

The object of this invention is to provide a kind of computational efficiency high, and noise is there is to the Saden that formula recursion bayes method of very strong robustness.

To achieve these goals, technical scheme of the present invention is: a kind of Saden that formula recursion bayes method is provided, and wherein, you formula recursion Bayesian Estimation method carry out this Saden according to the following steps:

Step (1) initialization

Make the initial probability of all patterns to be identified all equate, even p is (η ₁| ε ⁽⁰⁾)=p (η ₂| ε ⁽⁰⁾)=...=p (η _n| ε ⁽⁰⁾)=1/N, wherein η _ibe i pattern, ε ⁽⁰⁾for initial characteristics amount, p (η _i| ε ⁽⁰⁾) be η _iinitial probability, the total number that N is pattern; Recursion counter k=1 is set;

Step (2) recursion calculates to obtain correct pattern

Step (2.1) judges that whether the maximum of posterior probability corresponding to all patterns is less than threshold value, judges max{p (η _i| ε ^(k-1)) whether <threshold sets up (threshold is threshold value), is to go to step (2.2); Otherwise go to step (3);

Step (2.2) is extracted and is obtained the corresponding characteristic quantity of all patterns;

Step (2.3) utilizes following formula to calculate the posterior probability of all patterns;

$\hat{p}\left({\mathrm{\&eta;}}_i\right|{\mathrm{\&epsiv;}}^{\left(k\right)})=\frac{p\left(e_i^{\left(k\right)}\right|{\mathrm{\&eta;}}_i)p\left({\mathrm{\&eta;}}_i\right|{\mathrm{\&epsiv;}}^{(k-1)})}{\underset{j<i}{\mathrm{\&Sigma;}}p\left(e_j^{\left(k\right)}\right|{\mathrm{\&eta;}}_j)\hat{p}\left({\mathrm{\&eta;}}_j\right|{\mathrm{\&epsiv;}}^{\left(k\right)})+\underset{j\&GreaterEqual;i}{\mathrm{\&Sigma;}}p\left(e_j^{\left(k\right)}\right|{\mathrm{\&eta;}}_j)p\left({\mathrm{\&eta;}}_j\right|{\mathrm{\&epsiv;}}^{(k-1)})}$

Wherein, for the corresponding characteristic quantity of all N pattern, i.e. residual error vector; pattern η while being the k time recursion _iposterior probability, p (η _i| ε ^(k-1)) be prior probability; be pattern η in the k time recursion _ithe conditional probability of corresponding residual error, i.e. likelihood function numerical value; the meaning of middle summation number is that the span of j is 1≤j≤N and j<i; the meaning of middle summation number is that the span of j is 1≤j≤N and j>=i;

Step (2.4) utilizes following formula to be normalized calculating to the posterior probability of all patterns;

$p\left({\mathrm{\&eta;}}_i\right|{\mathrm{\&epsiv;}}^{\left(k\right)})=\hat{p}\left({\mathrm{\&eta;}}_i\right|{\mathrm{\&epsiv;}}^{\left(k\right)})/{\mathrm{\&Sigma;}}_{j=1}^N\hat{p}\left({\mathrm{\&eta;}}_j\right|{\mathrm{\&epsiv;}}^{\left(k\right)})$

Wherein, p (η _i| ε ^(k)) pattern η while being the k time recursion _inormalization posterior probability values.

Step (2.5) makes k=k+1, goes to step (2.1);

The pattern of step (3) posterior probability maximum is correct pattern.

More than be the calculation procedure of your formula recursion bayes method of Saden, the method has general adaptability for all pattern recognitions that contain Uncertainty measurement.Compared with basic recursion bayes method, your formula recursion bayes method of Saden has used in time the posterior probability of the pattern just having obtained in each step of recursion, and research shows that the recognition efficiency of such recursion mode is higher, and robustness is better.

Another object of the present invention is to provide the application of Saden that formula recursion bayes method in state estimation, in the time carrying out power network topology misidentification or load tap changer location estimation with your formula recursion bayes method of Saden, its concrete calculation procedure is as follows:

Step (1) initialization

Form all possible power network topology pattern or all possible load tap changer position; Make the initial probability of all power network topology patterns to be identified or all tap joint position of transformer all equate, even wherein be the initial probability of i power network topology pattern or load tap changer position, N is the total number of power network topology pattern or load tap changer position; Recursion counter k=1 is set;

Step (2) recursion calculates to obtain correct power network topology pattern or load tap changer position

Step (2.1) judges whether the maximum of the corresponding posterior probability of all power network topology patterns or load tap changer position is less than threshold value, i.e. judgement whether (threshold is threshold value) sets up, and is to go to step (2.2); Otherwise go to step (3);

Step (2.2) input actual measurements, virtual measurement and pseudo-measurement, these measurement amounts form measurement equation and are

z＝h(x)+τ

Wherein, z ∈ R ^mfor measuring vector, comprise node inject meritorious measure, node injects idle measurement, branch road meritoriously measures, branch road is idle measurement and node voltage measure; These measurement amounts are generally got substantial amount and are measured, if substantial amount is measured disappearance, replace with pseudo-measurement, pseudo-ly measure employing and carry out pattern in the superpose random number of 10% normal distribution of the result of upper once state estimation; In measurement vector, should add 0 injection virtual measurement simultaneously; X ∈ R ⁿit is the state variable including node voltage amplitude and phase angle; H (x) is for state vector is to the Nonlinear Mapping that measures vector; τ～N (0, R) is error in measurement; for error in measurement variance matrix.

According to above measurement equation, carry out iteration by following update equation, until state estimation convergence can obtain state vector estimated value

x _k+1＝x _k+G(x _k)H ^T(x _k)R ^-1(z-h(x _k))

Wherein, for Jacobian matrix, G (x _k)=(H ^t(x _k) R ^-1h (x _k)) ^-1.

According to following formula, the independent residual error vector calculating under all power network topology patterns or load tap changer position respectively $e_i^{\left(k\right)}(i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,N)$

$e_i^{\left(k\right)}=z-h\left(\hat{x}\right)$

According to following formula, the independent varivance matrix C calculating under all power network topology patterns or load tap changer position respectively _f,i(i=1,2 ..., N)

$C_{f,i}^{-1}=\mathrm{diag}({\mathrm{\&sigma;}}_v^2,{\mathrm{\&sigma;}}_{p_i}^2,{\mathrm{\&sigma;}}_{q_i}^2,{\mathrm{\&sigma;}}_{p_b}^2,{\mathrm{\&sigma;}}_{q_b}^2)$

C _{f, i}(i=1,2 ..., N) in from [H ^tr ^-1h] ^-1the respective items of diagonal element obtains, C _f,i(i=1,2 ..., N) in every can be from matrix H [H ^tr ^-1h] ^-1h ^tthe respective items of diagonal element obtains.

In the time carrying out power network topology misidentification, above carried out state estimation is the whole network state estimation; And in the time carrying out load tap changer location estimation, the state estimation of more than carrying out be transformer branch road and near branch road local state estimate.

Step (2.3) utilizes following formula to calculate all topological structure of electric or the corresponding posterior probability of tap joint position;

${\hat{p}}_i^{\left(k\right)}=\frac{\exp (-{\left(e_i^{\left(k\right)}\right)}^T{C}_{f,i}{e}_i^{\left(k\right)}/2)p_i^{(k-1)}}{\underset{j<i}{\mathrm{\&Sigma;}}\exp (-{\left(e_j^{\left(k\right)}\right)}^T{C}_{f,j}{e}_j^{\left(k\right)}/2){\hat{p}}_j^{\left(k\right)}+\underset{j\&GreaterEqual;i}{\mathrm{\&Sigma;}}\exp (-{\left(e_j^{\left(k\right)}\right)}^T{C}_{f,j}{e}_j^{\left(k\right)}/2)p_j^{(k-1)}}$

Wherein, the corresponding characteristic quantity of pattern i, i.e. residual error vector while being the k time recursion; the posterior probability of pattern i while being the k time recursion, for prior probability; be the conditional probability of the corresponding residual error of pattern i in the k time recursion, i.e. likelihood function numerical value; the meaning of middle summation number is that the span of j is 1≤j≤N and j<i; the meaning of middle summation number is that the span of j is 1≤j≤N and j>=i;

Step (2.4) utilizes following formula to be normalized calculating to the posterior probability of all patterns;

$p_i^{\left(k\right)}={\hat{p}}_i^{\left(k\right)}/{\mathrm{\&Sigma;}}_{j=1}^N{\hat{p}}_j^{\left(k\right)}$

Wherein, the normalization posterior probability values of pattern i when the k time recursion.

Step (2.5) makes k=k+1, goes to step (2.1);

The pattern of step (3) posterior probability maximum is correct power network topology pattern or load tap changer position.

The Saden that formula recursion bayes method that the present invention proposes, belongs to a kind of new mode identification method.Compared with existing recursion bayes method, your computational efficiency of formula recursion bayes method of Saden is high, and noise is had to very strong robustness.You have a wide range of applications Saden by formula recursion bayes method in electric power system, can obtain accurately and fast correct topological structure of electric or correct load tap changer position based on your formula recursion bayes method of Saden.

Brief description of the drawings

The schematic diagram of your formula recursion Bayes mode identification method of Fig. 1 Saden;

Power network topology misidentification method or the load tap changer location estimation method schematic diagram of Fig. 2 based on your formula recursion bayes method of Saden;

The winding diagram of Fig. 3 test macro;

Topological mode recognition result based on basic recursion bayes method when Fig. 4 noise is 1%;

Topological mode recognition result based on your formula recursion bayes method of Saden when Fig. 5 noise is 1%;

Topological mode recognition result based on basic recursion bayes method when Fig. 6 noise is 6%;

Topological mode recognition result based on your formula recursion bayes method of Saden when Fig. 7 noise is 6%;

Topological mode recognition result based on basic recursion bayes method when Fig. 8 noise is 10%;

Topological mode recognition result based on your formula recursion bayes method of Saden when Fig. 9 noise is 10%;

Embodiment

In order to describe the technology contents, structural feature of your formula recursion bayes method of Saden that the present invention proposes and the application in state estimation thereof in detail, below in conjunction with execution mode and coordinate accompanying drawing to be described further.

Fig. 7 is the schematic diagram of your formula recursion Bayes mode identification method of Saden; In the time carrying out power network topology misidentification or load tap changer location estimation with your formula recursion bayes method of Saden, schematic diagram as shown in Figure 2.

The present invention utilizes anglist to test the validity of the topology error identification method based on your formula recursion Bayesian Estimation method of Saden with the power distribution network of two 11kV of distribution system (U.K.Generic Distribution System, UKGDS).This system comprises two networks, and two networks connect by normal open switch, and wherein network 1 comprises 26 computing nodes, 25 branch roads, 13 loads and a generator, and network 2 comprises 13 computing nodes, 13 branch roads and 8 loads.Network parameter, load parameter and generator parameter can obtain on the http://monaco.eee.strath.ac.uk/ukgds/ of website.The winding diagram of this network as shown in Figure 3.

Because the load that wasted power is little is little on the impact of state estimation result, losing large load has larger impact to state estimation result, therefore, when the large load of loss, thinks that change has occurred operation of power networks pattern; In addition, moving the branch breaking that determines or closure by protective device is also considered to operation of power networks pattern change has occurred.By analysis, the network 1 shown in Fig. 3 is had to operation of power networks pattern possible in 5, as shown in table 1.

Table 1 network 1 possible 5 in topological mode

In this example, pattern 1 is correct topological mode, gets threshold value threshold=0.9.Below adopt respectively your formula recursion bayes method of Saden that basic recursion bayes method and the present invention propose to carry out the topological operational mode of estimation network 1, the estimated result by two kinds of methods of examination contrast when the measurement noise in various degree.

1) estimated result when 1% noise

In the time that measurement noise is 1%, while adopting basic recursion bayes method, the change curve of the posterior probability of all topological mode as shown in Figure 4.As seen from Figure 4, through 100 recursion, the posterior probability of correct topological mode 1 reaches 0.9, and basic recursion bayes method has obtained correct estimated result.But the recursion number of times of the needs of basic recursion bayes method is many.

Now, while adopting your the formula recursion bayes method of Saden of invention herein, the change curve of the posterior probability of all topological mode as shown in Figure 5.As seen from Figure 5, through 66 recursion, the posterior probability of correct topological mode 1 reaches 0.9, and you have obtained correct estimated result by formula recursion bayes method equally Saden, and its efficiency is apparently higher than basic recursion bayes method.

2) estimated result when 6% noise

In the time that measurement noise increases to 6%, adopt all topological mode that basic recursion bayes method obtains posterior probability change curve as shown in Figure 6, visible, along with the increase of measurement noise, the performance of basic recursion bayes method has worsened a lot, be that basic recursion bayes method needs recursion just can identify many times correct topological operational mode, its recursion efficiency declines greatly.

Now, while adopting your the formula recursion bayes method of Saden of invention herein, the change curve of the posterior probability of all topological mode as shown in Figure 7.As seen from Figure 7, although measurement noise has increased, but your performance of formula recursion bayes method of Saden is substantially unaffected, the recursion number of times that your formula recursion bayes method of Saden needs is far smaller than basic recursion bayes method, and the former has good recursion efficiency and stronger robustness.

3) estimated result when 10% noise

In the time that measurement noise continues to increase to 10%, adopt all topological mode that basic recursion bayes method obtains posterior probability change curve as shown in Figure 8, visible, now basic recursion bayes method None-identified goes out correct topological mode,

Now, while adopting your the formula recursion bayes method of Saden of invention herein, the change curve of the posterior probability of all topological mode as shown in Figure 9.As seen from Figure 9, now your formula of Saden recursion bayes method can identify correct topological mode, and its recognition efficiency is not affected, and has shown very strong robustness.

Can be found out by above estimated result, the efficiency of your formula of Saden recursion bayes method that the present invention proposes far above with basic recursion bayes method, and measurement noise is had to very strong robustness, the latter's robustness is very poor.

Above disclosed is only preferred embodiment of the present invention, the interest field that certainly can not limit the present invention with this, and the equivalent variations of therefore doing according to the claims in the present invention, still belongs to the scope that the present invention is contained.

Claims (1)

1. the power network topology misidentification method based on your formula recursion Bayesian Estimation of Saden, its concrete calculation procedure is as follows:

Step (1) initialization

Form all possible power network topology pattern; Make the posterior probability of all power network topology patterns to be identified all equate, even wherein " ... " representative the span of i is 1≤i≤N, be the posterior probability of i power network topology pattern, N is the total number of power network topology pattern; Recursion counter k=1 is set;

Step (2) recursion calculates to obtain correct power network topology pattern

Step (2.1) judges whether the maximum of the corresponding posterior probability of all power network topology patterns is less than threshold value, i.e. judgement whether set up, be to go to step (2.2); Otherwise go to step (3); Threshold is threshold value;

Step (2.2) input actual measurements, virtual measurement and pseudo-measurement, these measurement amounts form measurement equation and are

z＝h(x)+τ

Wherein, z ∈ R ^mfor measuring vector, comprise node inject meritorious measure, node injects idle measurement, branch road meritoriously measures, branch road is idle measurement and node voltage measure; These measurement amounts are generally got substantial amount and are measured, if substantial amount is measured disappearance, replace with pseudo-measurement, pseudo-ly measure employing in the superpose random number pattern of 10% normal distribution of the result of upper once state estimation; In measurement vector, should add 0 injection virtual measurement simultaneously; X ∈ R ⁿit is the state variable including node voltage amplitude and phase angle; H (x) is for state vector is to the Nonlinear Mapping that measures vector; τ～N (0, R) is error in measurement; for error in measurement variance matrix,

According to above measurement equation, carry out iteration by following update equation, until state estimation convergence can obtain state vector estimated value

x _k+1＝x _k+G(x _k)H ^T(x _k)R ^-1(z-h(x _k))

Wherein, for Jacobian matrix, G (x _k)=(H ^t(x _k) R ^-1h (x _k)) ^-1;

According to following formula, the independent residual error vector calculating under all power network topology patterns respectively and varivance matrix C _f,i

$e_i^{\left(k\right)}=z-h\left(\hat{x}\right)$

According to following formula, the independent varivance matrix C calculating under all power network topology patterns respectively _f,i

$C_{f,i}^{-1}=\mathrm{diag}({\mathrm{\&sigma;}}_v^2,{\mathrm{\&sigma;}}_{p_i}^2,{\mathrm{\&sigma;}}_{q_i}^2,{\mathrm{\&sigma;}}_{p_b}^2,{\mathrm{\&sigma;}}_{q_b}^2)$

C _f,iin from [H ^tr ^-1h] ^-1the respective items of diagonal element obtains, C _f,iin every can be from matrix H [H ^tr ^-1h] ^-1h ^tthe respective items of diagonal element obtains;

In the time carrying out power network topology misidentification, above carried out state estimation is the whole network state estimation;

Step (2.3) utilizes following formula to calculate the corresponding initial posterior probability of all topological structure of electric;

${\hat{p}}_i^{\left(k\right)}=\frac{\exp (-{\left(e_i^{\left(k\right)}\right)}^T{C}_{f,i}{e}_i^{\left(k\right)}/2)p_i^{(k-1)}}{\underset{j<i}{\mathrm{\&Sigma;}}\exp (-{\left(e_j^{\left(k\right)}\right)}^T{C}_{f,j}{e}_j^{\left(k\right)}/2){\hat{p}}_j^{\left(k\right)}+\underset{j\&GreaterEqual;i}{\mathrm{\&Sigma;}}\exp (-{\left(e_j^{\left(k\right)}\right)}^T{C}_{f,j}{e}_j^{\left(k\right)}/2)p_j^{(k-1)}}$

Wherein, the corresponding residual error vector of pattern i while being the k time recursion, wherein the span of i is 1≤i≤N; the initial posterior probability of pattern i while being the k time recursion, it is the posterior probability of pattern i after the k-1 time recursion; be the conditional probability of the corresponding residual error of pattern i in the k time recursion, i.e. likelihood function numerical value; the meaning of middle summation number is that the span of j is 1≤j≤N and j<i; the meaning of middle summation number is that the span of j is 1≤j≤N and j>=i;

Step (2.4) utilizes following formula to be normalized calculating to the initial posterior probability of all patterns;

$p_i^{\left(k\right)}={\hat{p}}_i^{\left(k\right)}/{\mathrm{\&Sigma;}}_{j=1}^N{\hat{p}}_j^{\left(k\right)}$

Wherein, the posterior probability of pattern i after the k time recursion;

Step (2.5) makes k=k+1, goes to step (2.1);

The pattern of step (3) posterior probability maximum is correct power network topology pattern, and identification finishes.