CN105512502B - One kind is based on the normalized weight function the least square estimation method of residual error - Google Patents
One kind is based on the normalized weight function the least square estimation method of residual error Download PDFInfo
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Abstract
The invention discloses one kind to be based on the normalized weight function the least square estimation method of residual error, belongs to dispatching automation of electric power systems field.The method of the present invention utilizes computer, pass through program, the SCADA data for any time section that input data terminal collects first, network structure and parameter information simultaneously initialize, then the bus admittance matrix of calculating network, form zero injection equality constraint equation, then voltage magnitude measurement equation is considered, injecting power measurement equation, branch power measurement equation, using the amplitude of node voltage and phase angle as state variable, calculate corresponding residual error, Jacobian matrix and weight function, final updating state variable, carry out convergence judgement, to realize the state estimation of power grid.The present invention can effectively suppress bad leverage measurement, have strong Robustness least squares and good convergence, and computational efficiency is very high, has good future in engineering applications.
Description
Technical field
The invention belongs to dispatching automation of electric power systems field, and in particular to one kind is based on the normalized weight function of residual error most
A young waiter in a wineshop or an inn multiplies method for estimating state.
Background technology
Power system state estimation is the important component of Energy Management System, is that operation power system other application is soft
The basis of part, its result directly affect intelligent analysis and the decision-making of dispatching of power netwoks.From Schweppe in 1970 first by state
Since estimation introduces electric system, domestic and foreign scholars and engineering staff have carried out state estimation a large amount of, in-depth study and reality
Trample, various method for estimating state occur during this.
At present, weighted least square method (Weighted least squares, WLS) is answered as in state estimation
With one of most commonly used method, its model is simple, is easy to solve, but does not have anti-interference, therefore robust state occurs
Estimation.Existing robust state estimation method includes exponential type object function robust state estimate (Maximum
Exponential square, MES), the weight function the least square estimation based on standardized residual inverse and index add
Weigh least square robust state estimate (Exponential function weighted least squares, EFWLS)
Deng.But these methods be in the calculation since the calculating of standardized residual can make each iteration consumption plenty of time, so as to influence
Their applications in systems in practice.
Based on this, the method for once having proposition can improve computational efficiency to a certain extent, but solution procedure is complicated, to some
Large-scale power system computation amount is too big, therefore is difficult to obtain extensive utilization.Also it is proposed that the state of index weight function is estimated
Calculating method (Exponential least absolute value, E-LAV), improves computational efficiency, but resists bad lever amount
The ability of survey is relatively low.Therefore, it is necessary to provide one kind for Power system state estimation to meet computational efficiency requirement and good
The method of estimation of robustness, to meet the needs of Practical Project utilization.
The content of the invention
The purpose of the present invention is the deficiency for standing state Estimation Study method, proposes that a kind of computational efficiency is high, restrains
Property it is good and can effectively suppress that bad regulations and parameters measures based on the normalized weight function the least square estimation method of residual error
(Residuals normalized weighted least squares,RNWLS)。
Realize the technical scheme is that：One kind is based on the normalized weight function the least square estimation side of residual error
Method, using computer, passes through program, the first SCADA data for any time section that input data terminal collects, network knot
Structure and parameter information simultaneously initialize, then the bus admittance matrix of calculating network, form zero injection equality constraint equation, then comprehensive
Close and consider voltage magnitude measurement equation, injecting power measurement equation, branch power measurement equation, with the amplitude of node voltage and phase
Angle is state variable, calculates corresponding residual error, Jacobian matrix and weight function, final updating state variable, carries out convergence and sentence
It is disconnected, to realize the state estimation of power grid.The method comprises the following steps that：
(1) basic data and initialization are inputted
1) basic data is inputted
The data acquisition of power grid and supervisor control, that is, SCADA data, network knot under any time section are inputted first
Structure and parameter information；The SCADA data of the power grid includes node voltage amplitude, node injecting power and branch power；Institute
Stating network structure and parameter information includes resistance, reactance, susceptance, rated voltage and the power reference of network element；The net
Network element includes circuit and transformer.
2) parameter initialization
M rank unit matrixs R is set-1, the unit matrix R-1Diagonal entry is all 1, and off diagonal element is all 0, m
For measurement variable number actual in state estimation；The initial value of node voltage amplitude is perunit value 1 when setting state estimation, phase
Angle is 0；The impedance of the network element and the equal reduction of susceptance are perunit value；Node injecting power and branch power also reduction
To perunit value；It is 40~60 to initialize maximum iteration Tmax；Convergence precision ε is 10-3~10-5, and during weight calculation it is small
The detection threshold value r of residual errormin=0.002, and iterations time=1 is set.
(2) calculate node admittance matrix
After the completion of (1) step, formula (1) calculate node admittance matrix Y is utilized：
In formula, the diagonal entry of bus admittance matrix is expressed as Yii, the off diagonal element expression of bus admittance matrix
For Yij, it is calculated by above formula；yijBranch impedance z between node i and node jijInverse；Symbol j ∈ i represent section
Point j and node i are connected directly, and situation when should also include when node i has ground branch j=0.
(3) zero injection equality constraint equation is formed
After the completion of (2) step, choose in measurement neither generator node is nor the node of load bus, will described in
The injecting power of node is expressed as zero injection equality constraint equation：
C (x)=0 (2)
In formula, c (x) is the measurement equation of zero injection node, its node represented injection active power and reactive power value
It can be calculated by formula (3)；X is the quantity of state of n dimensions, and n is state variable number actual in state estimation.The calculating of c (x)
Formula is：
In formula, Pi' the active power injected for node i；Qi' the reactive power injected for node i；uiFor the voltage of node i
Amplitude, ujFor the voltage magnitude of node j；θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj,
Represent node i and the phase difference of voltage of node j；Gij、BijElement between corresponding node i and j respectively in bus admittance matrix
Real and imaginary parts；N is the node number of power grid.
(4) residual error, Jacobian matrix and weight function are calculated
After the completion of (3) step, with power grid interior joint voltage magnitude, except zero-suppress injection node after node injecting power and
Branch power is measurement, calculates residual error, Jacobian matrix and the weight function of measurement in power grid.Specific steps content is as follows：
I residual error) is calculated
The parameter of each measurement in state estimation is calculated based on formula (4)：
R=z-h (x) (4)
In formula, z is the measurement of m dimensions, and m is measurement variable number actual in state estimation；X be n dimension quantity of state, n
It is state variable number actual in state estimation.H (x) is measurement equation, including the corresponding measurement equation of injecting power, circuit
The corresponding measurement equation of branch power, the corresponding measurement equation of transformer branch power and the corresponding measurement equation of node voltage；h
(x) can be calculated by formula (5)-formula (8)；R is measurement residuals.
Be removed the corresponding measurement equation of node injecting power after zero injection node based on formula (5)：
In formula, PiFor the active power of node i injection；QiFor the reactive power of node i injection；uiFor the voltage amplitude of node i
Value, ujFor the voltage magnitude of node j；θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, table
Show the phase difference of voltage of node i and node j；Gij、BijRespectively in bus admittance matrix between corresponding node i and j element reality
Portion and imaginary part.
The corresponding measurement equation of circuit branch road power is obtained based on formula (6)：
In formula, Pij、QijThe respectively active power of circuit branch road node i side, reactive power, Pji、QjiFor branch node j
Active power, the reactive power of side.uiFor the voltage magnitude of node i, ujFor the voltage magnitude of node j；θiFor the voltage of node i
Phase angle, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node i and the phase difference of voltage of node j.G is line conductance,
B is line admittance, ycFor line-to-ground susceptance.
The corresponding measurement equation of transformer branch power is obtained based on formula (7)：
In formula, Pijk、QijkThe respectively active power of transformer branch node i side, reactive power, Pjik、QjikRespectively
Active power, the reactive power of transformer branch node j sides.uiFor the voltage magnitude of node i, ujFor the voltage magnitude of node j.
θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node i and the voltage phase angle of node j
Difference.K is the non-standard no-load voltage ratio of transformer：J is standard side, no-load voltage ratio 1, and i is non-standard side, no-load voltage ratio K；bTFor transformer standard side
Susceptance.
The corresponding measurement equation of node voltage is obtained based on formula (8)：
Ui=ui (8)
In formula, Ui、uiIt is represented as the voltage magnitude of node i.
II Jacobian matrix) is calculated
Jacobian matrix H and C are formed based on formula (9)-formula (19).Wherein H is the Jacobian matrix of measurement, including
The Jacobian matrix element that injecting power, circuit branch road power, transformer branch power and node voltage amplitude are formed.C
It is the Jacobian matrix of zero injection equality constraint, the Jacobian matrix element group formed by the injecting power of zero injection node
Into.
For injecting power, its Jacobian matrix element is formed based on formula (9) and (10)：
In formula, PiFor the active power of node i injection；QiFor the reactive power of node i injection.uiFor the voltage amplitude of node i
Value, ujFor the voltage magnitude of node j.θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, table
Show the phase difference of voltage of node i and node j.Gij、BijRespectively in bus admittance matrix between corresponding node i and j element reality
Portion and imaginary part.Gii、BiiThe real and imaginary parts of element respectively in bus admittance matrix at corresponding node i on leading diagonal.
For circuit i sides branch power, its Jacobian matrix element is formed based on formula (11) and (12)：
In formula, Pij、QijActive power, reactive power for circuit branch road node i side.uiFor the voltage magnitude of node i, uj
For the voltage magnitude of node j.θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node
The phase difference of voltage of i and node j.G is line conductance, and b is line admittance, ycFor line-to-ground susceptance.
For circuit j sides branch power, its Jacobian matrix element is formed based on formula (13) and (14)：
In formula, Pji、QjiFor active power, the reactive power of circuit branch road node j sides.uiFor the voltage magnitude of node i, uj
For the voltage magnitude of node j.θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node
The phase difference of voltage of i and node j.G is line conductance, and b is line admittance, ycFor line-to-ground susceptance.
For transformer i sides branch power, its Jacobian matrix element is formed based on formula (15) and (16)：
In formula, Pijk、QijkActive power, reactive power for transformer branch node i side.uiFor the voltage amplitude of node i
Value, ujFor the voltage magnitude of node j.θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, θij=θi-θj, represent
The phase difference of voltage of node i and node j.K is the non-standard no-load voltage ratio of transformer：J is standard side, and no-load voltage ratio 1, i is non-standard side, is become
Than for K；bTFor the susceptance of transformer standard side.
For transformer j sides branch power, its Jacobian matrix element is formed based on formula (17) and (18)：
In formula, Pjik、QjikFor active power, the reactive power of transformer branch node j sides.uiFor the voltage amplitude of node i
Value, ujFor the voltage magnitude of node j.θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, θij=θi-θj, represent
The phase difference of voltage of node i and node j.K is the non-standard no-load voltage ratio of transformer：J is standard side, and no-load voltage ratio 1, i is non-standard side, is become
Than for K；bTFor the susceptance of transformer standard side.
For node voltage, its Jacobian matrix element is formed based on formula (19)：
In formula, Ui、uiIt is the voltage magnitude of node i, ujFor the voltage magnitude of node j, θiFor the voltage phase angle of node i,
θjFor the voltage phase angle of node j.
Zero injection equality constraint Jacobian matrix beWherein c (x) is the measurement equation of zero injection node,
It represents the injection active power and reactive power value of node；X is the quantity of state of n dimensions, and n is that state actual in state estimation becomes
Measure number.
III weight function) is calculated
The diagonal matrix element of weight function W is calculated based on formula (20), calculation formula is：
In formula, rminFor the detection threshold value of small residual error, 0.002, w is takeni *To measure the weight function of i,To measure i's
Fixed weight；riTo measure the residual error of i.
(5) state variable renewal and convergence judge
I) state variable updates
After the completion of (4) step, the correction amount x of state variable is calculated according to formula (21)(time), then more new state change
Amount, obtains state variable newly value x(time+1)=x(time)+Δx(time), time=time+1.
In formula, time is calculating iterations；x(time)For the time times iteration when quantity of state；W is diagonal for weight function
Battle array, its diagonal element are equal to weight function, i.e. Wii=wi *。For the Jacobian matrix of measurement, HTFor its transposition.c
(x(time)) it is that iterative value is x(time)When zero injection equality constraint,It is the Jacobean matrix of zero injection equality constraint
Battle array, CTFor its transposition.z-h(x(time)) expression iterative value is x(time)When residual error；λ(time)For the time times iteration when glug
Bright day multiplier vector.
Ii) convergence judges
As the correction amount x of state variable(time)Meet max (| Δ x(time)|) ＜ ε, then terminate to iterate to calculate, output knot
Fruit；When max (| Δ x(time)|) >=ε and iterations time >=Tmax, then stop iteration, output " does not restrain！”.
When max (| Δ x(time)|) >=ε and iterations time ＜ Tmax, iterations time is increased by 1, return (3)
Step, is iterated to calculate again.
The present invention is after adopting the above technical scheme, mainly have the following effects：
Compared with the weight coefficient assignment mode of the prior art, the present invention is using the normalized weight function of residual error is based on, repeatedly
Weight need to be only changed during generation, the influence of bad data can be eliminated automatically.Robustness is good, and can effectively press down
Make harmful effect of the bad leverage measurement to estimated result.
The present invention introduces threshold value criterion when weight function calculates, i.e., for residual error, less than the measurement of threshold value, its weight is equal
It is taken as 1, and residual error is more than the measurement of threshold value its weight and is respectively less than 1.So as to take full advantage of the effective information in measurement, keep away
Exempt to cause state estimation not restrain because the fluctuation of measurement weight is too big, therefore greatly improved bad data and error in measurement
Convergence and numerical stability when coexisting.
The present invention, directly using the residual error of measurement as variable, avoids the meter of standardized residual when weight function calculates
Calculate, therefore substantially increase calculating speed, save and calculate the time, be extremely suitable for the application of engineering reality.
The additional aspect and advantage of the present invention will be set forth in part in the description, and will partly become from the following description
Obtain substantially, or recognized by the practice of the present invention.
Brief description of the drawings
Fig. 1 is the flow diagram based on the normalized weight function the least square estimation method of residual error of the present invention；
Fig. 2 is the network parameter of 3 node systems and measures distribution；
Fig. 3 is the wiring diagram of IEEE-9 node systems.
Embodiment
The invention will be further described with reference to the accompanying drawings and examples, but should not be construed the above-mentioned theme of the present invention
Scope is only limitted to following embodiments.Without departing from the idea case in the present invention described above, known according to ordinary skill
Knowledge and customary means, make various replacements and change, should all include within the scope of the present invention.
As indicated with 1, a kind of comprising the following steps that based on the normalized weight function the least square estimation method of residual error：
(1) basic data and initialization are inputted
1) basic data is inputted
SCADA data, network structure and the parameter information of power grid under any time section are inputted first.That is input power grid exists
Node voltage amplitude, node injecting power and branch power under any time section, network structure and parameter information are net
Resistance, reactance and the susceptance parameter and rated voltage, power reference of network element (including circuit, transformer).
2) parameter initialization
According to " Power system control and stability " one written by Anderson P M and Fouad A A
Normal data of the text on IEEE-9 node systems, input network structure, load data and relevant parameter information.With Load flow calculation
Gained node voltage amplitude, node injecting power, branch power are measurement, it is assumed that without bad data in measurement, and normal amount
Survey and survey error with a certain amount, which meets normal distribution.M rank unit matrixs R is set-1(i.e. m ranks unit matrix R-1
Diagonal entry be all 1,0) off diagonal element is all, m is actual measurement variable number in state estimation, is 57；Put
The initial value of node voltage amplitude is perunit value 1 during state estimation, and phase angle is 0；The impedance of network element, the equal reduction of susceptance are
Perunit value；Also reduction to perunit value, reference power is set to 100MVA for node injecting power and branch power；And weight calculation
When small residual error detection threshold value rmin=0.002, and iterations time=1 is set.For initialization maximum iteration and
The selection of convergence precision uses, and the present embodiment preferred Tmax=40, ε=10-4。
(2) calculate node admittance matrix
After the completion of (1) step, the bus admittance matrix of the network is calculated, calculation formula is the formula (1) in technical solution.
According to the network structure and parameter information of IEEE-9 node systems, according to the formula (1) in technical solution, calculate
Bus admittance matrix Y to the network is：
(3) zero injection equality constraint equation is formed
After the completion of (2) step, neither generator node is nor the node of load bus, it is injected in selection measurement
Power injects equality constraint equation as zero, according to the formula (2) in technical solution, is illustrated with the result of the 1st iterative calculation,
Zero injecting power c (x), which is calculated, is：
(4) residual error, Jacobian matrix and weight function are calculated
After the completion of (3) step, with power grid interior joint voltage magnitude, except zero-suppress injection node after node injecting power, with
And branch power is measurement, residual error, Jacobian matrix and the weight function of measurement in power grid are calculated, calculation formula is technical side
Formula (4)-formula (20) in case.
I residual error) is calculated
According to the definition of residual error and measurement equation, according to the formula (4) in technical solution-formula (8), with the 1st iteration
The result citing of calculating, the residual error r that measurement is calculated are：
II Jacobian matrix) is calculated
According to the definition of Jacobian matrix element, Jacobean matrix is calculated according to the formula (9) in technical solution-formula (19)
Battle array H and C, is illustrated, Jacobian matrix H and C, which is calculated, is with the result of the 1st iterative calculation：
III weight function) is calculated
According to the definition of weight function, according to the formula (20) in technical solution, illustrated with the result of the 1st iterative calculation,
Weight function W, which is calculated, is：
(5) state variable renewal and convergence judge
I) state variable updates
After the completion of (4) step, the correction amount x of state variable is calculated according to the formula (21) in technical solution(time), so
After update state variable, obtain state variable and be newly worth, i.e.,：x(time+1)=x(time)+Δx(time), time=time+1.
Illustrated with the result of the 1st iterative calculation, by the formula (21) in technical solution, repairing for state variable is calculated
Positive quantity Δ x(1)For：
Ii) convergence judges
As the correction amount x of state variable(time)Meet max (| Δ x(time)|) ＜ ε, then terminate to iterate to calculate, output knot
Fruit；When max (| Δ x(time)|) >=ε and iterations time >=Tmax, then stop iteration, output " does not restrain！”.When max (| Δ
x(time)|) >=ε and iterations time ＜ Tmax, iterations time is increased by 1, return to (3) step, carry out iteration meter again
Calculate.
Illustrated with the result of the 1st iterative calculation, at this time, ε=10-4, Tmax=40, max (| Δ x(1)|)=0.16419
＞ ε, time=1 ＜ Tmax.Judged according to convergence, carry out following steps：Time=time+1=2, returns to (3) step, weight
Newly it is iterated calculating.
According to the step of above, meet the condition of convergence after iteration 3 times, at this time max (| Δ x(3)|)=7.4934E-05＜ ε,
The estimated result of state variable is as shown in the table：
1 IEEE-9 node system state variable estimated results of table
Experiment effect comparative analysis
The advantages of to make those skilled in the art more fully understand the present invention and understanding the present invention compared with the prior art, Shen
Ask someone further to be explained in conjunction with specific embodiments.
(1) comparison of robustness
Inventor by the present invention RNWLS compared with other state estimators, to test the Robustness least squares of RNWLS.
1) 3 node system
By taking 3 node systems shown in 2 as an example, No. 1 measurement is leverage measurement, and is arranged to bad data, other measuring values take
True value (Load flow calculation value), wherein P, Q represent active and reactive power respectively.With four kinds of state estimators (RNWLS, EFWLS,
E-LAV, WLS) simulation analysis are carried out to above-mentioned example, it is as a result as follows：
The comparison of each method estimated result in 23 node system of table
Wherein, the data of boldface letter mark are measurement bad data.By above-mentioned result of calculation as it can be seen that handling bad lever amount
In the ability of survey, the result of the method for the present invention RNWLS and EFWLS estimation illustrates that it can estimate exactly closest to trend true value
The time of day of system is counted out, and No. 1 bad leverage measurement can be identified and exclude automatically；And the wattful power of E-LAV estimations
Rate is closer to trend true value, and reactive power then differs larger；WLS estimated results differ maximum with trend true value, show that it is anti-not
The ability of good leverage measurement is limited, is unable to estimate out the time of day of system.
2) IEEE-118 node systems
For robustness of the analysis the method for the present invention RNWLS in larger system, using IEEE-118 node systems as
Example, respectively to bad data ratio for 0%~10% totally 11 kinds of situations test.Wherein normal measure is surveyed with a certain amount
Error, and the error meets normal distribution；Bad data is corresponding proportion voltage magnitude, active power, idle work(in trend true value
The opposite number of rate value (wherein, active power, reactive power set bad data at maximum).With four kinds of state estimators
(RNWLS, EFWLS, E-LAV, WLS) carries out above-mentioned example simulation analysis, as a result as follows：
3 IEEE-118 node system difference method of estimation voltage magnitude application conditions of table
Wherein, Δ UmaxWith Δ UmeanThe maximum estimated error and averaged power spectrum error of node voltage amplitude are represented respectively, are examined
It is 0 to consider as the voltage phase angle of the electric network swim information interior joint acquired in SCADA, thus only to the amplitude of node voltage into
Row simulation analysis.By above-mentioned result of calculation as it can be seen that during bad data is from 0%~10% change, E-LAV and WLS institutes
It is larger to obtain resultant error；And the estimated result of RNWLS and EFWLS, its error show in larger system all close to 0,
Both methods of estimation can exclude bad data, obtain accurate estimated result, and robustness is good.
(2) comparison of computational efficiency
Inventor is in order to carry out efficiency comparison, by taking IEEE-118 node systems as an example, it is assumed that without bad data in measurement, and
Normal measure surveys error with a certain amount, which meets normal distribution.With four kinds of state estimators (RNWLS, EFWLS,
E-LAV, WLS) simulation analysis are carried out to above-mentioned example, obtain averagely each iteration time and total evaluation time is as shown in the table：
The computational efficiency of each method compares in 4 IEEE-118 node systems of table
By above-mentioned result of calculation as it can be seen that the method for the present invention RNWLS and EFWLS estimation convergence are more preferable, but for average every
For secondary iteration time, WLS estimations are most short, but it does not have Robustness least squares；EFWLS estimations are most long；RNWLS estimations are estimated with E-LAV
Meter is placed in the middle.
In conclusion RNWLS estimations proposed by the invention are similar to EFWLS estimations, robustness on robustness
Well, and harmful effect of the bad leverage measurement to result can effectively be suppressed；And it is similar to E-LAV estimations in computational efficiency,
Computational efficiency is high, and convergence is good.So as to which active set has suffered standing state method of estimation on robustness and computational efficiency
Advantage, is extremely suitable for the application of Practical Project.
Claims (1)
1. one kind is based on the normalized weight function the least square estimation method of residual error, using computer, by program, realize
The state estimation of power grid, it is characterised in that：The specific steps of the method include herein below；
(1) basic data and initialization are inputted
1) basic data is inputted
Input first data acquisition and the supervisor control, that is, SCADA data of power grid under any time section, network structure and
Parameter information；The SCADA data of the power grid includes node voltage amplitude, node injecting power and branch power；The net
Network structure and parameter information include resistance, reactance, susceptance, rated voltage and the power reference of network element；The network element
Part includes circuit and transformer；
2) parameter initialization
M rank unit matrixs R is set-1, the unit matrix R-1Diagonal entry is all 1, and it is state that off diagonal element, which is all 0, m,
Actual measurement variable number in estimation；The initial value of node voltage amplitude is perunit value 1 when setting state estimation, and phase angle is
0；The impedance of the network element and the equal reduction of susceptance are perunit value；Node injecting power and branch power also reduction to perunit
Value；It is 40~60 to initialize maximum iteration Tmax；Convergence precision ε is 10-3~10-5, and small residual error during weight calculation
Detect threshold value rmin=0.002, and iterations time=1 is set；
(2) calculate node admittance matrix
After the completion of (1) step, formula (1) calculate node admittance matrix Y is utilized；
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In formula, the diagonal entry of bus admittance matrix is expressed as Yii, the off diagonal element of bus admittance matrix is expressed as Yij,
It is calculated by above formula；yijBranch impedance z between node i and node jijInverse；Symbol j ∈ i represent node j and
Node i is connected directly, and situation when should also include when node i has ground branch j=0；
(3) zero injection equality constraint equation is formed
After the completion of (2) step, choose in measurement neither generator node is nor the node of load bus, by the node
Injecting power as zero injection equality constraint equation, be expressed as；
C (x)=0 (2)
In formula, c (x) is the measurement equation of zero injection node, its node represented injection active power and reactive power value can be by
Formula (3) is calculated；X is the quantity of state of n dimensions, and n is state variable number actual in state estimation；The calculation formula of c (x)
For；
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>P</mi>
<mi>i</mi>
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<mo>=</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
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<mi>G</mi>
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<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
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</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>B</mi>
<mrow>
<mi>i</mi>
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<mi>sin&theta;</mi>
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</mtr>
<mtr>
<mtd>
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<mrow>
<mo>(</mo>
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<mi>G</mi>
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<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>B</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
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<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, P 'iFor the active power of node i injection；Q′iFor the reactive power of node i injection；uiFor the voltage amplitude of node i
Value, ujFor the voltage magnitude of node j；θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, table
Show the phase difference of voltage of node i and node j；Gij、BijRespectively in bus admittance matrix between corresponding node i and j element reality
Portion and imaginary part；N is the node number of power grid；
(4) residual error, Jacobian matrix and weight function are calculated
After the completion of (3) step, with power grid interior joint voltage magnitude, except the node injecting power and branch after the injection node that zero-suppresses
Power is measurement, calculates residual error, Jacobian matrix and the weight function of measurement in power grid；Specific steps content is as follows；
I residual error) is calculated
The parameter of each measurement in state estimation is calculated based on formula (4)；
R=z-h (x) (4)
In formula, z is the measurement of m dimensions, and m is measurement variable number actual in state estimation；X is the quantity of state of n dimensions, and n is shape
Actual state variable number in state estimation；H (x) is measurement equation, including the corresponding measurement equation of injecting power, circuit branch road
The corresponding measurement equation of power, the corresponding measurement equation of transformer branch power and the corresponding measurement equation of node voltage；h(x)
It can be calculated by formula (5)-formula (8)；R is measurement residuals；
Be removed the corresponding measurement equation of node injecting power after zero injection node based on formula (5)；
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<mtable>
<mtr>
<mtd>
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<msub>
<mi>P</mi>
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<mi>u</mi>
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<mi>Q</mi>
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<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
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<mo>-</mo>
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<mi>B</mi>
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<mi>i</mi>
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<mi>i</mi>
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</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
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<mo>(</mo>
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In formula, PiFor the active power of node i injection；QiFor the reactive power of node i injection；uiFor the voltage magnitude of node i, uj
For the voltage magnitude of node j；θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node
The phase difference of voltage of i and node j；Gij、BijRespectively in bus admittance matrix between corresponding node i and j element real part and void
Portion；
The corresponding measurement equation of circuit branch road power is obtained based on formula (6)；
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>P</mi>
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<mi>i</mi>
<mi>j</mi>
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</msub>
<mo>=</mo>
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<mi>u</mi>
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<mn>2</mn>
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<mi>g</mi>
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<mi>u</mi>
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<mi>u</mi>
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<mi>g</mi>
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<mi>cos&theta;</mi>
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<mi>sin&theta;</mi>
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</msup>
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</msub>
<mrow>
<mo>(</mo>
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<mi>g</mi>
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<mi>b</mi>
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<mi>sin&theta;</mi>
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<mi>y</mi>
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<mi>u</mi>
<mi>i</mi>
</msub>
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<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
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<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
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</mrow>
</msub>
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</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
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</mrow>
</mrow>
In formula, Pij、QijThe respectively active power of circuit branch road node i side, reactive power, Pji、QjiRespectively branch node j
Active power, the reactive power of side；uiFor the voltage magnitude of node i, ujFor the voltage magnitude of node j；θiFor the voltage of node i
Phase angle, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node i and the phase difference of voltage of node j；G is line conductance,
B is line admittance, ycFor line-to-ground susceptance；
The corresponding measurement equation of transformer branch power is obtained based on formula (7)；
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>P</mi>
<mrow>
<mi>j</mi>
<mi>k</mi>
</mrow>
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<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
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<mi>u</mi>
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</msub>
<msub>
<mi>sin&theta;</mi>
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<msub>
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<mi>u</mi>
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</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
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</mrow>
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</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>P</mi>
<mrow>
<mi>j</mi>
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</msub>
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<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
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<msub>
<mi>b</mi>
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</msub>
<msup>
<msub>
<mi>u</mi>
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</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Pijk、QijkThe respectively active power of transformer branch node i side, reactive power, Pjik、QjikRespectively transformation
Active power, the reactive power of device branch node j sides；uiFor the voltage magnitude of node i, ujFor the voltage magnitude of node j；θiFor
The voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node i and the phase difference of voltage of node j；K
For the non-standard no-load voltage ratio of transformer：J is standard side, no-load voltage ratio 1, and i is non-standard side, no-load voltage ratio K；bTFor the electricity of transformer standard side
Receive；
The corresponding measurement equation of node voltage is obtained based on formula (8)；
Ui=ui (8)
In formula, Ui、uiIt is represented as the voltage magnitude of node i；
II Jacobian matrix) is calculated
Jacobian matrix H and C are formed based on formula (9)-formula (19)；Wherein H is the Jacobian matrix of measurement, including is injected
The Jacobian matrix element that power, circuit branch road power, transformer branch power and node voltage amplitude are formed；C is zero
The Jacobian matrix of equality constraint is injected, the Jacobian matrix element formed by the injecting power of zero injection node forms；
For injecting power, its Jacobian matrix element is formed based on formula (9) and (10)；
<mrow>
<mo>{</mo>
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
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<mrow>
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<msub>
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<mfrac>
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<msub>
<mi>u</mi>
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<mrow>
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<msub>
<mi>G</mi>
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<msub>
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</mrow>
</mrow>
</mtd>
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<mtr>
<mtd>
<mrow>
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<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
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<mo>=</mo>
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<msub>
<mi>B</mi>
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<mi>i</mi>
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<mi>Q</mi>
<mi>i</mi>
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</mtr>
<mtr>
<mtd>
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<mrow>
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<msub>
<mi>P</mi>
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<msub>
<mi>u</mi>
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<mfrac>
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<mo>&part;</mo>
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<mi>P</mi>
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<mrow>
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<mi>B</mi>
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<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mfrac>
<mrow>
<mo>(</mo>
<mo>-</mo>
<msub>
<mi>B</mi>
<mrow>
<mi>i</mi>
<mi>i</mi>
</mrow>
</msub>
<msup>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msub>
<mi>Q</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>G</mi>
<mrow>
<mi>i</mi>
<mi>i</mi>
</mrow>
</msub>
<msup>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msub>
<mi>P</mi>
<mi>i</mi>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>G</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>B</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>G</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>B</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, PiFor the active power of node i injection；QiFor the reactive power of node i injection；uiFor the voltage magnitude of node i, uj
For the voltage magnitude of node j；θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node
The phase difference of voltage of i and node j；Gij、BijRespectively in bus admittance matrix between corresponding node i and j element real part and void
Portion；Gii、BiiThe real and imaginary parts of element respectively in bus admittance matrix at corresponding node i on leading diagonal；
For circuit i sides branch power, its Jacobian matrix element is formed based on formula (11) and (12)；
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<mi>g</mi>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mn>2</mn>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>b</mi>
<mo>+</mo>
<msub>
<mi>y</mi>
<mi>c</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Pij、QijThe respectively active power of circuit branch road node i side, reactive power；uiFor the voltage magnitude of node i, uj
For the voltage magnitude of node j；θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node
The phase difference of voltage of i and node j；G is line conductance, and b is line admittance, ycFor line-to-ground susceptance；
For circuit j sides branch power, its Jacobian matrix element is formed based on formula (13) and (14)；
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mi>g</mi>
<mo>+</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mn>2</mn>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>b</mi>
<mo>+</mo>
<msub>
<mi>y</mi>
<mi>c</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>g</mi>
<mi> </mi>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>-</mo>
<mi>b</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Pji、QjiRespectively active power, the reactive power of circuit branch road node j sides；uiFor the voltage magnitude of node i, uj
For the voltage magnitude of node j；θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, and θij=θi-θj, represent node
The phase difference of voltage of i and node j；G is line conductance, and b is line admittance, ycFor line-to-ground susceptance；
For transformer i sides branch power, its Jacobian matrix element is formed based on formula (15) and (16)；
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>2</mn>
<msup>
<mi>K</mi>
<mn>2</mn>
</msup>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>&theta;</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>16</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Pijk、QijkThe respectively active power of transformer branch node i side, reactive power；uiFor the voltage amplitude of node i
Value, ujFor the voltage magnitude of node j；θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, θij=θi-θj, represent
The phase difference of voltage of node i and node j；K is the non-standard no-load voltage ratio of transformer：J is standard side, and no-load voltage ratio 1, i is non-standard side, is become
Than for K；bTFor the susceptance of transformer standard side；
For transformer j sides branch power, its Jacobian matrix element is formed based on formula (17) and (18)；
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>P</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>17</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mn>2</mn>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>cos&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>j</mi>
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>K</mi>
</mfrac>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<msub>
<mi>b</mi>
<mi>T</mi>
</msub>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>18</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Pjik、QjikRespectively active power, the reactive power of transformer branch node j sides；uiFor the voltage amplitude of node i
Value, ujFor the voltage magnitude of node j；θiFor the voltage phase angle of node i, θjFor the voltage phase angle of node j, θij=θi-θj, represent
The phase difference of voltage of node i and node j；K is the non-standard no-load voltage ratio of transformer, and j is standard side, and no-load voltage ratio 1, i is non-standard side, is become
Than for K；bTFor the susceptance of transformer standard side；
For node voltage, its Jacobian matrix element is formed based on formula (19)；
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>U</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>U</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>i</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>U</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msub>
<mi>U</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>&theta;</mi>
<mi>j</mi>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>19</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Ui、uiIt is the voltage magnitude of node i, ujFor the voltage magnitude of node j, θiFor the voltage phase angle of node i, θjFor
The voltage phase angle of node j；
Zero injection equality constraint Jacobian matrix beWherein c (x) is the measurement equation of zero injection node, its table
Show the injection active power and reactive power value of node；X is the quantity of state of n dimensions, and n is state variable actual in state estimation
Number；
III weight function) is calculated
The diagonal matrix element of weight function W is calculated based on formula (20), calculation formula is：
<mrow>
<msup>
<msub>
<mi>w</mi>
<mi>i</mi>
</msub>
<mo>*</mo>
</msup>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>|</mo>
<msub>
<mi>r</mi>
<mi>i</mi>
</msub>
<mo>|</mo>
<mo>&le;</mo>
<msub>
<mi>r</mi>
<mi>min</mi>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<msub>
<mi>R</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mfrac>
<mrow>
<mo>|</mo>
<mn>1</mn>
<mo>/</mo>
<msub>
<mi>r</mi>
<mi>i</mi>
</msub>
<mo>|</mo>
</mrow>
<mrow>
<mo>|</mo>
<mn>1</mn>
<mo>/</mo>
<msub>
<mi>r</mi>
<mi>min</mi>
</msub>
<mo>|</mo>
</mrow>
</mfrac>
<mo>=</mo>
<msup>
<msub>
<mi>R</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mfrac>
<msub>
<mi>r</mi>
<mi>min</mi>
</msub>
<mrow>
<mo>|</mo>
<msub>
<mi>r</mi>
<mi>i</mi>
</msub>
<mo>|</mo>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>|</mo>
<msub>
<mi>r</mi>
<mi>i</mi>
</msub>
<mo>|</mo>
<mo>></mo>
<msub>
<mi>r</mi>
<mi>min</mi>
</msub>
</mrow>
</mtd>
</mtr>
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</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>20</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, rminFor the detection threshold value of small residual error, 0.002 is taken；wi *To measure the weight function of i；To measure the fixed power of i
Weight；riTo measure the residual error of i；
(5) state variable renewal and convergence judge
I) state variable updates
After the completion of (4) step, the correction amount x of state variable is calculated according to formula (21)(time), state variable is then updated, is obtained
To the new value x of state variable(time+1)=x(time)+Δx(time), time=time+1；
<mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<mi>&Delta;x</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mi>i</mi>
<mi>m</mi>
<mi>e</mi>
<mo>)</mo>
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</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<mi>&lambda;</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mi>i</mi>
<mi>m</mi>
<mi>e</mi>
<mo>)</mo>
</mrow>
</msup>
</mtd>
</mtr>
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</mfenced>
<mo>=</mo>
<msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<mi>H</mi>
<mi>T</mi>
</msup>
<mi>W</mi>
<mi>H</mi>
</mrow>
</mtd>
<mtd>
<msup>
<mi>C</mi>
<mi>T</mi>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>C</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mfenced open = "[" close = "]">
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<mtd>
<msup>
<mi>H</mi>
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<mo>-</mo>
<mi>h</mi>
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<mtd>
<mo>-</mo>
<mi>c</mi>
<mo>(</mo>
<msup>
<mi>x</mi>
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<mo>(</mo>
<mi>t</mi>
<mi>i</mi>
<mi>m</mi>
<mi>e</mi>
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</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>21</mn>
<mo>)</mo>
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In formula, time is calculating iterations；x(time)For the time times iteration when quantity of state；W is weight function diagonal matrix, its
Diagonal element is equal to weight function, i.e. Wii=wi *；For the Jacobian matrix of measurement, HTFor its transposition；c(x(time)) it is that iterative value is x(time)When zero injection equality constraint,It is the Jacobian matrix of zero injection equality constraint,
CTFor its transposition；z-h(x(time)) expression iterative value is x(time)When residual error；λ(time)For the time times iteration when glug it is bright
Day multiplier vector；
Ii) convergence judges
As the correction amount x of state variable(time)Meet max (| Δ x(time)|) ＜ ε, then terminate to iterate to calculate, export result；When
max(|Δx(time)|) >=ε and iterations time >=Tmax, then stop iteration, output " does not restrain！”；
When max (| Δ x(time)|) >=ε and iterations time ＜ Tmax, iterations time is increased by 1, return to (3) step,
Again iterated to calculate.
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| CN107016489A (en) * | 2017-03-09 | 2017-08-04 | 中国电力科学研究院 | A kind of electric power system robust state estimation method and device |
| CN108460547A (en) * | 2018-05-16 | 2018-08-28 | 山东大学 | Cogeneration system combined heat and power method for estimating state and system |
| EP3579366A1 (en) * | 2018-06-08 | 2019-12-11 | Siemens Aktiengesellschaft | Method and apparatus for control of an electric grid |
| CN109255171B (en) * | 2018-08-29 | 2023-09-05 | 深圳十沣科技有限公司 | Method for automatically judging convergence of numerical simulation calculation |
| CN110943445A (en) * | 2018-09-21 | 2020-03-31 | 中国电力科学研究院有限公司 | Robust state estimation method and device for flexible direct current and alternating current hybrid power grid |
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