CN107133406A - A kind of method for fast searching on power system steady state voltage stability domain border - Google Patents
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Abstract
A kind of method for fast searching on power system steady state voltage stability domain border, including：Determine Static Voltage Stability Region Boundary general mathematical model；Determine initial static voltage stability domain boundary point；The general mathematical model on initialized static voltage stability domain border；Predict neighboring static voltage stability domain boundary pointCorrect future positionObtain and initial static voltage stability domain boundary point z0Adjacent Static Voltage Stability Region Boundary check pointBy check pointMake initial value circulation and use said process, search for next adjacent Static Voltage Stability Region Boundary point, in cyclic process, s in simultaneous gradient matrix and extra increased equation is replaced withUntil power growing direction angle beta is more than 90 ° of each appearance and once terminated less than 0 with power growing direction angle beta.The SVSR borders that the present invention is built have higher precision, and the fast search on power system steady state voltage stability domain border in two dimension and higher-dimension active injection space can be achieved.
Description
Technical field
Problem is searched for the present invention relates to a kind of Static Voltage Stability Region Boundary.More particularly to a kind of power system static electricity
The method for fast searching on voltage stabilization field border.
Background technology
Regional power grid interconnection scale constantly expands, high permeability regenerative resource large-scale grid connection, power electronic equipment are wide
General application, while strengthening power system ability to transmit electricity, improving Operation of Electric Systems flexibility and economy, also brings work(
The problems such as rate injection fluctuation, trend distribution randomness, Operation of Electric Systems risk is exacerbated, system voltage stability is brought
It is difficult to the hidden danger estimated[1]-[5].Therefore, high permeability regenerative resource is grid-connected and power system power electronics background for research
Under voltage stability be of great significance[6]-[7]。
At present, continuous tide (continuation power flow, CPF) method[8]It is still research power system static electricity
The basic skills of stability is pressed, this method core concept is to carry out pointwise iteration according to the power growing direction of determination, calculates system
The peak load nargin united under the premise of voltage stability is met, by load margin come the voltage stability of assessment system.By
In algorithm realization easily, and with higher computational accuracy, thus in practical power systems obtained extensively should
With[9]-[11].But the large-scale application of the large-scale grid connection and Demand Side Response technology with regenerative resource, regenerative resource and
The distinctive intermittent uncertainty that system power growing direction is added with randomness of Demand Side Response, relies solely on load pre-
Survey and power growing direction determined by Unit Combination calculates obtained load margin and can not truly reflect that the voltage of system is steady
It is qualitative.To the real voltage stability of assessment system, the power growing direction being likely to occur need to be directed to, repetition calls CPF to count
The load margin of calculation system, this certainly will bring heavy computation burden, be unfavorable for the efficiency of static system Voltage Stability Evaluation.Electricity
The Static Voltage Stability Region of Force system[12](static voltage stability region, SVSR) is that description determines network
Under topological structure and parameter, system has the operation area of static electric voltage stability., can accurate evaluation by constructed SVSR
Go out voltage stability of the system under uncertain power growing direction, it is to avoid conventional voltage Method of Stability Analysis is facing
System load and power supply exert oneself uncertainty when deficiency, be it is a kind of analyze and assess Power System Voltage Stability reliable side
Method[13]。
Build SVSR keys be accurately, fast search its inhibited stably, the method on calculating SVSR borders mainly includes
Fitting process[14]-[15]And approximation method[16]-[18]Two kinds.Fitting process is developed by conventional voltage Method of Stability Analysis, according to
The characteristics of SVSR borders are mainly made up of saddle node bifurcation (saddle node bifurcation, saddle node bifurcation) point, from
Ground state trend is set out, and repetition calls CPF to calculate the saddle node bifurcation point under different capacity growing direction, eventually through what is searched for
Saddle node bifurcation point constitutes SVSR borders[14]-[15].Such method lays particular emphasis on using saddle node bifurcation point to build SVSR borders,
With higher precision.But it is still CPF to search for the core calculations algorithm of saddle node bifurcation, thus when searching for whole SVSR borders
Substantial amounts of interative computation can be brought, off-line calculation amount is huge.Approximation method is no longer clicked through to the saddle node bifurcation on SVSR boundary points
Row point by point search, but speed is built to improve SVSR by the Local approximation to SVSR borders.Document [16]-[18] are by right
SVSR boundary topology characteristic research, it is found that good linear characteristic is presented in SVSR local boundaries, can be carried out using hyperplane approximately,
The part analysis expression formula for being capable of approximate expression SVSR borders is derived on this basis.Such method can according to actual needs,
A certain hyperplane is quickly generated under the premise of computational accuracy is met, with approximate substitution SVSR local boundary.The hyperplane can be retouched
The local topology characteristic on SVSR borders is stated, but there is larger conservative, it is impossible to ensures the global precision on SVSR borders, and not
Chosen with key parameter influences larger to SVSR boundary topologies structure, it is difficult to the description of analytic uniform formula and acquisition versatility knot
By.At present, SVSR boundary topology characteristics are probed into, on the basis of computational accuracy and computational efficiency is taken into account, it is still SVSR to build SVSR
The difficult point of research.
The content of the invention
The technical problems to be solved by the invention are to provide a kind of the quick of power system steady state voltage stability domain border and searched
Suo Fangfa.
The technical solution adopted in the present invention is：A kind of fast search side on power system steady state voltage stability domain border
Method, makes full use of on Static Voltage Stability Region Boundary incidence relation between adjacent saddle node bifurcation, lifts search efficiency, specific bag
Include following steps：
1) Static Voltage Stability Region Boundary general mathematical model is determined；
2) initial static voltage stability domain boundary point is determined；
3) general mathematical model on initialized static voltage stability domain border；
4) neighboring static voltage stability domain boundary point is predicted
5) future position is correctedObtain and initial static voltage stability domain boundary point z0Adjacent Static Voltage Stability Region side
Boundary's check point
6) by check pointMake initial value circulation and use step 4)~step 5) the next adjacent Static Voltage Stability Region of search
Boundary point, in cyclic process, by step 4) (3) step simultaneous gradient matrix and extra increased equation in s replace withUntil power growing direction angle beta is less than 0 and power growing direction angle beta is more than 90 ° of each appearance once, terminate.
Step 1) include：
(1) by continuous tide equation and Static Voltage Stability Region Boundary criterion simultaneous, the given power growth side of construction description
To d0The mathematical modeling of corresponding Static Voltage Stability Region Boundary point is as follows：
In formula, x is the vector of static system state variable, and g (x)=0 is power system conventional Load Flow equation；λ is is
System load margin, d0For power growing direction, L is the corresponding left eigenvector of Load Flow Jacobian Matrix zero eigenvalue, JT(x) it is
The transposed matrix of Load Flow Jacobian Matrix；
(2) determine that the general mathematical model of Static Voltage Stability Region Boundary is as follows：
In formula, η and γ are newly-increased power parameter, and power growing direction d is worked as in description respectively0In node i, the active note of j two dimensions
Enter when changing in space, the active power variable quantity of corresponding Static Voltage Stability Region Boundary point on i, j node, eiAnd ej
The vector of node i and the change of j active injections power unit, e are represented respectivelyiAnd ejLength and d0Identical vector, and respectively only
It is 1 to have ith and jth element, and remaining element is all 0；λ0d0It is operating point in power growing direction d0Under load margin.
Step 2) include：
(1) basic data, including Ground flow state variable, system topology are obtained；
(2) according to given power growing direction d0, initial static voltage stability domain border is determined by continuous power flow equation
The corresponding state variable x of point0With load margin λ0, so as to obtain initial static voltage stability domain boundary point.
Step 3) include：
(1) trend Jacobian matrix J (x at initial static voltage stability domain boundary point are determined0) the corresponding left side of zero eigenvalue
Characteristic vector L0If increasing power parameter initial value γ newly0=η0=0；
(2) the general mathematical model initial value z on formula initialized static voltage stability domain border is determined0=[x0,L0,γ0, η0]T,
In formula, x0For the corresponding state variable of initial static voltage stability domain boundary point.
Step 4) include：
(1) general mathematical model of solution Static Voltage Stability Region Boundary is as follows to the gradient matrix of each variable partial derivative：
(2) the extra increase equation below on the basis of gradient matrix, to specify some component in tangent vector
Size and Orientation,
Define power growing direction angle as follows：
In formula,ForIn correspond to newly-increased power parameter η component；E takes ± 1；When e takes+1,Point to
The side of power growing direction angle beta increase, when e takes -1 to point to the side that power growing direction angle beta reduces；J(x0) it is initial
Trend Jacobian matrix, J at Static Voltage Stability Region Boundary pointT(x) it is the transposed matrix of Load Flow Jacobian Matrix, L0For J (x0)
The corresponding left eigenvector of zero eigenvalue；
(3) simultaneous gradient matrix and extra increased equation such as following formula, determine initial static voltage stability domain boundary point
Tangent vectorSuch as following formula：
In formula, s=[0 ... 0 1], E=[0 ... 0 e]T, the two length withIt is equal；
(4) step-length σ, prediction and initial static voltage stability domain boundary point z are set0Adjacent Static Voltage Stability Region Boundary
PointSuch as following formula：
In formula,For the tangent vector after normalization.
Step 5) include：
(1) determine to pass through future positionAnd and tangent vectorVertical hyperplane, such as following formula：
Z is free variable in formula；
(2) correction equation is provided, with future positionFor initial value, using Newton Algorithm check pointIt is as follows：
A kind of method for fast searching on power system steady state voltage stability domain border of the present invention, can be achieved two dimension and higher-dimension
The fast search on power system steady state voltage stability domain border in active injection space, overcomes approximation method and is building SVSR
There is the deficiency of conservative in border, while when avoiding global precise search SVSR borders, saddle is iterated to calculate one by one using CPF methods
Save the computation burden of bifurcation point.The SVSR borders that the method for the present invention is built have higher precision, the relative Static Electro based on CPF
The computational efficiency of voltage stabilization field construction method, the structure power system steady state voltage stability domain calculating time is greatly reduced, and significantly carries
The high structure efficiency of voltage stability domain of electric power system.
Brief description of the drawings
Fig. 1 is the fast search diagram on SVSR borders；
Fig. 2 is a kind of flow chart of the method for fast searching on power system steady state voltage stability domain border of the invention；
Fig. 3 is WECC-9 measurement system diagrams；
Fig. 4 is the SVSR for the active consumption that WECC-9 system coordinates axle is load bus；
Fig. 5 is the calculation error comparison diagram on SVSR borders.
Embodiment
A kind of the quick of power system steady state voltage stability domain border of the present invention is searched with reference to embodiment and accompanying drawing
Suo Fangfa is described in detail.
The following description and drawings fully show specific embodiments of the present invention, to enable those skilled in the art to
Put into practice them.Other embodiments can include structure, logic, electric, process and other changes.Embodiment
Only represent possible change.Unless explicitly requested, otherwise single component and function are optional, and the order operated can be with
Change.The part of some embodiments and feature can be included in or replace part and the feature of other embodiments.This hair
The scope of bright embodiment includes the gamut of claims, and claims is all obtainable equivalent
Thing.Herein, these embodiments of the invention can individually or generally be represented that this is only with term " invention "
For convenience, and if in fact disclosing the invention more than one, it is not meant to automatically limit the scope of the application to appoint
What single invention or inventive concept.
The Static Voltage Stability Region of power system is the hyperspace that all saddle node bifurcation points are surrounded in system, system
It is the stable necessary condition of maintenance voltage to operate in the Static Voltage Stability Region built by border of saddle node bifurcation point, therefore
The search on SVSR borders can be equivalent to the search of saddle node bifurcation point.
As shown in Figure 1 and Figure 2, the method for fast searching on a kind of power system steady state voltage stability domain border of the invention, its
It is characterised by, makes full use of on Static Voltage Stability Region Boundary incidence relation between adjacent saddle node bifurcation, lifts search efficiency,
Specifically include following steps：
1) Static Voltage Stability Region Boundary general mathematical model is determined；Including：
(1) by continuous tide equation and Static Voltage Stability Region Boundary criterion simultaneous, the given power growth side of construction description
To d0The mathematical modeling of corresponding Static Voltage Stability Region Boundary point is as follows：
In formula, x is the vector of static system state variable, and g (x)=0 is power system conventional Load Flow equation；λ is is
System load margin, d0For power growing direction, L is the corresponding left eigenvector of Load Flow Jacobian Matrix zero eigenvalue, JT(x) it is
The transposed matrix of Load Flow Jacobian Matrix；
(2) the general mathematical model such as following formula of Static Voltage Stability Region Boundary is determined：
In formula, η and γ are newly-increased power parameter, and power growing direction d is worked as in description respectively0In node i, the active note of j two dimensions
Enter when changing in space, the active power variable quantity of corresponding Static Voltage Stability Region Boundary point on i, j node, eiAnd ej
The vector of node i and the change of j active injections power unit, e are represented respectivelyiAnd ejLength and d0Identical vector, and respectively only
It is 1 to have ith and jth element, and remaining element is all 0；λ0d0It is operating point in power growing direction d0Under load margin.
2) initial static voltage stability domain boundary point is determined；Including：
(1) basic data, including Ground flow state variable, system topology are obtained；
(2) according to given power growing direction d0, initial static voltage stability domain border is determined by continuous power flow equation
The corresponding state variable x of point0With load margin λ0, so as to obtain initial static voltage stability domain boundary point.
3) general mathematical model on initialized static voltage stability domain border；Including：
(1) trend Jacobian matrix J (x at initial static voltage stability domain boundary point are determined0) the corresponding left side of zero eigenvalue
Characteristic vector L0If increasing power parameter initial value γ newly0=η0=0；
(2) the general mathematical model initial value z on formula initialized static voltage stability domain border is determined0=[x0,L0,γ0, η0]T,
In formula, x0For the corresponding state variable of initial static voltage stability domain boundary point.
4) neighboring static voltage stability domain boundary point is predictedIncluding：
(1) general mathematical model of solution Static Voltage Stability Region Boundary is as follows to the gradient matrix of each variable partial derivative：
(2) the extra increase equation below on the basis of gradient matrix, to specify some component in tangent vector
Size and Orientation,
Define power growing direction angle as follows：
In formula, Δ η |zForIn correspond to newly-increased power parameter η component；E takes ± 1；When e takes+1, Δ z |zPoint to
The side of power growing direction angle beta increase, when e takes -1 to point to the side that power growing direction angle beta reduces；J(x0) it is initial
Trend Jacobian matrix, J at Static Voltage Stability Region Boundary pointT(x) it is the transposed matrix of Load Flow Jacobian Matrix, L0For J (x0)
The corresponding left eigenvector of zero eigenvalue；
(3) simultaneous gradient matrix and extra increased equation such as following formula, determine initial static voltage stability domain boundary point
Tangent vectorSuch as following formula：
In formula, s=[0 ... 0 1], E=[0 ... 0 e]T, the two length withIt is equal；
(4) step-length σ, prediction and initial static voltage stability domain boundary point z are set0Adjacent Static Voltage Stability Region Boundary
PointSuch as following formula：
In formula,For the tangent vector after normalization.
5) future position is correctedObtain and initial static voltage stability domain boundary point z0Adjacent Static Voltage Stability Region side
Boundary's check pointIncluding：
(1) determine to pass through future positionAnd and tangent vectorVertical hyperplane, such as following formula：
Z is free variable in formula；
(2) correction equation is provided, with future positionFor initial value, using Newton Algorithm check pointIt is as follows：
6) by check pointMake initial value circulation and use step 4)~step 5) the next adjacent Static Voltage Stability Region of search
Boundary point, in cyclic process, by step 4) (3) step simultaneous gradient matrix and extra increased equation in s replace withUntil power growing direction angle beta is less than 0 and power growing direction angle beta is more than 90 ° of each appearance once, terminate.
Using check pointSearch in next adjacent Static Voltage Stability Region Boundary point, seek check pointTangent vector
When, by step 4) s is replaced with the simultaneous gradient matrix of (3) step and extra increased equationIt is i.e. next adjacent quiet
The augmentation gradient matrix of state voltage stability domain boundary point prediction link Chinese style (6) can directly use a Static Voltage Stability Region Boundary
Point calibration link Chinese style (10) Jacobian matrix is substituted.By that analogy, until power growing direction angle beta is less than 0, terminate from initial
Static Voltage Stability Region Boundary point reduces the process that Static Voltage Stability Region Boundary point is searched in direction to β.Further, return to initial
Parameter e takes -1 in saddle node bifurcation point, E, then to β augment directions searches for Static Voltage Stability Region Boundary point by above-mentioned steps, directly
It is more than 90 ° to β.
Instantiation is given below：
This example is exemplified by searching for the node system SVSR borders of WECC3 machines 9, to verify Static Voltage Stability Region (SVSR)
The validity of border method for fast searching, the bus test system of WECC3 machines 9 is as shown in Figure 3.
Selection load bus 5 and 7 is voltage stabilization key node, in the two dimension using the active injection of node 5 and 7 as reference axis
Using the method search SVSR borders of the present invention in burden with power space.As shown in figure 4, using ground state as starting point, setting is just
Beginning power growing direction d0=[Δ S2,ΔS3,ΔS4,ΔS5,ΔS6,ΔS7,ΔS8,ΔS9]T=[0,0,0,0.9,0,1,0,
0]T, unbalanced power amount is by generator compensation, wherein Δ S where balance nodes caused by power growthiFor the multiple work(of node i
Rate, using CPF along d0Initial SNB points 9 in Fig. 4 are followed the trail of to obtain, its coordinate is (2.323,2.091).According to initial SNB correlation behaviors
Variable x0Calculate L0, initialize z0=[x0,L0,γ0, η0]T, z0Detailed numerical value is shown in Table θ in 1, table 1iAnd ViRespectively node i
Voltage phase angle and amplitude.
Table 1 is based on initial SNB using relevant parameter result of calculation during prediction-next SNB of correction search
Corresponding unit vector e is set according to selected key node 5,75、e7, select in prediction step σ=0.3, E and join
Number e=1, calculates the tangent vector Δ z of SNB points 90=[Δ x0,ΔL0,Δγ0, Δ η0]T, wherein Δ γ0=-0.920 0,
Δη0=1.000 0, to Δ z0There is Δ γ after normalization0=-0.663 5, Δ η0=0.720 3.Obtain first future positionIts state variable detailed numerical value is as shown in table 1.
WillBring formula (10) into be solved, obtain state variable at SNB points 8In detail
It the results are shown in Table 1.WithThe next SNB points of fast method continuation are carried using the present invention as initial value, can obtain SNB points 7,6,5,4,
3rd, 2,1,0, its coordinate and γ and η are as shown in table 2 in the concrete numerical value of each prediction-trimming process.
The β of table 2 reduces the SNB search results under direction
| Numbering | ηpre | γpre | ηcor | γcor | Coordinate |
| 7 | 0.4423 | -0.4130 | 0.4188 | -0.4182 | (2.7421,1.6728) |
| 6 | 0.6206 | -0.6370 | 0.6167 | 0.6421 | (2.9401,1.4489) |
| 5 | 0.8103 | -0.8705 | 0.8062 | -0.8756 | (3.1295,1.2154) |
| 4 | 0.9907 | -1.1133 | 0.9861 | -1.1182 | (3.3095,0.9728) |
| 3 | 1.1609 | -1.3649 | 1.1559 | -1.3696 | (3.4792,0.7214) |
| 2 | 1.3198 | -1.6248 | 1.3144 | -1.6292 | (3.6377,0.4618) |
| 1 | 1.4667 | -1.8922 | 1.4608 | -18963 | (3.7841,0.1947) |
| 0 | 1.6006 | -2.1663 | 1.5942 | -2.1698 | (3.9176,-0.0788) |
Its midpoint 0 is located in Fig. 4 beyond first quartile, there is β≤0, indicates empty in active injection since initial SNB points
Between reduce direction search SVSR critical points to β in first quartile and terminate.Now need to come back to initial SNB points, in active injection
In the first quartile of space SVSR critical points are searched for β augment directions.Initial SNB points information is thus read in again, and formula (10) is set
Middle parameter e is -1, and first prediction and check point are obtained through above-mentioned prediction-bearing calibrationWith(midpoint 10 of corresponding diagram 4), it is counted
Calculate gained state variable and be shown in Table 1, follow-up calculating process is ibid.When calculating to SNB points 19, there are β >=90 °, terminate to calculate, SNB
Occurrence of the coordinate and γ and η of point 10 to 19 in each prediction-trimming process is as shown in table 3.By all SNB points one by one
Connection can obtain the SVSR borders in two-dimentional active injection power space.
SNB search results under the β augment directions of table 3
Further Fig. 5 compared for building SVSR computational accuracy using the method and CPF of the present invention.In theory, SVSR sides
There is series of SN-striking B points composition on boundary, and for any one SVSR boundary points, system load flow Jacobian matrix is unusual, minimal characteristic
It is worth for 0, therefore, institute's extracting method of the present invention can be judged by minimal eigenvalue and the distance between 0 as obtained by judging actual calculate
SVSR precision is built with CPF.Obviously, in addition to SNB points 9, the precision for searching for gained SNB points using institute's extracting method of the present invention is remote
Higher than using the SNB points obtained by CPF trackings.Using the SNB points 3,5,6,7,10,11,12,13,14 obtained by CPF trackings in figure
Minimal eigenvalue with 17 is all higher than 10-3, its minimum value 0.110 × 10-3Appear in SNB points 16, maximum 6.81 × 10-3Go out
Present point SNB points 18, averaging of income minimal eigenvalue is 2.63 × 10-3；And use obtained by institute's extracting method of the present invention calculating most
Small characteristic value average value is 7.25 × 10-4, and calculate obtained by all minimal eigenvalues be respectively less than 1.00 × 10-3, compared with CPF meters
Calculate result precision higher.
Using institute's extracting method of the present invention and calculating time (the calculating platform CPU Intel on CPF search gained SVSR borders
Core i5-4210H, dominant frequency 2.9GHz, internal memory 8GB) it is respectively 13.9263s and 1.6402s.As a result show：Search for SVSR sides
In boundary during identical quantity SNB points, method of the invention takes time-consuming 1/10th of about CPF.
The method of the present invention has taken into account the precision and computation burden on SVSR borders at the same time, and high efficiency obtains high precision electro
Force system Static Voltage Stability Region Boundary.
The above embodiments are merely illustrative of the technical scheme of the present invention and are not intended to be limiting thereof, although with reference to above-described embodiment pair
The present invention is described in detail, and those of ordinary skill in the art can still enter to the embodiment of the present invention
Row modification or equivalent substitution, these any modifications or equivalent substitution without departing from spirit and scope of the invention, in application
Within pending claims of the invention.
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Claims (6)
1. the method for fast searching on a kind of power system steady state voltage stability domain border, it is characterised in that make full use of Static Electro
Incidence relation between adjacent saddle node bifurcation on voltage stabilization field border, lifts search efficiency, specifically includes following steps：
1) Static Voltage Stability Region Boundary general mathematical model is determined
2) initial static voltage stability domain boundary point is determined；
3) general mathematical model on initialized static voltage stability domain border；
4) neighboring static voltage stability domain boundary point is predicted
5) future position is correctedObtain and initial static voltage stability domain boundary point z0Adjacent Static Voltage Stability Region Boundary school
On schedule
6) by check pointMake initial value circulation and use step 4)~step 5) the next adjacent Static Voltage Stability Region Boundary of search
Point, in cyclic process, by step 4) (3) step simultaneous gradient matrix and extra increased equation in s replace with
Until power growing direction angle beta is less than 0 and power growing direction angle beta is more than 90 ° of each appearance once, terminate.
2. a kind of method for fast searching on power system steady state voltage stability domain border according to claim 1, its feature
It is, step 1) include：
(1) by continuous tide equation and Static Voltage Stability Region Boundary criterion simultaneous, the given power growing direction d of construction description0Institute
The mathematical modeling of corresponding Static Voltage Stability Region Boundary point is as follows：
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<msub>
<mi>&lambda;d</mi>
<mn>0</mn>
</msub>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>J</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>&CenterDot;</mo>
<mi>L</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>L</mi>
<mi>T</mi>
</msup>
<mo>&CenterDot;</mo>
<mi>L</mi>
<mo>-</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, x is the vector of static system state variable, and g (x)=0 is power system conventional Load Flow equation；λ is negative for system
Lotus nargin, d0For power growing direction, L is the corresponding left eigenvector of Load Flow Jacobian Matrix zero eigenvalue, JT(x) it is trend
The transposed matrix of Jacobian matrix；
(2) determine that the general mathematical model of Static Voltage Stability Region Boundary is as follows：
<mrow>
<mi>&Phi;</mi>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>&lambda;</mi>
<mn>0</mn>
</msub>
<msub>
<mi>d</mi>
<mn>0</mn>
</msub>
<mo>+</mo>
<msub>
<mi>&eta;e</mi>
<mi>i</mi>
</msub>
<mo>+</mo>
<msub>
<mi>&gamma;e</mi>
<mi>j</mi>
</msub>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>J</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>&CenterDot;</mo>
<mi>L</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>L</mi>
<mi>T</mi>
</msup>
<mo>&CenterDot;</mo>
<mi>L</mi>
<mo>-</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, η and γ are newly-increased power parameter, and power growing direction d is worked as in description respectively0In node i, the two-dimentional active injection spaces of j
During interior change, the active power variable quantity of corresponding Static Voltage Stability Region Boundary point on i, j node, eiAnd ejDifference table
Show the vector of node i and the change of j active injections power unit, eiAnd ejLength and d0Identical vector, and only have i-th respectively
It is 1 with j-th of element, remaining element is all 0；λ0d0It is operating point in power growing direction d0Under load margin.
3. a kind of method for fast searching on power system steady state voltage stability domain border according to claim 1, its feature
It is, step 2) include：
(1) basic data, including Ground flow state variable, system topology are obtained；
(2) according to given power growing direction d0, determine that initial static voltage stability domain boundary point pair should by continuous power flow equation
State variable x0With load margin λ0, so as to obtain initial static voltage stability domain boundary point.
4. a kind of method for fast searching on power system steady state voltage stability domain border according to claim 1, its feature
It is, step 3) include：
(1) trend Jacobian matrix J (x at initial static voltage stability domain boundary point are determined0) the corresponding left feature of zero eigenvalue to
Measure L0If increasing power parameter initial value γ newly0=η0=0；
(2) the general mathematical model initial value z on formula initialized static voltage stability domain border is determined0=[x0,L0,γ0,η0]T,
In formula, x0For the corresponding state variable of initial static voltage stability domain boundary point.
5. a kind of method for fast searching on power system steady state voltage stability domain border according to claim 1, its feature
It is, step 4) include：
(1) general mathematical model of solution Static Voltage Stability Region Boundary is as follows to the gradient matrix of each variable partial derivative：
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<mi>&Phi;</mi>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>z</mi>
</mrow>
</mfrac>
<msub>
<mo>|</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>J</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msub>
<mi>e</mi>
<mi>i</mi>
</msub>
</mtd>
<mtd>
<msub>
<mi>e</mi>
<mi>j</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>&lsqb;</mo>
<msup>
<mi>J</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<msub>
<mi>L</mi>
<mn>0</mn>
</msub>
<mo>&rsqb;</mo>
<msub>
<mo>|</mo>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>J</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<msubsup>
<mi>L</mi>
<mn>0</mn>
<mi>T</mi>
</msubsup>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
(2) extra increase equation below, the size to specify some component in tangent vector on the basis of gradient matrix
And direction,
<mrow>
<mi>&Delta;</mi>
<mi>&eta;</mi>
<msub>
<mo>|</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
</msub>
<mo>=</mo>
<mi>e</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
Define power growing direction angle as follows：
<mrow>
<mi>&beta;</mi>
<mo>=</mo>
<mi>arctan</mi>
<mfrac>
<mrow>
<msub>
<mi>&lambda;</mi>
<mn>0</mn>
</msub>
<msub>
<mi>&Delta;P</mi>
<mi>j</mi>
</msub>
<mo>+</mo>
<mi>&gamma;</mi>
</mrow>
<mrow>
<msub>
<mi>&lambda;</mi>
<mn>0</mn>
</msub>
<msub>
<mi>&Delta;P</mi>
<mi>i</mi>
</msub>
<mo>+</mo>
<mi>&eta;</mi>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula,ForIn correspond to newly-increased power parameter η component；E takes ± 1；When e takes+1,Point to power
The side of growing direction angle beta increase, when e takes -1 to point to the side that power growing direction angle beta reduces；J(x0) it is initial static
Trend Jacobian matrix, J at voltage stability domain boundary pointT(x) it is the transposed matrix of Load Flow Jacobian Matrix, L0For J (x0) zero spy
The corresponding left eigenvector of value indicative；
(3) simultaneous gradient matrix and extra increased equation such as following formula, determine the tangential of initial static voltage stability domain boundary point
AmountSuch as following formula：
<mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<mi>&Phi;</mi>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>z</mi>
</mrow>
</mfrac>
<msub>
<mo>|</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>s</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&CenterDot;</mo>
<mi>&Delta;</mi>
<mi>z</mi>
<msub>
<mo>|</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
</msub>
<mo>=</mo>
<mi>E</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>&Delta;</mi>
<mi>z</mi>
<msub>
<mo>|</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
</msub>
<mo>=</mo>
<msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<mi>&Phi;</mi>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>z</mi>
</mrow>
</mfrac>
<msub>
<mo>|</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>s</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mi>E</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, s=[0 ... 0 1], E=[0 ... 0 e]T, the two length withIt is equal；
(4) step-length σ, prediction and initial static voltage stability domain boundary point z are set0Adjacent Static Voltage Stability Region Boundary point
Such as following formula：
<mrow>
<msubsup>
<mi>z</mi>
<mn>1</mn>
<mrow>
<mi>p</mi>
<mi>r</mi>
<mi>e</mi>
</mrow>
</msubsup>
<mo>=</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mover>
<mi>z</mi>
<mo>&OverBar;</mo>
</mover>
<msub>
<mo>|</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
</msub>
<mo>&CenterDot;</mo>
<mi>&sigma;</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula,For the tangent vector after normalization.
6. a kind of method for fast searching on power system steady state voltage stability domain border according to claim 1, its feature
It is, step 5) include：
(1) determine to pass through future positionAnd and tangent vectorVertical hyperplane, such as following formula：
<mrow>
<mi>&Delta;</mi>
<msup>
<mover>
<mi>z</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<msub>
<mo>|</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
</msub>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>-</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>&sigma;</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
Z is free variable in formula；
(2) correction equation is provided, with future positionFor initial value, using Newton Algorithm check pointIt is as follows：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>&Phi;</mi>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>&Delta;</mi>
<msup>
<mover>
<mi>z</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<msub>
<mo>|</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
</msub>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>-</mo>
<msub>
<mi>z</mi>
<mn>0</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>&sigma;</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
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| CN107968399A (en) * | 2017-12-04 | 2018-04-27 | 国网辽宁省电力有限公司 | A kind of method of fast search Static Voltage Stability Region Boundary |
| CN108376981A (en) * | 2018-03-16 | 2018-08-07 | 东北电力大学 | Power system steady state voltage stability domain border search method containing flexible direct current |
| WO2019033820A1 (en) * | 2017-08-17 | 2019-02-21 | 东北电力大学 | Optimization model for quick boundary searching for static voltage stability region of electric power system |
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| CN107529644B (en) * | 2017-10-10 | 2020-08-21 | 东北电力大学 | Linear approximation method for static voltage stability domain boundary of power system |
| CN107529644A (en) * | 2017-10-10 | 2018-01-02 | 东北电力大学 | A kind of power system steady state voltage stability domain border linear approximation method |
| CN107947199A (en) * | 2017-12-01 | 2018-04-20 | 东北电力大学 | The searching method on the electric system thermal stability security domain border based on Optimized model |
| CN107947199B (en) * | 2017-12-01 | 2021-01-15 | 东北电力大学 | Optimization model-based method for searching thermal stability security domain boundary of electric power system |
| CN107968399A (en) * | 2017-12-04 | 2018-04-27 | 国网辽宁省电力有限公司 | A kind of method of fast search Static Voltage Stability Region Boundary |
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| CN116756198B (en) * | 2023-05-08 | 2024-10-25 | 万合智慧(北京)能源科技有限公司 | Rapid searching method for static voltage stability domain boundary of power system |
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