CN102799777A - Full characteristic spectrum tracking method for time delay power system based on optimization procedure - Google Patents

Abstract

The invention discloses a full characteristic spectrum tracking method for a time delay power system based on an optimization procedure. The method comprises the following steps of: tracking a full characteristic spectrum locus of the time delay power system according to the prediction-correction thought, gradually increasing time delay of the system from zero time delay, performing prediction by using the calculation results of the front two steps, and correcting the prediction result by solving an optimization model to realize solution of all characteristic spectrum loci of the time delay system. The invention provides the method capable of tracking the time delay change loci of complete characteristic spectrums of the time delay system for the first time, and analysis of complex dynamic behaviors of the time delay system is realized by accurately solving the full characteristic spectrum loci of the time delay system. The method can be used for researching the change rule of the characteristic value of the time delay system, and can also be used for scientifically assessing the control effect of a closed-loop controller for the time delay system.

Description

The full characteristic spectrum method for tracing of time-lag power system based on optimizing process

Technical field

The present invention relates to the full characteristic spectrum method for tracing of a kind of time-lag power system, particularly relate to the full characteristic spectrum method for tracing of a kind of time-lag power system based on optimizing process.

Background technology

At occurring in nature, system's development in future trend had both depended on current state, and also relevant with past state, this type phenomenon is called time lag.The time lag phenomenon extensively is present in each link of electric system, but traditional control signal is mainly taken from local measuring equipment, and time lag is very little, does not consider usually ^[2]But under the wide area environment, the time lag that a distant place measures link is very obvious, therefore studies time lag the influence of stability of power system is of great practical significance.

Existing method to the time lag system stability analysis can be divided three classes substantially:

(1) time-domain-simulation method ^[1-2]: its adopts numerical integration algorithm directly the time lag system dynamic equation to be carried out integration finding the solution its complete running orbit, and then judges the stable situation of system.This method can only provide the judge whether system stablizes under special scenes, and can't provide such as useful informations such as system stability degree, time lag the permitted maximum range, system features values for the operations staff ^[3]

(2) based on the theoretical method of Lyapunov ^[4,5]: it is separated the Lyapunov function through row and comes directly to judge system stability.Difference according to institute's array function can be divided into Lyapunov-Razumikhin type method ^[6]With Lyapunov-Krasovskii type method ^[7]Because it is its stable necessary condition that there is the Lyapunov function in time lag system, so these class methods can't be eliminated its result's conservative property fully.

(3) based on the analytical approach of linear time lag system stability theory ^[1,8-16]: it is extended to linear time lag system with comparatively ripe in theory linear system stability analysis theory, comes the stability state of research nonlinear systems with delay under special scenes indirectly.Because it can deeply disclose some essential laws of time lag system, and the time lag system design of Controller is had fine directive significance, therefore receive a lot of concerns in the recent period, formed many practical approaches thus.Adopting the Smith prediction device to offset the time lag influence of control loop like document [8-9], is that constant or its Changing Pattern are known but the prerequisite of its application is a time lag; Document [10-11] adopts the Rekasius conversion that the secular equation of time lag system is transformed to polynomial equation, finds the solution what surmount avoiding, and then realizes finding the solution the time lag system critical eigenvalue; Document [12] is converted into the Lambert-W function with the item that surmounts of time lag system secular equation, adopts Lambert-W tool implementation finding the solution the system core eigenwert then; Document [13-14] then is converted into the analysis to an Optimization Model with finding the solution of time lag system critical eigenvalue; Purpose is to avoid handling the item that surmounts of time lag secular equation; Document [15] is further used it for the tracking to the critical eigenvalue curve, and document [16] is then used it for the calculating of electric system time lag microvariations stable region.

As everyone knows, time lag system belongs to typical infinite dimensional system ^[17], its dynamic perfromance is complicated unusually ^[1-2,4-5]Its dynamic behaviour is in close relations with its eigenwert again; But time-domain-simulation method and Lyapunov method all can't directly be found the solution the time lag system eigenwert, and based on the existing analytical approach of linear time lag system stability theory, all can only realize asking for the time lag system critical eigenvalue without exception ^[8-16]Because said method all can not accurately be followed the tracks of the complete characteristic spectrum variation track of time lag system, also just can't disclose the many complicated dynamic behaviour of Dynamic System with Delays.

Summary of the invention

The objective of the invention is to overcome the above-mentioned deficiency of existing method, a kind of full characteristic spectrum method for tracing of the time-lag power system based on optimizing process of following the trail of the full characteristic spectrum of time-lag power system is provided.

The technical scheme that the present invention adopted: the full characteristic spectrum method for tracing of a kind of time-lag power system based on optimizing process comprises the steps:

The 1st step: to containing the power system dynamic states model of time lag link as follows

$\left\{\begin{array}{c}\stackrel{\·}{x}=f(x,y,x_{\mathrm{\τ}1},y_{\mathrm{\τ}1},x_{\mathrm{\τ}2},y_{\mathrm{\τ}2},...,x_{\mathrm{\τk}},y_{\mathrm{\τk}})\\ 0=g(x,y)\\ 0=g(x_{\mathrm{\τi}},y_{\mathrm{\τi}}),i=\mathrm{1,2},...,k\end{array}\right.$

Wherein: x ∈ R ⁿ, y ∈ R ^mBe respectively state variable and algebraically variable; (x _{τ i}, y _{τ i}) :=[x (t-τ _i), y (t-τ _i)] be time lag state variable and time lag algebraically variable; τ=[τ ₁, τ ₂..., τ _k] be the time lag vector, at system balancing point (x ₀, y ₀) locate to carry out linearization, obtain the following depression of order Linear Time-delay differential equation:

$\mathrm{\&Delta;}\stackrel{\&CenterDot;}{x}={\mathrm{<figure-callout\; id="0"\; label="A"\; filenames="HDA00001881721300022.png,HDA00001881721300031.png"\; state="\{\{state\}\}">A</figure-callout>}}_0\mathrm{\&Delta;x}+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{\mathrm{\&Delta;x}}_{\mathrm{\&tau;i}}$

Its characteristic of correspondence equation is Δ (λ)=det (λ I-A)=0

Wherein: $A=A_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{e}^{-\mathrm{\&lambda;}\&CenterDot;{\mathrm{\&tau;}}_i}$

The 2nd step: the initialization of system features value trajectory track comprises:

(2.1) initialization of variable

The set up departments number N of system eigenwert track _λ=n, order

Wherein, Be called the time lag coefficient, correspondence

Wherein

Be k eigenwert of system,

With

Be respectively

The real part of pairing right proper vector and imaginary part, () ^TBe corresponding transpose of a matrix; If follow the trail of counter initial value i=1, establish time lag coefficient initial value

If the tracking step-length initial value of k eigenwert track

(2.2) calculate the starting point of system features value trajectory track

When time lag initial value τ=0, time lag system deteriorates to following linear differential equation, and system features value number equals matrix A _BDimension:

$\mathrm{\&Delta;}\stackrel{\&CenterDot;}{x}=(A_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i)\mathrm{\&Delta;x}=A_B\&CenterDot;\mathrm{\&Delta;x}$

Find the solution A _BCharacteristic spectrum λ _B=[λ _B1, λ _B2..., λ _Bn] and corresponding right proper vector V _B=[v _B1, _B2..., v _Bn], thereby obtain eigenwert trajectory track starting point $X_k^{}$ ${}_1^{}=[{\mathrm{\&lambda;}}_{\mathrm{Bk}},{\left(v_{\mathrm{Bk}}^r\right)}^T,{\left(v_{\mathrm{Bk}}^{\mathrm{\&omega;}}\right)}^T],k=\mathrm{1,2},...,n.$

The 3rd step: with the growth of following the trail of counter i, starting from scratch increases gradually

Find the solution one by one $X_k^i=[{\mathrm{\&lambda;}}_k^i,{\left(v_{\mathrm{Ki}}^r\right)}^T,{\left(v_{\mathrm{Ki}}^{\mathrm{\&omega;}}\right)}^T],k=\mathrm{1,2},...,n,$ Comprise:

(3.1) judge if whether be the repeated root or the conjugate character value of computation of characteristic values, then directly utilize to have got and change (3.6) step after result of calculation is upgraded

; Changeing (3.2) step if not continues;

(3.2) make

and also as follows next step result to be asked predicted:

$\left\{\begin{array}{cc}{\tilde{X}}_k^{i+1}=X_k^i& i=1\\ {\tilde{X}}_k^{i+1}=\frac{(h_k^i+h_k^{i-1})X_k^i-X_k^{i-1}{h}_k^i}{h_k^{i-1}}& i>1\end{array}\right.$

(3.3) be initial value with

, find the solution following Optimization Model

min?f(λ _i)

$s.t.A^r\&CenterDot;v_i^r-A^{\mathrm{\&omega;}}\&CenterDot;v_i^{\mathrm{\&omega;}}={\mathrm{\&lambda;}}_i^r\&CenterDot;v_i^r-{\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;v_i^{\mathrm{\&omega;}}$

$A^r\&CenterDot;v_i^{\mathrm{\&omega;}}+A^{\mathrm{\&omega;}}\&CenterDot;v_i^r={\mathrm{\&lambda;}}_i^r\&CenterDot;v_i^{\mathrm{\&omega;}}+{\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;v_i^r$

${\left(v_i^r\right)}^T\&CenterDot;v_i^r+{\left(v_i^{\mathrm{\&omega;}}\right)}^T\&CenterDot;v_i^{\mathrm{\&omega;}}=1.0$

If calculate convergence, corresponding

changes (3.5) step then to get

; Otherwise, change (3.4) step and revise calculating step-length;

(3.4) judge

if; This eigenwert track calculates and stops, and changes (3.6) step; If not, press following formula correction

back and change (3.2) step retry:

$h_k^i=\max (h_k^i*\mathrm{\&beta;},h_{\min })$

Wherein: β is the correction factor less than 1.0, h _MinIt is predefined minimum step;

(3.5) judge whether to need to increase to calculate step-length, if not, change (3.6) step; If, then next step calculating step-length to be revised by following formula, (3.6) step is changeed in the back:

$h_k^{i+1}=\min (h_k^i*\mathrm{\&alpha;},h_{\max })$

Wherein: α is the correction factor greater than 1.0, h _MaxIt is predefined maximum step-length;

(3.6) judge k>=N _λIf,, change the 4th step and continue; Otherwise, make changeing the calculating that next eigenwert is continued in (3.1) behind the k=k+1;

The 4th step: according to

K=1,2 ..., N _λResult of calculation, judge whether system the vibration fork of dieing out occurs, and promptly a pair of conjugate character value becomes a real character value after meeting on the real axis, if think that in tracing algorithm after this will there be a pair of true weight root in system, at this moment N _λValue and eigenvalue calculation series are constant, but for avoiding meaningless calculating, run into repeated root and only calculate once; If not, judging whether to occur the vibration fork that is born, promptly a real character value is split into a pair of conjugate character value, if, N after this then _λValue adds 1, and eigenwert increases calculating series, i.e. a conjugate moiety simultaneously; If not, change the 5th step over to;

The 5th step: judge that

is the predefined time lag coefficient maximal value that is used for the eigenwert trajectory track; If; Calculate and finish, preserve result of calculation; Otherwise, make i=i+1 change the 3rd step and continue.

Tracing algorithm adopts prediction-correcting mode to realize the tracking to the eigenwert track, and adopts the variable step algorithm to find the solution efficient with raising: when optimizing process is not restrained, reduce the step-length retry; When continuous 3 suboptimization processes restrain, increase step-length automatically; The conjugate character value is only calculated one because of occurring in pairs.

The full characteristic spectrum method for tracing of time-lag power system based on optimizing process of the present invention; Based on the predicted correction thinking the full characteristic spectrum of time-lag power system is followed the trail of; Begin from zero time lag, increase system's time lag gradually, and utilize preceding two step results to predict; Proofread and correct predicting the outcome through finding the solution an Optimization Model then, to realize finding the solution the whole eigenwerts of time lag system.This invention has provided the method that can follow the trail of time lag system complete characterization spectrum variation track first, through accurately finding the solution the full characteristic spectrum track of time lag system, and then realizes the analysis to the complicated dynamic behaviour of time lag system.Adopt method of the present invention, not only can study the Changing Pattern of time lag system eigenwert, also can be used for science assessment time lag system closed loop controller control effect.

Description of drawings

Fig. 1 is an eigenwert tracing algorithm principle schematic;

Fig. 2 is WSCC three machines nine node system synoptic diagram;

Fig. 3 is the synoptic diagram that utilizes three types of special forks that the inventive method finds

Wherein (a) is the fork of dieing out that vibrates, and (b) is vibration birth fork, (c) is the fork that distorts that vibrates;

System features when Fig. 4 is θ=60 ° is composed hysteresis curve map at any time;

The variation track figure of the eigenwert 1 when Fig. 5 is θ=60 °;

Eigenwert 2 variation track figure when Fig. 6 is θ=60 °;

The variation track figure of the eigenwert 3,4 when Fig. 7 is θ=60 °;

The variation track figure of the eigenwert 5,6 when Fig. 8 is θ=60 °;

The variation track figure of the eigenwert 7,8 when Fig. 9 is θ=60 °;

The variation track figure of the eigenwert 9,10 when Figure 10 is θ=60 °;

Increase the system change situation with

when Figure 11 is θ=60 ° and sum up figure;

Figure 12 be θ when getting different value eigenwert 2 with the change curve of time lag;

Figure 13 is θ eigenwert 3,4 change curves with time lag when getting different value;

Wherein,<situation of change before θ＜42 ° (b) is the situation of change of θ>=42 ° (a) to be 0 °;

Embodiment

Below in conjunction with embodiment and accompanying drawing the full characteristic spectrum method for tracing of the time-lag power system based on optimizing process of the present invention is made detailed description.

1. electric system Time-Delay model

The power system dynamic states model that contains the time lag link can be expressed as following form:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}=f(x,y,x_{\mathrm{\&tau;}1},y_{\mathrm{\&tau;}1},x_{\mathrm{\&tau;}2},y_{\mathrm{\&tau;}2},...,x_{\mathrm{\&tau;k}},y_{\mathrm{\&tau;k}})\\ 0=g(x,y)\\ 0=g(x_{\mathrm{\&tau;i}},y_{\mathrm{\&tau;i}}),i=\mathrm{1,2},...,k\end{array}\right.---\left(1\right)$

Wherein: x ∈ R ⁿ, y ∈ R ^mBe respectively state variable and algebraically variable; (x _{τ i}, y _{τ i}) :=[x (t-τ _i), y (t-τ _i)] be time lag state variable and time lag algebraically variable; τ=[τ ₁, τ ₂..., τ _k] be the time lag vector, τ _i>0 is i time lag component.For describing conveniently order

$\mathrm{\&tau;}=[{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA00001881721300011.png,HDA00001881721300012.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,k_{2}k...,k_k]\&CenterDot;\left|\right|\mathrm{\&tau;}\left|\right|=[{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA00001881721300011.png,HDA00001881721300012.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA00001881721300011.png,HDA00001881721300032.png"\; state="\{\{state\}\}">k</figure-callout>}}_2,...,k_k]\&CenterDot;\widetilde{\mathrm{\&tau;}}---\left(2\right)$

Wherein,

is normaliztion constant.Further, at system balancing point (x ₀, y ₀) locate can get following incremental model to formula (1) linearization:

$\left\{\begin{array}{c}\mathrm{\&Delta;}\stackrel{\&CenterDot;}{x}={\tilde{A}}_0\mathrm{\&Delta;x}+{\tilde{B}}_0\mathrm{\&Delta;y}+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}({\tilde{A}}_i{\mathrm{\&Delta;x}}_{\mathrm{\&tau;i}}+{\tilde{B}}_i{\mathrm{\&Delta;y}}_{\mathrm{\&tau;i}})\\ 0={\tilde{C}}_0\mathrm{\&Delta;x}+{\tilde{D}}_0\mathrm{\&Delta;<figure-callout\; id="0"\; label="y"\; filenames="HDA00001881721300022.png,HDA00001881721300031.png"\; state="\{\{state\}\}">y</figure-callout>}\\ \begin{array}{c}0={\tilde{C}}_i{\mathrm{\&Delta;x}}_{\mathrm{\&tau;i}}+{\tilde{D}}_i{\mathrm{\&Delta;y}}_{\mathrm{\&tau;i}},i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,k\end{array}\end{array}\right.---\left(3\right)$

Wherein:

is when

is nonsingular; But the algebraic equation in the subtractive (3), thus the following depression of order Linear Time-delay differential equation (ODE) obtained:

$\mathrm{\&Delta;}\stackrel{\&CenterDot;}{x}=A_0\mathrm{\&Delta;x}+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{\mathrm{\&Delta;x}}_{\mathrm{\&tau;i}}---\left(4\right)$

Wherein: $A_0={\tilde{A}}_0-{\tilde{B}}_0\&CenterDot;{\tilde{D}}_0^{-1}\&CenterDot;{\tilde{C}}_0---\left(5\right)$

$A_i={\tilde{A}}_i-{\tilde{B}}_i\&CenterDot;{\tilde{D}}_i^{-1}\&CenterDot;{\tilde{C}}_i,i=\mathrm{1,2},...,k---\left(6\right)$

Formula (4) characteristic of correspondence equation is:

△(λ)＝det(λI-A)＝0 （7）

Wherein: $A=A_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{e}^{-\mathrm{\&lambda;}\&CenterDot;{\mathrm{\&tau;}}_i}---\left(8\right)$

2. ultimate principle of the present invention

Definition by eigenwert can be known, if λ i is eigenwert, i.e. a λ of time lag system formula (4) _i∈ λ is that of formula (7) separates, and then has following relation to exist:

A·v _i=λ _i·v _i (9)

||v _i||=1.0 (10)

In the formula, V _iBe and λ _iCorresponding right proper vector.Further order:

${\mathrm{\&lambda;}}_i={\mathrm{\&lambda;}}_i^r+{\mathrm{j\&lambda;}}_i^{\mathrm{\&omega;}}---\left(11\right)$

$v_i=v_i^r+{\mathrm{jv}}_i^{\mathrm{\&omega;}}---\left(12\right)$

A=A ^r+jA ^ω (13)

Wherein: A ^r, A ^ωBe respectively the real part and the imaginary part of matrix A, it is following to embody formula:

$A^r=\mathrm{Re}\left(A\right)=A_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{e}^{-{\mathrm{\&lambda;}}_i^r\&CenterDot;{\mathrm{\&tau;}}_i}\cos ({\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;{\mathrm{\&tau;}}_i)---\left(14\right)$

$A^{\mathrm{\&omega;}}=\mathrm{Im}\left(A\right)=-\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{\text{e}}^{-{\mathrm{\&lambda;}}_i^r\&CenterDot;{\mathrm{\&tau;}}_i}\sin ({\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;{\mathrm{\&tau;}}_i)---\left(15\right)$

Then can formula (9) be rewritten as corresponding real matrix operational form.Further, if λ _iBe the time lag system eigenwert, then it must separating for following Optimization Model:

min?f(λ _i) (16)

$s.t.A^r\&CenterDot;v_i^r-A^{\mathrm{\&omega;}}\&CenterDot;v_i^{\mathrm{\&omega;}}={\mathrm{\&lambda;}}_i^r\&CenterDot;v_i^r-{\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;v_i^{\mathrm{\&omega;}}---\left(17\right)$

$A^r\&CenterDot;v_i^{\mathrm{\&omega;}}+A^{\mathrm{\&omega;}}\&CenterDot;v_i^r={\mathrm{\&lambda;}}_i^r\&CenterDot;v_i^{\mathrm{\&omega;}}+{\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;v_i^r---\left(18\right)$

${\left(v_i^r\right)}^T\&CenterDot;v_i^r+{\left(v_i^{\mathrm{\&omega;}}\right)}^T\&CenterDot;v_i^{\mathrm{\&omega;}}=1.0---\left(19\right)$

Wherein, formula (17) and (18) are corresponding real part and imaginary-part operation formula of formula (9), and formula (19) is the equivalent form of value of formula (10), and the purpose that adopts above-mentioned distortion is in calculating process, to avoid occurring complex operation.F () is and λ _iRelevant scalar function.

The present invention adopts prediction-correction thinking to realize the tracking to time lag system eigenwert track; Then realize at correction link through solving-optimizing model (16)～(19); Its principle is as shown in Figure 1: tracing algorithm did not begin from

(system does not contain time lag); Can pass through the eigenwert of the direct solving system of classic method this moment, and it is made as starting point; Slowly increase

then; And, call trimming process subsequently and realize accurately finding the solution of this point by getting next point to be asked of prediction of result.

3. the full characteristic spectrum method for tracing of time-lag power system of the present invention

The 1st step: to containing the power system dynamic states model of time lag link as follows

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}=f(x,y,x_{\mathrm{\&tau;}1},y_{\mathrm{\&tau;}1},x_{\mathrm{\&tau;}2},y_{\mathrm{\&tau;}2},...,x_{\mathrm{\&tau;k}},y_{\mathrm{\&tau;k}})\\ 0=g(x,y)\\ 0=g(x_{\mathrm{\&tau;i}},y_{\mathrm{\&tau;i}}),i=\mathrm{1,2},...,k\end{array}\right.$

Wherein: x ∈ R ⁿ, y ∈ R ^mBe respectively state variable and algebraically variable; (x _{τ i}, y _{τ i}) :=[x (t-τ _i), y (t-τ _i)] be time lag state variable and time lag algebraically variable; τ=[τ ₁, τ ₂..., τ _k] be the time lag vector, at system balancing point (x ₀, y ₀) locate to carry out linearization, obtain the following depression of order Linear Time-delay differential equation:

$\mathrm{\&Delta;}\stackrel{\&CenterDot;}{x}=A_0\mathrm{\&Delta;x}+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{\mathrm{\&Delta;x}}_{\mathrm{\&tau;i}}$

Its characteristic of correspondence equation is △ (λ)=det (λ I-A)=0

Wherein: $A=A_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{e}^{-\mathrm{\&lambda;}\&CenterDot;{\mathrm{\&tau;}}_i}$

The 2nd step: the initialization of system features value trajectory track

(2.1) initialization of variable

The set up departments number N of system eigenwert track _λ=n, order Wherein,

Be called the time lag coefficient, correspondence

K=1,2 ..., n, wherein

Be k eigenwert of system,

With Be respectively

The real part of pairing right proper vector and imaginary part, () ^TBe corresponding transpose of a matrix; If follow the trail of counter initial value i=1, establish time lag coefficient initial value

K=1,2 ..., n establishes the tracking step-length initial value of k eigenwert track

(2.2) calculate the starting point of system features value trajectory track

When time lag initial value τ=0, time lag system deteriorates to following linear differential equation, and system features value number equals matrix A _BDimension:

$\mathrm{\&Delta;}\stackrel{\&CenterDot;}{x}=(A_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i)\mathrm{\&Delta;x}=A_B\&CenterDot;\mathrm{\&Delta;x}$

Find the solution A _BCharacteristic spectrum λ _B=[λ _B1, λ _B2..., λ _Bn] and corresponding right proper vector V _B=[v _B1, v _B2..., v _Bn], thereby obtain eigenwert trajectory track starting point

K=1,2 ..., n.

The 3rd step: with the growth of following the trail of counter i, starting from scratch increases gradually

Find the solution one by one $X_k^i=[{\mathrm{\&lambda;}}_k^i,{\left(v_{\mathrm{Ki}}^r\right)}^T,{\left(v_{\mathrm{Ki}}^{\mathrm{\&omega;}}\right)}^T],$ K=1,2 ..., n

Is (3.1) judging

the repeated root or the conjugate character value of computation of characteristic values? If then directly utilize to have got and change (3.6) step after result of calculation is upgraded

; Changeing (3.2) step if not continues.

(3.2) make and also as follows next step result to be asked predicted:

$\left\{\begin{array}{cc}{\tilde{X}}_k^{i+1}=X_k^i& i=1\\ {\tilde{X}}_k^{i+1}=\frac{(h_k^i+h_k^{i-1})X_k^i-X_k^{i-1}{h}_k^i}{h_k^{i-1}}& i>1\end{array}\right.$

(3.3) be initial value with

, find the solution following Optimization Model

min?f(λ _i)

$s.t.A^r\&CenterDot;v_i^r-A^{\mathrm{\&omega;}}\&CenterDot;v_i^{\mathrm{\&omega;}}={\mathrm{\&lambda;}}_i^r\&CenterDot;v_i^r-{\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;v_i^{\mathrm{\&omega;}}$

$A^r\&CenterDot;v_i^{\mathrm{\&omega;}}+A^{\mathrm{\&omega;}}\&CenterDot;v_i^r={\mathrm{\&lambda;}}_i^r\&CenterDot;v_i^{\mathrm{\&omega;}}+{\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;v_i^r$

${\left(v_i^r\right)}^T\&CenterDot;v_i^r+{\left(v_i^{\mathrm{\&omega;}}\right)}^T\&CenterDot;v_i^{\mathrm{\&omega;}}=1.0$

If calculate convergence, corresponding

changes (3.5) step then to get ; Otherwise, change (3.4) step and revise calculating step-length.

(3.4) judge if; This eigenwert track calculates and stops, and changes (3.6) step; If not, press following formula correction

back and change (3.2) step retry:

$h_k^i=\max (h_k^i*\mathrm{\&beta;},h_{\min })$

Wherein: β is the correction factor less than 1.0, h _MinIt is predefined minimum step.

(3.5) judge whether to need to increase to calculate step-length, if not, change (3.6) step; If, then next step calculating step-length to be revised by following formula, (3.6) step is changeed in the back:

$h_k^{i+1}=\min (h_k^i*\mathrm{\&alpha;},h_{\max })$

Wherein: α is the correction factor greater than 1.0, h _MaxIt is predefined maximum step-length.

(3.6) judge k>=N _λIf change the 4th step and continue; Otherwise, make changeing the calculating that next eigenwert is continued in (3.1) behind the k=k+1.

The 4th step: according to

K=1,2 ..., N _λResult of calculation, judge whether system the vibration fork (a pair of conjugate character value becomes a real character value after meeting on the real axis) of dieing out occurs, if think that in tracing algorithm after this will there be a pair of true weight root in system, at this moment N _λValue and eigenvalue calculation series are constant, but for avoiding meaningless calculating, run into repeated root and only calculate once; If not, judging whether to occur the vibration fork (a real character value is split into a pair of conjugate character value) that is born, if, N after this then _λValue adds 1, and eigenwert increases a calculating series (conjugate moiety) simultaneously; If not, change the 5th step over to.

The 5th step: judge (

is the predefined time lag coefficient maximal value that is used for the eigenwert trajectory track); If; Calculate and finish, preserve result of calculation; Otherwise, make i=i+1 change the 3rd step and continue.

Tracing algorithm adopts prediction-correcting mode to realize the tracking to the eigenwert track, in this process, for guaranteeing Algorithm Convergence and its counting yield of raising, adopts the variable step correction:

1) when optimizing process is not restrained, then reduce to calculate step-length and recomputate, see (3.4) step;

When 2) (the present invention is 3 times) optimizing process was restrained when continuously several times, tracing program increased step-length automatically, to improve counting yield, saw (3.5) step;

3) the conjugate character value is because of occurring in pairs, only needs to calculate one (like imaginary part greater than zero) and get final product, sees that (3.1) go on foot.

4. embodiment

The present invention carries out example and checking with WSCC-3 machine 9 node time-lag power systems as embodiment.

WSCC three machines nine node systems shown in Figure 3, generator 1 is thought of as infinite busbar, and generator 2,3 is dynamically described with the following five rank differential equations:

${\stackrel{\&CenterDot;}{\mathrm{\&delta;}}}_i={\mathrm{\&omega;}}_i-{\mathrm{\&omega;}}_s$

$2H\&CenterDot;{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_i=P_{\mathrm{mi}}-(E_{\mathrm{qi}}^{\&prime;}-X_{\mathrm{di}}^{\&prime;}\&CenterDot;I_{\mathrm{di}})I_{\mathrm{qi}}$

$-(E_{\mathrm{di}}^{\&prime;}+X_{\mathrm{qi}}^{\&prime;}\&CenterDot;I_{\mathrm{qi}})I_{\mathrm{di}}-D({\mathrm{\&omega;}}_i-{\mathrm{\&omega;}}_s)$

$T_{\mathrm{doi}}^{\&prime;}\&CenterDot;{\stackrel{\&CenterDot;}{E}}_{\mathrm{qi}}^{\&prime;}=-E_{\mathrm{qi}}^{\&prime;}-(X_{\mathrm{di}}-X_{\mathrm{di}}^{\&prime;})\&CenterDot;I_{\mathrm{di}}+E_{\mathrm{fdi}}$

$T_{\mathrm{qoi}}^{\&prime;}\&CenterDot;{\stackrel{\&CenterDot;}{E}}_{\mathrm{di}}^{\&prime;}=-E_{\mathrm{di}}^{\&prime;}+(X_{\mathrm{qi}}-X_{\mathrm{qi}}^{\&prime;})\&CenterDot;I_{\mathrm{qi}}$

$T_{\mathrm{Ai}}\&CenterDot;{\stackrel{\&CenterDot;}{E}}_{\mathrm{fdi}}=-E_{\mathrm{fdi}}+K_{\mathrm{Ai}}(V_{\mathrm{refi}}-V_{\mathrm{Gi}})$

I=2 wherein, 3.Adopt systematic parameter shown in the table 1, the load level of supposing the system is that 2.0p.u immobilizes, and we consider that there is time lag in the set end voltage measuring value of generator 2 and generator 3 here.

Table 1 three machines nine node system parameters

ωB	X d2	X′ d2	X q2	X′ q2	T′ d02	T′ q02	H 2
								377	0.8958	01198	0.8645	0.1969	60	0.54	64
D 2	T A2	K A2	V ref2	X d3	X′ d3	X q3	X′ q3
								0.05	0.02	50.0	1.1223	0.90	0.10	0.85	0.10
T′ d03	T′ q03	H 3	D 3	T A3	K A3	V ref3	P L5
								8.00	0.25	3.01	0.05	0.02	50.0	11.223	10
Q L5	P L7	Q L7	P L9	Q L9
								0.5	1.0	0.5	1.0	0.5

4.1 validation verification

For the correctness of checking the inventive method, at first use document [15] to find the solution this system at V to method _Ref=1.03 p.u and V _RefTime lag stability margin during=1.04 p.u; The sharp then eigenvalue spectrum that the inventive method elder generation solving system is arranged, and confirm its time lag stability margin according to the eigenwert track, two kinds of method result of calculations relatively be shown in table 2.

Table 2 is with two kinds of method solving system time lag stability margins

Be not difficult to find out that therefrom two kinds of result's unanimities that method is found the solution show that method that the present invention gives is correct.Although the resolution principle of two kinds of methods is close, original method is only calculated the critical eigenvalue with tracing system, and the inventive method is the full characteristic spectrum of solving system then, and latter's information is horn of plenty more.

4.2 three kinds of special fork forms utilizing the inventive method to find

There is abundant dynamic behaviour in time lag system as a kind of complicated dynamic system, utilizes the inventive method to carry out system features value trajectory track, has found three kinds of special fork forms (as shown in Figure 3):

1) vibration is die out, and (ODB, Fig. 3 a): at the ODB place, a pair of conjugate character value becomes a real character value to fork after meeting on the real axis.The unusual fork (SIB) of inducing of ODB and differential-algebraic equation (DAE) is similar, all is after fork occurs, and causes the system oscillation pattern to disappear.But after SIB takes place, will form two real character values by the conjugate character value, one along real axis to left movement, one moves right along real axis; And after the ODB generation, the conjugate character value only forms a real character value (also can think two repeated roots that value is identical certainly), and moves along a direction of real axis.Further can ODB be subdivided into two types of ODLB and ODRB, the former real character value behind fork along real axis to left movement, the latter then moves right along real axis behind fork, Fig. 3 a has been an example typical ODRB type fork.

2) vibration birth fork (OEB, Fig. 3 b): at the OEB place, a real character value will be split into a pair of conjugate character value.After this not only the eigenwert number of system adds 1, and new mode of oscillation (being determined by emerging conjugate character value) will appear in system.Comparison diagram 3a and Fig. 3 b are not difficult to find out that OEB can regard the inverse process of ODB as, and the former causes the eigenwert number to increase, and produces new mode of oscillation; The latter then causes the eigenwert decreased number, and follows the disappearance of mode of oscillation.

3) vibration distortion fork (OAB, Fig. 3 c): a pair of conjugate character value of system is before OAB occurs, and along with growth (left side) motion to the right of parameter, system stability increases constantly degenerate (improving) with parameter; And after fork occurred, it moved round about.Radical change has taken place at OAB point place in the dependence of system stability and parameter change.

ODB, OEB and OAB find in to the time lag system simulation study; ODB wherein and OEB fork; As the fork of the uniqueness in time lag system form, they can cause the increase and decrease of system features value number, in ODE that does not contain time lag and DAE system, can not occur; Reason is that the latter's eigenwert number needs to equate with the dimension of system dynamics parameter is strict, shows that thus the dynamic perfromance of time lag system will be more complicated.

4.3 eigenwert trajectory track example

Example to the front provides at first makes V _Ref=1.04p.u with

The eigenwert of studying time lag system in the case is the track of hysteresisization at any time.

1, starting point is set

When there is not time lag in system, find the solution its characteristic spectrum λ _BAs shown in table 3, by λ _BCorresponding with it right proper vector V _BThe common initial value that constitutes calculating.

System features spectrum when table 3 does not contain time lag

Numbering	Eigenwert	Numbering	Eigenwert
				1	-45.8187	6	-3.2323
2	-38.2955	7	-1.3932+11.6858i
				3	-12.1446+1.2822i	8	-1.3932-11.6858i
4	-12.1446-1.2822i	9	-0.1237+5.1744i
				5	-4.9756	10	-0.1237-5.1744i

2, the system features value trajectory track during θ=60 °

Get h ₀=0.1s, h _Max=h ₀, h _Min=0.01s, α=β ^-1=2, and get

Adopt tracing algorithm that the present invention gives, the system features value track during to θ=60 ° is followed the trail of, and the gained result is as shown in Figure 4, (for the conjugate character value, the real axis the first half of only drawing, same down).Therefrom can find out, adopt algorithm that the present invention gives, can follow the tracks of whole eigenvalue graph of system.Be further to disclose the complicated dynamic behaviour of time lag system, the partial feature value track of system is drawn separately like Fig. 5～shown in Figure 10.Wherein:

● the variation track of eigenwert 1

Eigenwert 1 is as shown in Figure 5 with the variation track that time lag increases; Therefrom can find out following characteristics: along with the increase of

; Eigenwert 1 at first develops left along real axis; When ; A vibration birth fork (OEB) has appearred in system; Eigenwert 1 is split into a pair of conjugate character value, and constantly motion to the right.

● the variation track of eigenwert 2

The variation track of eigenwert 2 is comparatively complicated; As shown in Figure 6; It is along with the increase of

at first develops to the right along real axis; And locate (A point among the figure at

; Near-19.5703) meet in the real character value that ODB fork back forms with eigenwert 3,4; Promptly divide a pair of conjugate character value subsequently, thereafter constantly motion to the right.The A point also is a kind of typical fork form, can regard unusual inverse process of inducing fork (SIB) as.

● the variation track of conjugate character value 3,4

Conjugate character value 3 and 4 variation track are also comparatively complicated; As shown in Figure 7; When

; ODB fork (at-13.383 places) appears in system, and it is a real character value that eigenwert 3 and 4 is melted, and moves to the left along real axis.Locate (the A point is same point among the A point among Fig. 7 and Fig. 6) at ; Be split into a pair of conjugate character value after meeting with eigenwert 2, and constantly motion to the right.During as ; This to the conjugate character value the B point (0.00445 ± j4.8384) almost with the imaginary axis tangent (having provided the enlarged drawing at this some place among Fig. 7), after this this is to conjugate character value moved beneath left again.

● the variation track of conjugate character value 5～10

The variation of eigenwert 5～10 does not cause complicated dynamic behaviour, provides explanation in the lump:

1) 5,6 two negative real character values of eigenwert; Growth along with

; Slowly move to the right along real axis, as shown in Figure 8.

2) track of eigenwert 7,8 is as shown in Figure 9; Motion is comparatively complicated: they are from initial point; Motion at first to the left; (1.775 ± j11.730) is back to right-hand rotation, and drawn a little ellipse, and the imaginary axis (D point among the figure) is located to pass in the back at

to arrive the C point.

3) eigenwert 9,10 tracks are shown in figure 10; They move at first to the right; When

, pass through the imaginary axis at the E point; Arrive the rightmost side at the F point subsequently; After this eigenwert is moved to the left; And when

, pass through the imaginary axis once more through the G point and reach its left side.

3, eigenwert is followed the trail of trifle as a result

Here we utilize Figure 11, and the main fork situation that θ=60 ° system features value increases with

is carried out brief summary:

1) when ; After eigenwert 3,4 is met; The ODB fork takes place in system, and the eigenwert number becomes 9 (2 pairs of conjugate complex eigenwerts and 5 real character values) by 10 (3 pairs of conjugate complex eigenwerts and 4 real character values).

2)

eigenwert 1 OEB occurs and is split into a pair of conjugate character value, and system features value number reverts to 10 (3 pairs of conjugate complex eigenwerts and 4 real character values) again.

3)

system features value becomes 4 pairs of conjugate complex eigenwerts and 2 real character values; Further; During as ; Eigenwert 9,10 is passed through the imaginary axis, and system the Hopf fork occurs and becomes instability; During to ; Eigenwert 9,10 is passed through the imaginary axis once more, and it is stable that system recovers microvariations again in Hopf fork back.

Other forks and dynamic behaviour

As a kind of complicated dynamic system, some other dynamic behaviours when present embodiment 3 machines 9 node systems are considered time lag.

● the OAB fork diverges and deposits cash with OEB and resembles

° beginning from θ=0 increases the corresponding track of value and computation of characteristic values 2 of θ gradually, and the gained result is shown in figure 12.When 0 ° < during θ≤2 °; Eigenwert 2 is along with the increase of

; At first move right along real axis; An OEB fork appears in system subsequently, this eigenwert is split into a pair of conjugate character value, and the conjugate character value is to left movement (illustrated among the figure θ=0 ° situation); < during θ＜42 °, this eigenwert is after the OEB fork takes place, and formed conjugate character value will move right and when 2 °.Definition by the front can know, system vibration distortion fork (OAB) occurred near θ=2 °.

Further; Behind θ >=42 °, eigenwert 2 will be met and will be split into a pair of conjugate character value immediately through the real character value of ODB fork back generation in the process that moves right with eigenwert 3,4; Its Changing Pattern has been done illustrated in detail at last joint through Fig. 6 and Fig. 7, repeats no more.

● the OAB fork diverges and deposits cash with ODB and resembles

° beginning from θ=0 increases the corresponding track of value and computation of characteristic values 3,4 of θ gradually, and the gained result shows shown in figure 13.< during θ＜42 °, eigenwert 3,4 is moved beneath left, cause system an ODB fork to occur when on real axis, meeting, and the real character value that after this forms is along real axis move to the right (Figure 13 a illustrated θ=0 °, 15 °, 30 ° situation) when 0 °.And behind θ >=42 °, after the ODB fork produced, formed real character value will be to left movement, and was split into a pair of conjugate character value after meeting with the eigenwert 2 in left side, and its process is just the same with the last situation that saves θ=60 °, repeats no more.

Be not difficult to find that near θ=42 °, an OAB has appearred in system equally, but different with Figure 12 scene by above-mentioned analysis, Figure 13 characteristic of correspondence value all is positioned on the real axis before and after OAB occurs.

The present invention has provided a kind of tracing algorithm of finding the solution the full characteristic spectrum of time-lag power system; And taken this in depth to analyze the situation of change of each eigenwert of concrete time lag system with time lag comprehensively; Find and example vibration fork (ODB), vibration fork (OEB) and vibration (OAB) the three types of special forks that diverge that distort that are born of dieing out; And ODB and OEB fork can cause the increase and decrease of system features value number; Be to exist in traditional non-time lag system, show that thus time lag system has more complicated dynamic phenomenon.

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Claims (2)

1. the full characteristic spectrum method for tracing of the time-lag power system based on optimizing process is characterized in that, comprises the steps: the 1st step: to containing the power system dynamic states model of time lag link as follows

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}=f(x,y,x_{\mathrm{\&tau;}1},y_{\mathrm{\&tau;}1},x_{\mathrm{\&tau;}2},y_{\mathrm{\&tau;}2},...,x_{\mathrm{\&tau;k}},y_{\mathrm{\&tau;k}})\\ 0=g(x,y)\\ 0=g(x_{\mathrm{\&tau;i}},y_{\mathrm{\&tau;i}}),i=\mathrm{1,2},...,k\end{array}\right.$

Wherein: x ∈ R ⁿ, y ∈ R ^mBe respectively state variable and algebraically variable; (x _{τ i}, y _{τ i}) :=[x (t-τ _i), y (t-τ _i)] be time lag state variable and time lag algebraically variable; τ=[τ ₁, τ ₂..., τ _k] be the time lag vector, at system balancing point (x ₀, y ₀) locate to carry out linearization, obtain the following depression of order Linear Time-delay differential equation:

$\mathrm{\&Delta;}\stackrel{\&CenterDot;}{x}=A_0\mathrm{\&Delta;x}+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{\mathrm{\&Delta;x}}_{\mathrm{\&tau;i}}$

Its characteristic of correspondence equation is △ (λ)=det (λ I-A)=0

Wherein: $A=A_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i{e}^{-\mathrm{\&lambda;}\&CenterDot;{\mathrm{\&tau;}}_i}$

The 2nd step: the initialization of system features value trajectory track comprises:

(2.1) initialization of variable

The set up departments number N of system eigenwert track _λ=n, order Wherein,

Be called the time lag coefficient, correspondence

K=1,2 ..., n, wherein Be k eigenwert of system,

With

Be respectively

The real part of pairing right proper vector and imaginary part, () ^TBe corresponding transpose of a matrix; If follow the trail of counter initial value i=1, establish time lag coefficient initial value

K=1,2 ..., n establishes the tracking step-length initial value of k eigenwert track

(2.2) calculate the starting point of system features value trajectory track

When time lag initial value τ=0, time lag system deteriorates to following linear differential equation, and system features value number equals matrix A _BDimension:

$\mathrm{\&Delta;}\stackrel{\&CenterDot;}{x}=(A_0+\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}{A}_i)\mathrm{\&Delta;x}=A_B\&CenterDot;\mathrm{\&Delta;x}$

Find the solution A _BCharacteristic spectrum λ _B=[λ _B1, λ _B2..., λ _Bn] and corresponding right proper vector V _B=[v _B1, v _B2..., v _Bn], thereby obtain eigenwert trajectory track starting point

K=1,2 ..., n.

The 3rd step: with the growth of following the trail of counter i;

k=1 is found the solution in the increase

gradually of starting from scratch one by one; 2; ...; N comprises:

Is (3.1) judging

the repeated root or the conjugate character value of computation of characteristic values? If then directly utilize to have got and change (3.6) step after result of calculation is upgraded

; Changeing (3.2) step if not continues;

(3.2) make

and also as follows next step result to be asked predicted:

$\left\{\begin{array}{cc}{\tilde{X}}_k^{i+1}=X_k^i& i=1\\ {\tilde{X}}_k^{i+1}=\frac{(h_k^i+h_k^{i-1})X_k^i-X_k^{i-1}{h}_k^i}{h_k^{i-1}}& i>1\end{array}\right.$

(3.3) be initial value with , find the solution following Optimization Model

min?f(λ _i)

$s.t.A^r\&CenterDot;v_i^r-A^{\mathrm{\&omega;}}\&CenterDot;v_i^{\mathrm{\&omega;}}={\mathrm{\&lambda;}}_i^r\&CenterDot;v_i^r-{\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;v_i^{\mathrm{\&omega;}}$

$A^r\&CenterDot;v_i^{\mathrm{\&omega;}}+A^{\mathrm{\&omega;}}\&CenterDot;v_i^r={\mathrm{\&lambda;}}_i^r\&CenterDot;v_i^{\mathrm{\&omega;}}+{\mathrm{\&lambda;}}_i^{\mathrm{\&omega;}}\&CenterDot;v_i^r$

${\left(v_i^r\right)}^T\&CenterDot;v_i^r+{\left(v_i^{\mathrm{\&omega;}}\right)}^T\&CenterDot;v_i^{\mathrm{\&omega;}}=1.0$

If calculate convergence, corresponding

changes (3.5) step then to get ; Otherwise, change (3.4) step and revise calculating step-length;

(3.4) judge

if; This eigenwert track calculates and stops, and changes (3.6) step; If not, press following formula correction

back and change (3.2) step retry:

$h_k^i=\max (h_k^i*\mathrm{\&beta;},h_{\min })$

Wherein: β is the correction factor less than 1.0, h _MinIt is predefined minimum step;

(3.5) judge whether to need to increase to calculate step-length, if not, change (3.6) step; If, then next step calculating step-length to be revised by following formula, (3.6) step is changeed in the back:

$h_k^{i+1}=\min (h_k^i*\mathrm{\&alpha;},h_{\max })$

Wherein: α is the correction factor greater than 1.0, h _MaxIt is predefined maximum step-length;

(3.6) judge k>=N _λIf,, change the 4th step and continue; Otherwise, make changeing the calculating that next eigenwert is continued in (3.1) behind the k=k+1;

The 4th step: according to

K=1,2 ..., N _λResult of calculation, judge whether system the vibration fork of dieing out occurs, and promptly a pair of conjugate character value becomes a real character value after meeting on the real axis, if think that in tracing algorithm after this will there be a pair of true weight root in system, at this moment N _λValue and eigenvalue calculation series are constant, but for avoiding meaningless calculating, run into repeated root and only calculate once; If not, judging whether to occur the vibration fork that is born, promptly a real character value is split into a pair of conjugate character value, if, N after this then _λValue adds 1, and eigenwert increases calculating series, i.e. a conjugate moiety simultaneously; If not, change the 5th step over to;

The 5th step: judge that

is the predefined time lag coefficient maximal value that is used for the eigenwert trajectory track; If; Calculate and finish, preserve result of calculation; Otherwise, make i=i+1 change the 3rd step and continue.

2. the full characteristic spectrum method for tracing of the time-lag power system based on optimizing process according to claim 1 is characterized in that:

Tracing algorithm adopts prediction-correcting mode to realize the tracking to the eigenwert track, and adopts the variable step algorithm to find the solution efficient with raising: when optimizing process is not restrained, reduce the step-length retry; When continuous 3 suboptimization processes restrain, increase step-length automatically; The conjugate character value is only calculated one because of occurring in pairs.

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