CN104505830A - Time-lag power system stability analysis method and device - Google Patents

Abstract

Provided are a time-lag power system stability analysis method and device. The time-lag power system stability analysis method comprises the steps of (A) acquiring time-lag system network structure parameters and electric generator frequency and a power angle in a system; (B) utilizing acquired data to establish an accident chain, combining the accident chain and Markov process to establish a time-lag Markov jump system state equation; (C) respectively establishing a time-lag stability criterion for improving a free-weighting matrix based on a Markov process transfer rate matrix and a Newton-Leibnitz's formula according to the time-lag Markov jump system state equation, conducting equivalence transformation on the time-lag stability criterion on the basis, utilizing a generalized eigenvalue method to solve a system time-lag stability upper limit; (D) outputting the system time-lag stability upper limit. By means of the time-lag power system stability analysis method and device, the time-lag stability upper limit can be effectively solved, and system stability is improved.

Description

Time-lag power system stability analytical method and device

Technical field

The present invention relates to power system analysis and control technology field, particularly relate to the stability analysis technology of time-lag power system.

Background technology

In engineering, many dynamical systems can be described by the differential equation of state variable Temporal Evolution.The phenomenon that between the state variable of wherein quite a few dynamical system, life period is delayed, namely the evolution trend of system not only depends on the current state of system, also depend on over the state in a certain moment or some moment, this kind of dynamical system is called as Dynamic System with Delays.In recent years, Dynamic System with Delays has become the primary study object in many fields, particularly in field of power, has caused pay close attention to widely the research of time-lag power system characteristic.On the other hand, stability is the most basic quality of electric power system, and for time-lag power system, its characteristic equation is the transcendental equation containing exponential function, has infinite multiple in principle, and therefore the distribution situation of its characteristic root is quite complicated.

The existence of time lag makes Power System Stability Analysis and control become more sophisticated, has become the root of system instability and degradation.Therefore, in the urgent need to analysing in depth the time lag stability of system, to improve the stability of system.

In time lag stability analysis, existing many theoretical results, are mainly divided into two large classes: 1) frequency domain method.Propose the method utilizing Rekasius conversion solving system time lag to stablize the upper limit, but the method needs the critical eigenvalue of hunting system in time lag space, amount of calculation is larger; The characteristic equation of time-lag system is converted into the purely imaginary eigenvalue of polynomial equation solving system by the imaginary axis, without the need to the substitution of variable of any centre, the time lag that effectively can solve single time-lag power system stablizes the upper limit, but the method is difficult to the calculating being applicable to large scale system.When system exists uncertainty and time lag changes in time, more than solve very difficult.Therefore, the technique study system time lags stability of frequency domain is adopted to have stronger limitation.2) time domain method.Respectively Finsler lemma, Park inequality, Moon inequality and Fridman generalized model transform method are combined with Lyapunov stability theory, the time lag relevant stable problem of analytical system; Propose free-form curve and surface (free-weightingmatrices, FWM) method, widen the thinking of time-lag system stability analysis further.Time domain method can process Time-varying time-delays problem effectively, but all there is conservative in various degree.

Summary of the invention

Given this, the object of the present invention is to provide a kind of time-lag power system stability analytical method and device, solve the difficult problem that traditional time lag method for analyzing stability is difficult to analyze electric power system generation cascading failure.

In order to realize this object, the technical scheme that the present invention takes is as follows.

A kind of time-lag power system stability analytical method, described method comprises step:

Generator frequency, merit angle in A, collection time-lag system network architecture parameters, system;

B, utilize image data to construct fault chains, and fault chains is combined with Markov process set up time lag Markov jump system state equation;

C, according to time lag Markov jump system state equation, the time lag stability criterion improving free-form curve and surface is constructed respectively based on Markov process transition rates and Newton-Leibniz Formula, on this basis, by time lag stability criterion equivalence transformation, generalized eigenvalue method solving system time lag is utilized to stablize the upper limit;

D, export described system time lags and stablize the upper limit.

Wherein said according to time lag Markov jump system state equation, the time lag stability criterion improving free-form curve and surface is constructed respectively based on Markov process transition rates and Newton-Leibniz Formula, on this basis, by time lag stability criterion equivalence transformation, the step utilizing generalized eigenvalue method solving system time lag to stablize the upper limit comprises:

C1, structure are considered the Liapunov-Krasovsky functional of markov saltus step and are solved its derived function along system, Markov process transition rates and Newton-Leibniz Formula is utilized to construct free claim respectively, be introduced in the weak infinitesimal operators of described Liapunov-Krasovsky functional, and whole Time-varying time-delays interval decomposed is become two subintervals, obtain time-lag system stability criterion;

C2, the time-lag system stability criterion equivalence transformation in step C1 is become to meet the canonical form that generalized eigenvalue method solves, solve time lag and stablize the upper limit.

Especially, utilize Markov process transition rates and Newton-Leibniz Formula to construct free claim respectively in step C1 to be:

${-x}^T\left(t\right)\underset{j=1}{\overset{S}{\mathrm{\Σ}}}{\mathrm{\π}}_{\mathrm{ij}}{W}_{j}x\left(t\right)=0,$

${-x}^T(t-h_t)\underset{j=1}{\overset{S}{\mathrm{\Σ}}}{\mathrm{\π}}_{\mathrm{ij}}{U}_{j}x(t-h_t)=0,$

${-x}^T(t-\stackrel{\‾}{h})\underset{j=1}{\overset{S}{\mathrm{\Σ}}}{\mathrm{\π}}_{\mathrm{ij}}{V}_{j}x(t-\stackrel{\‾}{h})=0,$

The equation of the improvement free-form curve and surface constructed by Newton-Leibniz Formula is:

$2{\mathrm{\ζ}}^T\left(t\right)N[x\left(t\right)-x(t-h_t)-{\∫}_{t-h_t}^t\stackrel{\·}{x}\left(s\right)\mathrm{ds}]=0,$

${2\mathrm{\ζ}}^T\left(t\right)L[x(t-h_t)-x(t-\stackrel{\‾}{h})-{\∫}_{t-\stackrel{\‾}{h}}^{t-h_t}\stackrel{\·}{x}\left(s\right)\mathrm{ds}]=0,$

${2\mathrm{\ζ}}^T\left(t\right)M[x\left(t\right)-x(t-\stackrel{\‾}{h})-{\∫}_{t-\stackrel{\‾}{h}}^{t}x\left(s\right)\mathrm{ds}]=0,$

Wherein, $\mathrm{\ζ}\left(t\right)={\left[\begin{array}{ccc}{x}^T& x^T(t-h_t)& x^T(t-\stackrel{\‾}{h})\end{array}\right]}^T,$

X (t) ∈ R ⁿthe state vector for time-lag power system,

for state vector x (t) is to the first derivative of time,

H _tfor time lag,

for time lag stablizes the upper limit,

μ is the maximum rate of change of time lag,

W, U and V for treating set matrix,

N, L and M are improvement free-form curve and surface,

π _ijfor the Markov transition probabilities entry of a matrix element of time-lag power system, refer to that system mode is in i state in t, and be in the transitional provavility density of j state in the t+ Δ moment,

Wherein time lag h _twith its first derivative satisfy condition:

$0\≤h_t\≤\stackrel{\‾}{h},$

${\stackrel{\·}{h}}_t\≤\mathrm{\μ}.$

In addition, solve time lag described in and stablize the upper limit for asking for optimization problem:

min d，

Its constraints is:

$\left[\begin{array}{ccc}{T}_1& 0& 0\\ *& T_2& 0\\ *& *& T_3\end{array}\right]<d\&CenterDot;\left[\begin{array}{ccc}{Z}_1& 0& 0\\ *& Z_2& 0\\ *& *& K\end{array}\right]$ With

$\left[\begin{array}{cccccc}{\mathrm{\&Omega;}}^{\&prime;}& N& L& M& {\stackrel{\&OverBar;}{A}}_i(Z_1+Z_2)& K\\ *& -T_1& 0& 0& 0& 0\\ *& *& -T_1& 0& 0& 0\\ *& *& *& {-T}_2& 0& 0\\ *& *& *& *& -(T_1+T_2)& 0\\ *& *& *& *& *& {-T}_3\end{array}\right]<0,$

Wherein, d is optimization aim, and

for time lag stablizes the upper limit,

${\stackrel{\&OverBar;}{A}}_i=\begin{array}{}\end{array}\left[\begin{array}{ccc}{A}_i& A_{\mathrm{di}}& 0\end{array}\right],$

A _ifor the state matrix of time-lag power system under operating condition i, and A _i∈ R ^{n × n},

A _difor the delay matrix of time-lag power system under operating condition i, and A _di∈ R ^{n × n},

T ₁, T ₂, T ₃for added martix, and ${-Z}_1/\stackrel{\&OverBar;}{h}<-T_1,-Z_2/\stackrel{\&OverBar;}{h}<{-T}_2,-K/\stackrel{\&OverBar;}{h}<{-T}_3,$

P, Q, R _i, K, Z _i, U _i, V _i, W _ibe and treat set matrix, and P=P ^t> 0, Q=Q ^t>=0, k=K ^t> 0, $Z_i=Z_i^T\&GreaterEqual;0,$ $U_i=U_i^T\&GreaterEqual;0,$ $V_i=V_i^T\&GreaterEqual;0$ And $W_i=W_i^T\&GreaterEqual;0,$

$\mathrm{\&Omega;}={\mathrm{\&Omega;}}^{\&prime;}+{\stackrel{\&OverBar;}{h}}_K,$

$\mathrm{\&Omega;}={\mathrm{\&Omega;}}_1+{\mathrm{\&Omega;}}_2+{\mathrm{\&Omega;}}_2^T,$

${\mathrm{\&Omega;}}_1=\left[\begin{array}{ccc}{P}_i{A}_i+A_i^T{P}_i+Q+R_i+\stackrel{\&OverBar;}{h}K+& P_i{A}_{\mathrm{di}}& 0\\ \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}(R_j+P_j-W_j)& -(1-\mathrm{\&mu;})Q-& 0\\ *& \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{U}_j& -R_i-\\ *& *& \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{V}_j\end{array}\right],$

Ω ₂＝[N+M L-N -L-M]，

π _ijfor the Markov transition probabilities entry of a matrix element of time-lag power system, refer to that system mode is in i state in t, and be in the transfer rate of j state in the t+ Δ moment, Δ is the variable quantity of moment t,

μ is the maximum rate of change of time lag,

S is limited mode set,

N, L and M are for improving free-form curve and surface.

A kind of time-lag power system stability analytical equipment, comprises data acquisition module, time lag Markov jump system generation module, the time lag upper limit solves module and result output module;

Described data acquisition module is used for generator frequency, merit angle in collection network structural parameters, system, and image data is sent to time lag Markov jump system generation module;

Described Markov jump system generation module utilizes the data configuration fault chains collected, and is combined with Markov process by fault chains and set up time lag Markov jump system state equation;

The described time lag upper limit solves module for according to time lag Markov jump system state equation, the time lag stability criterion improving free-form curve and surface is constructed respectively based on Markov process transition rates and Newton-Leibniz Formula, on this basis, by time lag stability criterion equivalence transformation, generalized eigenvalue method solving system time lag is utilized to stablize the upper limit;

Described result output module stablizes the upper limit for exporting described system time lags.

The described time lag upper limit solves module and comprises time-lag system stability criterion determining unit and standard solves unit, wherein:

Time-lag system stability criterion determining unit, for constructing Liapunov-Krasovsky functional of considering markov saltus step and solving its derived function along system, Markov process transition rates and Newton-Leibniz Formula is utilized to construct free claim respectively, be introduced in the weak infinitesimal operators of described Liapunov-Krasovsky functional, and whole Time-varying time-delays interval decomposed is become two subintervals, obtain time-lag system stability criterion;

Standard solves unit and becomes to meet for the time-lag system stability criterion equivalence transformation that time-lag system stability criterion determining unit obtained the canonical form that generalized eigenvalue method solves, and solves time lag and stablize the upper limit.

By adopting method and apparatus of the present invention, utilize Markov process transition rates and Newton-Leibniz Formula to construct improvement free-form curve and surface respectively and set up time lag markov change system stability criterion, and by generalized eigenvalue method, the system time lags upper limit is solved.Result shows, the method rationally can disclose the time lag stability of fault electric power system, and can effectively solve time lag and stablize the upper limit.

Accompanying drawing explanation

Fig. 1 is the structure chart of time-lag power system stability analytical equipment in embodiment of the present invention.

Fig. 2 is the scene topology diagram that an embodiment of the present invention is applied.

Fig. 3 is that an embodiment of the present invention is at different delayed time situation lower part generator relative merit angular motion state response curve.

Fig. 4 is that the present invention one implementation method is at different delayed time situation lower part generator relative merit angular motion state response curve

Embodiment

Below in conjunction with accompanying drawing, the present invention is elaborated.

The example embodiment that following discloses are detailed.But concrete structure disclosed herein and function detail are only the objects for describing example embodiment.

But should be appreciated that, the present invention is not limited to disclosed concrete example embodiment, but covers all modifications, equivalent and the alternative that fall within the scope of the disclosure.In the description to whole accompanying drawing, identical Reference numeral represents identical element.

Should be appreciated that, term "and/or" as used in this comprises one or morely relevant lists any of item and all combinations simultaneously.Should be appreciated that in addition, when parts or unit are called as " connection " or " coupling " to another parts or unit, it can be directly connected or coupled to miscellaneous part or unit, or also can there is intermediate member or unit.In addition, other words being used for describing relation between parts or unit should be understood according to identical mode (such as, " between " to " directly ", " adjacent " to " direct neighbor " etc.).

In order to introduce technical scheme of the present invention, first know-why of the present invention is described.

The general principle of time-lag power system stability analytical method provided by the invention and device is as follows:

First generator frequency, merit angle in time-lag system network architecture parameters, system is gathered.

When electric power system normally runs, circuit is with certain initial load, but after stopping transport, the load on this circuit will be transferred on All other routes, and these circuits may be caused in succession to transship or false protection when certain circuit is cut, generation cascading failure.The present invention adopts α, β, γ respectively as the evaluation index of the power flow changing rate of circuit, overload nargin and 3 aspects of the degree of coupling between faulty line and prediction circuit, and with index as intermediate link prediction index, this index determines the next stage faulty line of fault chains.

Assuming that system neutral road i breaks down, S _ifor the trend that circuit i passes through, then the calculating formula of α, β and γ is as follows:

${\mathrm{\&alpha;}}_{\mathrm{ij}}=\left|\frac{S_{j\left(t_b\right)}-S_{j\left(t_f\right)}}{S_{j\left(t_f\right)}}\right|---\left(1\right)$

${\mathrm{\&beta;}}_{\mathrm{ij}}=\left|\frac{S_{j\left(t_b\right)}}{S_{j\max }}\right|---\left(2\right)$

${\mathrm{\&gamma;}}_{\mathrm{ij}}=\left|\frac{S_{j\left(t_b\right)-}{S}_{j\left(t_f\right)}}{S_{i\left(t_f\right)}}\right|---\left(3\right)$

In formula, the trend of circuit i before expression accident, the trend of circuit j before expression accident, the trend of circuit j after expression circuit i has an accident, S _jmaxrepresent that circuit j allows the strength of current flow through.

Definition intermediate link prediction index for:

In formula (4), value larger, to be then subject to the impact of circuit i fault larger for non-fault line j, shows that circuit j becomes the possibility of next stage faulty line higher.Therefore, the state transition probability caused by power flow transfer can be calculated by following formula:

As can be seen from above-mentioned declarative procedure, the evolution of fault chains can be described as a series of evolution process with conditional probability event.In the links of fault chains development, the generation of every one-level is only relevant with upper level, and have nothing to do with the accident before system, the conditional probability distribution of its to-be only depends on current state, therefore, the evolution of fault chains belongs to Markov process, utilizes Markov Theory can describe the fault chains with randomness and correlation feature: to establish r _t=r (t) is system mode, it be value in finite aggregate S={1,2 ..., the homogeneous markov process of s}.For fault chains L=L ₁l ₂l ₃... L _n, by intermediate link L _ias the mode r of Stochastic Markov process _i=r (i), i.e. the adapter ring joint number of the corresponding fault chains L of limited mode S set, recycling formula (5) calculates Markov transition probabilities matrix π, sets up the Markov process based on fault chains.

Fault chains is combined with Markov process and obtains time lag Markov jump system model:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_{r\left(t\right)}\left(t\right)=A_{r\left(t\right)}\left(t\right)+A_{\mathrm{dr}\left(t\right)}x(t-h_t)\\ z_{r\left(t\right)}\left(t\right)=B_{r\left(t\right)}x\left(t\right)+B_{\mathrm{dr}\left(t\right)}x(t-h_t)\\ x_{r\left(t\right)}\left(t\right)={\mathrm{\&delta;}}_{r\left(t\right)}\left(t\right),t\&Element;[-\stackrel{\&OverBar;}{h},0]\end{array}\right.---\left(6\right)$

In formula (6), time lag h _tsatisfy condition:

$0\&le;h_t\&le;\stackrel{\&OverBar;}{h},{\stackrel{\text{\&CenterDot;}}{h}}_t<\mathrm{\&mu;}---\left(7\right)$

Wherein, x (t) ∈ R ⁿwith z (t) ∈ R ⁿstate vector and the output vector of system respectively, A _i, A _di, B _i, B _difor the known matrix of suitable dimension, h _tfor system time lags, h is that time lag stablizes the upper limit, and μ is the maximum rate of change of time lag, then the state transition probability of homogeneous markov process is:

$p_{\mathrm{ij}}=P(r_{t+\mathrm{\&Delta;}}=j|r_t=i)=\left\{\begin{array}{cc}{\mathrm{\&pi;}}_{\mathrm{ij}}\mathrm{\&Delta;}+o\left(\mathrm{\&Delta;}\right),& i\&NotEqual;j\\ 1+{\mathrm{\&pi;}}_{\mathrm{ii}}\mathrm{\&Delta;}+o\left(\mathrm{\&Delta;}\right),& i=j\end{array}\right.---\left(8\right)$

Wherein $\begin{array}{cc}\underset{\mathrm{\&Delta;}\&RightArrow;\&infin;}{\lim }\frac{o\left(\mathrm{\&Delta;}\right)}{\mathrm{\&Delta;}}=0& \end{array}(\mathrm{\&Delta;}>0)---\left(9\right)$

π _ijbe that system mode is in i state in t, and be in the transfer rate of j state in the t+ Δ moment, and Δ is the variable quantity of moment t, the higher order indefinite small that o (Δ) is moment t variation delta.

Be constructed as follows form Liapunov-Krasovsky functional thus:

$\begin{array}{c}V(x_t,t)=x^T\left(t\right)P_{i}x\left(t\right)+{\&Integral;}_{t-h_t}^t{x}^T\left(s\right)\mathrm{Qx}\left(s\right)\mathrm{ds}\\ +{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^t{x}^T\left(s\right)R_{i}x\left(s\right)\mathrm{ds}\\ +{\&Integral;}_{-\stackrel{\&OverBar;}{h}}^0{\&Integral;}_{t+\mathrm{\&theta;}}^t{x}^T\left(s\right)\mathrm{Kx}\left(s\right)\mathrm{dsd\&theta;}\\ +{\&Integral;}_{-\stackrel{\&OverBar;}{h}}^0{\&Integral;}_{t+\mathrm{\&theta;}}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)(Z_1+Z_2)\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{dsd\&theta;}\end{array}---\left(10\right)$

Wherein P, Q, R _i, K, Z _ibe and treat set matrix, and P=P ^t> 0, Q=Q ^t>=0, k=K ^t> 0, $Z_i=Z_i^T\&GreaterEqual;0.$

Relation is met between element in the markov transition rates π of the time-lag system that consideration formula (6) characterizes: ${\mathrm{\&pi;}}_{\mathrm{ii}}+\underset{i\&NotEqual;j}{\mathrm{\&Sigma;}}{\mathrm{\&pi;}}_{\mathrm{ij}}=0({\mathrm{\&pi;}}_{\mathrm{ij}}\&GreaterEqual;0,i\&NotEqual;j),$ Then for the matrix of any appropriate dimension $W_i=W_i^T\&GreaterEqual;0$ Have:

${-x}^T\left(t\right)\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{W}_{j}x\left(t\right)=0---\left(11\right)$

${-x}^T(t-h_t)\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{U}_{j}x(t-h_t)=0---\left(12\right)$

${-x}^T(t-\stackrel{\&OverBar;}{h})\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{V}_{j}x(t-\stackrel{\&OverBar;}{h})=0---\left(13\right)$

Wherein U _i, V _i, W _ibe and treat set matrix, and and

Meanwhile, from Newton-Leibniz Formula, for arbitrary suitable dimension matrix N, L and M, has following formula to set up:

$2{\mathrm{\&zeta;}}^T\left(t\right)N[x\left(t\right)-x(t-h_t)-{\&Integral;}_{t-h_t}^t\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}]=0---\left(14\right)$

${2\mathrm{\&zeta;}}^T\left(t\right)L[x(t-h_t)-x(t-\stackrel{\&OverBar;}{h})-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^{t-h_t}\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}]=0---\left(15\right)$

${2\mathrm{\&zeta;}}^T\left(t\right)M[x\left(t\right)-x(t-\stackrel{\&OverBar;}{h})-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^{t}x\left(s\right)\mathrm{ds}]=0---\left(16\right)$

Wherein, to V (x _t, t) get random process { x _t, r _t, the weak infinitesimal operators of t>=0} $\mathrm{\&Gamma;}\left[V(x_t,t)\right]=\frac{\mathrm{\&delta;V}}{\mathrm{\&delta;t}}+{\stackrel{\&CenterDot;}{x}}_t^T\frac{\mathrm{\&delta;V}}{\mathrm{\&delta;x}}{|}_{r_t=t}+\underset{j=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ii}}V(x_t,t,j,i),$ Then have:

$\begin{array}{c}\mathrm{\&Gamma;}\left[V(x_t,t,i)\right]={2x}^T\left(t\right)P_i\stackrel{\&CenterDot;}{x}\left(t\right)+x^T\left(t\right)\mathrm{Qx}\left(t\right)-(1-{\stackrel{\&CenterDot;}{h}}_t)x^T(t-h_t)\mathrm{Qx}(t-h_t)\\ +x^T\left(t\right)R_{i}x\left(t\right)-x^T(t-\stackrel{\&OverBar;}{h})+\stackrel{\&OverBar;}{h}{x}^T\left(t\right)\mathrm{Kx}\left(t\right)-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^t{x}^T\left(s\right)\mathrm{Kx}\left(s\right)\mathrm{ds}\\ +\stackrel{\&OverBar;}{h}{\stackrel{\&CenterDot;}{x}}^T\left(t\right)(Z_1+Z_2)\stackrel{\&CenterDot;}{x}\left(t\right){\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)(Z_1+Z_2)\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}\\ +\underset{j=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{x}^T\left(t\right)P_{j}x\left(t\right)+{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^t{x}^T\left(s\right)\left(\underset{j=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{R}_j\right)x\left(s\right)d\end{array}---\left(17\right)$

In order to effectively reduce the conservative of this patent method, the free claim (formula (11)-Shi (13)) built by markov transition rates π is introduced on formula (17) right side, and the free claim (formula (14)-Shi (16)) to be constructed by Newton-Leibniz Formula, simultaneously by whole Time-varying time-delays interval [0, h] be decomposed into two subintervals [0, h], [h, h] (h > 0), arrangement can obtain:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\left(x\right)\&le;x^T\left(t\right)(P_i{A}_i+A_i^T{P}_i+Q+R_i+\stackrel{\&OverBar;}{h}K+\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}(P_j-W_j))x\left(t\right)\\ +{2x}^T\left(t\right)P_i{A}_{\mathrm{di}}x(t-h_t)-x^T(t-h_t)[(1-\mathrm{\&mu;})Q+\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{U}_j]x^T(t-h_t)\\ -x^T(t-\stackrel{\&OverBar;}{h})[R_i+\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{V}_j]x(t-\stackrel{\&OverBar;}{h})+\stackrel{\&OverBar;}{h}{\stackrel{\&CenterDot;}{x}}^T\left(t\right)(Z_1+Z_2)\stackrel{.}{x}\left(t\right)-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z_2\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}\\ -{\&Integral;}_{t-h_t}^t{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z_1\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^{t-h_t}{\stackrel{\&CenterDot;}{x}}^T\left(s\right)Z_1\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}+{2\mathrm{\&zeta;}}^T\left(t\right)N[x\left(t\right)-x(t-h_t)-{\&Integral;}_{t-h_t}^t\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}]\\ +{2\mathrm{\&zeta;}}^T\left(t\right)L[x(t-h_t)-x(t-\stackrel{\&OverBar;}{h})-{\&Integral;}_{t-h_t}^t\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}]+{2\mathrm{\&zeta;}}^T\left(t\right)M[x\left(t\right)-x(t-\stackrel{\&OverBar;}{h})-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^t\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}]\\ \&le;{\mathrm{\&zeta;}}^T\left(t\right)\left(t\right)[\mathrm{\&Phi;}+{\mathrm{\&Phi;}}_v]\mathrm{\&zeta;}\left(t\right)-{\&Integral;}_{t-h_t}^t{\mathrm{\&Delta;}}_1\mathrm{ds}-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^t{\mathrm{\&Delta;}}_2\mathrm{ds}-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^{t-h_t}{\mathrm{\&Delta;}}_3\mathrm{ds}\\ <{\mathrm{\&zeta;}}^T\left(t\right)\left(t\right)[\mathrm{\&Phi;}+{\mathrm{\&Phi;}}_v]\mathrm{\&zeta;}\left(t\right)\end{array}---\left(18\right)$

In formula 18:

$\begin{array}{c}{\mathrm{\&Phi;}}_v=\stackrel{\&OverBar;}{h}\&CenterDot;{\stackrel{\&OverBar;}{A}}^T(Z_1+Z_2)\stackrel{\&OverBar;}{A}+\stackrel{\&OverBar;}{h}\&CenterDot;N{Z}_1^{-1}{N}^T\\ +\stackrel{\&OverBar;}{h}\&CenterDot;L{Z}_1^{-1}{L}^T+\stackrel{\&OverBar;}{h}\&CenterDot;M{Z}_2^{-1}{M}^T\end{array}---\left(19\right)$

${\mathrm{\&Delta;}}_1=[{\mathrm{\&zeta;}}^T\left(t\right)N+\stackrel{\&CenterDot;}{x}\left(t\right)Z_1]Z_1^{-1}[N^T\mathrm{\&zeta;}\left(t\right)+Z_1\stackrel{\&CenterDot;}{x}\left(t\right)]---\left(20\right)$

${\mathrm{\&Delta;}}_2=[{\mathrm{\&zeta;}}^T\left(t\right)M+\stackrel{\&CenterDot;}{x}\left(t\right)Z_2]Z_2^{-1}[M^T\mathrm{\&zeta;}\left(t\right)+Z_2\stackrel{\&CenterDot;}{x}\left(t\right)]---\left(21\right)$

${\mathrm{\&Delta;}}_3=[{\mathrm{\&zeta;}}^T\left(t\right)L+\stackrel{\&CenterDot;}{x}\left(t\right)Z_1]Z_1^{-1}[L^T\mathrm{\&zeta;}\left(t\right)+Z_1\stackrel{\&CenterDot;}{x}\left(t\right)]---\left(22\right)$

Wherein, N, L and M are for improving free-form curve and surface.

As Φ+Φ _v≤ 0, based on Schur complement fixed reason, the time-lag system stability criterion that formula (6) characterizes can be obtained as follows:

Given scalar and μ, if there is P=P ^t> 0, Q=Q ^t>=0, k=K ^t> 0, $Z_i=Z_i^T\&GreaterEqual;0,$ $U_i=U_i^T\&GreaterEqual;0,$ $V_i=V_i^T\&GreaterEqual;0$ And $W_i=W_i^T\&GreaterEqual;0$ (i=1,2 ... S), N=[N ₁n ₂n ₃] ^t, L=[L ₁l ₂l ₃] ^tand M=[M ₁m ₂m ₃] ^t, following formula (23) is set up, then markov saltus step time-lag system (6) Stochastic stable of the formula that satisfies condition (7):

$\left[\begin{array}{ccccc}\mathrm{\&Omega;}& \stackrel{\&OverBar;}{h}N& \stackrel{\&OverBar;}{h}L& \stackrel{\&OverBar;}{h}M& \stackrel{\&OverBar;}{h}\&CenterDot;{\stackrel{\&OverBar;}{A}}_i(Z_1+Z_2)\\ *& -\stackrel{\&OverBar;}{h}{Z}_1& 0& 0& 0\\ *& *& -\stackrel{\&OverBar;}{h}{Z}_1& 0& 0\\ *& *& *& -\stackrel{\&OverBar;}{h}{Z}_2& 0\\ *& *& *& *& -\stackrel{\&OverBar;}{h}(Z_1+Z_2)\end{array}\right]<0---\left(23\right)$

Wherein:

$\mathrm{\&Omega;}={\mathrm{\&Omega;}}_1+{\mathrm{\&Omega;}}_2+{\mathrm{\&Omega;}}_2^T,$

${\mathrm{\&Omega;}}_1=\left[\begin{array}{ccc}{P}_i{A}_i+A_i^T{P}_i+Q+R_i+\stackrel{\&OverBar;}{h}K+& P_i{A}_{\mathrm{di}}& 0\\ \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}(R_j+P_j-W_j)& -(1-\mathrm{\&mu;})Q-& 0\\ *& \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{U}_j& -R_i-\\ *& *& \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{V}_j\end{array}\right],$

Ω ₂＝[N+M L-N -L-M]，

${\stackrel{\&OverBar;}{A}}_i=\left[\begin{array}{ccc}{A}_i& A_{\mathrm{di}}& 0\end{array}\right],$

i＝{1,2，…，N}，

Wherein Ω, Ω ₁, Ω ₂, it is all intermediate variable.A _ifor the state matrix of the time-lag power system under operating condition i, and A _i∈ R ^{n × n}, A _difor the delay matrix of the time-lag power system under operating condition i, and A _di∈ R ^{n × n}.

Whether the MATRIX INEQUALITIES that formula (23) characterizes only can decision-making system be stablized, and system time lags cannot be obtained stablize the information such as the upper limit, considering that time lag stablizes solving of the upper limit is a convex optimization problem with linear inequality constraint, present the form of generalized eigenvalue, therefore, the present invention proposes to utilize the time lag of generalized eigenvalue method computing system to stablize the upper limit.Due to the generalized eigenvalue form that formula (23) is not standard, need to carry out necessary process, meanwhile, notice that formula (23) all the other four column elements except first row, all containing h, therefore utilize matrix $\mathrm{diag}\{I,1/\stackrel{\&OverBar;}{h\&CenterDot;}I,1/\stackrel{\&OverBar;}{h\&CenterDot;}I,1/\stackrel{\&OverBar;}{h}\&CenterDot;I,1/\stackrel{\&OverBar;}{h}\&CenterDot;I\}$ Premultiplication and right multiplier (23), obtain:

$\left[\begin{array}{ccccc}\mathrm{\&Omega;}& N& L& M& {\stackrel{\&OverBar;}{A}}_i(Z_1+Z_2)\\ *& -Z_1/\stackrel{\&OverBar;}{h}& 0& 0& 0\\ *& *& -Z_1/\stackrel{\&OverBar;}{h}& 0& 0\\ *& *& *& -Z_2/\stackrel{\&OverBar;}{h}& 0\\ *& *& *& *& -(Z_1+Z_2)/\stackrel{\&OverBar;}{h}\end{array}\right]<0---\left(24\right)$

Then, make mend character according to Schur, can obtain:

$\left[\begin{array}{cccccc}{\mathrm{\&Omega;}}^{\&prime;}& N& L& M& {\stackrel{\&OverBar;}{A}}_i(Z_1+Z_2)& K\\ *& -Z_1/\stackrel{\&OverBar;}{h}& 0& 0& 0& 0\\ *& *& -Z_1/\stackrel{\&OverBar;}{h}& 0& 0& 0\\ *& *& *& -Z_2/\stackrel{\&OverBar;}{h}& 0& 0\\ *& *& *& *& -(Z_1+Z_2)/\stackrel{\&OverBar;}{h}& 0\\ *& *& *& *& *& K/\stackrel{\&OverBar;}{h}\end{array}\right]<0---\left(25\right)$

Order ${-Z}_1/\stackrel{\&OverBar;}{h}<-T_1,-Z_2/\stackrel{\&OverBar;}{h}<{-T}_2,-K/\stackrel{\&OverBar;}{h}<{-T}_3$ And $d=1/\stackrel{\&OverBar;}{h},$ :

$\left[\begin{array}{ccc}{T}_1& 0& 0\\ *& T_2& 0\\ *& *& T_3\end{array}\right]<d\&CenterDot;\left[\begin{array}{ccc}{Z}_1& 0& 0\\ *& Z_2& 0\\ *& *& K\end{array}\right]---\left(26\right)$

In formula (25), by-T ₁,-T ₂,-T ₃replace respectively obtain:

$\left[\begin{array}{cccccc}{\mathrm{\&Omega;}}^{\&prime;}& N& L& M& {\stackrel{\&OverBar;}{A}}_i(Z_1+Z_2)& K\\ *& -T_1& 0& 0& 0& 0\\ *& *& -T_1& 0& 0& 0\\ *& *& *& {-T}_2& 0& 0\\ *& *& *& *& -(T_1+T_2)& 0\\ *& *& *& *& *& {-T}_3\end{array}\right]<0---\left(27\right)$

By formula (24)-(27), time lag stablizes upper limit h problem can be converted into following optimization problem:

$\stackrel{\min }{s,t\left(26\right),}\stackrel{d}{\left(27\right)}---\left(28\right)$

By solving-optimizing problem (28), and be constraint with formula (26) and formula (27), finally, utilize h=1/d can derive the time lag of electric power system in cascading failure situation and stablize the upper limit.

Therefore embodiments of the present invention comprise a kind of time-lag power system stability analytical method, and described method comprises step:

Generator frequency, merit angle in A, collection time-lag system network architecture parameters, system;

B, utilize image data to construct fault chains, and fault chains is combined with Markov process set up time lag Markov jump system state equation;

C, according to time lag Markov jump system state equation, the time lag stability criterion improving free-form curve and surface is constructed respectively based on Markov process transition rates and Newton-Leibniz Formula, on this basis, by time lag stability criterion equivalence transformation, generalized eigenvalue method solving system time lag is utilized to stablize the upper limit;

D, export described system time lags and stablize the upper limit.

Wherein, described according to time lag Markov jump system state equation, construct the time lag stability criterion based on the improvement free-form curve and surface constructed respectively by Markov process transition rates and Newton-Leibniz Formula, on this basis, by time lag stability criterion equivalence transformation, the step utilizing generalized eigenvalue method solving system time lag to stablize the upper limit comprises:

C1, construct a class and consider the Liapunov-Krasovsky functional of markov saltus step and solve its derived function along system, Markov process transition rates and Newton-Leibniz Formula is utilized to construct free claim respectively, be introduced in the weak infinitesimal operators of described Liapunov-Krasovsky functional, and whole Time-varying time-delays interval decomposed is become two subintervals, obtain time-lag system stability criterion;

C2, the time-lag system stability criterion equivalence transformation in step C1 is become to meet the canonical form that generalized eigenvalue method solves, solve time lag and stablize the upper limit.

Especially, utilize Markov process transition rates and Newton-Leibniz Formula to construct free claim respectively in step C1 to be:

${-x}^T\left(t\right)\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{W}_{j}x\left(t\right)=0,$

${-x}^T(t-h_t)\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{U}_{j}x(t-h_t)=0,$

${-x}^T(t-\stackrel{\&OverBar;}{h})\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{V}_{j}x(t-\stackrel{\&OverBar;}{h})=0,$

The equation of the improvement free-form curve and surface constructed by Newton-Leibniz Formula is:

$2{\mathrm{\&zeta;}}^T\left(t\right)N[x\left(t\right)-x(t-h_t)-{\&Integral;}_{t-h_t}^t\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}]=0,$

${2\mathrm{\&zeta;}}^T\left(t\right)L[x(t-h_t)-x(t-\stackrel{\&OverBar;}{h})-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^{t-h_t}\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}]=0,$

${2\mathrm{\&zeta;}}^T\left(t\right)M[x\left(t\right)-x(t-\stackrel{\&OverBar;}{h})-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^{t}x\left(s\right)\mathrm{ds}]=0,$

Wherein, $\mathrm{\&zeta;}\left(t\right)={\left[\begin{array}{ccc}{x}^T& x^T(t-h_t)& x^T(t-\stackrel{\&OverBar;}{h})\end{array}\right]}^T,$

X (t) ∈ R ⁿthe state vector for time-lag power system,

for state vector x (t) is to the first derivative of time,

for time lag stablizes the upper limit,

μ is the maximum rate of change of time lag,

W, U and V for treating set matrix,

N, L and M are improvement free-form curve and surface,

π _ijfor the Markov transition probabilities entry of a matrix element of time-lag power system, refer to that system mode is in i state in t, and be in the transfer rate of j state in the t+ Δ moment,

Wherein time lag h _twith its first derivative satisfy condition:

$0\&le;h_t\&le;\stackrel{\&OverBar;}{h},$

${\stackrel{\&CenterDot;}{h}}_t\&le;\mathrm{\&mu;}.$

In addition, solve time lag described in and stablize the upper limit for asking for optimization problem:

min d，

Its constraints is:

$\left[\begin{array}{ccc}{T}_1& 0& 0\\ *& T_2& 0\\ *& *& T_3\end{array}\right]<d\&CenterDot;\left[\begin{array}{ccc}{Z}_1& 0& 0\\ *& Z_2& 0\\ *& *& K\end{array}\right]$ With

$\left[\begin{array}{cccccc}{\mathrm{\&Omega;}}^{\&prime;}& N& L& M& {\stackrel{\&OverBar;}{A}}_i(Z_1+Z_2)& K\\ *& -T_1& 0& 0& 0& 0\\ *& *& -T_1& 0& 0& 0\\ *& *& *& {-T}_2& 0& 0\\ *& *& *& *& -(T_1+T_2)& 0\\ *& *& *& *& *& {-T}_3\end{array}\right]<0,$

Wherein, d is optimization aim, and

for time lag stablizes the upper limit,

X (t) ∈ R ⁿthe state vector for time-lag power system,

for state vector x (t) is to the first derivative of time,

${\stackrel{\&OverBar;}{A}}_i=\begin{array}{}\end{array}\left[\begin{array}{ccc}{A}_i& A_{\mathrm{di}}& 0\end{array}\right],$

A _ifor the state matrix of the time-lag power system under operating condition i, and A _i∈ R ^{n × n},

A _difor the delay matrix of the time-lag power system under operating condition i, and A _di∈ R ^{n × n},

T ₁, T ₂, T ₃for added martix, and ${-Z}_1/\stackrel{\&OverBar;}{h}<-T_1,-Z_2/\stackrel{\&OverBar;}{h}<{-T}_2,-K/\stackrel{\&OverBar;}{h}<{-T}_3,$

P, Q, R _i, K, Z _i, U _i, V _i, W _ibe and treat set matrix, and P=P ^t> 0, Q=Q ^t>=0, k=K ^t> 0, $Z_i=Z_i^T\&GreaterEqual;0,$ $U_i=U_i^T\&GreaterEqual;0,$ $V_i=V_i^T\&GreaterEqual;0$ And $W_i=W_i^T\&GreaterEqual;0,$

$\mathrm{\&Omega;}={\mathrm{\&Omega;}}^{\&prime;}+{\stackrel{\&OverBar;}{h}}_K,$

$\mathrm{\&Omega;}={\mathrm{\&Omega;}}_1+{\mathrm{\&Omega;}}_2+{\mathrm{\&Omega;}}_2^T,$

${\mathrm{\&Omega;}}_1=\left[\begin{array}{ccc}{P}_i{A}_i+A_i^T{P}_i+Q+R_i+\stackrel{\&OverBar;}{h}K+& P_i{A}_{\mathrm{di}}& 0\\ \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}(R_j+P_j-W_j)& -(1-\mathrm{\&mu;})Q-& 0\\ *& \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{U}_j& -R_i-\\ *& *& \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{V}_j\end{array}\right],$

Ω ₂＝[N+M L-N -L-M]，

π _ijfor the Markov transition probabilities entry of a matrix element of time-lag power system, refer to that system mode is in i state in t, and be in the transfer rate of j state in the t+ Δ moment, Δ is the variable quantity of moment t,

μ is the maximum rate of change of time lag,

S is limited mode set,

N, L and M are for improving free-form curve and surface.

In order to realize time-lag power system stability analytical method of the present invention, a kind of time-lag power system stability analytical equipment is also comprised in embodiment of the present invention, as shown in Figure 1, described time-lag power system stability analytical equipment comprises data acquisition module, time lag Markov jump system generation module, the time lag upper limit solve module and result output module;

Described data acquisition module is used for generator frequency, merit angle in collection network structural parameters, system, and image data is sent to time lag Markov jump system generation module;

Described Markov jump system generation module utilizes the data configuration fault chains collected, and is combined with Markov process by fault chains and set up time lag Markov jump system state equation;

The described time lag upper limit solves module for according to time lag Markov jump system state equation, the time lag stability criterion improving free-form curve and surface is constructed respectively based on Markov process transition rates and Newton-Leibniz Formula, on this basis, by time lag stability criterion equivalence transformation, generalized eigenvalue method solving system time lag is utilized to stablize the upper limit;

Described result output module stablizes the upper limit for exporting described system time lags.

The described time lag upper limit solves module and comprises time-lag system stability criterion determining unit and standard solves unit, wherein: time-lag system stability criterion determining unit, consider the Liapunov-Krasovsky functional of markov saltus step for constructing a class and solve its derived function along system, Markov process transition rates and Newton-Leibniz Formula is utilized to construct free claim respectively, be introduced in the weak infinitesimal operators of described Liapunov-Krasovsky functional, and whole Time-varying time-delays interval decomposed is become two subintervals, obtain time-lag system stability criterion,

Standard solves unit and becomes to meet for the time-lag system stability criterion equivalence transformation that time-lag system stability criterion determining unit obtained the canonical form that generalized eigenvalue method solves, and solves time lag and stablizes the upper limit.

Below by way of specific embodiment, technique effect of the present invention is described, it will be understood by those skilled in the art that this specific embodiment is only exemplary, and not mean to cause any restriction to protection scope of the present invention.

Figure 2 shows that the New England-New York interconnected systems of IEEE16 machine 68 node.This system can be divided into 5 large regions: region 1,2 and 3 such as to be respectively at the valve system, and region 4 is New York system, and region 5 is New England's system.Generator adopts 6 rank detailed models, and excitation adopts IEEE-DC1 type excitation, and load model adopts WECC load model, the permanent burden with power of 50%, the permanent reactive impedance load of 50%, the dynamic load of 20%.

Electric power system fault is mostly line fault, considers the importance of interregional interconnection, the triggering link using fault of interconnected transmission line as fault chains.In the region interconnection of IEEE16 machine 68 node system, the trend of the interconnection 1-2 between region 4 and region 5 and the interconnection 46-49 between region 3 and region 4 is respectively 160.47MW, 208.47MW, close to the maximum delivery capacity of circuit, therefore the triggering link of interconnection 1-2 and interconnection 46-49 respectively as different fault chains is chosen, until fault chains generates end when system generation constant amplitude, increasing amplitude low frequency oscillation.According to the index mentioned in foregoing teachings of the present invention, prediction fault chains is:

L1={ circuit 1-2 → circuit 3-4 → circuit 2-3}

L2={ circuit 46-49 → circuit 32-33 → circuit 31-38}

Set up the Markov process of corresponding fault chains L1 and L2, system mode r _t=r (t) difference value is in finite aggregate S _m=1,2,3}, m=1, and 2, its transition probability matrix is:

${\mathrm{\&pi;}}_{L_1}=\left[\begin{array}{ccc}-0.3886& 0.3385& 0.0501\\ 0.0659& -0.1612& 0.0953\\ 0.1807& 0.0005& -0.1812\end{array}\right]$

${\mathrm{\&pi;}}_{L_2}=\left[\begin{array}{ccc}-0.2788& 0.2602& 0.0186\\ 0.0020& -0.0978& 0.0958\\ 0.0022& 0.0016& -0.0038\end{array}\right]$

For fault chains L1, by the state matrix A of 16 machine systems after Schur depression of order _iwith delay matrix A _di(i=1,2,3) substitute into formula (27) and formula (28), and utilize LMI method to try to achieve time lag to stablize the upper limit .

For feasibility and the validity of checking institute of the present invention extracting method, based on H2/H ∞ control method, damping controller design is carried out to IEEE 16 machine 68 node system, and time lag is set to 0ms respectively, 40ms, 82.9ms and 120ms, observe the relative merit angular difference dynamic response curve between generator 8-15 and generator 1-16, as shown in Figure 3.As seen from the figure, when time lag is less than, time lag is stable above prescribes a time limit, and system is all in stable state in fault chains evolution, and energy damping inter-area oscillations rapidly; When time lag equals time lag, stable upper system is in underdamping state in the initial link of fault chains in limited time, but along with the differentiation of fault chains, system returns to stable state gradually; When time lag, to be greater than time lag stable upper in limited time, and along with the generation of fault chains, system is progressively labile state by underdamping state development.

Table 1 and table 2 list the damping ratio of relative merit angular difference curve in the final link of fault chains of generator 8-15 and the generator 1-16 utilizing prony algorithm to obtain respectively.As seen from table, when system generation cascading failure, when time lag reaches 82.9ms, the damping ratio of merit angular difference curve is respectively 8.49% and 9.62%, although still there is certain damping, its damping ratio is down to less than 10%, cannot meet control overflow.It can thus be appreciated that, fault chains is combined with Markov process, the method of the time-lag power system stability of embodiment of the present invention is practical, and meanwhile, the time lag utilizing time-lag power system stability method of the present invention to try to achieve stablizes the upper limit and actual value is comparatively close.

Damping ratio under table 1 16 machine system G8 and each Slack time of G15 merit angular difference

Damping ratio under table 2 16 machine system G1 and each Slack time of G16 merit angular difference

For fault chains L2, according to transition probability matrix with known coefficient matrices A _i, A _di(i=1,2,3), solve time lag by generalized eigenvalue method and stablize the upper limit .For checking this method validity, time lag size is set to 0ms respectively, 50ms, 101.4ms and 150ms, observes the relative merit angular difference dynamic response curve between generator 4-13 and generator 7-14, as shown in Figure 4.

As seen from Figure 4, when time lag is less than, time lag is stable above prescribes a time limit, and the links in fault chains evolution all can at 20s internal damping inter-area oscillations; When time lag increases to 101.4ms by 0ms, damping weakens thereupon, and generator all occurs certain swing relative to merit angular difference curve in each link; When time lag is increased to 150ms, there is constant amplitude and increasing oscillation in generator relative merit angular difference curve, system is in instability status.

Table 3 and table 4 sets forth the damping ratio of relative merit angular difference curve in the final link of fault chains of generator 4-13 and the generator 7-14 utilizing prony algorithm to obtain.As seen from table, the damping ratio of system when time lag is 0ms is respectively 17.59% and 15.75%, and damping ratio when time lag is 40ms is respectively 11.81% and 11.54%, and in two kinds of situations, system all has larger damping, is in stable state.But when time lag reaches 101.4ms, the damping ratio of merit angular difference curve is all down to less than 10%, and be respectively 5.16% and 5.92%, system is in critical stable state.It can thus be appreciated that, fault chains is combined with Markov process, rationally can not only disclose the time lag stability of fault electric power system, and can effectively solve time lag and stablize the upper limit.

Damping ratio under table 3 16 machine system G4 and each Slack time of G13 merit angular difference

Damping ratio under table 4 16 machine system G7 and each Slack time of G14 merit angular difference

It should be noted that; above-mentioned execution mode is only the present invention's preferably embodiment; can not limiting the scope of the invention be understood as, not depart under concept thereof of the present invention, all protection scope of the present invention is belonged to modification to any minor variations that the present invention does.

Claims (6)

1. a time-lag power system stability analytical method, described method comprises step:

Generator frequency, merit angle in A, collection time-lag system network architecture parameters, system;

B, utilize image data to construct fault chains, and fault chains is combined with Markov process set up time lag Markov jump system state equation;

C, according to time lag Markov jump system state equation, the time lag stability criterion improving free-form curve and surface is constructed respectively based on Markov process transition rates and Newton-Leibniz Formula, on this basis, by time lag stability criterion equivalence transformation, generalized eigenvalue method solving system time lag is utilized to stablize the upper limit;

D, export described system time lags and stablize the upper limit.

2. the time-lag power system stability analytical method described in claim 1, it is characterized in that, described according to time lag Markov jump system state equation, the time lag stability criterion improving free-form curve and surface is constructed respectively based on Markov process transition rates and Newton-Leibniz Formula, on this basis, by time lag stability criterion equivalence transformation, the step utilizing generalized eigenvalue method solving system time lag to stablize the upper limit comprises:

C1, structure are considered the Liapunov-Krasovsky functional of markov saltus step and are solved its derived function along system, Markov process transition rates and Newton-Leibniz Formula is utilized to construct free claim respectively, be introduced in the weak infinitesimal operators of described Liapunov-Krasovsky functional, and whole Time-varying time-delays interval decomposed is become two subintervals, obtain time-lag system stability criterion;

C2, the time-lag system stability criterion equivalence transformation in step C1 is become to meet the canonical form that generalized eigenvalue method solves, solve time lag and stablize the upper limit.

3. the time-lag power system stability analytical method described in claim 2, is characterized in that, utilizes Markov process transition rates and Newton-Leibniz Formula to construct free claim to be respectively in step C1:

$-x^T\left(t\right)\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{W}_{j}x\left(t\right)=0,$

$-x^T(t-h_t)\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{U}_{j}x(t-h_t)=0,$

$-x^T(t-\stackrel{\&OverBar;}{h})\underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{V}_{j}x(t-\stackrel{\&OverBar;}{h})=0,$

The equation of the improvement free-form curve and surface constructed by Newton-Leibniz Formula is:

$2{\mathrm{\&zeta;}}^T\left(t\right)N[x\left(t\right)-x(t-h_t)-{\&Integral;}_{t-h_t}^t\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}]=0,$

$2{\mathrm{\&zeta;}}^T\left(t\right)L[x(t-h_t)-x(t-\stackrel{\&OverBar;}{h})-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^{t-h_t}\stackrel{\&CenterDot;}{x}\left(s\right)\mathrm{ds}]=0,$

$2{\mathrm{\&zeta;}}^T\left(t\right)M[x\left(t\right)-x(t-\stackrel{\&OverBar;}{h})-{\&Integral;}_{t-\stackrel{\&OverBar;}{h}}^{t}x\left(s\right)\mathrm{ds}]=0,$

Wherein, $\mathrm{\&zeta;}\left(t\right)={\left[\begin{array}{ccc}{x}^T& x^T(t-h_t)& x^T(t-\stackrel{\&OverBar;}{h})\end{array}\right]}^T,$

X (t) ∈ R ⁿthe state vector for time-lag power system,

for state vector x (t) is to the first derivative of time,

H _tfor time lag,

for time lag stablizes the upper limit,

μ is the maximum rate of change of time lag,

W, U and V for treating set matrix,

N, L and M are improvement free-form curve and surface,

π _ijfor the Markov transition probabilities entry of a matrix element of time-lag power system, refer to that system mode is in i state in t, and be in the transitional provavility density of j state in the t+ Δ moment,

Wherein time lag h _twith its first derivative satisfy condition:

$0\&le;h_t\&le;\stackrel{\&OverBar;}{h},$

${\stackrel{\&CenterDot;}{h}}_t\&le;\mathrm{\&mu;}.$

4. the time-lag power system stability analytical method described in claim 2, is characterized in that, described in solve time lag and stablize the upper limit for asking for optimization problem:

min d，

Its constraints is:

$\left[\begin{array}{ccc}{T}_1& 0& 0\\ *& T_2& 0\\ *& *& T_3\end{array}\right]<d\&CenterDot;\left[\begin{array}{ccc}{Z}_1& 0& 0\\ *& Z_2& 0\\ *& *& K\end{array}\right]$ With

$\left[\begin{array}{cccccc}{\mathrm{\&Omega;}}^{\&prime;}& N& L& M& {\stackrel{\&OverBar;}{A}}_i(Z_1+Z_2)& K\\ *& -T_1& 0& 0& 0& 0\\ *& *& -T_1& 0& 0& 0\\ *& *& *& -T_2& 0& 0\\ *& *& *& *& -(T_1+T_2)& 0\\ *& *& *& *& *& -T_3\end{array}\right]<0,$

Wherein, d is optimization aim, and

for time lag stablizes the upper limit,

${\stackrel{\&OverBar;}{A}}_i=\left[\begin{array}{ccc}{A}_i& A_{\mathrm{di}}& 0\end{array}\right],$

A _ifor the state matrix of time-lag power system under operating condition i, and A _i∈ R ^{n × n},

A _difor the delay matrix of time-lag power system under operating condition i, and A _di∈ R ^{n × n},

T ₁, T ₂, T ₃for added martix, and $-Z_1/\stackrel{\&OverBar;}{h}<-T_1,-Z_2/\stackrel{\&OverBar;}{h}<-T_2,-K/\stackrel{\&OverBar;}{h}<-T_3,$

P, Q, R _i, K, Z _i, U _i, V _i, W _ibe and treat set matrix, and P=P ^t> 0, Q=Q ^t>=0, R _i=R _i ^t>=0, K=K ^t> 0, Z _i=Z _i ^t>=0, U _i=U _i ^t>=0, V _i=V _i ^t>=0 and W _i=W _i ^t>=0,

$\mathrm{\&Omega;}={\mathrm{\&Omega;}}^{\&prime;}+\stackrel{\&OverBar;}{h}K,$

$\mathrm{\&Omega;}={\mathrm{\&Omega;}}_1+{\mathrm{\&Omega;}}_2+{\mathrm{\&Omega;}}_2^T,$

${\mathrm{\&Omega;}}_1=\left[\begin{array}{ccc}{P}_i{A}_i+{A_i}^T{P}_i+Q+R_i+\stackrel{\&OverBar;}{h}K+& & \\ \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}(R_j+P_j-W_j)& P_i{A}_{\mathrm{di}}& 0\\ & -(1-\mathrm{\&mu;})Q-& \\ *& \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{U}_j& 0\\ & & -R_i-\\ *& *& \underset{j=1}{\overset{S}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_{\mathrm{ij}}{V}_j\end{array}\right],$

Ω ₂＝[N+M L-N -L-M]，

π _ijfor the Markov transition probabilities entry of a matrix element of time-lag power system, refer to that system mode is in i state in t, and be in the transfer rate of j state in the t+ Δ moment, Δ is the variable quantity of moment t,

μ is the maximum rate of change of time lag,

S is limited mode set,

N, L and M are for improving free-form curve and surface.

5. a time-lag power system stability analytical equipment, comprises data acquisition module, time lag Markov jump system generation module, the time lag upper limit solves module and result output module;

Described data acquisition module is used for generator frequency, merit angle in collection network structural parameters, system, and image data is sent to time lag Markov jump system generation module;

Described Markov jump system generation module utilizes the data configuration fault chains collected, and is combined with Markov process by fault chains and set up time lag Markov jump system state equation;

The described time lag upper limit solves module for according to time lag Markov jump system state equation, the time lag stability criterion improving free-form curve and surface is constructed respectively based on Markov process transition rates and Newton-Leibniz Formula, on this basis, by time lag stability criterion equivalence transformation, generalized eigenvalue method solving system time lag is utilized to stablize the upper limit;

Described result output module stablizes the upper limit for exporting described system time lags.

6. the time-lag power system stability analytical equipment described in claim 5, is characterized in that the described time lag upper limit solves module and comprises time-lag system stability criterion determining unit and standard solves unit, wherein:

Time-lag system stability criterion determining unit, for constructing Liapunov-Krasovsky functional of considering markov saltus step and solving its derived function along system, Markov process transition rates and Newton-Leibniz Formula is utilized to construct free claim respectively, be introduced in the weak infinitesimal operators of described Liapunov-Krasovsky functional, and whole Time-varying time-delays interval decomposed is become two subintervals, obtain time-lag system stability criterion;

Standard solves unit and becomes to meet for the time-lag system stability criterion equivalence transformation that time-lag system stability criterion determining unit obtained the canonical form that generalized eigenvalue method solves, and solves time lag and stablize the upper limit.