CN104933636A - Multi-working-condition power system stability analysis method and device - Google Patents

Abstract

The invention relates to a multi-working-condition power system stability analysis method and device. The device comprises a data acquisition module, a Markov multi-working-condition system generation module, a stability analysis module and a result output module, wherein the data acquisition module is used for acquiring network structure parameters, power flow before and after a system accident, and a generator frequency and a power angle in a system; the Markov multi-working-condition system generation module is used for establishing a discrete Markov multi-working-condition power system model; the stability analysis model is used for determining a stability criterion and judging the system stability; and the result output module is used for outputting a system stability result judged by the stability analysis model. Through adoption of the multi-working-condition power system stability analysis method and device, the stability of a power system under multiple working conditions can be effectively analyzed. Moreover, the method and device are easy to implement; judgments can be made easily; and the goodness of fit with the practical situation is high.

Description

Multi-state method for analyzing stability of power system and device

Technical field

The present invention relates to stability of power system analysis technical field, relate in particular to multi-state stability of power system analysis technical field.

Background technology

Along with the quick growth of electricity needs and the increasingly sophisticated of grid structure, the multi-state stable problem causing topologies change to cause due to power equipment and element fault has become to be affected power system safety and stability and runs one of the most key problem, therefore, the stability studied for the change of electric system multi-state has very important realistic meaning.

In prior art, some technical schemes are had for the analysis of cascading failure and prevention technique, mainly concentrates on the aspects such as cascading failure searching method, method for establishing model and methods of risk assessment.Specifically, (1) cascading failure searching method has: based on determinacy criterion, utilizes analytical method select line and generator outage situation, and calculates the fault mode causing electrical network generation cascading failure; Based on Monte Carlo probability sampling, utilize stochastic simulation method search system fault mode.(2) method for establishing model has: carry out abstract based on Complex System Theory to electric power networks and simplify, build operating mode bang path (Operational Path Analysis, OPA) model, CASCADE model and branching process model, describes cascading failure; Based on Complex Networks Theory, from network structure angle, the ability to bear of research network for various attack and the possibility of cascading failure generation, establish ER model, small-world network model, Watts tectonic model etc., analyze cascading failure in power system.(3) methods of risk assessment has: from load, power supply, electrical network three aspects, and the impact that machine, cutting load and off-the-line cause cascading failure is cut in assessment; Exploitation right repeated factor has considered various factors, reflects the impact of cascading failure on whole electric system.

But, but seldom having in prior art can effectively point factorial cascading failure and the technical scheme of the multi-state stability of power system problem caused, this makes the multi-state electric system caused because of cascading failure easily occur unstability, brings huge economic loss thus.

Summary of the invention

In view of this, object of the present invention is intended to overcome in prior art the shortcoming being difficult to analyze the multi-state stability of power system problem caused because of cascading failure, provides a kind of multi-state method for analyzing stability of power system and device.First, in the uncertain action of consideration protection/isolating switch, under the prerequisite of the random occurrences such as system hardware inefficacy, based on the set of power flow transfer theoretical certainty annuity operating mode, and calculate transition probability matrix between each operating mode, on this basis, set up the discrete Markov multi-state electric power system model considering cascading failure, structure is containing the Lyapunov function of this model, utilize process of iteration to derive and meet the Robust Stochastic Stability criterion of interference attenuation degree γ, and by solving LMI (Linear Matrix Inequalities, LMI) the feasibility problems judgement system stability in.In a specific embodiment, correctness and the validity of multi-state method for analyzing stability of power system of the present invention and device is demonstrated based on the multi-state Power System Time Domain Simulation Under of cascading failure.

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

A kind of multi-state stability of power system analytical equipment, described device comprises the data acquisition module, markov multi-state system generation module, stability analysis module and the result output module that are linked in sequence, wherein,

Image data for generator frequency and merit angle in trend, system before and after collection network structural parameters, systematic failures, and is sent to markov multi-state system generation module by data acquisition module;

The data that markov multi-state system generation module is used for according to gathering set up discrete Markov multi-state electric power system model;

Stability analysis module, for building the Lyapunov function containing markov multi-state electric power system model, determines stability criterion, and judgement system stability;

Result output module is used for the system stability result that output stability analysis module differentiates.

After described stability analysis module construction Lyapunov function, the Robust Stochastic Stability criterion meeting the requirement of interference attenuation degree is as described stability criterion to utilize process of iteration to determine.

Or after described stability analysis module determination stability criterion, carry out judgement system stability by the feasibility problems asking for planning solving.

A kind of multi-state method for analyzing stability of power system, comprises step:

Generator frequency and merit angle in trend, system before and after A, collection network structural parameters, systematic failures;

B, according to gather data set up discrete Markov multi-state electric power system model;

C, structure, containing the Lyapunov function of discrete Markov multi-state electric power system model, determine stability criterion, and judgement system stability;

D, output system stability distinguishing result.

Wherein, described discrete Markov multi-state electric power system model is:

$\left\{\begin{array}{c}{x}_{k+1}=\left(A\right(s_k)+{\mathrm{\ΔA}}_k\left(s_k\right))x_k+B\left(s_k\right)u_k+G\left(s_k\right){\mathrm{\ω}}_k\\ z_k=C\left(s_k\right)x\left(t\right)+D\left(s_k\right)u_k+L\left(s_k\right){\mathrm{\ω}}_k\end{array}\right.,$

Wherein x _k∈ R ⁿfor state vector,

U _k∈ R ^pfor control inputs vector,

Z _k∈ R ^rfor controlling output vector,

Process noise ω _kfor zero mean noise sequence,

S (t), t>=0} is at finite space S={1,2 ..., the Markov chain of value in l}, each operating condition that corresponding cascading failure may exist, its state probability p _ijfor:

$\mathrm{Pr}\{s_{k+1}=j|s_k=i\}=p_{ij},\underset{j=1}{\Σ}{p}_{ij}=1(p_{ij}\≥0),$

Δ A _k(s _k) be uncertain parameter, meet: Δ A _k(s _k)=H _if _k(i) M _i,

Wherein H _iand M _ifor known matrix, real matrix F _ki () reflects system Structure with uncertain parameters information, satisfy condition: $F_k^T\left(i\right)F_k\left(i\right)\≤I.$

In addition, described structure comprises containing the Lyapunov function of discrete Markov multi-state electric power system model:

V _k(M _k(s _k),s _k)＝tr[M _k(s _k)P(s _k)](P(s _k)>0)，

State correlation matrix M _k(s _k) be: wherein,

P=Q ^-1, Q is positive definite symmetric matrices undetermined.

Wherein, describedly determine that stability criterion comprises:

Disturbance dough softening γ meets:

$\underset{k=0}{\overset{N}{\Σ}}E\[\left|\right|z_k|{|}^2\]\≤{\mathrm{\γ}}^2\underset{k=0}{\overset{N}{\Σ}}E\[|\left|{\mathrm{\ω}}_k\right|{|}^2\],$

Wherein N is the infinitely-great positive integer of convergence,

Then described multi-state electric system robust convergency.

Or, describedly determine that stability criterion comprises:

Work as u _k=0, ω _k=0, the uncertain Δ A allowed if all _imeet:

$\underset{N\→\mathrm{\∞}}{\lim }E\left\{\underset{k=0}{\overset{N}{\Σ}}\right|x_k{|}^2|x_0,s_0\}\≤\tilde{M}(x_0,s_0),$

Then described multi-state electric system robust convergency.

The method of judgement system stability comprises:

The feasibility problems asking for planning solving carrys out judgement system stability, and wherein said planning solving is:

$\left[\begin{array}{cccc}-Q_i& 0& Q_i{A}_i^T{\mathrm{\&Omega;}}_i& Q_i{C}_i^T\\ *& -{\mathrm{\&gamma;}}^{2}I& G_i^T{\mathrm{\&Omega;}}_i& L_i^T\\ *& *& -\mathrm{\&psi;}& 0\\ *& *& *& -I\end{array}\right]<0,$

${\mathrm{\&Omega;}}_i=\left[\begin{array}{cccc}\sqrt{p_{i1}}I& \sqrt{p_{i2}}I& \mathrm{...}& \sqrt{p_{il}}I\end{array}\right],$

ψ＝diag(Q ₁,Q ₂,…,Q _l)，

Wherein, for each s (t)=i ∈ S, note A (s _k), B (s _k), G (s _k), C (s _k), D (s _k), L (s _k) be respectively A _i, B _i, G _i, C _i, D _i, L _i,

If there is one group of positive definite symmetric matrices Q _i>0, i ∈ S, makes described planning solving set up, then as u (t) ≡ 0, and described multi-state electric system robust convergency, and meet the requirement of the disturbance dough softening.

By adopting multi-state method for analyzing stability of power system of the present invention and device, following technique effect can be realized.

(1) based on power flow transfer theory, electric system is divided into different operating conditions, sets up Discrete Markov Model accordingly, therefore, it is possible to stability of power system under effectively analyzing multi-state.

(2) the multi-state stability problem of electric system is converted into feasibility Solve problems, without the need to obtaining running track, therefore multi-state method for analyzing stability of power system of the present invention and device easily realize, and differentiate effectively simple.

(3) to time-dependent system under single operating mode, may stablize when considering multi-state change; Otherwise, to systems stabilisation under single operating mode, may unstability when considering multi-state change.The stability criterion of multi-state method for analyzing stability of power system of the present invention and device mates these actual conditions more, therefore accurate and effective more.

Accompanying drawing explanation

Fig. 1 is the structural representation of multi-state stability of power system analytical equipment in the specific embodiment of the invention.

Fig. 2 is for an application scenarios---the 16 machine 68 node configuration of power networks that multi-state method for analyzing stability of power system and device in the specific embodiment of the invention adopt are described.

Fig. 3 to adopt in the specific embodiment of the invention generator G1-G13 relative merit angular motion state response diagram after multi-state method for analyzing stability of power system and device.

Fig. 4 to adopt in the specific embodiment of the invention generator relative merit angular motion state response diagram after multi-state method for analyzing stability of power system and device.

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.).

Before introducing the specific embodiment of the present invention, first introduce the principle of multi-state method for analyzing stability of power system of the present invention and device, and analyze in conjunction with technical scheme of the present invention and calculate.

When electric 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.Suppose line status r _jonly comprise 0,1 two state, represent operation and the disconnection of circuit respectively, introduce α _ij, β _ij, γ _ijthe power flow changing rate of circuit j after excising respectively as m level faulty line i, the overload order of severity and the evaluation index of coupled relation between change of power flow and circuit original trend, be expressed as follows respectively:

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

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

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

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 _{j max}represent that circuit j allows the strength of current flow through.

Introduce weight factor ω _j, it represents trend proportion in total trend on circuit j:

${\mathrm{\&omega;}}_j=\frac{\left|S_{j\left(t_b\right)}\right|}{\underset{k=1}{\overset{1}{\&Sigma;}}\left|S_{i\left(t_b\right)}\right|}---\left(4\right)$

Under m level fault, the state transition probability taking into account relay protection device incorrect operation can be expressed as:

p _ij＝p _{ij_ori}[(1-p _r)(1-p _c)+p _err] (5)

$p_{ij\_ori}=\frac{D_{ij}}{\underset{j=1}{\overset{l}{\&Sigma;}}{D}_{ij}-D_{ii}}---\left(6\right)$

D _ij＝ω _kD _iα _ijβ _ijγ _ij(7)

$D_i=\left|\frac{S_i}{S_{imax}}\right|---\left(8\right)$

In formula: p _rfor the tripping probability of circuit overload protection, p _cfor circuit overload protects the tripping probability of corresponding isolating switch; p _errfor the disconnection probability of circuit under relay protection device incorrect operation state; p _{ij_ori}for disregarding the state transition probability of relay protection device incorrect operation.

Due in cascading failure, after every level fault occurs, the power flow transfer amount of circuit is only relevant to upper level fault network structural parameters, and therefore, the calculating of line status transition probability is also only relevant to upper level failure condition.Based on this, discrete Markov system model can be adopted considering that modeling is carried out in the multi-state electric system of cascading failure.

Consider that the multi-state electric power system model of cascading failure can adopt discrete Markov jump system to be expressed as:

$\left\{\begin{array}{c}{x}_{k+1}=\left(A\right(s_k)+{\mathrm{\&Delta;A}}_k\left(s_k\right))x_k+B\left(s_k\right)u_k+G\left(s_k\right){\mathrm{\&omega;}}_k\\ z_k=C\left(s_k\right)x\left(t\right)+D\left(s_k\right)u_k+L\left(s_k\right){\mathrm{\&omega;}}_k\end{array}\right.---\left(9\right)$

Wherein, x _k∈ R ⁿit is state vector; u _k∈ R ^pit is control inputs vector; z _k∈ R ^rcontrol output vector; Process noise ω _kfor zero mean noise sequence.S (t), t>=0} is at finite space S={1,2 ..., the Markov chain of value in l}, each operating condition that may exist in its corresponding cascading failure, describes the evolutionary process of system condition with fault, its state probability p _ijbe expressed as:

$\begin{array}{cc}\mathrm{Pr}\{s_{k+1}=j|s_k=i\}=p_{ij}& \underset{j=1}{\&Sigma;}{p}_{ij}=1(p_{ij}\&GreaterEqual;0)\end{array}---\left(10\right)$

For each s (t)=i ∈ S, note A (s _k), B (s _k), G (s _k), C (s _k), D (s _k), L (s _k) be expressed as A _i, B _i, G _i, C _i, D _i, L _i, and uncertain parameter meets matching condition:

ΔA _k(s _k)＝ΔA _k(i)＝H _iF _k(i)M _i(11)

Wherein, H _iand M _ifor known matrix, real matrix F (i, t) has reacted the structural information of system uncertain parameter, meets:

$F_k^T\left(i\right)F_k\left(i\right)\&le;I---\left(12\right)$

Formula (9)-(12) establish the multi-state electric power system model considering cascading failure, and the present invention will further according to this model inference stability criterion, and the stability under analytic system multi-state situation of change.

For the multi-state electric system that formula (9) represents, work as u _k=0, ω _k=0, the uncertain Δ A allowed if all _imeet:

$\underset{N\&RightArrow;\mathrm{\&infin;}}{\lim }E\left\{\underset{k=0}{\overset{N}{\&Sigma;}}\right|x_k{|}^2|x_0,s_0\}\&le;\tilde{M}(x_0,s_0)---\left(13\right)$

The then power train completely robust convergency that describes of through type (9).

, if there is one group of positive definite symmetric matrices Q in prerequisite 1: given normal number γ >0 _i>0, i ∈ S, makes to set up as next group MATRIX INEQUALITIES:

$\left[\begin{array}{cccc}-Q_i& 0& Q_i{A}_i^T{\mathrm{\&Omega;}}_i& Q_i{C}_i^T\\ *& -{\mathrm{\&gamma;}}^{2}I& G_i^T{\mathrm{\&Omega;}}_i& L_i^T\\ *& *& -\mathrm{\&psi;}& 0\\ *& *& *& -I\end{array}\right]<0---\left(14\right)$

${\mathrm{\&Omega;}}_i=\left[\begin{array}{cccc}\sqrt{p_{i1}}I& \sqrt{p_{i2}}I& \mathrm{...}& \sqrt{p_{il}}I\end{array}\right]---\left(15\right)$

ψ＝diag(Q ₁,Q ₂,…,Q _l) (16)

Then as u (t) ≡ 0, by the electric system robust convergency described by formula (9), and meet disturbance dough softening γ, that is:

$\underset{k=0}{\overset{N}{\&Sigma;}}E\&lsqb;\left|\right|z_k|{|}^2\&rsqb;\&le;{\mathrm{\&gamma;}}^2\underset{k=0}{\overset{N}{\&Sigma;}}E\&lsqb;|\left|{\mathrm{\&omega;}}_k\right|{|}^2\&rsqb;---\left(17\right)$

Illustrate: from formula (14), for i ∈ S, following planning solving is set up:

$\left[\begin{array}{cc}-Q_i& Q_i{A}_i^T{\mathrm{\&Omega;}}_i\\ *& -\mathrm{\&psi;}\end{array}\right]<0---\left(18\right)$

Therefore, under only need determining the condition that formula (18) is set up, the electric system Stochastic stable described by formula (9), can illustrate that the Stochastic stable criterion in prerequisite 1 is set up.

Order by left and right for formula (18) each multiply matrix diag (P ₁, I) have:

$\left[\begin{array}{cc}-P_i& A_i^T{\mathrm{\&Omega;}}_i\\ *& -\mathrm{\&Sigma;}\end{array}\right]<0---\left(19\right)$

Wherein,

$\mathrm{\&Sigma;}=diag(P_1^{-1},P_2^{-1},...,P_l^{-1})---\left(20\right)$

, from formula (19), there is scalar δ >0, following formula set up in application Shu Er complement fixed reason:

$A_i^T{\mathrm{\&Omega;}}_i{\mathrm{\&Sigma;}}^{-1}{\mathrm{\&Omega;}}_i^T{A}_i-P_i+\mathrm{\&delta;}I<0---\left(21\right)$

State correlation matrix M _k(s _k) be:

$M_k\left(s_k\right)=E\&lsqb;x_k\left(s_k\right)x_k^T\left(s_k\right)\&rsqb;---\left(22\right)$

Structure Lyapunov function:

V _k(M _k(s _k),s _k)＝tr[M _k(s _k)P(s _k)](P(s _k)>0) (23)

Then for i ∈ S, have:

$V_{k+1}\left(M_{k+1}\right(i),i)=tr\&lsqb;A_i{M}_k\left(i\right)A_i^T{P}_i\&rsqb;=tr\&lsqb;M_k\left(i\right)A_i{\mathrm{\&Omega;}}_i{\&Sigma;}^{-1}{\mathrm{\&Omega;}}_i^T{A}_i^T\&rsqb;<tr\&lsqb;M_k\left(i\right)P_i\&rsqb;=V_k\left(M_k\right(i),i)---\left(24\right)$

Can be obtained by formula (24):

$\frac{E\{V_{k+1}\left(M_{k+1}\right(s_{k+1}),s_{k+1})|x_k,s_k\}-V_k\left(M_k\right(s_k),s_k)}{V_k\left(M_k\right(s_k),s_k)}=\frac{E\{V_{k+1}\left(M_{k+1}\right(s_{k+1}),s_{k+1})|x_k,s_k\}}{V_k\left(M_k\right(s_k),s_k)}=\mathrm{\&alpha;}-1<-\underset{s_k\&Element;S}{\min }\left(\frac{{\mathrm{\&lambda;}}_{\min }(\&Sigma;)}{{\mathrm{\&lambda;}}_{\min }\left(P\right(s{k}_{}))}\right)---\left(25\right)$

Wherein,

$\mathrm{\&alpha;}=\frac{E\{V_{k+1}\left(M_{k+1}\right(s_{k+1}),s_{k+1})|x_k,s_k\}}{V_k\left(M_k\right(s_k),s_k)}---\left(26\right)$

Consider:

$\begin{array}{c}\frac{E\{V_{k+1}\left(M_{k+1}\right(s_{k+1}),s_{k+1})|x_k,s_k\}}{V_k\left(M_k\right(s_k),s_k)}>0\\ \underset{s_k\&Element;S}{\min }\left(\frac{{\mathrm{\&lambda;}}_{\min }(\&Sigma;)}{{\mathrm{\&lambda;}}_{\min }\left(P\right(s_k))}\right)>0\end{array}---\left(27\right)$

Therefore, α satisfies condition 1> α >0, and:

E{V _k+1(M _k+1(s _k+1),s _k+1)|x _k,s _k}

<αV _k(M _k(s _k),s _k) (28)

By 0 to k, iteration is carried out to formula (28), has:

E{V _k+1(M _k+1(s _k+1),s _k+1)|x ₀,s ₀}<α ^kV ₀(M ₀(s ₀),s ₀) (29)

Can be obtained by formula (29):

$\begin{array}{c}E\{\underset{k=0}{\overset{N}{\&Sigma;}}{V}_k\left(M_k\right(s_k),s_k)|x_0,s_0\}<(1+\mathrm{\&alpha;}+...+{\mathrm{\&alpha;}}^N)V_0\left(M_0\right(s_0),s_0)\\ <\frac{1}{1-\mathrm{\&alpha;}}{V}_0\left(M_0\right(s_0),s_0)\end{array}---\left(30\right)$

From formula (30):

$E\left\{\underset{k=0}{\overset{N}{\&Sigma;}}\right|\left|x_k\right|{|}^2|x_0,s_0\}<\frac{\underset{s_k\&Element;S}{min}{\mathrm{\&lambda;}}_{min}\left(P\right(s_k))}{1-\mathrm{\&alpha;}}{V}_0\left(M_0\right(s_0),s_0)---\left(31\right)$

Note:

$\tilde{N}(x_0,s_0)=\frac{\underset{s_k\&Element;S}{min}{\mathrm{\&lambda;}}_{min}\left(P\right(s_k))}{1-\mathrm{\&alpha;}}{V}_0\left(M_0\right(s_0),s_0)---\left(32\right)$

Can be obtained by formula (31) and formula (32):

$\underset{N\&RightArrow;\mathrm{\&infin;}}{\lim }E\left\{\underset{k=0}{\overset{N}{\&Sigma;}}\right|\left|x_k\right|{|}^2|x_0,s_0\}<\tilde{N}(x_0,s_0)---\left(33\right)$

The i.e. electric system Stochastic stable that describes of formula (9), the system when ω (t) ≠ 0 that the following describes has dough softening γ.

To formula (14) premultiplication and right multiply matrix diag (P _i, I, I, I), can as lower inequality:

$\left[\begin{array}{cccc}-P_i& 0& A_i^T{\mathrm{\&Omega;}}_i& C_i^T\\ *& -{\mathrm{\&gamma;}}^{2}I& G_i^T{\mathrm{\&Omega;}}_i& L_i^T\\ *& *& -\mathrm{\&Sigma;}& 0\\ *& *& *& -I\end{array}\right]<0---\left(34\right)$

Application Shu Er complement fixed reason, formula (14) can be equivalent to:

$\begin{array}{c}\left[\begin{array}{c}{A}_i^T{\mathrm{\&Omega;}}_i{\mathrm{\&Sigma;}}^{-1}{\mathrm{\&Omega;}}_i^T{A}_i-P_i+C_i^T{C}_i\\ *\end{array}\right.\\ \left.\begin{array}{c}{A}_i^T{\mathrm{\&Omega;}}_i{\mathrm{\&Sigma;}}^{-1}{\mathrm{\&Omega;}}_i^T{G}_i+C_i^T{L}_i\\ G_i^T{\mathrm{\&Omega;}}_i{\mathrm{\&Sigma;}}^{-1}{\mathrm{\&Omega;}}_i^T{G}_i+L_i^T{L}_i-{\mathrm{\&gamma;}}^{2}I\end{array}\right]<0\end{array}---\left(35\right)$

Structure Lyapunov function:

$V_k(x_k,s_k)=x_k^T{P}_i{x}_k,(P>0,i\&Element;S)---\left(36\right)$

Definition target function is:

$J_N=E\&lsqb;\underset{k=0}{\overset{N}{\&Sigma;}}\left(\right|\left|z_k\right|{|}^2-{\mathrm{\&gamma;}}^2\left|\right|{\mathrm{\&omega;}}_k\left|{|}^2\right)\&rsqb;---\left(37\right)$

Can be obtained by formula (37):

$\begin{array}{c}{J}_N=E\left\{\underset{k=0}{\overset{N}{\&Sigma;}}\left(\right|\left|z_k\right|{|}^2+E\&lsqb;V_{k+1}(x_{k+1},s_{k+1})\right|x_k,s_k\&rsqb;\\ -V_k(x_k,s_k)-{\mathrm{\&gamma;}}^2\left|\right|{\mathrm{\&omega;}}_k\left|{|}^2\right)\}-E\&lsqb;V_{N+1}(x_{N+1},s_{N+1})\&rsqb;\\ \&le;E\left\{\underset{k=0}{\overset{N}{\&Sigma;}}\left(\right|\left|z_k\right|{|}^2+E\&lsqb;V_{k+1}(x_{k+1},s_{k+1})\right|x_k,s_k\&rsqb;\\ -V_k(x_k,s_k)-{\mathrm{\&gamma;}}^2\left|\right|{\mathrm{\&omega;}}_k\left|{|}^2\right)\}\end{array}---\left(38\right)$

For i ∈ S, can be obtained by formula (35):

$\begin{array}{c}{J}_N\&le;E\{\underset{k=0}{\overset{N}{\&Sigma;}}{\&lsqb;C_i{x}_k+L_i{\mathrm{\&omega;}}_k\&rsqb;}^T\&lsqb;C_i{x}_k+L_i{\mathrm{\&omega;}}_k\&rsqb;\\ +E\&lsqb;{(A_i{x}_k+G_i{\mathrm{\&omega;}}_k)}^{T}P\left(s_{k+1}\right)(A_i{x}_k+G_i{\mathrm{\&omega;}}_k)\\ -x_x^{T}P\left(s_k\right)x_k|x_k,s_k=i\&rsqb;-{\mathrm{\&gamma;}}^2{\mathrm{\&omega;}}_k^T{\mathrm{\&omega;}}_k\}\\ =\left[\begin{array}{cc}{x}_x^T& {\mathrm{\&omega;}}_k^T\end{array}\right]\left[\begin{array}{c}{A}_i^T{\mathrm{\&Omega;}}_i{\mathrm{\&Sigma;}}^{-1}{\mathrm{\&Omega;}}_i^T{A}_i-P_i+C_i^T{C}_i\\ *\end{array}\right.\\ \left.\begin{array}{c}{A}_i^T{\mathrm{\&Omega;}}_i{\mathrm{\&Sigma;}}^{-1}{\mathrm{\&Omega;}}_i^T{G}_i+C_i^T{L}_i\\ G_i^T{\mathrm{\&Omega;}}_i{\mathrm{\&Sigma;}}^{-1}{\mathrm{\&Omega;}}_i^T{G}_i+L_i^T{L}_i-{\mathrm{\&gamma;}}^{2}I\end{array}\right]\left[\begin{array}{c}{x}_k\\ {\mathrm{\&omega;}}_k\end{array}\right]\\ <0\end{array}---\left(39\right)$

$\underset{k=0}{\overset{N}{\&Sigma;}}E\&lsqb;\left|\right|z_k|{|}^2\&rsqb;\&le;{\mathrm{\&gamma;}}^2\underset{k=0}{\overset{N}{\&Sigma;}}E\&lsqb;|\left|{\mathrm{\&omega;}}_k\right|{|}^2\&rsqb;---\left(40\right)$

Namely the multi-state electric system that formula (9) describes meets disturbance dough softening γ requirement.

In addition, the stability criterion of the multi-state electric system described by formula (9) can be converted into following optimization problem and represent:

$\begin{array}{c}\underset{Q_i,\mathrm{\&gamma;}}{\min }t\\ s.t.\left(14\right)-\left(16\right)\end{array}---\left(41\right)$

Feasibility problems method for solving is utilized to solve planning solving, if there is t<0 and meet positive definite matrix P shown in prerequisite 1 _i>0, then the multi-state electric system robust convergency that describes of formula (9), and meet disturbance dough softening γ requirement; Otherwise system is unstable.

Therefore, as shown in Figure 1, disclose a kind of multi-state stability of power system analytical equipment in the specific embodiment of the invention, described device comprises the data acquisition module, markov multi-state system generation module, stability analysis module and the result output module that are linked in sequence, wherein

Image data for generator frequency and merit angle in trend, system before and after collection network structural parameters, systematic failures, and is sent to markov multi-state system generation module by data acquisition module;

The data that markov multi-state system generation module is used for according to gathering set up discrete Markov multi-state electric power system model;

Stability analysis module, for building the Lyapunov function containing markov multi-state electric power system model, determines stability criterion, and judgement system stability;

Result output module is used for the system stability result that output stability analysis module differentiates.

Therefore, electric system can be divided into different operating conditions based on power flow transfer theory by multi-state stability of power system analytical equipment of the present invention, sets up Discrete Markov Model accordingly, therefore, it is possible to stability of power system under effectively analyzing multi-state.

Wherein, after described stability analysis module construction Lyapunov function, the Robust Stochastic Stability criterion meeting the requirement of interference attenuation degree is as stability criterion to utilize process of iteration to determine.

And after described stability analysis module determination stability criterion, carry out judgement system stability by the feasibility problems asking for planning solving.

Corresponding with aforementioned multi-state stability of power system analytical equipment, also disclose a kind of multi-state method for analyzing stability of power system in the specific embodiment of the invention, it comprises step:

Generator frequency and merit angle in trend, system before and after A, collection network structural parameters, systematic failures;

B, according to gather data set up discrete Markov multi-state electric power system model;

C, structure, containing the Lyapunov function of discrete Markov multi-state electric power system model, determine stability criterion, and judgement system stability;

D, output system stability distinguishing result.

Wherein said discrete Markov multi-state electric power system model is:

$\left\{\begin{array}{c}{x}_{k+1}=\left(A\right(s_k)+{\mathrm{\&Delta;A}}_k\left(s_k\right))x_k+B\left(s_k\right)u_k+G\left(s_k\right){\mathrm{\&omega;}}_k\\ z_k=C\left(s_k\right)x\left(t\right)+D\left(s_k\right)u_k+L\left(s_k\right){\mathrm{\&omega;}}_k\end{array}\right.,$

X _k∈ R ⁿfor state vector,

U _k∈ R ^pfor control inputs vector,

Z _k∈ R ^rfor controlling output vector,

Process noise ω _kfor zero mean noise sequence,

S (t), t>=0} is at finite space S={1,2 ..., the Markov chain of value in l}, each operating condition that corresponding cascading failure may exist, its state probability p _ijfor:

$\mathrm{Pr}\{s_{k+1}=j|s_k=i\}=p_{ij},\underset{j=1}{\&Sigma;}{p}_{ij}=1(p_{ij}\&GreaterEqual;0),$

Δ A _k(s _k) be uncertain parameter, meet: Δ A _k(s _k)=H _if _k(i) M _i,

Wherein H _iand M _ifor known matrix, real matrix F _ki () reflects system Structure with uncertain parameters information, satisfy condition: $F_k^T\left(i\right)F_k\left(i\right)\&le;I.$

In addition, the Lyapunov function of described structure discrete Markov multi-state electric power system model comprises:

V _k(M _k(s _k),s _k)＝tr[M _k(s _k)P(s _k)](P(s _k)>0)，

State correlation matrix M _k(s _k) be: wherein,

P=Q ^-1, Q is positive definite symmetric matrices undetermined.

Describedly determine that stability criterion comprises:

Disturbance dough softening γ meets:

$\underset{k=0}{\overset{N}{\&Sigma;}}E\&lsqb;\left|\right|z_k|{|}^2\&rsqb;\&le;{\mathrm{\&gamma;}}^2\underset{k=0}{\overset{N}{\&Sigma;}}E\&lsqb;|\left|{\mathrm{\&omega;}}_k\right|{|}^2\&rsqb;,$

Wherein N is the infinitely-great positive integer of convergence,

Then described multi-state electric system robust convergency.

Or describedly determine that stability criterion is expressed as:

Work as u _k=0, ω _k=0, the uncertain Δ A allowed if all _imeet:

$\underset{N\&RightArrow;\mathrm{\&infin;}}{\lim }E\left\{\underset{k=0}{\overset{N}{\&Sigma;}}\right|x_k{|}^2|x_0,s_0\}\&le;\tilde{M}(x_0,s_0),$

Then described multi-state electric system robust convergency.

In addition, the method for judgement system stability comprises:

The feasibility problems asking for planning solving carrys out judgement system stability, and wherein said planning solving is:

$\left[\begin{array}{cccc}-Q_i& 0& Q_i{A}_i^T{\mathrm{\&Omega;}}_i& Q_i{C}_i^T\\ *& -{\mathrm{\&gamma;}}^{2}I& G_i^T{\mathrm{\&Omega;}}_i& L_i^T\\ *& *& -\mathrm{\&psi;}& 0\\ *& *& *& -I\end{array}\right]<0,$

${\mathrm{\&Omega;}}_i=\left[\begin{array}{cccc}\sqrt{p_{i1}}I& \sqrt{p_{i2}}I& \mathrm{...}& \sqrt{p_{il}}I\end{array}\right],$

ψ＝diag(Q ₁,Q ₂,…,Q _l)，

Wherein, for each s (t)=i ∈ S, note A (s _k), B (s _k), G (s _k), C (s _k), D (s _k), L (s _k) be respectively A _i, B _i, G _i, C _i, D _i, L _i,

If there is one group of positive definite symmetric matrices Q _i>0, i ∈ S, makes described planning solving set up, then as u (t) ≡ 0, and described multi-state electric system robust convergency, and meet the requirement of the disturbance dough softening.

Below by way of a more concrete application scenarios, the multi-state electric system self-adaptation control method of the specific embodiment of the invention and the technique effect of device are described.

The present embodiment carries out the explanation of technical scheme based on IEEE16 machine 68 node New England-New York interacted system, and the structure of this electric system as shown in Figure 2.

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%.

If initial trigger fault is circuit 4-14, fault is out of service, and its load is transferred to residue circuit, utilizes formula (1)-(8) to calculate the state transition probability p of residue circuit _ij, this probability is after fault at the corresponding levels in cascading failure occurs and remains the probability that subordinate's fault occurs circuit.After giving the stoppage in transit of circuit 4-14 fault in following table, the state transition probability of part circuit, due to when prime failure path is consistent, transition probability is larger, the probability that this circuit becomes next stage cascading failure circuit is also larger, therefore, chooses the system condition that the larger part line fault of state transition probability is corresponding, as the state set of discrete Markov system, less state transition probability is regarded as without line fault.

In addition, the rear system condition set of circuit 4-14 fault stoppage in transit is contained in following table.

After circuit 4-14 fault is stopped transport, be denoted as S={1,2 ..., 8}, the state matrix under utilizing modal analysis method to obtain the corresponding operating mode of 16 machine system, then by SMA method depression of order, the minimum damping ratio situation after depression of order under each operating mode is as shown in the table:

From this table, there is single unstable period 3 and operating mode 6 in system.But when system changes to operating mode 1 by normal operating condition, then when changing to operating mode 3 or operating mode 6, multi-state unstability does not occur, as shown in Figure 3, the relative merit angular difference between generator G1 and G13, tends towards stability after the short time waves relatively.Known by planning solving feasibility problems method for solving, the planning solving constructed by prerequisite 1 is set up, and there is positive definite matrix P _i>0, therefore, when considering that multi-state changes, electric system robust convergency, consistent with simulation results.

Show thus, for the multi-state system instability that cascading failure impact is lower, utilize eigenvalue Method etc. only for the method for analyzing stability of single operating mode, possibly correctly cannot to judge system stability, even occur diametrically opposite conclusion.

In another kind of situation, if initial trigger fault is circuit 1-31, fault is out of service, and its load is transferred to residue circuit, utilizes formula (1)-(8) to calculate residue line status transition probability p _ij.After following table gives the stoppage in transit of circuit 1-31 fault, the state transition probability of part circuit, choose the system condition that the larger part line fault of state transition probability is corresponding, as the state set of discrete Markov system, less state transition probability is regarded as without line fault.

Correspondingly, after circuit 1-31 fault is stopped transport, stability during system subordinate circuit generation cascading failure changes, and selected operating mode set is as shown in the table, is denoted as S={1,2,3,4}.

Systematic state transfer probability matrix is:

$p=\left[\begin{array}{cccc}0.0155& 0.5957& 0.3784& 0.0105\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],$

State matrix under utilizing modal analysis method to obtain the corresponding operating mode of electric system in Fig. 2, and utilize SMA method depression of order, the minimum damping ratio situation after depression of order under each operating mode is as shown in the table:

In table, each operating mode of system is single stable operating mode.But, when system changes to operating mode 1 by normal operating condition, then when changing to other operating mode, system unstability, as shown in Figure 4, the relative merit angular difference unstability between generator G1 and G13 and G1 and G11.Known by the feasibility problems method for solving of planning solving, the planning solving constructed by prerequisite 1 is set up, but, owing to there is nonpositive definite matrix P _i, therefore, when considering that multi-state changes, system robust is unstable at random, consistent with simulation results.Show thus, for the multi-state system instability under cascading failure impact, the single operating mode of system is all stablized when can not ensure that multi-state changes and is still stablized.

Can being found by the analysis of above embodiment, by adopting multi-state method for analyzing stability of power system of the present invention and device, following technique effect can be realized.

(1) based on power flow transfer theory, electric system is divided into different operating conditions, sets up Discrete Markov Model accordingly, therefore, it is possible to stability of power system under effectively analyzing multi-state.

(2) the multi-state stability problem of electric system is converted into feasibility Solve problems, without the need to obtaining running track, therefore multi-state method for analyzing stability of power system of the present invention and device operand little, differentiate simple effectively.

(3) to time-dependent system under single operating mode, may stablize when considering multi-state change; Otherwise, to systems stabilisation under single operating mode, may unstability when considering multi-state change.The stability criterion of multi-state method for analyzing stability of power system of the present invention and device mates these actual conditions more.

It should be noted that; above-mentioned embodiment is only the present invention's preferably embodiment; the restriction to distance protection scope of the present invention can not be understood as, not depart under concept thereof of the present invention, with modification, distance protection scope of the present invention is all belonged to any subtle change that the present invention does.

Claims (9)

1. a multi-state stability of power system analytical equipment, described device comprises the data acquisition module, markov multi-state system generation module, stability analysis module and the result output module that are linked in sequence, wherein,

Image data for generator frequency and merit angle in trend, system before and after collection network structural parameters, systematic failures, and is sent to markov multi-state system generation module by data acquisition module;

The data that markov multi-state system generation module is used for according to gathering set up discrete Markov multi-state electric power system model;

Stability analysis module, for building the Lyapunov function containing markov multi-state electric power system model, determines stability criterion, and judgement system stability;

Result output module is used for the system stability result that output stability analysis module differentiates.

2. according to the multi-state stability of power system analytical equipment described in claim 1, it is characterized in that, after described stability analysis module construction Lyapunov function, the Robust Stochastic Stability criterion meeting the requirement of interference attenuation degree is as described stability criterion to utilize process of iteration to determine.

3. according to the multi-state stability of power system analytical equipment described in claim 1, it is characterized in that, after described stability analysis module determination stability criterion, carry out judgement system stability by the feasibility problems asking for planning solving.

4. a multi-state method for analyzing stability of power system, comprises step:

Generator frequency and merit angle in trend, system before and after A, collection network structural parameters, systematic failures;

B, according to gather data set up discrete Markov multi-state electric power system model;

C, structure, containing the Lyapunov function of discrete Markov multi-state electric power system model, determine stability criterion, and judgement system stability;

D, output system stability distinguishing result.

5. according to the multi-state method for analyzing stability of power system described in claim 4, it is characterized in that, described discrete Markov multi-state electric power system model is:

$\left\{\begin{array}{c}{x}_{k+1}=\left(A\right(s_k)+{\mathrm{\&Delta;A}}_k\left(s_k\right))x_k+B\left(s_k\right)u_k+G\left(s_k\right){\mathrm{\&omega;}}_k\\ z_k=C\left(s_k\right)x\left(t\right)+D\left(s_k\right)u_k+L\left(s_k\right){\mathrm{\&omega;}}_k\end{array}\right.,$

Wherein x _k∈ R ⁿfor state vector,

U _k∈ R ^pfor control inputs vector,

Z _k∈ R ^rfor controlling output vector,

Process noise ω _kfor zero mean noise sequence,

S (t), t>=0} is at finite space S={1,2 ..., the Markov chain of value in l}, each operating condition that corresponding cascading failure may exist, its state probability p _ijfor:

$\mathrm{Pr}\{s_{k+1}=j|s_k=i\}=p_{ij},\underset{j=1}{\&Sigma;}{p}_{ij}=1(p_{ij}\&GreaterEqual;0),$

Δ A _k(s _k) be uncertain parameter, meet: Δ A _k(s _k)=H _if _k(i) M _i,

Wherein H _iand M _ifor known matrix, real matrix F _ki () reflects system Structure with uncertain parameters information, satisfy condition:

6. according to the multi-state method for analyzing stability of power system described in claim 5, it is characterized in that, described structure comprises containing the Lyapunov function of discrete Markov multi-state electric power system model:

V _k(M _k(s _k),s _k)＝tr[M _k(s _k)P(s _k)](P(s _k)>0)，

State correlation matrix M _k(s _k) be: wherein,

P=Q ^-1, Q is positive definite symmetric matrices undetermined.

7. according to the multi-state method for analyzing stability of power system described in claim 5, it is characterized in that, describedly determine that stability criterion comprises:

Disturbance dough softening γ meets:

$\underset{k=0}{\overset{N}{\&Sigma;}}E\&lsqb;\left|\right|z_k|{|}^2\&rsqb;\&le;{\mathrm{\&gamma;}}^2\underset{k=0}{\overset{N}{\&Sigma;}}E\&lsqb;|\left|{\mathrm{\&omega;}}_k\right|{|}^2\&rsqb;,$

Wherein N is the infinitely-great positive integer of convergence,

Then described multi-state electric system robust convergency.

8. according to the multi-state method for analyzing stability of power system described in claim 5, it is characterized in that, describedly determine that stability criterion comprises:

Work as u _k=0, ω _k=0, the uncertain Δ A allowed if all _imeet:

$\underset{N\&RightArrow;\mathrm{\&infin;}}{\lim }E\left\{\underset{k=0}{\overset{N}{\&Sigma;}}\right|x_k{|}^2|x_0,s_0\}\&le;\tilde{M}(x_0,s_0),$

Then described multi-state electric system robust convergency.

9. according to the multi-state method for analyzing stability of power system described in claim 5, it is characterized in that, the method for judgement system stability comprises:

The feasibility problems asking for planning solving carrys out judgement system stability, and wherein said planning solving is:

$\left[\begin{array}{cccc}-Q_i& 0& Q_i{A}_i^T{\mathrm{\&Omega;}}_i& Q_i{C}_i^T\\ *& -{\mathrm{\&gamma;}}^{2}I& G_i^T\mathrm{\&Omega;}& L_i^T\\ *& *& -\mathrm{\&psi;}& 0\\ *& *& *& -I\end{array}\right]<0,$

${\mathrm{\&Omega;}}_i=\left[\begin{array}{cccc}\sqrt{p_{i1}}I& \sqrt{p_{i2}}I& \mathrm{...}& \sqrt{p_{il}}I\end{array}\right],$

ψ＝diag(Q ₁,Q ₂,…,Q _l)，

Wherein, for each s (t)=i ∈ S, note A (s _k), B (s _k), G (s _k), C (s _k), D (s _k), L (s _k) be respectively A _i, B _i, G _i, C _i, D _i, L _i,

If there is one group of positive definite symmetric matrices Q _i>0, i ∈ S, makes described planning solving set up, then as u (t) ≡ 0, and described multi-state electric system robust convergency, and meet the requirement of the disturbance dough softening.