CN109659982B - Method for judging time lag dependence stability of new energy power system - Google Patents
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Abstract
The invention discloses a method for judging time-lag dependence stability of a new energy power system, which is based on a Wirtinger inequality and a Markov model and is used for analyzing the maximum time-lag stability margin born by the power system. Firstly, establishing a new energy power system model dependent on time lag based on a Markov model, and establishing the system model as a discrete Markov hopping linear system model; then constructing a Lyapunov functional aiming at the established model, and adopting a Wirtinger inequality to carry out scaling in the derivation process of the functional so as to reduce the conservative property of the criterion; finally, the obtained criterion is expressed by a group of Linear Matrix Inequalities (LMI). By the method, the safety and the stability of the operation of the power grid are improved.
Description
Technical Field
The invention belongs to an electric power system, and particularly relates to a method for judging time lag dependence stability of a new energy electric power system.
Background
In an electric power system, main indexes for measuring the quality of electric energy comprise voltage deviation, frequency deviation, harmonic waves, voltage fluctuation and flicker, three-phase voltage unbalance degree and the like. In the operation of an electric power system, it is required that at any time the power delivered by the system and the power consumed by the load should be balanced and the frequency at the nominal value, while changes in the power plant output and changes in the load cause changes in the frequency. Any sudden disturbance in the power system may cause a change in frequency and the system load may change from time to time and many situations are unpredictable and uncontrolled. Therefore, in order to ensure that the power system can maintain a stable load frequency, an effective load frequency control system is required, and when the power grid system is subjected to frequency disturbance and frequency deviation occurs, the frequency of the power system should be maintained or quickly restored to be within a stable range. The new energy represented by wind energy, solar energy, ocean energy and biomass energy is greatly influenced by the geographical position and the natural environment, and has high volatility and randomness, so that the new energy can be merged into a power grid and cannot ensure the stability of the output power of the new energy like conventional generator sets such as thermal power, hydropower and the like. At present, the output power generated by new energy cannot be accurately predicted, and the fluctuation of the power can bring significant influence on the electric energy quality and the economic operation of a power system. Therefore, in a sense, when the new energy is incorporated into the power grid, the new energy unit is an interference of the power system, which has randomness and volatility, and the stability of the load frequency of the power grid caused by the new energy being incorporated into the power grid is a problem worthy of research and attention.
With the access of new energy sources mainly represented by wind energy and photovoltaic to a traditional power plant, the new energy sources gradually replace traditional energy sources, so that the use pressure of the traditional energy sources is relieved, but due to the intermittency and randomness of the capacity change of the new energy sources and the fluctuation and unpredictability of the power generation of the new energy sources, the stability of a power system is inevitably affected by large-scale new energy source grid connection, and a new challenge is provided for the load frequency control of the power system.
On the basis, as the network of the power system is continuously enlarged, and problems of network congestion, packet loss and the like exist, a time lag problem inevitably occurs in the power system. It has been shown that even a small time lag may have an effect on power system stability. Therefore, the maximum time lag which can be borne by the power system is considered, and the method has very important significance for safe and stable operation of the power grid.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problems in the prior art, the invention provides a method for judging time lag dependence stability of a new energy power system.
The technical scheme is as follows: a method for judging time lag dependence stability of a new energy power system is based on
The Wirtinger inequality and Markov model comprise the following steps:
(1) establishing a load frequency control model depending on time lag in a new energy power system;
(2) converting the load frequency control model of the new energy power system into a new energy power system model based on a Markov model;
(3) constructing a Lyapunov functional, scaling the Lyapunov functional through a Wirtinger inequality, further judging the progressive stability of the system, and then judging the H of the system∞Judging the stability;
(4) and solving the maximum time lag which can be borne by the new energy power system.
Further, the establishment of the load frequency control model in the step (1) includes the following steps:
in a traditional power system model, the influence of wind energy on the frequency in the system model is added, the influence is regarded as an external random disturbance, and the external random disturbance is added to a disturbance term of the system model, wherein the following expression is shown:
wherein ω (t) ═ Δ Pd+ΔPwind+ΔPtie，ΔPdIs the load deviation; delta PwindOutputting the deviation for the wind driven generator; delta PtieRegional tie line bias.
Further, the step (2) is established as a Markov model based on a Markov random theory, and comprises the steps of analyzing different state reactions of the new energy system under different modes, wherein the expression is as follows:
x(k+1)＝Ax(k)+BK(dr(k))Cx(k-dr(k))+Fω(k)，
in the formula: x is the number ofi(t)＝[Δfi ΔPmi ΔPvi∫ACEi ΔPtie-i]T，
x(t)＝[x1(t) x2(t) x3(t) … xn(t)]T，
ui(t)＝-KPACEi-KI∫ACEi，u(t)＝[u1(t) u2(t) u3(t) … un(t)]T，
yi(t)＝[ACEi∫ACEi]T，ωi(t)＝ΔPdi+ΔPwindi+ΔPtie-i，
ω(t)＝[ω1(t) ω2(t) ω3(t) … ωn(t)]T，Bd＝diag[B1 B2 … Bn]，
TijThe synchronization coefficient of a connecting line between the ith layer area and the jth layer area; in the ith layer region,. DELTA.PdiIs the load deviation; delta PmiIs the generator mechanical power variation; delta PviIs the valve position deviation; delta PwindiOutputting the deviation for the wind driven generator; Δ fiIs a frequency deviation; miIs the moment of inertia; diThe damping coefficient of the generator; t isgiIs the time constant of the regulator; t ischiIs the time constant of the turbine; riIs the rotation speed; beta is aiIs a frequency deviation factor.
Further, constructing a Lyapunov functional in the step (3), and scaling by adopting a Wirtinger inequality in a derivation process of the functional, wherein the scaling specifically comprises the following steps:
(31) constructing a Lyapunov functional, wherein the expression of the Lyapunov functional is as follows:
V1(k)＝xT(k)Px(k)
wherein P isr，Q1r，Q2r，SrIs a positive definite matrix with proper dimension;
(32) and scaling the derivative of the functional by adopting a Wirtinger inequality, wherein the expression is as follows:
(33) judging the gradual stability of the new energy power system:
the consumption equation is established under the condition that the specified attenuation level lambda is more than 0, and the expression is as follows:
to the system H∞And (5) judging the stability.
Furthermore, in the method, the stability of the new energy power system is judged based on the characteristic value, the characteristic value and the maximum time lag which can be borne by the system are solved, and the stability judgment of the new energy system comprises the following modes:
for a given controller gain KrAnd scalar λ, if a positive definite matrix P existsr，Q1r，Q2r，SrAnd a matrix X of appropriate dimensions1，X2，X3，X4，YrFor all r ═ 0.. L, the following inequality holds true, and the time-lag new energy power system is H at attenuation level λ > 0∞And (3) stabilizing:
minμ
constraint conditions are as follows:
in the formula:
Σ22r＝-Q1r-8Sr+X1+X2-X3-X4，Σ31r＝X1-X2+X3-X4，
Σ32＝-2Sr-X1+X2+X3-X4，Σ42＝6Sr+X3+X4，Σ43＝-2X3+2X4，
Σ51＝2X2+2X4，Σ52＝6Sr+X3+X4，dM＝μ-1/2，
firstly, establishing a frequency load frequency control model of a traditional power system as a new energy power system frequency load frequency control model dependent on time lag based on a Markov model, and establishing the system model as a discrete Markov hopping linear system model; then constructing a Lyapunov functional according to the established model, and adopting a Wirtinger inequality to carry out scaling in the derivation process of the functional so as to reduce the conservative property of the criterion; and consequent progressive stability of its system and H∞The stability was analyzed; and finally, the nonlinear coupling term is converted into a solution eigenvalue problem due to the existence of the nonlinear coupling term, so that a stability criterion III is obtained.
Has the advantages that: the method has the advantages that the Wirtinger inequalities are used for scaling, the conservatism of the system is greatly reduced, meanwhile, the inequalities with nonlinear coupling are converted into the inequalities which can be directly solved by an LMI tool box, the solvability of the inequalities is greatly improved, and meanwhile, the finally obtained criterion can also be used for solving the maximum time lag margin which can be born by the system. By the method, the safety and the stability of the operation of the power grid are improved.
Drawings
FIG. 1 is a schematic flow diagram of the process of the present invention;
FIG. 2 is a block diagram of a skew power system according to the present invention;
FIG. 3 is a diagram of a random Markov transition of the present invention;
FIG. 4 is a diagram illustrating the frequency error response of the power system under random load disturbance according to the present invention;
fig. 5 is a graph of ACE signal response of a power system under random load disturbance according to the invention.
Detailed Description
For the purpose of illustrating the technical solutions disclosed in the present invention in detail, the following description is further provided with reference to the accompanying drawings and specific embodiments.
Example 1
As shown in FIGS. 1-3, the invention firstly establishes a power system model as a discrete Markov jump linear system model, constructs a new Lyapunov functional, and then reduces the conservative property of the result by utilizing Wirtinger inequality scaling skill. The progressive stability of the system is first analyzed, and then the H of the system∞And analyzing the stability, finally converting the problem which is difficult to solve the global optimal solution in the obtained inequality into a characteristic value solving problem, and solving the maximum time lag which can be borne by the system.
The method for judging the stability of the new energy power system under the Wirtinger inequality comprises the following steps:
(1) establishing time-lag electric power system model
x(k+1)＝Ax(k)+BK(ds)Cx(k-ds)+Fω(k)
Wherein: x is the number ofi(t)＝[Δfi ΔPmi ΔPvi∫ACEi ΔPtie-i]T，x(t)＝[x1(t) x2(t) x3(t) …xn(t)]T，
ui(t)＝-KPACEi-KI∫ACEi，u(t)＝[u1(t) u2(t) u3(t) … un(t)]T，
yi(t)＝[ACEi∫ACEi]T，ωi(t)＝ΔPdi+ΔPwindi+ΔPtie-i，
ω(t)＝[ω1(t) ω2(t) ω3(t) … ωn(t)]T，Bd＝diag[B1 B2 … Bn]，
Compared with the traditional load frequency control model of the power system, the load frequency control model is established as a load frequency control model based on Markov model discrete time delay dependence, new energy wind energy is added into the system model, the wind energy is regarded as external disturbance of the system model, and the discrete Markov model is utilized to analyze the system states under different modes.
(2) Giving a stability determination condition
For a given controller gain KrAnd scalar λ, if a positive definite matrix P existsr，Q1r，Q2r，SrAnd a matrix X of appropriate dimensions1，X2，X3，X4，YrFor all r ═ 0.. L, the following inequality holds true, then the time-lapse new energy power system is H at attenuation level λ > 0∞And (3) stabilizing:
minμ
s.t.
in the formula:
Σ22r＝-Q1r-8Sr+X1+X2-X3-X4，Σ31r＝X1-X2+X3-X4，
Σ32＝-2Sr-X1+X2+X3-X4，Σ42＝6Sr+X3+X4，Σ43＝-2X3+2X4，
Σ51＝2X2+2X4，Σ52＝6Sr+X3+X4，dM＝μ-1/2，
(3) using a Linear Matrix Inequality (LMI) tool box in Matlab, giving an initial controller gain, solving the judgment condition given in the step (2), and if the judgment condition has a solution, judging the time delay dr(k) Time lag power system under condition is H∞And (4) the product is stable.
Example 2
The method for judging the stability of the new energy power system under the Wirtinger inequality comprises the following specific processes:
(1) establishment of new energy power system model dependent on time lag based on Markov model
In general, a power system can be described by a set of differential algebraic equations, which are linearized around the system operating point, and the final system can be expressed as:
wherein:
xi(t)＝[Δf iΔPmi ΔPvi∫ACEi ΔPtie-i]T，x(t)＝[x1(t) x2(t) x3(t) … xn(t)]T，
ui(t)＝-KPACEi-KI∫ACEi，u(t)＝[u1(t) u2(t) u3(t) … un(t)]T，
yi(t)＝[ACEi∫ACEi]T，y(t)＝[y1(t) y2(t) y3(t) … yn(t)]T，
ωi(t)＝ΔPdi+ΔPwindi+ΔPtie-i，ω(t)＝[ω1(t) ω2(t) ω3(t) … ωn(t)]T，
Fd＝[F1 F2 … Fn]。Tijis the synchronization coefficient of the tie between the ith layer region and the jth layer region. In the ith layer region,. DELTA.PdiIs the load deviation; delta PmiIs the generator mechanical power variation; delta PviIs the valve position deviation; delta PwindiOutputting the deviation for the wind driven generator; Δ fiIs a frequency deviation; miIs the moment of inertia; diThe damping coefficient of the generator; t isgiIs the time constant of the regulator; t ischiIs the time constant of the turbine; riIs the rotation speed; beta is aiIs a frequency deviation factor.
ACE in ith layer areaiThe signal may be expressed as:
ACEi＝βiΔfi+ΔPtie-i (2)
the PI controller of the ith layer area is designed as follows:
and can obtain:
u(t)＝KCx(t) (4)
wherein: ki＝[-KPi -KIi]，K＝[KT 0 KT 1 KT 2 … KT L]T。
Discretizing the system model (1) and the controller (4) can obtain the following models:
due to time lag dkDue to the transmission delay caused by the transmission of the far-end signal, the following results can be obtained:
u(k)＝K(dk)Cx(k-dk) (6)
the discrete power system model can therefore be written as:
x(k+1)＝Ax(k)+BK(dk)Cx(k-dk)+Fω(k) (7)
the whole system is added into the network, the network is modeled into a stochastic process described by Markov, and the description method is as follows:
P[rs(k+1)＝j|rs(k)＝i]＝pij
where i, j represents a change from one modality to another. In this network model, we use the time lag in the system as a description object, that is: the network state describes the probability of change from the delay of one time to the delay of the next time, so that there is 0 ≦ i, j ≦ dM,
Wherein d isMRepresents the maximum value of the finite delay and also represents that the number of network nodes contained in the network is L + 1.
In summary, the Markov model-based new energy power system model with time lag is as follows:
x(k+1)＝Ax(k)+BK(dr(k))Cx(k-dr(k))+Fω(k) (8)
(2) method for judging stability of new energy power system under Wirtinger inequality
The method provided by the invention adopts a brand-new inequality-Wirtinger inequality and utilizes the Lyapunov theory to analyze the system stability, so that the conservation of the obtained result can be greatly reduced. First, three important arguments are given for the proposed method of the present invention.
Lemma one (Schur complement lemma of matrix): if three matrices are known for each of the three matrices,
Z3then, then
If and only if
And 2, leading to a second principle: for a given positive definite matrix N > 0, the scalar 0 ≦ h1≤h2And vector function eta [ -h2h1]→Rn×nThe following inequality is satisfied:
wherein:
η(s)＝x(s+1)-x(s)
Ω1＝x(k-h1)-x(k-h2)
and (3) leading three: for a given positive definite matrix R> 0, matrix W1，W2And a scalar α ∈ (0,1), defining the function Θ (α, R) for all ξ:
if matrix X is present, let
Then the following inequality holds:
criterion one is as follows: for a given controller gain KrIf a positive definite matrix P is presentr，Q1r，Q2r，SrAnd a matrix X of appropriate dimensions1，X2，X3，X4For all r-0, L holds the following inequality, the lag new energy power system (8) is progressively stable when ω (k) is 0:
wherein:
Σ22r＝-Q1r-8Sr+X1+X2-X3-X4，Σ31r＝X1-X2+X3-X4，
Σ32＝-2Sr-X1+X2+X3-X4，Σ42＝6Sr+X3+X4Σ43＝-2X3+2X4，
Σ51＝2X2+2X4，Σ52＝6Sr+X3+X4，
and (3) proving that:
structure Lyapunov functional
V1(k)＝xT(k)Px(k)
Wherein P isr，Q1r，Q2r，SrIs a positive definite matrix with appropriate dimensions.
Derivation of v (k) yields:
ΔV(k)＝ΔV1(k)+ΔV2(k)+ΔV3(k) (11)
wherein:
wherein:
using the Wirtinger inequality given in the third lemma and in combination with the second lemma, the following inequality can be obtained:
wherein:
the following augmentation matrices are defined:
and: e.g. of a cylinder1＝[1 0 0 0 0]，e2＝[0 1 0 0 0]，
e3＝[0 0 1 0 0]，e4＝[0 0 0 1 0]，e5＝[0 0 0 0 1]。
The integration of (11) and (12) can obtain:
wherein:
by using the theorem one, the formula (9) can be obtained. If pirIf both the expression (10) and < 0 are satisfied, Δ V (k) < ξT(k)Πrξ(k)＜-ε||ξ(k)||2< 0 holds for any small ε > 0. Therefore, the time-lag new energy power system (8) is gradually stable when omega (k) is 0, and the criterion is verified.
Criterion two: for a given controller gain KrAnd scalar λ, if a positive definite matrix P existsr，Q1r，Q2r，SrAnd a matrix X of appropriate dimensions1，X2，X3，X4For all r ═ 0.. L, the following inequality holds true, then the time-lapse new energy power system is H at attenuation level λ > 0∞And (3) stabilizing:
wherein:
Σ22r＝-Q1r-8Sr+X1+X2-X3-X4，Σ31r＝X1-X2+X3-X4，
Σ32＝-2Sr-X1+X2+X3-X4，Σ42＝6Sr+X3+X4Σ43＝-2X3+2X4，
Σ51＝2X2+2X4，Σ52＝6Sr+X3+X4，
and (3) proving that:
having a consumption equation at a defined attenuation level λ > 0
When J is less than or equal to 0, the time-lag power system is H under the attenuation level lambda is more than 0∞And (4) the product is stable. Therefore, J ≦ 0 should be solved. Since y (k) is cx (k), equation (17) can be simplified to the following form:
xT(k)CTCx(k)-λ2ωT(k)ω(k)+ΔV(k)＜0 (18)
defining an amplification matrix as
And:
e6＝[1 0 0 0 0 0]，e7＝[0 1 0 0 0 0]，e8＝[0 0 1 0 0 0]，
e9＝[0 0 0 1 0 0]，e10＝[0 0 0 0 1 0]，e11＝[0 0 0 0 0 1]。
the following inequality can be obtained:
wherein
In that
When the theorem one is used, the formula (15) can be obtained. Finally, if both (15) and (16) can be satisfied, x can be obtainedT(k)CTCx(k)-λ2ωT(k)ω(k)+ΔV(k)＜-ε||ξ1(k)||2< 0 is true. Therefore, at attenuation level λ > 0, the time-lag new energy power system (8) is H∞Stable, the criterion is determined by two.
Note: notably due to the fact that22rIn the presence of a non-linear coupling term
Equation (15) therefore cannot be solved directly using the Linear Matrix Inequality (LMI) toolbox, and it is also difficult to find a globally optimal solution. The following reasoning gives the upper bound d for the maximum time lag in solving equation (15)MThe method of (1).
Due to the nonlinear coupling term in the formula (15)
So that equation (15) can be translated into a eigenvalue problem. Equation (15) can therefore be written as the following inequality:
wherein:
introducing a new variable Yr> 0, then (20) can be described as:
and because of
So that criterion three can be obtained.
Criterion three: for a given controller gain KrAnd scalar λ, if a positive definite matrix P existsr，Q1r，Q2r，SrAnd with a matrix X of appropriate dimensions1，X2，X3，X4，YrFor all r ═ 0.. L, the following inequality holds true, then the time-lapse new energy power system is H at attenuation level λ > 0∞And (3) stabilizing:
minμ
s.t.
in the formula: dM＝μ-1/2。
An embodiment of the present invention is described below:
as shown in fig. 1, in the multi-region power system load frequency control structure, a transition probability matrix of a Markov model is:
the coefficient items in the dual-region power system model are set as shown in the following table.
When the power system controller gain is set to:
the maximum time lag upper bound that the electric power system can bear at this moment can be obtained according to the inference as follows: dM＝2.2961。
Fig. 4 and 5 show the frequency error response of the dual-area power system under random load disturbance and the response of the ACE signal, respectively. The simulation result shows that under the condition of random disturbance, the frequency error response of the system and the response of the ACE signal both tend to be stable before the next disturbance occurs. In summary, in this case, the power system (8) is H when the stability index λ is 0.1∞And (4) the product is stable.
Claims (3)
1. A method for judging time lag dependence stability of a new energy power system is characterized by comprising the following steps: the method is based on a Wirtinger inequality and a Markov model, and comprises the following steps:
(1) establishing a load frequency control model depending on time lag in a new energy power system;
(2) convert new energy power system load frequency control model into Markov-based
A new energy power system model of the model; establishing a Markov model based on a Markov random theory, analyzing different state reactions of the new energy system under different modes, wherein the expression is as follows:
x(k+1)＝Adx(k)+BdK(dr(k))Cdx(k-dr(k))+Fdω(k)，
in the formula: x is the number ofi(t)＝[Δfi ΔPmi ΔPvi ∫ACEi ΔPtie-i]T，
x(t)＝[x1(t) x2(t) x3(t)…xn(t)]T，
ui(t)＝-KPiACEi-KIi∫ACEi，u(t)＝[u1(t) u2(t) u3(t)…un(t)]T，
yi(t)＝[ACEi ∫ACEi]T，ωi(t)＝ΔPdi+ΔPwindi+ΔPtie-i，
ω(t)＝[ω1(t) ω2(t) ω3(t)…ωn(t)]T，Bd＝diag[B1 B2…Bn]，
ACEi＝βiΔfi+ΔPtie-i，Ki＝[-KPi -KIi]，K＝[KT 0 KT 1 KT 2…KT L]T；
TijThe synchronization coefficient of a connecting line between the ith layer area and the jth layer area; in the ith layer region,. DELTA.PdiIs the load deviation; delta PmiIs the generator mechanical power variation; delta PviIs the valve position deviation; delta PwindiOutputting the deviation for the wind driven generator; Δ fiIs a frequency deviation; miIs the moment of inertia; diIs the generator damping coefficient; ggiIs the time constant of the regulator; t is a unit ofchiIs the time constant of the turbine; riIs the rotation speed; beta is a betaiIs a frequency deviation factor;
(3) constructing a Lyapunov functional, scaling the Lyapunov functional through a Wirtinger inequality, further judging the progressive stability of the system, and then judging the H of the system∞Judging the stability; constructing a Lyapunov functional, and scaling by adopting a Wirtinger inequality in the derivation process of the functional, wherein the method specifically comprises the following steps:
(31) constructing a Lyapunov functional, wherein the expression of the Lyapunov functional is as follows:
V1(k)＝xT(k)Prx(k)
wherein P isr，Q1r，Q2r，SrIs a positive definite matrix with proper dimension; dMIs the maximum value in the finite delay;
(32) and scaling the derivative of the functional by adopting a Wirtinger inequality, wherein the expression is as follows:
wherein the content of the first and second substances,
(33) judging the gradual stability of the new energy power system:
the consumption equation is established under the condition that the specified attenuation level lambda is more than 0, and the expression is as follows:
to the system H∞Judging the stability;
(4) adjusting the gain of the initial controller according to the stability judgment;
(5) and solving the maximum time lag which can be borne by the new energy power system.
2. The method for determining the time-lag dependence stability of the new energy power system according to claim 1, wherein: the establishment of the load frequency control model in the step (1) comprises the following steps:
in a traditional power system model, the influence of wind energy on the frequency in the system model is added, the influence is regarded as an external random disturbance, and the external random disturbance is added to a disturbance term of the system model, wherein the following expression is shown:
wherein ω (t) ═ Δ Pd+ΔPwind+ΔPtie，ΔPdIs the load deviation; delta PwindOutputting the deviation for the wind driven generator; delta PtieRegional tie line bias.
3. The method for determining the time-lag dependence stability of the new energy power system according to claim 1, wherein: in the method, the stability of the new energy power system is judged based on the characteristic value, the characteristic value and the maximum time lag which can be borne by the system are solved, and the stability judgment of the new energy power system comprises the following steps:
for a given controller gain KrAnd scalar λ, if a positive definite matrix P existsr，Q1r，Q2r，SrAnd a matrix X of appropriate dimensions1，X2，X3，X4，YrFor all r ═ 0, 1., L, the following inequality holds true, and the time lag new energy power system is H at attenuation level λ > 0∞And (3) stabilizing:
minμ
constraint conditions are as follows:
in the formula:
∑22r＝-Q1r-8Sr+X1+X2-X3-X4，∑31r＝X1-X2+X3-X4，
∑32r＝-2Sr-X1+X2+X3-X4，∑42r＝6Sr+X3+X4，∑43r＝-2X3+2X4，
∑51r＝2X2+2X4，∑52r＝6Sr+X3+X4，
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