CN105958476B - Time-lag power system stability sex determination method based on Wirtinger inequality - Google Patents

Time-lag power system stability sex determination method based on Wirtinger inequality Download PDF

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CN105958476B
CN105958476B CN201610299335.7A CN201610299335A CN105958476B CN 105958476 B CN105958476 B CN 105958476B CN 201610299335 A CN201610299335 A CN 201610299335A CN 105958476 B CN105958476 B CN 105958476B
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孙永辉
李宁
卫志农
孙国强
张世达
郭敏
秦晨
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Hohai University HHU
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]

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Abstract

The invention discloses a kind of time-lag power system stability sex determination method based on Wirtinger inequality, the maximum delay margin that can bear for analyzing power system.This method comprises the following steps that：First, the electric power system model for considering time-delay is established.Then, for institute's established model structure Lyapunov functionals, scaling is carried out by using Wirtinger inequality during the derivation of functional, to reduce the conservative of criterion.Finally gained criterion is represented with one group of LMI (LMI).

Description

Time-lag power system stability sex determination method based on Wirtinger inequality
Technical field
The present invention relates to a kind of time-lag power system stability sex determination method based on Wirtinger inequality, it is applied to Solve the latency issue in interconnected electric power system wide-area control strategy, belong to technical field of power systems.
Background technology
With the continuous expansion of power network scale, simple can not meet wanting for electric network performance by local control information Ask.Therefore network stability control strategy is progressively intended to global control from local control, particularly WAMS in recent years (WAMS) fast development, make power system overall control strategy is implemented as possibility.
On this basis, modern power systems controller will utilize local signal and remote signaling from global angle Wide-area controller is designed as feedback signal.Due to the introducing of remote signaling, the delay of signal will become inevitable, grind Study carefully and show, even if the time lag of very little may all have an impact to stability of power system.Therefore consider that power system can bear Maximum time lag, for power network safe and stable operation tool be of great significance.
Research on time lag system at present, there are a variety of method for solving, frequently with method for solving be construction Lyapunov functionals, based on Lyapunov Theory of Stability, system stability criterion is obtained, finally by LMI (LMI) delay margin is solved.But the criterion obtained by the above method, really to the Lyapunov functionals after derivation Adequate condition obtained by scaling, therefore cause gained criterion that there is certain conservative.So reducing Conservative Property turns into The emphasis of this method research, and one of difficult point.
The content of the invention
Goal of the invention：For problems of the prior art, the present invention proposes a kind of based on Wirtinger inequality Time-lag power system stability sex determination method, brand-new Lyapunov functionals are constructed first, time lag lower limit is not zero and taken into account In criterion, then using Wirtinger inequality scaling skills, the conservative of result is reduced.
Technical scheme：A kind of time-lag power system stability sex determination method based on Wirtinger inequality, including it is as follows Step：
(1) the time-lag power system model for including wide-area control loop is establishedIn formula： h1≤d(t)≤h2,μ≤1.Wherein 0≤h1＜ h2, μ is constant；D (t) is system delay；For at the beginning of system mode Value.
(2) stable decision condition is given：
If positive definite matrix P ∈ R be present4n×4n；Positive definite matrix Qi∈Rn×n, i=1,2,3；Positive definite matrix Zj∈Rn×n, j=1, 2；Matrix Xk∈R2n×2n, k=1,2 set up following MATRIX INEQUALITIES, then time-lag power system is asymptotically stable：
Wherein：
G3=e1-e2, G4=e1+e2-2e5, G5=e2-e4, G6=e2+e4-2e7
G7=e3-e2, G8=e3+e2-2e6, G9=e2-e4, G10=e2+e4-2e7
el=[0l-1 I 07-l], l=1 ..., 7
For matrix A, He (A)=A+AT.Wherein I represents unit matrix.
(3) utilize linear matrix (LMI) tool box in Matlab judge given time lag d (t) whether meet step (2) to The decision condition gone out, if satisfied, then can determine that the time-lag power system under the conditions of delay d (t) is asymptotically stable.
Time-lag power system modelIn formula：X=[x1 xc]T,x1For system state variables；xcFor the state variable of controller；A1For system State matrix；B1For system input matrix；AcFor the state matrix of control system；CcFor the output matrix of control system；C1To be System output matrix；BcFor the input matrix of control system.
Brief description of the drawings
The time-lag power system stability sex determination method flow diagram that Fig. 1 is carried for the present invention；
Fig. 2 is the time-lag power system comprising WADC；
Fig. 3 is the regional power system of four machine two；
Fig. 4 is the regional power system response diagram of four machine two in the case of d (t)=0ms；
Fig. 5 is the regional power system response diagram of four machine two in the case of d (t)=110ms.
Embodiment
With reference to specific embodiment, the present invention is furture elucidated, it should be understood that these embodiments are merely to illustrate the present invention Rather than limitation the scope of the present invention, after the present invention has been read, various equivalences of the those skilled in the art to the present invention The modification of form falls within the application appended claims limited range.
(1) foundation of Power System Delay model
Under normal circumstances, power system can be described by one group of differential algebraic equations, linear to its near system operating point Change, final system is represented by：
In formula：x1For system state variables；A1For systematic observation matrix；B1For system input matrix；C1Square is exported for system Battle array；U is system control input, and y is system control output.
The control strategy of system uses the control of the wide-area damping control based on WAMS, wherein wide area damping control (WADC) Input signal processed contains system remote signaling, and its state equation is represented by：
In formula：xcFor the state variable of controller；AcFor the state matrix of control system；BcFor the input square of control system Battle array；CcFor the output matrix of control system；ucInputted for control system, ycExported for control system.DcFor scalar, reflect Export ycWith inputting ucBetween direct correlation.
Fig. 2 gives the time-lag power system structural relation figure comprising wide-area control loop, and the delay d (t) therein that becomes is Transmission delay caused by remote signaling transmits.Graph of a relation according to Fig. 2, it can obtain：
Wherein：h1≤d(t)≤h2,
Further, time-lag power system model is represented by：
In formula：
(2) power system delay Dependent Stability Criterion method
During solving time lag stability criterion, conventional scaling skill is to utilize Jensen inequality.Although this method Conservative that is feasible but adding result.Institute's extracting method of the present invention is entered using a kind of brand-new inequality-Wirtinger inequality Row scaling, the conservative of acquired results can be substantially reduced.First, provide that institute's extracting method of the present invention is used two important to draw Reason.
Lemma 1：For given positive definite matrix M ＞ 0, with lower inequality for the continuously differentiable function x on section [a, b] All set up：
Wherein：ξ1=x (b)-x (a),
Lemma 2：For given positive definite matrix R ＞ 0, matrix W1,W2With scalar ce ∈ (0,1), define for all ξ, Function Θ (α, R)：
If there is matrix X, makeSo set up with lower inequality：
Construct following Lyapunov functionals：
In formula：P∈R4n×4n；Qi∈Rn×n, i=1,2,3；Zj∈Rn×n, j=1,2,
If el∈R7n×n, el=[0l-1 I 07-l], l=1 ..., 7 be piecemeal coordinates matrix, can obtain following stability criteria：
Criterion：If positive definite matrix P ∈ R be present4n×4n；Positive definite matrix Qi∈Rn×n, i=1,2,3；Positive definite matrix Zj∈Rn×n, J=1,2；Matrix Xk∈R2n×2n, k=1,2 set up following MATRIX INEQUALITIES, then time-lag power system is asymptotically stable：
Wherein：
G3=e1-e2, G4=e1+e2-2e5, G5=e2-e4, G6=e2+e4-2e7
G7=e3-e2, G8=e3+e2-2e6, G9=e2-e4, G10=e2+e4-2e7
ei=[0i-1 I 07-i], i=1 ..., 7
So system (4) Asymptotic Stability.
For matrix A, He (A)=A+AT.Wherein I represents unit matrix.
Prove：
Carrying out derivation for the Lyapunov functionals in criterion can obtain：
Wherein：
Then last two of the Lyapunov functionals (7) after derivation are handled using quoting 1,2, with Exemplified by, anotherProcessing method is identical, and concrete operations are as follows：
It can be obtained using lemma 1 respectively for formula (9)：
Wherein：
ξ11=G3ξ1, ξ12=G4ξ1, ξ21=G5ξ1, ξ22=G6ξ1
For above formula, can be obtained using lemma 2：
AnotherUsing same processing method：
Go to obtain in the Lyapunov functionals that two after processing are updated to after derivation：
System Asymptotic Stability is, it is necessary to make required Lyapunov Functional derivations be less than 0, criterion must be demonstrate,proved.
Note：Inequality (6) in criterion dependent on d (t) andDirectly it can not be solved using LMI tool boxes.But Be inequality (6) be on d (t) andConvex function, as long as so making above-mentioned inequality in d (t)=h1,On all into immediately Can.
One embodiment of the present of invention is described below：
The regional power system of four machine two selects ω as shown in figure 3, be provided with wide area damping control on No. 1 generator13Make For controller feedback signal.Wide area damping control routine lead-lag WADC, is shown below：
Wherein：Tw=10s, T1=0.324s, T2=0.212s
In h1In the case of=0, μ=0, the regional power system of four machine two is obtained in K using institute's extracting method of the present inventiona=10 with KaThe stability margin of system is respectively 346.4ms and 101.0ms when=22.
Three phase short circuit fault occurs at bus 3 for setting system, continues 200ms.By in Fig. 4 without being under the conditions of time lag System response, it can be seen that wide-area controller WADC optimizes the performance of system, eliminates internal oscillator.System is given in Fig. 5 The middle system response existed in the case of 110ms delays, it can be seen that in the case of delay 110ms, Ka=10 systems are stable, and Ka=22 systems are unstable, and divergent trend is presented.This also complies with institute's extracting method of the present invention and tries to achieve KaTime lag under the conditions of=22 Stability margin 101.0ms.

Claims (2)

  1. A kind of 1. time-lag power system stability sex determination method based on Wirtinger inequality, it is characterised in that including as follows Step：
    (1) the time-lag power system model for including wide-area control loop is establishedIn formula：h1≤d (t)≤h2,μ≤1；
    (2) stable decision condition is given：
    If positive definite matrix P ∈ R be present4n×4n；Positive definite matrix Qi∈Rn×n, i=1,2,3；Positive definite matrix Zj∈Rn×n, j=1,2；Square Battle array Xk∈R2n×2n, k=1,2 set up following MATRIX INEQUALITIES, then time-lag power system is asymptotically stable：
    <mrow> <msub> <mi>&amp;Phi;</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>&amp;Phi;</mi> <mn>0</mn> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mn>2</mn> </msub> </mfrac> <msubsup> <mi>&amp;Gamma;</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mi>&amp;Phi;</mi> <mn>2</mn> </msub> <msub> <mi>&amp;Gamma;</mi> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>h</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> </mrow> </mfrac> <msubsup> <mi>&amp;Gamma;</mi> <mn>2</mn> <mi>T</mi> </msubsup> <msub> <mi>&amp;Phi;</mi> <mn>3</mn> </msub> <msub> <mi>&amp;Gamma;</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>0</mn> </mrow>
    <mrow> <msub> <mi>&amp;Phi;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>Z</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>X</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <msub> <mover> <mi>Z</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>&gt;</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>&amp;Phi;</mi> <mn>3</mn> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>Z</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>X</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <msub> <mover> <mi>Z</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>&gt;</mo> <mn>0</mn> </mrow>
    Wherein：
    <mrow> <msub> <mi>&amp;Gamma;</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>G</mi> <mn>3</mn> <mi>T</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>G</mi> <mn>4</mn> <mi>T</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>G</mi> <mn>5</mn> <mi>T</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>G</mi> <mn>6</mn> <mi>T</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow>
    G3=e1-e2, G4=e1+e2-2e5, G5=e2-e4, G6=e2+e4-2e7
    <mrow> <msub> <mi>&amp;Gamma;</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>G</mi> <mn>7</mn> <mi>T</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>G</mi> <mn>8</mn> <mi>T</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>G</mi> <mn>9</mn> <mi>T</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>G</mi> <mn>10</mn> <mi>T</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow>
    G7=e3-e2, G8=e3+e2-2e6, G9=e2-e4, G10=e2+e4-2e7
    <mrow> <msub> <mover> <mi>Q</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>-</mo> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mn>0</mn> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mover> <mi>Q</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mn>0</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mn>0</mn> <mrow> <mn>3</mn> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
    <mrow> <msub> <mover> <mi>Q</mi> <mo>^</mo> </mover> <mn>3</mn> </msub> <mo>=</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>Q</mi> <mn>3</mn> </msub> <mo>,</mo> <mo>-</mo> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mover> <mi>d</mi> <mo>&amp;CenterDot;</mo> </mover> </mrow> <mo>)</mo> <msub> <mi>Q</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mn>0</mn> <mrow> <mn>5</mn> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mover> <mi>Z</mi> <mo>^</mo> </mover> <mo>=</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> <mo>&amp;times;</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mn>0</mn> <mrow> <mn>3</mn> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;times;</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mn>0</mn> <mrow> <mn>3</mn> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mover> <mi>Z</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mn>1</mn> </msub> <mo>,</mo> <mn>3</mn> <msub> <mi>Z</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mover> <mi>Z</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>Z</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>3</mn> <msub> <mi>Z</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>e</mi> <mn>1</mn> <mi>T</mi> </msubsup> </mtd> <mtd> <mrow> <mi>d</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msubsup> <mi>e</mi> <mn>5</mn> <mi>T</mi> </msubsup> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <mi>d</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>-</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo>)</mo> <msubsup> <mi>e</mi> <mn>6</mn> <mi>T</mi> </msubsup> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>d</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> <msubsup> <mi>e</mi> <mn>7</mn> <mi>T</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow>
    <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <msub> <mi>Ae</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>A</mi> <mi>d</mi> </msub> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <mrow> <msubsup> <mi>e</mi> <mn>1</mn> <mi>T</mi> </msubsup> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mover> <mi>d</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <msubsup> <mi>e</mi> <mn>2</mn> <mi>T</mi> </msubsup> </mrow> </mtd> <mtd> <mrow> <msubsup> <mi>e</mi> <mn>3</mn> <mi>T</mi> </msubsup> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mover> <mi>d</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <msubsup> <mi>e</mi> <mn>2</mn> <mi>T</mi> </msubsup> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <msubsup> <mi>e</mi> <mn>4</mn> <mi>T</mi> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mover> <mi>d</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <msubsup> <mi>e</mi> <mn>2</mn> <mi>T</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow>
    el=[0l-1 I 07-l], l=1 ..., 7
    For matrix A, He (A)=A+AT；Wherein I represents unit matrix；
    (3) linear matrix (LMI) tool box in Matlab is utilized to judge whether given time lag d (t) meets what step (2) provided Decision condition, if satisfied, then can determine that the time-lag power system under the conditions of delay d (t) is asymptotically stable.
  2. 2. the time-lag power system stability sex determination method based on Wirtinger inequality as claimed in claim 1, it is characterised in that Time-lag power system modelIn formula：X=[x1 xc]T, x1For system state variables；xcFor the state variable of controller；A1For systematic observation matrix；B1For system input matrix；AcFor control The state matrix of system processed；CcFor the output matrix of control system；C1For system output matrix；BcFor the input square of control system Battle array.
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