Internal cavity risk early warning method for small signal stability domain of time-delay power system

Abstract

The invention discloses a time-delay electric power system small signal stability domain internal cavity risk early warning method comprising the following steps: S1, obtaining electric power parameters, and establishing an electric power system model; s2, locally linearizing the model to obtain a characteristic equation; s3, Rekaius transformation is carried out on an exponential term in the characteristic equation; s4, solving the stable critical time lag of the small signal based on the transformed characteristic equation; s5, whether degradation Hopf bifurcation occurs under the critical time lag is judged, and if yes, an early warning signal is sent out; if not, the early warning signal is not sent out. The method has the beneficial effects that whether the risk of cavity occurrence exists in the small signal stability domain of the system is judged by detecting whether the system can generate degradation Hopf bifurcation under the small signal stability critical time lag, and if the risk of cavity occurrence exists in the small signal stability domain, an early warning signal is sent out in time.

Description

Technical Field

[0001] The present invention relates to the technical field of power system risk early warning, and in particular to a method for early warning of internal void risks in a small signal stability domain of a time-lag power system. Background Art

[0002] The power system is a typical time-delay dynamic system. Various system measurement and control signals exhibit significant time delays in the transmission, exchange, and processing of these signals within the network. As an inherent variable in power systems, time delay is a significant factor leading to control failure, operational degradation, and instability. Even small time delays can render even high-performance controllers in systems that do not account for time delay ineffective, potentially causing catastrophic failures. Therefore, system time delays cannot be simply ignored; the impact of time delay variations on the overall system stability region must be evaluated. The loss of small-signal stability in power systems can be attributed to bifurcation behavior in the system's dynamics. Therefore, bifurcation theory can be used to study small-signal stability. When a Hopf bifurcation fails to meet the transversality condition, the system undergoes a degenerate Hopf bifurcation. This degenerate Hopf bifurcation results in non-persistent bifurcations in the system, leading to a hole effect in the stable region and further deteriorating the system's stable operating conditions.

[0003] Prior art research on time-delay power systems primarily focuses on solving the time-delayed, differential-algebraic equations of the power system using mathematical methods such as the Lyapunov functional and the Hessenberg transform, thereby analyzing the impact of time delay on the system's small-signal stability at a specific equilibrium point. However, these methods cannot identify whether holes will appear within the system's small-signal stability domain under a specific time delay. This presents a problem: the inability to identify holes within the system's small-signal stability domain under a specific time delay.

[0004] For example, a "Lyapunov stability analysis method for time-delay power systems" disclosed in Chinese patent literature has a publication number of CN103227467A and an application date of April 19, 2013. This invention establishes a constrained time-delay differential equation model for a power system: the system states are rearranged with the states without time delay effects placed first and the states with time delay effects placed last, thereby obtaining a constrained time-delay differential equation model corresponding to the original time-delay system. The new stability criterion based on the constrained time-delay differential equation model effectively reduces the dimension of the time-delay differential equation and has higher computational efficiency. However, there is a problem in that it cannot identify whether voids will appear within the small-signal stability domain of the system under a certain time delay. Summary of the Invention

[0005] In view of the deficiency of existing technologies in not being able to identify whether voids will occur within the small-signal stability domain of the system under a certain time delay, the present invention proposes a method for early warning of void risks within the small-signal stability domain of a time-delay power system. By detecting whether degenerate Hopf bifurcation will occur in the system under the critical time delay of small-signal stability, it is judged whether there is a risk of voids within the small-signal stability domain of the system. If there is a void risk within the small-signal stability domain, an early warning signal is issued in time.

[0006] The following is a technical solution of the present invention, a method for early warning of internal void risk in the small signal stability domain of a time-delay power system, comprising the following steps:

[0007] S1. Obtain power parameters and establish a power system model;

[0008] S2, locally linearize the model to obtain the characteristic equation;

[0009] S3, perform Rekasius transformation on the exponential term in the characteristic equation;

[0010] S4. Calculate the critical time lag of small signal stability based on the transformed characteristic equation;

[0011] S5. Determine whether a degenerate Hopf bifurcation occurs under the critical time lag. If so, issue a warning signal; if not, do not issue a warning signal.

[0012] In this solution, power parameters are acquired and a power system model is established. The model is locally linearized to obtain a first locally linearized model without time delay. Based on the first locally linearized model, a second locally linearized model with time delay is derived. The characteristic equation is obtained based on the second locally linearized model, and the characteristic equation is modified based on the conjugate pure imaginary eigenvalue. The exponential term in the modified characteristic equation is subjected to a Rekasius transform. The critical time delay for small-signal stability is determined based on the transformed characteristic equation. The criterion for degenerate Hopf bifurcation is defined. The solution for the critical time delay for small-signal stability determines whether a Hopf bifurcation or a degenerate Hopf bifurcation occurs at this time delay. If a degenerate Hopf bifurcation occurs, indicating a risk of a hole within the system's small-signal stability region, a warning signal is issued, alerting system operators to monitor operating parameters to avoid falling into unstable hole regions. This method can determine whether there is a risk of a hole within the system's small-signal stability region. If a hole risk is present within the small-signal stability region, a warning signal is issued promptly.

[0013] As a preference, in S1, the power system model expression is as follows:

[0014]

[0015] Where x is the system state variable, and x∈R n ; y is the system algebraic variable, and x∈Rm ; k and τ are parameters and time lag respectively, and

[0016] Preferably, in S2, locally linearizing the model to obtain a characteristic equation comprises the following steps:

[0017] S21: Create the first local linearized model when there is no time lag;

[0018] S22: Based on the first local linearization model, deriving a second local linearization model in the presence of a time lag;

[0019] S23: Obtain a characteristic equation based on the second local linearization model.

[0020] Preferably, in S21, the first local linearization model expression is as follows:

[0021]

[0022] In the formula, the vector of the state variable micro-increment Δx∈R n , vector of algebraic variable increments Δy∈R m ;L n is an n×n identity matrix; And A∈R n×n , B∈R n×m , C∈R m×n , D∈R m×m .

[0023] Preferably, in S22, the second local linearization model is expressed as follows:

[0024]

[0025] Where τ is the time delay, and the sparse matrix J1∈R (n+m)×(n+m) , since the rank of matrix J1 is very low, J1 can be decomposed to obtain J1 = GH, where G∈R (n+m)×l , H∈R l×(n+m) , and l<<n+m.

[0026] Preferably, in S23, the characteristic equation is expressed as follows:

[0027] det(λE-Je -λτ GH)=0 (4).

[0028] Preferably, in S3, performing a Rekasius transformation on the exponential term in the characteristic equation comprises the following steps:

[0029] S31: Based on the conjugate pure imaginary eigenvalue, the characteristic equation is modified. The expression of the modified characteristic equation is as follows:

[0030]

[0031] Where, τ c is the same as λ * =±jω ci The corresponding small signal stability critical time lag;

[0032] S32: Perform Rekasius transformation on the exponential term in the modified characteristic equation. The expression is as follows:

[0033]

[0034] Where, T c is the transformation coefficient to be determined, and when λ=λ * There is only one T c Corresponding to it.

[0035] As a preference, the specific steps of S4 are as follows:

[0036] According to formula (5), there exists a characteristic vector υ that satisfies:

[0037]

[0038] and ||υ||≠0, combining equations (6) and (7), we can obtain:

[0039] υ=-jω ci T c υ-2jω ci T c (jω ci EJ-GH) -1 GHυ (8)

[0040] Substituting formula (6) into formula (7) yields:

[0041] Hυ / Τ c =-jω ci (E+2H(jω ci EJ-GH) -1 G)Hυ (9)

[0042] Let the matrix s(ω ci )=-jω ci (E+2H(jω ci EJ-GH) -1 G), then Hυ and 1 / T c They are respectively the matrix S(ω ci )’s eigenvectors and eigenvalues; if and only if S(ω ci )When there is a purely real eigenvalue other than 0, there is a purely imaginary eigenvalue λ * ; So 1 / Tc Substituting into the following formula, we can obtain the critical time delay of small signal stability, which is expressed as follows:

[0043] τ c =2[tan -1 (ω ci ·T c )±m·π] / ω ci ,m=0,1,2,3… (10).

[0044] Preferably, in S5, determining whether a degenerate Hopf bifurcation occurs under the critical time lag comprises the following steps: replacing jω in equation (7) with ci Expressed as λ, equation (7) can be transformed into the following form:

[0045] λEυ=(J+e -λτ GH)υ (11)

[0046] The derivative of the eigenvalue λ with respect to the time lag τ can be calculated from formula (11):

[0047]

[0048] Where u is the left eigenvector of the characteristic matrix in equation (4) relative to λ, u H represents the conjugate transpose of u;

[0049] According to equations (8), (10) and (11), the conjugate imaginary root λ can be obtained * , relative to λ * The left eigenvalue vector u and the right eigenvector υ of τ c ,λ * Substituting , u and υ into formula (12), we can obtain Define ρ = dRe(λ) / dτ as the criterion for degenerate Hopf bifurcation, when τ = τ c ,λ=λ * When ρ≠0, a Hopf bifurcation occurs at this time lag; if ρ=0, a degenerate Hopf bifurcation occurs at this time lag.

[0050] The beneficial effects of the present invention are:

[0051] 1. Based on the Rekasius transform and degenerate Hopf bifurcation criterion, it is detected whether the system will experience degenerate Hopf bifurcation under the critical time delay of small signal stability, thereby effectively judging whether there is a risk of voids in the small signal stability domain of the system;

[0052] 2. If a void risk is detected within the small signal stability domain, an early warning signal will be issued in a timely manner to remind system operators to monitor operating parameters to avoid falling into unstable void areas, or to take measures to prevent further increases in time delays, thereby ensuring the stability of the power system in actual operation. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 The present invention provides a flow chart of a method for early warning of internal void risks in a small-signal stability domain of a time-lag power system.

[0054] Figure 2 A three-node power system structure diagram of a method for early warning of internal void risks in a small-signal stability domain of a time-lag power system according to the present invention.

[0055] Figure 3 A diagram of the division of the small signal stability domain of a power system under different time delays of a method for early warning of internal void risks in the small signal stability domain of a time-delay power system according to the present invention. DETAILED DESCRIPTION

[0056] In order to further elaborate on the technical solution of the present invention, the method of the present invention will be described in more detail below in conjunction with the drawings in the embodiments of the present application. It should be noted that the embodiments combined are only part of the embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0057] It should also be noted that, in order to avoid obscuring the present invention due to unnecessary factors, the drawings only show structures closely related to the present invention, while details that are not closely related to the present invention are ignored.

[0058] Example:

[0059] This example uses Figure 2 The three-node power system structure is shown.

[0060] like Figure 1 As shown, a method for early warning of internal void risk in the small signal stability domain of a time-delay power system includes the following steps:

[0061] S1. Obtain power parameters and establish a power system model.

[0062] S2. Locally linearize the model to obtain the characteristic equation.

[0063] S3. Perform Rekasius transformation on the exponential term in the characteristic equation.

[0064] S4. Calculate the critical time lag of small signal stability based on the transformed characteristic equation.

[0065] S5. Determine whether a degenerate Hopf bifurcation (DHB) occurs under the critical time lag. If DHB occurs, issue a warning signal.

[0066] The power parameters are obtained and a power system model is established. The model is locally linearized to obtain a first locally linearized model without time delay. Based on the first locally linearized model, a second locally linearized model with time delay is derived. The characteristic equation is obtained based on the second locally linearized model, and the characteristic equation is modified based on the conjugate pure imaginary eigenvalue. The exponential term in the modified characteristic equation is subjected to a Rekasius transform. The critical time delay for small-signal stability is determined based on the transformed characteristic equation. The criterion for degenerate Hopf bifurcation is defined. By solving the critical time delay for small-signal stability, it is determined whether a Hopf bifurcation or a degenerate Hopf bifurcation occurs under this time delay. If a degenerate Hopf bifurcation occurs, there is a risk of a void within the system's small-signal stability region, and a warning signal is issued. This alerts system operators to monitor operating parameters to avoid falling into unstable void regions. The system can determine whether there is a risk of a void within the system's small-signal stability region, and if so, a warning signal is issued promptly.

[0067] In step S1, the power system parameters are obtained. The power parameters are used to participate in the model calculation. The power parameters can be obtained through sensors and other equipment. A nonlinear time-delay-differential-algebraic equation model of the power system is established. According to the system structure diagram, the system time-delay-differential-algebraic equation model can be obtained as follows:

[0068]

[0069] In order to ensure the reliability of the power system, the generator adopts AVR excitation control. Because the control parameters of the excitation control circuit are taken from the remote bus, the generator terminal voltage U t There is a time lag τ in the measured value, so after the time lag U t Change to U t (t-τ). In formula (A), U L ∠δ L is the load bus voltage, δ is the generator rotor phase angle, S m is the generator rotor slip, E′ q is the q-axis potential of the generator rotor, E fd is the excitation electromotive force of the generator. P, Q are the active and reactive power provided by the network to the load. U ref is the reference voltage. d ,i q is the rotor current. The meanings of the system parameters are as follows: Y i ∠θ i is the admittance and phase angle of the i-th branch, ω Bis the system frequency; d is the generator rotor damping coefficient; P m P is the output power of the generator prime mover. ld , Q ld , P0, Q0 are the WALVE load model powers; p1, p2, p3, q1, q2 are the WALVE load model control coefficients; K A ,T A is the generator AVR link parameter; x d , x′ d , x q is the rotor winding impedance.

[0070] Then the formula (A) is systematically organized into the following form:

[0071]

[0072] Where x is the system state variable, and x=[δ L ,U L ,δ,S m ,E' q ,E' fd ]; y is the system algebraic variable, and y=[P,Q,a1,a2,i d ,i q ,U t ]; k and τ are the system parameters and time delay respectively, and

[0073] In step S2, the model is locally linearized to obtain the system characteristic equation. In order to obtain the local linearized model of the time-delay power system, the local linearized model of the system at the equilibrium point when there is no time delay is first obtained:

[0074]

[0075] Among them, the vector of state variable micro-increment Δx∈R 6 , vector of algebraic variable increments Δy∈R 6 ;L n is a 6×6 identity matrix; And A∈R 6×6 , B∈R 6×6 , C∈R 6×6 , D∈R 6×6 .

[0076] According to the above formula, after considering the time delay τ, the local linearization model of the system is obtained as follows:

[0077]

[0078] Where, the sparse matrix J1∈R 12×12Since the rank of matrix J1 is very low, J1 can be decomposed to obtain J1 = GH, where G∈R 12×l , H∈R l×12 , and l<<12.

[0079] Therefore, after taking into account the time lag, the characteristic equation of the system is:

[0080] det(λE-Je -λτ GH)=0 (E)

[0081] In step S3, the exponential term in (E) is subjected to Rekasius transformation. First, it is assumed that as the time delay τ changes, there may be one or more pairs of conjugate pure imaginary eigenvalues ​​on the imaginary axis of the system, that is, λ * =±jω ci , and i=1,2,3,…. Therefore, the system characteristic equation after taking into account the time lag can be further expressed as:

[0082]

[0083] Among them, τ c is the same as λ * =±jω ci The corresponding critical time lag of small signal stability of the system. Then, the exponential term in the system characteristic equation is transformed by Rekasius to obtain:

[0084]

[0085] Among them, T c is the transformation coefficient to be determined, and when λ=λ * There is only one T c Corresponding to it.

[0086] In step S4, the critical time delay of small signal stability of the system is obtained based on the transformed characteristic equation. According to formula (F), there exists a characteristic vector υ that satisfies:

[0087]

[0088] and ||υ||≠0. Combining equations (G) and (H), we can obtain:

[0089] υ=-jω ci T c υ-2jω ci T c (jω ci EJ-GH) -1 GHυ (I)

[0090] Substituting formula (G) into formula (H) yields:

[0091] Hυ / Τ c =-jω ci (E+2H(jω ci EJ-GH) -1 G)Hυ (J)

[0092] Let the matrix s(ω ci )=-jω ci (E+2H(jω ci EJ-GH) -1 G), then Hυ and 1 / T c They are respectively the matrix S(ω ci )’s eigenvectors and eigenvalues. Therefore, if and only if S(ω ci ) has a purely real eigenvalue other than 0, the system (A) has a purely imaginary eigenvalue λ * Therefore, 1 / T c Substituting the following formula, we can obtain the critical time delay of small signal stability of the system:

[0093] τ c =2[tan -1 (ω ci ·T c )±m·π] / ω ci ,m=0,1,2,3… (K)

[0094] In step S5, based on the degenerate Hopf bifurcation (DHB) criterion, it is determined whether the system has a DHB at the critical time delay. If a DHB occurs, an internal cavity warning signal is issued.

[0095] Replace jω in formula (H) ci Expressed as λ, formula (G) can be transformed into the following form:

[0096] λEυ=(J+e -λτ GH)υ (L)

[0097] The derivative of the eigenvalue λ with respect to the time lag τ can be calculated from formula (L):

[0098]

[0099] Where u is the left eigenvector of the characteristic matrix in formula (E) relative to λ, u H represents the conjugate transpose of u. According to formula (I), formula (K) and formula (L), the conjugate imaginary root λ of the system can be obtained * , relative to λ * The left eigenvalue vector u and the right eigenvector υ. c ,λ * Substituting , u and υ into formula (M), we can obtain the DHB criterion: When τ=τc ,λ=λ * When ρ≠0, the system undergoes Hopf bifurcation at this time lag; if ρ=0, the system undergoes degenerate Hopf bifurcation (DHB) at this time lag.

[0100] Figure 3 For this example system, m -d plane small signal stability region division diagram. The area enclosed by different types of curves and the horizontal axis in the figure, as well as the internal area enclosed by different closed curves centered on the DHB point, are all small signal unstable areas of the system under different time delays, and the other areas are small signal stable areas. When the time delay τ = τ c =24.83ms, The system experiences DHB. As the time delay increases, a hole centered at the DHB point will appear inside the small signal stability region, such as Figure 3 This is shown in the gray area, which increases the risk of system instability under this time lag and worsens the system's operating environment. Therefore, after detecting a DHB in the system using the DHB criterion, the system issues a warning signal for a hole within the small signal stability region, reminding operators to monitor operating parameters to avoid falling into the unstable hole region or to take measures to prevent further increases in the time lag, thereby ensuring the stability of the power system in actual operation.

[0101] It should be noted that the contents not described in detail in the embodiments of the present invention belong to the prior art known to those skilled in the art.

[0102] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in the present invention should be included in the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be based on the scope of protection of the claims.

Claims

1. A method for early warning of internal void risk in the small signal stability region of a time-delay power system, characterized in that: The following steps are involved: S1. Obtain power parameters and establish a power system model; S2, locally linearize the model to obtain the characteristic equation; S3, perform Rekasius transformation on the exponential term in the characteristic equation; S4. Calculate the critical time lag of small signal stability based on the transformed characteristic equation; S5. Determine whether a degenerate Hopf bifurcation occurs under the critical time lag. If so, issue a warning signal; if not, do not issue a warning signal.

2. The method for early warning of internal void risk in the small signal stability region of a time-delay power system according to claim 1 is characterized in that: In S1, the power system model expression is as follows: Where x is the system state variable, and x∈R n ; y is the system algebraic variable, and x∈R m ; k and τ are parameters and time lag respectively, and 3. The method for early warning of internal void risk in the small signal stability region of a time-delay power system according to claim 1, characterized in that: In S2, the model is locally linearized to obtain the characteristic equation, which includes the following steps: S21: Create the first local linearized model when there is no time lag; S22: Based on the first local linearization model, deriving a second local linearization model in the presence of a time lag; S23: Obtain a characteristic equation based on the second local linearization model.

4. A method for early warning of internal void risk in the small signal stability region of a time-delay power system according to claim 1 or 3, characterized in that: In S21, the first local linearization model expression is as follows: In the formula, the vector of the state variable micro-increment Δx∈R n , vector of algebraic variable increments Δy∈R m ; L n is an n×n identity matrix; And A∈R n×n , B∈R n×m , C∈R m ×n , D∈R m×m .

5. The method for early warning of internal void risk in the small signal stability region of a time-delay power system according to claim 4, characterized in that: In S22, the second local linearization model expression is as follows: Where τ is the time delay, and the sparse matrix J1∈R (n+m)×(n+m) , since the rank of matrix J1 is very low, J1 can be decomposed to obtain J1 = GH, where G∈R (n+m)×l , H∈R l×(n+m) , and l<<n+m.

6. The method for early warning of internal void risk in the small signal stability region of a time-delay power system according to claim 5, characterized in that: In S23, the characteristic equation is expressed as follows: det(λE-Je -λτ GH)=0 (4)。 7. The method for early warning of internal void risk in the small signal stability region of a time-delay power system according to claim 1, characterized in that: In S3, the exponential term in the characteristic equation is subjected to a Rekasius transformation, which includes the following steps: S31: Based on the conjugate pure imaginary eigenvalue, the characteristic equation is modified. The expression of the modified characteristic equation is as follows: Where, τ c is the same as λ * =±jω ci The corresponding small signal stability critical time lag; S32: Perform Rekasius transformation on the exponential term in the modified characteristic equation. The expression is as follows: Where, T c is the transformation coefficient to be determined, and when λ=λ * There is only one T c Corresponding to it.

8. The method for early warning of internal void risk in the small signal stability region of a time-delay power system according to claim 7, characterized in that: The specific steps of S4 are as follows: According to formula (5), there exists a characteristic vector υ that satisfies: and ||υ||≠0, combining equations (6) and (7), we can obtain: υ=-jω ci T c u-2jω ci T c (y) ci EJ-GH) -1 GHU (8) Substituting formula (6) into formula (7) yields: Hυ / Τ c -jω ci (E+2H(jω ci EJ-GH) -1 G)Hυ (9) Let the matrix s(ω ci )=-jω ci (E+2H(jω ci EJ-GH) -1 G), then Hυ and 1 / T c They are respectively the matrix S(ω ci )’s eigenvectors and eigenvalues; If and only if S(ω ci )When there is a purely real eigenvalue other than 0, there is a purely imaginary eigenvalue λ * ; So 1 / T c Substituting into the following formula, we can obtain the critical time delay of small signal stability, which is expressed as follows: t c =2[tan -1 (oh ci ·T c )±m·π] / ω ci ,m=0,1,2,3… (10).

9. The method for early warning of internal void risk in the small signal stability region of a time-delay power system according to claim 7, characterized in that: In S5, determining whether a degenerate Hopf bifurcation occurs at the critical time lag includes the following steps: In formula (7), jω ci Expressed as λ, equation (7) can be transformed into the following form: λEυ=(J+e -λτ GH)υ (11) The derivative of the eigenvalue λ with respect to the time lag τ can be calculated from formula (11): Where u is the left eigenvector of the characteristic matrix in equation (4) relative to λ, u H represents the conjugate transpose of u; According to equations (8), (10) and (11), the conjugate imaginary root λ can be obtained * , relative to λ * The left eigenvalue vector u and the right eigenvector υ of τ c ,λ * Substituting , u and υ into formula (12), we can obtain Define ρ = dRe(λ) / dτ as the criterion for degenerate Hopf bifurcation, when τ = τ c ,λ=λ * When ρ≠0, a Hopf bifurcation occurs at this time lag; if ρ=0, a degenerate Hopf bifurcation occurs at this time lag.

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