JP4543192B1 - Transient stability limit value calculation method, transient stability limit value calculation device, and program - Google Patents

Abstract

を、多次元状態変数ｘ^０が、電力系統が故障した後に回復可能となる時間と電力系統が故障した後に回復不可能となる時間との臨界となる臨界故障除去時間を定数として含み、且つ、多次元状態変数ｘ^０から多次元状態変数ｘ^ｍ＋１に至る軌跡が、電力系統方程式に基づく所定の関数の支配的不安定平衡点である特異点を通過する、という条件の下で最小化し、目的関数を最小化する多次元状態変数ｘ^０に基づいて、助変数の過渡安定度限界値を算出する。
【選択図】 図２A transient stability limit value calculation method for calculating a transient stability limit value of an auxiliary variable for specifying an operation state between a generator and a load before a power system fails, comprising an objective function

, As a constant, the multidimensional state variable x ⁰ includes a critical failure removal time that is critical between a time that can be recovered after a power system failure and a time that cannot be recovered after a power system failure, and Minimize under the condition that the trajectory from the multidimensional state variable x ⁰ to the multidimensional state variable x ^{m + 1} passes through the singular point that is the dominant unstable equilibrium point of the predetermined function based on the power system equation. based on the multi-dimensional state variables x ⁰ that minimizes the function to calculate the transient stability limit of the parametric.
[Selection] Figure 2

Description

  The present invention relates to a transient stability limit value calculation method, a transient stability limit value calculation device, and a program.

  When a fault such as a ground fault occurs in a part of the power grid in which the generator and the load are connected by a transmission line, the power system can be recovered and cannot be recovered in the time that can be spent to eliminate the fault. There is a critical value (critical failure removal time) that divides the possibility (see, for example, Patent Document 1). For example, when a fault occurs at a point PA illustrated in FIG. 7 and the fault is removed within a critical fault removal time at a point PB on the fault trajectory 10, the phase angle δ and the angular frequency ω of the generator are The phase angle δ and the angular frequency ω of the generator are on the locus 40 when the failure is removed at the point PD on the failure locus 10 beyond the critical failure removal time. Diverges at (irrecoverable). This figure is a schematic diagram showing the nonlinear phenomenon of the one-machine infinite bus system without braking as a locus on the δ-ω phase plane. In the figure, the timing at which the fault is removed at the point PC (between the point PB and the point PD) gives the critical fault elimination time, and the end point PE of the critical locus 30 starting from this point PC is the dominant unstable equilibrium. It corresponds to a point (that is, a singular point).

  By the way, the critical locus 30 illustrated in FIG. 7 is displaced on the δ-ω phase plane according to the operation state between the generator and the load before the failure occurrence at the point PA. The operating state is, for example, active power supplied from a generator to a load. The critical trajectory 30 is displaced toward the trajectory 20 indicating that the recoverable state is smaller as the active power is smaller under a constant failure removal time. On the other hand, the critical trajectory 30 is more likely to be unrecoverable as the active power is larger. Displace to the side. Here, when a failure occurs in the actual power system, the protection relay operates and the failure is removed. In other words, in the case of an electric power system in which the protection relay operates in a certain time after the failure occurs and the failure is removed, the system operator can recover and not recover the power system under the certain failure removal time. The limit value of active power (transient stability limit value) that divides the possibility becomes important.

  An example of a processing procedure for obtaining the transient stability limit value of the active power before the failure will be described with reference to FIG. In addition, the figure is a flowchart which shows an example of the procedure in which the information processing apparatus for power system operation | use calculates | requires the transient stability limit value of active power.

First, the information processing apparatus fixes the critical fault removal time to τ _F representing, for example, the operation time of the protection relay (S900), sets the active power to a certain value (S901), and indicates the state of the power system. A function g representing a state before and during failure of the same system with the vector x as a variable and a function f representing a state after the failure removal of the power system with x as a variable are specified (S902), and the function g An objective function with x as a variable is generated from the power system equation based on (x) and the function f (x) (S903). The objective function means a known function in general such that x that minimizes the objective function is a solution of the power system equation described above. Further, the above-described τ _F and active power are included in the above-described function g (x).

  Next, the information processing apparatus obtains a vector x that minimizes the objective function (S904), and determines whether or not the locus of x is the critical locus 30 (FIG. 7) (that is, the active power set in step S901). Whether or not is a transient stability limit value) is determined (S905). When it is determined that the active power is not the transient stability limit value (S905: NO), the information processing apparatus sets the active power to another value in step S901, and executes the processing after step S902 again. On the other hand, when it is determined that the active power is the transient stability limit value (S905: YES), the information processing apparatus ends the process.

JP 2007-53836 A

  The transient stability limit calculation method described above is a trial-and-error process in which the active power is reset, the power system equation is solved, and whether or not the active power shows the transient stability limit value is determined based on the solution. It is a method by. For this reason, the calculation of solving the power system equation (that is, for example, minimizing the above-described objective function) must be repeated until the active power converges to the transient stability limit value.

  Note that the above is not limited to the active power supplied from the generator before the failure to the load, but widely, the transient stability limit value of the auxiliary variable that specifies the operation state between the generator and the load before the failure occurs. It is a problem related to the calculation of.

  The present invention has been made in view of such a problem, and the object of the present invention is to set a transient stability limit value of an auxiliary variable for specifying an operation state between a generator and a load before the power system fails. It is to calculate efficiently.

として定義される電力系統方程式と、を用いて

として定義される誤差ベクトルを用いて

として定義される目的関数を、前記多次元状態変数ｘ^０が、前記電力系統が故障した後に回復可能となる時間と前記電力系統が故障した後に回復不可能となる時間との臨界となる臨界故障除去時間を定数として含み、且つ、前記多次元状態変数ｘ^０から前記多次元状態変数ｘ^ｍ＋１に至る前記軌跡が、前記電力系統方程式に基づく所定の関数の支配的不安定平衡点である特異点を通過する、という条件の下で最小化し、前記目的関数を最小化する前記多次元状態変数ｘ^０に基づいて、前記助変数の過渡安定度限界値を算出することを特徴とする過渡安定度限界値算出方法である。 The invention for solving the above-mentioned problem is a transient stability for calculating a transient stability limit value of an auxiliary variable for specifying an operation state between a generator and a load in the power system before the power system fails. This is a limit value calculation method, which is a multidimensional state variable x ⁰ that is a function of the auxiliary variable, and a multidimensional state variable x ^{m + 1} that indicates the end point of a trajectory starting from the multidimensional state variable x ⁰ (m is an integer) A plurality of multidimensional state variables x ^k (1 ≦ k ≦ m: k is an integer) discretized between the multidimensional state variables x ⁰ and x ^{m + 1,} and the multidimensional state variables x ^{0 to} x Euclidean distance ε between multi-dimensional state variables x ^k and x ^{k + 1} adjacent to each other in ^{m + 1} ,

Using the power system equation defined as

With an error vector defined as

The objective function is defined as, the multi-dimensional state variable x ⁰ is the critical failure time and the electric power system to be recoverable after the power system has failed is critical and time satisfying unrecoverable after failure A singular point including a removal time as a constant and the trajectory from the multidimensional state variable x ⁰ to the multidimensional state variable x ^{m + 1} is a dominant unstable equilibrium point of a predetermined function based on the power system equation A transient stability limit value of the auxiliary variable is calculated based on the multidimensional state variable x ⁰ that minimizes the objective function and minimizes the objective function. This is a limit value calculation method.

According to this transient stability limit value calculation method, the transient stability limit value of the auxiliary variable that specifies the operation state between the generator and the load in the power system is converted into the multidimensional state variables x ⁰ and x ^{m + 1} described above. it can be obtained from the multi-dimensional state variable x ⁰ that minimizes the objective function described above under conditions according to. In other words, since the substantial calculation can be performed in two steps, such as calculating the auxiliary variable that gives the transient stability limit value from the multidimensional state variable x ⁰ that minimizes the objective function and minimizes the objective function. Resetting this until the variable reaches the transient stability limit value is more computationally efficient than the trial and error method of repeatedly solving the power system equation.

In the transient stability limit value calculation method, the multidimensional state variable x ⁰ includes an r-order expression (r ≧ 1: r is an integer) of the auxiliary variable, and when a fault is removed in a critical state of the power system. It is preferable that the variable vector indicates the state of

  In the transient stability limit value calculation method, the singular point is a singular point based on a synchronization force coefficient matrix K of a generator i (1 ≦ i ≦ n) linked to the power system. Also good.

但し、

として定義されるｎ次元正方行列である、ことが好ましい。 In the transient stability limit value calculation method, the synchronization force coefficient matrix K uses the acceleration power Pa _i and the phase angle δ _i of the generator i,

However,

It is preferably an n-dimensional square matrix defined as

但し、

として定義されるｎ次元正方行列である、ことが好ましい。 Further, in such transient stability limit value calculation method, the synchronizing power coefficient matrix K, by using the acceleration torque Ta _i and the phase angle [delta] _i of the generator i,

However,

It is preferably an n-dimensional square matrix defined as

In the transient stability limit value calculation method, the singular point is obtained by setting the multiplication result of the first condition that the synchronization force coefficient matrix K is an irregular matrix and the synchronization force coefficient matrix K to zero. It is preferable that the direction of the n-dimensional eigenvector v to be performed is a singular point that satisfies both the second condition in which the direction of the n-dimensional vector ω in which the angular frequencies ω _i are arranged coincides with the second condition.

が成立し、
前記第２の条件として、スカラー係数ｋｓを用いて、

が成立する特異点である、ことが好ましい。 In the transient stability limit value calculation method, the singular point uses, as the first condition, the synchronization force coefficient matrix K and a scalar q representing the size of the n-dimensional eigenvector v,

Is established,
Using the scalar coefficient ks as the second condition,

It is preferable that the singular point holds.

として定義される誤差ベクトルμ^ｍ＋１と、正の対角要素を有する正方の対角行列Ｗと、を用いて

として定義される二乗誤差関数を用いて

として定義される関数を最小化する前記多次元状態変数ｘ^０を算出する、ことが好ましい。 In the transient stability limit value calculation method, the multidimensional state variable x ^{m + 1} is a variable vector indicating the singular point, and is based on the first and second conditions.

Using an error vector μ ^{m + 1} defined as and a square diagonal matrix W with positive diagonal elements

Using the square error function defined as

It is preferable to calculate the multidimensional state variable x ⁰ that minimizes the function defined as

として定義される電力系統方程式と、を用いて

として定義される誤差ベクトルを用いて

として定義される目的関数を、前記多次元状態変数ｘ^０が、前記電力系統が故障した後に回復可能となる時間と前記電力系統が故障した後に回復不可能となる時間との臨界となる臨界故障除去時間を定数として含み、且つ、前記多次元状態変数ｘ^０から前記多次元状態変数ｘ^ｍ＋１に至る前記軌跡が、前記電力系統方程式に基づく所定の関数の特異点を通過する、という条件の下で最小化する第１の情報処理部と、前記目的関数を最小化する前記多次元状態変数ｘ^０に基づいて、前記助変数の過渡安定度限界値を算出する第２の情報処理部と、を備えたことを特徴とする過渡安定度限界値算出装置である。 Further, the invention for solving the above-mentioned problem is a transient calculation for calculating a transient stability limit value of an auxiliary variable for specifying an operation state between a generator and a load in the power system before the power system fails. A stability limit value calculation device, which is a multidimensional state variable x ⁰ that is a function of the auxiliary variable, and a multidimensional state variable x ^{m + 1} (m is a value indicating the end point of a trajectory starting from the multidimensional state variable x ^0. An integer), a plurality of multidimensional state variables x ^k (1 ≦ k ≦ m: k is an integer) discretized between the multidimensional state variables x ⁰ and x ^{m + 1,} and the multidimensional state variable x ^0. Euclidean distance ε between multi-dimensional state variables x ^k and x ^{k + 1} that are adjacent to each other in x ^{m + 1} ,

Using the power system equation defined as

With an error vector defined as

The objective function is defined as, the multi-dimensional state variable x ⁰ is the critical failure time and the electric power system to be recoverable after the power system has failed is critical and time satisfying unrecoverable after failure Under the condition that the removal time is included as a constant, and the trajectory from the multidimensional state variable x ⁰ to the multidimensional state variable x ^{m + 1} passes through a singular point of a predetermined function based on the power system equation. a first information processing unit to minimize in, said the objective function based on the multidimensional state variables x ⁰ that minimizes, the second information processing unit for calculating the transient stability limit of the parametric, A transient stability limit value calculating device characterized by comprising:

として定義される電力系統方程式と、を用いて

として定義される誤差ベクトルを用いて

として定義される目的関数を、前記多次元状態変数ｘ^０が、前記電力系統が故障した後に回復可能となる時間と前記電力系統が故障した後に回復不可能となる時間との臨界となる臨界故障除去時間を定数として含み、且つ、前記多次元状態変数ｘ^０から前記多次元状態変数ｘ^ｍ＋１に至る前記軌跡が、前記電力系統方程式に基づく所定の関数の特異点を通過する、という条件の下で最小化する第１の手順と、前記目的関数を最小化する前記多次元状態変数ｘ^０に基づいて、前記助変数の過渡安定度限界値を算出する第２の手順と、を実行させるプログラムである。 Further, the invention for solving the above problem is information for calculating a transient stability limit value of an auxiliary variable for specifying an operation state between a generator and a load in the power system before the power system fails. The processing device includes a multidimensional state variable x ⁰ that is a function of the auxiliary variable, a multidimensional state variable x ^{m + 1} (m is an integer) that indicates an end point of a trajectory starting from the multidimensional state variable x ⁰ , and the multi Among a plurality of multidimensional state variables x ^k (1 ≦ k ≦ m: k is an integer) discretized between the dimensional state variables x ⁰ and x ^{m + 1,} and among the multidimensional state variables x ^{0 to} x ^{m + 1} Euclidean distance ε between adjacent multidimensional state variables x ^k and x ^{k + 1} ,

Using the power system equation defined as

With an error vector defined as

The objective function is defined as, the multi-dimensional state variable x ⁰ is the critical failure time and the electric power system to be recoverable after the power system has failed is critical and time satisfying unrecoverable after failure Under the condition that the removal time is included as a constant, and the trajectory from the multidimensional state variable x ⁰ to the multidimensional state variable x ^{m + 1} passes through a singular point of a predetermined function based on the power system equation. a first step in minimizing in the objective function based on the multidimensional state variables x ⁰ that minimizes the program to be executed and a second step of calculating the transient stability limit of the parametric and It is.

  It is possible to efficiently calculate the transient stability limit value of the auxiliary variable that specifies the operation state between the generator and the load before the power system fails.

Schematic diagram showing examples of a plurality of nonlinear phenomena corresponding to a plurality of active powers (that is, a plurality of auxiliary variables λ) in the power system of the present embodiment by a plurality of trajectories on the δ-ω phase plane of the generator. It is. It is a schematic diagram which displays an example of the discretized multidimensional state variable which shows the state of the electric power grid | system in which the auxiliary variable (lambda) of this Embodiment becomes a transient stability limit value. It is a conceptual diagram which shows the model of the one machine infinite bus system of this Embodiment. It is a graph which shows an example of the power phase difference angle curve of the one machine infinite bus system before failure of this embodiment. It is a flowchart which shows an example of the procedure which calculates | requires the transient stability limit value of the auxiliary variable (lambda) of this Embodiment. It is a block diagram which shows the structural example of the transient stability limit value calculation apparatus of this Embodiment. It is the model which represented the nonlinear phenomenon of the one machine infinite bus system without braking with the locus in the δ-ω phase plane. It is a flowchart which shows an example of the procedure in which the information processing apparatus for electric power system operation | use calculates | requires the transient stability limit value of active power.

=== How to calculate the transient stability limit value ===
The transient stability limit value calculation method of the present embodiment will be described with reference to FIGS. FIG. 1 shows an example of a plurality of nonlinear phenomena corresponding to a plurality of active powers (ie, a plurality of auxiliary variables λ) in the power system 1 by a plurality of trajectories on the δ-ω phase plane of the generator 2. It is a schematic diagram. FIG. 2 is a schematic diagram showing an example of discretized multidimensional state variables x ^{0 to} x ^{m + 1} indicating the state of the power system 1 in which the auxiliary variable λ becomes the transient stability limit value. FIG. 3 is a conceptual diagram showing a model of a one-machine infinite bus system. FIG. 4 is a graph showing an example of the power phase difference curve of the one machine infinite bus system before the failure. FIG. 5 is a flowchart showing an example of a procedure for obtaining the transient stability limit value of the auxiliary variable λ.

Hereinafter, the power system equation of the present embodiment will be described based on the power system 1 of the one-machine infinite bus system illustrated in FIG. When the power system equation plurality of generators is applied to the associated lines, such as by increasing the number of generators dimension vector of the multidimensional state variable x ^k, for example to be described later, extend the same equations easily it can.

  However, regarding the solution of the power system equation under a predetermined constraint condition, the handling of the singular point S described later differs between the case of a one-machine infinite bus system and the system of a plurality of generators linked together. Therefore, these two cases will be described separately.

ここで、δ^ｋは、発電機２の時刻ｔ^ｋにおける位相角を表わし、ω^ｋは、発電機２の時刻ｔ^ｋにおける角周波数を表わす。これらδ^ｋ及びω^ｋの座標は、慣性中心座標系又はそれ以外の座標系の何れでもよい。ｘ_ｓｙｓ ^ｋは、電力系統１の時刻ｔ^ｋにおけるその他の多次元状態変数、例えば不図示のＳＶＣ（Static Var Compensator）やＳＶＧ（Static Var Generator）等の制御器の多次元状態変数を表わす。 <<< One machine infinite bus system >>>
<Power system equation>
The power system 1 illustrated in FIG. 3 is a one-machine infinite bus system, and the power source 2 side bus 201 and the load 4 side bus 401 have two power transmission lines 301 and 302 (“power transmission line 3”). Are collectively connected). The multidimensional state variable x ^k (0 ≦ k ≦ m + 1, where k and m are integers) representing the state of the power system 1 at discrete times t ^k is a vector represented by the following “Expression 1”.

Here, [delta] ^k, represents the phase angle at time t ^k of the generator 2, the omega ^k, represents the angular frequency at time t ^k of the generator 2. The coordinates of δ ^k and ω ^k may be either an inertial center coordinate system or any other coordinate system. x _sys ^k represents another multidimensional state variable of the power system 1 at time t ^k , for example, a multidimensional state variable of a controller such as an unillustrated SVC (Static Var Compensator) or SVG (Static Var Generator).

ここで、Ｅ^ｋはＡＶＲのモデルの状態変数、Ｐｍ^ｋはガバナのモデルの状態変数であり、双方ともに時刻ｔ^ｋにおける状態を表わしている。例えばＥ^ｋはＡＶＲに関して採用するモデルの次数(次元)のベクトルであり、一次元のＡＶＲモデルを使用する場合は、Ｅ^ｋはスカラー変数（１次元）となる。Ｐｍ^ｋは、同様にガバナモデルに相当する状態変数である。 Incidentally, the power system 1, AVR (Automatic Voltage Regulator), governor (Governor), if provided with a PSS (Power System Stabilizer) etc. (all not shown), the multi-dimensional state variable ^{x k} at time ^{t k,} It becomes a vector represented by the following “Equation 2”.

Here, E ^k is a state variable of the AVR model, and Pm ^k is a state variable of the governor model, and both represent the state at time t ^k . For example, E ^k is a vector of the order (dimension) of the model adopted for AVR, and when a one-dimensional AVR model is used, E ^k is a scalar variable (one dimension). Similarly, Pm ^k is a state variable corresponding to the governor model.

Power system equations multidimensional state variable x ^k of the same strain 1 after removal of the fault occurring in the power system 1, the motion equation of "Formula 3" below based on a non-linear function f after fault clearance. This failure is, for example, a ground fault of a part of the transmission line 3 (for example, the transmission line 301).

ここで、「式４」の左辺のＸ_ＬＰは、後述する臨界故障除去時間τ_Ｆを定数として含み、助変数λの関数である故障前の多次元状態変数ｘ_ｐｒｅ（λ）を初期値とする関数を表わしている。また、「式４」の右辺のａ_０、ａ_１、ａ_２、ａ_３は、それぞれ、助変数λの０次項、１次項、２次項、３次項の係数を表わしている。 In addition, the power system equation in failure representing the multidimensional state variable x _pre before the failure in the power system 1 to the multidimensional state variable x ⁰ of the power system 1 at the time t ⁰ indicating the time of failure removal is a well-known transient. It can be solved by stability simulation. For example, as shown in “Expression 4”, the multidimensional state variable x ⁰ is expanded with a polynomial of a scalar value auxiliary variable λ that specifies the operation state between the generator 2 and the load 4 before the failure of the power system 1. it can.

Here, X _{LP on} the left side of “Expression 4” includes a critical failure removal time τ _F, which will be described later, as a constant, and a multidimensional state variable x _pre (λ) before failure, which is a function of the auxiliary variable λ, is an initial value. Represents the function to perform. Further, a ₀ , a ₁ , a ₂ , and a _{3 on} the right side of “Expression 4” represent the coefficients of the 0th, 1st, 2nd, and 3rd order terms of the auxiliary variable λ, respectively.

When the auxiliary variable λ is a scalar value corresponding to the active power P supplied from the generator 2 to the load 4, the auxiliary variable λ and the multidimensional state variable x ⁰ are certainly “ The fact that the correlation is expressed by the equation (4) will be described. For example, the operating state of the power system 1 before the failure is expressed as “Equation 5” by the auxiliary variable λ using P ₀ representing the initial amount of the state and P ₁ representing the amount of change in the state. it can.

On the other hand, the active power P and the phase difference angle δ ′ between the generator 2 and the load 4 (that is, the difference between the phase angle of the generator 2 and the phase angle of the load 4) are generally shown in the power phase difference curve in FIG. With the correlation shown. In this phase difference curve, the active power P is a function (P (λ)) of the auxiliary variable λ expressed by “Expression 5”, and the phase difference angle δ ′ of the generator 2 and the load 4 is the phase of the generator 2. Correlation with the angle δ and the angular frequency ω (which is the time derivative of the in-phase angle δ). As illustrated in FIG. 4, the power phase difference angle curve is symmetric about the phase difference angle δ _c ′ that gives the maximum value P _MAX of the active power P. That is, for example valid 'if drops "(<P in), a stable equilibrium point of the power system 1, the phase difference angle [delta] _1" from [delta] ₁ power from P "' P direction to decrease to (<[delta] _1") On the other hand, the unstable equilibrium point of the electric power system 1 is displaced in a direction in which the phase difference angle increases from δ ₂ ″ to δ ₂ ′ (> δ ₂ ″).

From the above, the phase angle δ of the generator 2 has a correlation indicated by the auxiliary variable λ and the phase difference angle curve of FIG. That is, it can be said that the multidimensional state variable x _pre constituted by the phase angle δ and the angular frequency ω of the generator 2 before the failure is certainly a function (x _pre (λ)) of the auxiliary variable λ. Therefore, the multidimensional state variable x ⁰ represented by the function X _LP having the multidimensional state variable x _pre (λ) as an initial value and the auxiliary variable λ have a correlation represented by the above-described “Expression 4”.

ここで、「式６」は、時刻ｔ^ｋにおける多次元状態変数ｘ^ｋを構成する位相角δ^ｋ及び角周波数ω^ｋの動揺を表わし、「式７」は、例えば、電力系統１の母線２０１、４０１の電流、電圧、有効電力、無効電力等に関する条件を与えたり、電力系統１の構成機器等の特性に関する条件を与えたりする。 <General form of equation>
The above-described power system equation of “Equation 3” is a differential equation “Equation 6” and an algebraic equation using a multidimensional dependent variable y ^k of the multidimensional state variable x ^k as a more general expression. It is represented by “Expression 7”.

Here, “Expression 6” represents the fluctuation of the phase angle δ ^k and the angular frequency ω ^k constituting the multidimensional state variable x ^k at time t ^k , and “Expression 7” is, for example, the bus 201 of the power system 1 , 401, conditions regarding the current, voltage, active power, reactive power, etc., and conditions regarding the characteristics of the components of the power system 1 are given.

(Expression by analytic function)
When the multidimensional dependent variable y ^k can be expressed by an analytical function p of the multidimensional state variable x ^k based on “Expression 7”, the right side of “Expression 6” can be transformed as “Expression 8”. That is, “Expression 6” and “Expression 7” representing the general form of the power system equation are reduced to “Expression 3” described above.

(Expression by numerical function)
On the other hand, if the multidimensional dependent variable y ^k of can not be represented by analytic functions of the multi-dimensional state variable x ^k, relationship between multidimensional state variable x ^k and multidimensional dependent variable y ^k based on "Equation 7" Can be obtained numerically, and the obtained x ^k and y ^k can be applied to “Expression 6” to numerically determine the left side of “Expression 6” meaning (dx ^k / dt ^k ). Further, the analysis can be performed by linearizing the nonlinear function f after the failure removal in the above-described “Expression 3”. In this case, the Jacobian matrix (linearization system matrix) A is defined as “Equation 9” relating to the function f or “Equation 10” relating to the functions h and s.

<Method for solving power system equations>
The transient stability limit value calculation method of the present embodiment uses a multidimensional state variable that is a solution of the power system equation of “Equation 3” described above as a critical trajectory 33 (see FIG. 1) after fault removal of the power system 1. calculated as x ^k, from a multidimensional state variables x ⁰ at the start point of the resulting critical path 33, for example using the "equation 4" described above, between the generator 2 before the failure of the power system 1 and the load 4 The transient stability limit value of the auxiliary variable λ for specifying the operation state is calculated. Incidentally, "Equation 4" is a variable vector indicating a singular point S of the predetermined function multidimensional state variables x ^{m + 1} is based on "Equation 3" of the "end-point of the critical path 33 shown by the" formula 11 "below And a constraint condition when the critical locus 33 is obtained.

助変数λの過度安定度限界値を求めるにあたって、図１を参照することによって、この過渡安定度限界値を与える助変数λと、多次元状態変数ｘ^ｋの軌跡との関係の概略を把握できる。

In obtaining the transient stability limit value of the auxiliary variable λ, the outline of the relationship between the auxiliary variable λ giving the transient stability limit value and the trajectory of the multidimensional state variable x ^k can be grasped by referring to FIG. .

When the auxiliary variable λ is less than the transient stability limit value, if the failure is removed at the point PB1 on the failure locus 11 after the time τ _{F has} elapsed since the failure occurred at the point PA1, the phase angle δ and the angle of the generator 2 The frequency ω converges to the point PA1 via the locus 21 (recoverable). After the same time τ _{F has} elapsed from the failure at the point PA2, the failure is removed at the point PB2 on the failure locus 12, and the phase angle δ and the angular frequency ω of the generator 2 converge to the point PA2 via the locus 22 (recoverable). ) Is the same. However, the above is an example that does not complicate the explanation. Strictly speaking, since the switching state of the switch (not shown) is usually different before and after the failure, the same operating state (stable equilibrium point) is not achieved. . Generally, it converges to an operating point (stable equilibrium point after failure) slightly deviated from the operating point before failure (stable equilibrium point before failure).

If the auxiliary variable λ exceeds the transient stability limit value, the phase of the generator 2 can be detected even if the failure is removed at the point PD4 on the failure locus 14 after the same time τ _{F has} elapsed since the failure occurred at the point PA4. The angle δ and the angular frequency ω diverge on the locus 44 (unrecoverable). Similarly, the failure is removed at the point PD5 on the failure locus 15 after the lapse of the same time τ _F from the failure at the point PA5, and the phase angle δ and the angular frequency ω of the generator 2 diverge on the locus 45 (unrecoverable). It is.

In FIG. 1, the above-described failure removal points PB1, PB2, PC3, PD4, and PD5 having the same variable removal time λ but the same failure removal time τ (τ _F ) A multidimensional state variable x ⁰ (= X _LP (τ _F ; (x _pre (λ)))) expressed by the above-described “Expression 4” is given.

Here, when the auxiliary variable λ is the transient stability limit value, when the failure is removed from the point PA3 on the failure locus 13 at the same timing (time τ _F ) as described above, the critical locus 33 starting from the point PC3. The end point PE3 of this corresponds to the dominant unstable equilibrium point.

Hereinafter, a specific solution of the above-described “Formula 3” will be described. By applying the trapezoidal formula approximation to the nonlinear equation of “Expression 3”, the following “Expression 12” is established.

As illustrated as the critical path 33 in FIG. 2, x ⁰ corresponding to the start point PC3 of the critical path 33, a vector of multi-dimensional state variable of the failure is removed giving the limit value of the parametric lambda, the critical X ^{m + 1} corresponding to the end point PE3 of the trajectory 33 is a vector of multidimensional state variables corresponding to the dominant unstable equilibrium state. However, the time difference (t ^{k + 1} −t ^k ) on the right side increases infinitely as the multidimensional state variable x ^k satisfying “Expression 12” approaches the end point PE3 that is the dominant unstable equilibrium point. In the form, the Euclidean distance ε between two adjacent multidimensional state variables x ^{k + 1} and x ^k illustrated in FIG. 2 is defined by the following “Equation 13”.

By using such Euclidean distance ε, “Expression 12” is converted to “Expression 14”. The Euclidean distance ε is always constant without becoming infinite even when approaching the dominant unstable equilibrium point PE3.

ここで、（式１６）におけるμ^ｋ’は、誤差ベクトルμ^ｋを転置したベクトルを意味する。 0 on the right side of “Expression 14” represents a zero vector, and “Expression 14” means (m + 1) multi-dimensional simultaneous equations. That is, obtaining the solution of the multiple simultaneous equations of “Expression 14” is equivalent to obtaining the multidimensional state variable x ^k indicating equally spaced points without directly dealing with infinite time. It should be noted that the right side of the (m + 1) multiple simultaneous equations shown in “Expression 14” is ideally all 0 (scalar value), but in reality, this is due to the approximation of the trapezoidal formula described above. Numerical error occurs. Therefore, in order to obtain a solution of the multiple simultaneous equations of “Expression 14”, first, as shown in “Expression 15”, the left side of “Expression 14” is defined as an error vector μ ^k , and then “Expression 16”. As shown, the sum of the magnitudes (norms) of the error vectors μ ^k is defined as the objective function O.

Here, μ ^{k ′} in (Expression 16) means a vector ^obtained by transposing the error vector μ ^k .

ここで、「式１８」は前述した「式４」と略同一であり、「式１９」は前述した「式１１」と同一であるが、「式１７」の制約条件を表わす式として改めてここに記載した。 The problem of finding the solution of the multiple simultaneous equations of “Expression 14” is a multidimensional state variable x ^k (0 ≦ k ≦ m + 1) that minimizes the objective function O defined by “Expression 16”, Euclidean distance ε, limit value As an optimization problem for obtaining an auxiliary variable λ that gives The critical fault removal time τ is included in the multidimensional state variable x ⁰ as a constant τ _F given by, for example, the operation time of the protection relay of the power system 1, and the multidimensional state variable x ⁰ is, for example, described above. This is a function of the auxiliary variable λ expressed by “Expression 4”. From the above, the variables x ^{0 to} x ^{m + 1} and ε and the auxiliary variable λ are optimized by “Expression 17” under the constraints of “Expression 18” and “Expression 19”.

Here, “Equation 18” is substantially the same as “Equation 4” described above, and “Equation 19” is the same as “Equation 11” described above. It was described in.

“Equation 18” may be obtained in advance by any known method. For example, the multidimensional state variable x ⁰ (= X _LP (τ _F ; x _pre (λ))) is expressed by the rth-order expression (r is the r The coefficient of each term can be obtained in advance by performing a well-known transient stability simulation according to the following procedures 1 and 2.

(Procedure 1) Two multidimensional state variables x ⁰ _ 1 and x ⁰ _ 2 are obtained from two different auxiliary variables λ ₁ and λ ₂ (λ ₁ <λ ₂ ) using “Expression 20”. A well-known transient stability simulation is performed with the multidimensional state variables x ⁰ _ 1 and x ⁰ _ 2 as the starting points, and the stability of the power system 1 is determined. Such a simulation is repeated until, for example, auxiliary variables λ ₁ and λ _{2 such} that the power system 1 is stable with the auxiliary variable λ ₁ and the power system 1 is unstable with the auxiliary variable λ ₂ are obtained.
(Procedure 2) A well-known transient stability simulation for determining the coefficients a _{0 to} a _r is performed a number of times corresponding to the dimension r of “Expression 20”. However, when r is 2 or less, the coefficient is obtained only from the procedure 1 described above.

<<< System in which multiple generators are linked >>>
<Power system equation>
As described above, for example, by increasing the dimension of the vector of the multidimensional state variable x ^k (0 ≦ k ≦ m + 1, where k and m are integers) by the number of generators, the power system equation is made infinite by one machine. The system can be easily extended from a bus system to a system (not shown) in which a plurality of generators (for example, n units, where n is an integer of 2 or more) are linked.

<Method for solving power system equations>
In the case of a system in which a plurality of generators are linked, the singularity given as S in the above-described “Equation 19” is derived from the synchronization force of the plurality of generators described below.

ここで、Ｍ_ｉは発電機ｉの慣性定数を表わし、Ｐａ_ｉ ^ｋ及びＴａ_ｉ ^ｋは、時刻ｔ^ｋにおける発電機ｉの加速電力及び加速トルクをそれぞれ表わす。 Among the electric power system equations expressed by “Expression 3” described above, the oscillation equation using the phase angle δ _i ^k and the angular frequency ω _i ^k of the generator i (not shown) at the time t ^k is expressed by the following “Expression 21”. Or “formula 22” (1 ≦ i ≦ n, where i and n are integers).

Here, _{M i} represents the inertia constant of the generator i, _Pa ^{i k} and _Ta ^{i k} represents the acceleration power and acceleration torque of the generator i at time ^{t k,} respectively.

Note that multi-dimensional state variable x ⁰ through x ^{m + 1} in the line where a plurality of generators are coordinated, the dimension of the vector is different from the multi-dimensional state variable x ⁰ through x ^{m + 1} of the above-described "Equation 3" and "Formula 19" However, the mathematical expressions can be handled in the same form.

Synchronizing power of the generator i is either defined by the differential coefficient _dPa ⁱ k / dt ^k acceleration power _Pa ^{i k} for the phase angle .delta.i ^k at time ^{t k,} or, for the phase angle [delta] _i ^k at time ^{t k} It is defined by the differential coefficient dTa _i ^k / dt ^k of the acceleration torque Ta _i ^k . Then, the singular point S is specifically obtained based on the synchronization force coefficient matrix K (“Expression 23”) which is an n-dimensional square matrix having these differential coefficients as elements.

或いは、Ｓとして与えられる特異点は、「式２５」で表わされる複数の発電機の同期化力係数行列Ｋに基づいて求められる。

However, the element group L1 of the first row and the second to (n−1) th columns, the element group L2 of the nth row and the second to (n−1) th columns, the second to The element group M1 in the (n-1) th row and the first column, the element group M2 in the second to (n-1) th row and the nth column, the second to (n-1) th row, and the second to the second. The element group O1 in the (n−1) column is as shown in “Expression 24” below. Pa _i and δ _j (1 ≦ i ≦ n, 1 ≦ j ≦ n, where i, j, and n are integers) are the acceleration power of the generator i and the phase angle of the generator j, respectively.

Alternatively, the singularity given as S is obtained based on the synchronization force coefficient matrix K of a plurality of generators expressed by “Equation 25”.

However, the element group L3 of the first row and the second to (n−1) th columns, the element group L4 of the nth row and the second to (n−1) th columns, the second to The element group M3 in the (n-1) th row and the first column, the element group M4 in the second to (n-1) th row and the nth column, the second to (n-1) th row, and the second to the second element. The element group O2 in the (n−1) column is as shown in “Expression 26” below. Ta _i and δ _j (1 ≦ i ≦ n, 1 ≦ j ≦ n, where i, j, and n are integers) are the acceleration torque of the generator i and the phase angle of the generator j, respectively.

The singular point S based on the synchronization force coefficient matrix K given by the above “Expression 23” or “Expression 25” includes the first condition in which the matrix K is an irregular (singular) matrix, and the zero eigenvalue of the matrix K. And the second condition in which the direction of the n-dimensional eigenvector v corresponding to (that is, the multiplication result with K is zero) coincides with the direction of the n-dimensional vector ω in which the angular frequency ω _i of the generator _i is arranged is established. It is a singular point.

ここで、「式２７」又は「式２８」が成立することは、同期化力係数行列Ｋのゼロ固有値に相当する（即ち、Ｋとの乗算結果をゼロとする）ｎ次元固有ベクトルｖを用いた以下の「式２９」及び「式３０」が成立することと等価である。

ここで、ｎ次元固有ベクトルｖは、「式２９」で定められるように、その方向のみが意味を持ち、その大きさは任意である。よって、「式３０」で定められるように、ｎ次元固有ベクトルｖの大きさを、スカラーｑ（例えば１）によって表わしている。 The necessary and sufficient condition for satisfying the first condition is that the following “Expression 27” or “Expression 28” is satisfied.

Here, the fact that “Expression 27” or “Expression 28” is satisfied corresponds to the zero eigenvalue of the synchronization force coefficient matrix K (that is, the multiplication result with K is set to zero), and the n-dimensional eigenvector v is used. This is equivalent to the following “Equation 29” and “Equation 30” being satisfied.

Here, the n-dimensional eigenvector v is meaningful only in its direction, and its size is arbitrary, as defined by “Expression 29”. Therefore, as defined by “Expression 30”, the magnitude of the n-dimensional eigenvector v is represented by a scalar q (for example, 1).

The necessary and sufficient condition for satisfying the second condition is that the following “Expression 31” is satisfied using the scalar coefficient ks. Alternatively, “Expression 31” may be modified in consideration of the center of inertia ω ₀ to be “Expression 32”.

  From the above, the condition of the singular point S is that all of the above-described “formula 29”, “formula 30”, and “formula 31” (or “formula 32”) are satisfied. Here, the unknowns are an n-dimensional eigenvector v, an n-dimensional vector ω, and a scalar coefficient ks.

また、「式３３」の二乗誤差関数を前述した「式１６」に加算して、以下の「式３７」を定義する。 The error vector μ ^{m + 1} of the multidimensional state variable x ^{m + 1} at the end point of the critical locus is formally rewritten as the following “33”, and each component of the error vector μ ^{m + 1} is rewritten to the right side of the above-mentioned “Expression 29”, Assuming that the right side of “Expression 30” and the right side of “Expression 31” are “Expression 29”, “Expression 30”, and “Expression 31”, “Expression 34”, “Expression 35”, and “Expression This is equivalent to the fact that the error vector μ ^{m + 1} in “Expression 33” having “36” as a component is a 0 vector.

Further, the following “Expression 37” is defined by adding the square error function of “Expression 33” to “Expression 16” described above.

Here, W is a square diagonal matrix whose diagonal elements are positive (for example, a unit matrix). Note that multi-dimensional state variable x ⁰ through x ^{m + 1} The first term of the "formula 37" is different dimension vectors with multidimensional state variables x ⁰ through x ^{m + 1} of the "formula 16" described above, the formula Can be handled identically in form.

Minimizing the objective function O defined by “Expression 16” under the condition relating to the singular point that the error vector μ ^{m + 1 of} “Expression 33” is a 0 vector is newly defined by “Expression 37”. This is equivalent to minimizing the objective function E to be performed. That is, x ^{1 to} x ^{m + 1} , ε, λ, v, and ks are optimized by “Expression 38” under the constraints of “Expression 39”.

In the optimization problem of “Expression 38”, the above-described “Expression 29”, “Expression 30”, and “Expression 31” are all used as the condition for the singular point S, and the multidimensional state variable at the end point of the critical locus is used. the error vector mu ^{m + 1} of x m ^{+ 1,} "formula 33", "formula 34", "formula 35", and defined by "formula 36", but is not limited thereto. For example, the error vector μ ^{m + 1} is obtained using the equivalent condition of the singular point S obtained by eliminating the n-dimensional eigenvector v and the scalar coefficient ks from the above-described “formula 29”, “formula 30”, and “formula 31”. Can also be defined.

Here, the vector ω in “Expression 31” described above is replaced with a vector (dδ / dt).

を通じて、ｘ^１乃至ｘ^ｍ＋１、ε、λを最適化してもよい。尚、この「式４２」では、最適化の対象であるｎ次元固有ベクトルｖ及びスカラー係数ｋｓが消去されているという点で、前述した「式３８」と異なる。 Under the constraints of these “Expression 40” and “Expression 41” and “Expression 39” described above, the following “Expression 42”

X ^{1 to} x ^{m + 1} , ε, λ may be optimized. The “expression 42” is different from the above-described “expression 38” in that the n-dimensional eigenvector v and the scalar coefficient ks to be optimized are deleted.

<<< Procedure for obtaining transient stability limit value of auxiliary variable λ >>>
With reference to FIG. 5, an example of a processing procedure for obtaining the transient stability limit value of the auxiliary variable λ that specifies the operating state between the generator and the load (for example, representing active power or the like) will be described.

First, the information processing apparatus 100, which will be described later, fixes the critical fault removal time to τ _F that represents, for example, the operation time of the protection relay (S100), and multi-dimensional state variable x (that is, x ^{0 to} x ^{m + 1)} ) And the function f of the multidimensional state variable x representing the state after fault removal of the power system (S101), and from the power system equation based on the function f (x), the objective function O (x) or E ( x) is generated (102).

Next, the information processing apparatus 100 obtains the multidimensional state variable x, the Euclidean distance ε, the n-dimensional eigenvector v, and the scalar coefficient ks that minimize the objective function O (x) or E (x) (S103). from x ⁰ of the multi-dimensional state variable x, obtaining the parametric λ give transient stability limit value (S104). Incidentally, the coefficient of each term of the case (the r, an integer of 1 or more) r following equation multidimensional state variables x ⁰ example parametric λ is approximated by function, as described above, by well-known transient stability simulation It is requested in advance.

According to this transient stability limit value calculation method, the transient stability limit value of the auxiliary variable λ for specifying the operation state between the generator and the load in the power system is converted into the multidimensional state variable x (x ⁰ and x ⁰ ). objective function O (x) or E (x) is can be determined from ^{x 0} to minimize under the conditions according to the ^{m + 1).} That is, substantial computation to minimize the objective function (step S103 described above), to calculate the parametric λ give transient stability limit value from x ⁰ that minimizes the same objective function (step S104 described above), Therefore, the calculation efficiency is better than the method based on trial and error in which the power system equation is repeatedly solved by setting it again until the auxiliary variable λ reaches the transient stability limit value.

In the embodiment described above, but the constraint that the end point is singular point S of the critical path (path leading from the multi-dimensional state variable x ⁰ multidimensional state variable x ^{m + 1)} have been imposed, limited to Is not to be done. For example, a constraint condition that this critical trajectory passes through the singular point S may be imposed.

=== Transient stability limit value calculation device ===
A configuration example of the information processing apparatus (transient stability limit value calculation apparatus) 100 according to the present embodiment will be described with reference to FIG. FIG. 2 is a block diagram illustrating a configuration example of the information processing apparatus 100.

  The information processing apparatus 100 includes a CPU 101, a display device 102, an input device 103, a memory 104, and a storage device 105, and is used by, for example, a power system operator.

The CPU 101 specifies the operation state between the generator and the load (for example, representing the active power), and the above-described steps S100 to S104 (FIG. 5) which are processing procedures for obtaining the transient stability limit value of the auxiliary variable λ. ). In particular, the constraint condition that the multidimensional state variable x ⁰ indicating the start point of the critical trajectory includes the critical fault removal time as a constant τ _F and the end point of the critical trajectory (multidimensional state variable x ^{m + 1} ) is the singular point S. The function of the CPU 101 in step S103 (FIG. 5) described above that minimizes the objective function O (x) or E (x) below corresponds to the first information processing unit. Further, based on the multi-dimensional state variables x ⁰ that minimizes the objective function O (x) or E (x), that calculates the transient stability limit of the parametric lambda, step S104 described above (Figure 5) The function of the CPU 101 corresponds to the second information processing unit.

  The display device 102 is, for example, a liquid crystal display that displays a system diagram of the power system, a transient stability limit value, and the like so that the system operator can view it.

The input device 103 is, for example, a keyboard or a mouse for the system operator to input information such as the critical fault removal time τ _F.

  The memory 104 stores, for example, data used for the processing of the CPU 101 described above.

The storage device 105 stores, for example, a program for causing the CPU 101 to execute the above-described steps S100 to S104 (FIG. 5), which is a processing procedure for obtaining the transient stability limit value of the auxiliary variable λ ( For example, a DVD or a CD) or a magnetic disk (for example, an MO or a floppy disk). In particular, in the program, the multidimensional state variable x ⁰ indicating the start point of the critical trajectory includes the critical fault removal time as a constant τ _F and the end point of the critical trajectory (multidimensional state variable x ^{m + 1} ) is the singular point S. Step S103 (FIG. 5) described above that minimizes the objective function O (x) or E (x) under the constraint conditions corresponds to the first procedure. Further, in the program, step S104 based on the multi-dimensional state variables x ⁰ that minimizes the objective function O (x) or E (x), described above as calculating the transient stability limit of the parametric lambda (Figure 5 ) Corresponds to the second procedure.

In the embodiment described above, but the constraint that the end point is singular point S of the critical path (path leading from the multi-dimensional state variable x ⁰ multidimensional state variable x ^{m + 1)} have been imposed, limited to Is not to be done. For example, a constraint condition that this critical trajectory passes through the singular point S may be imposed.

=== Other Embodiments ===
The above-described embodiment is intended to facilitate understanding of the present invention, and is not intended to limit the present invention. The present invention is changed and improved without departing from the gist thereof, and the present invention includes equivalents thereof.

  For example, instead of the above-described power system equation of “Expression 3”, the above-described “Expression 6” which is a differential equation and the above-mentioned “Expression 7” which is an algebraic equation are described as a more general expression of the equation. And may be used.

In this case, the above-described “Expression 7” is set as the constraint condition. That is, “Expression 43” representing the square error function of the error vector μ _ADD ^k defined by the following “Expression 44” is added to the aforementioned objective function E (x) and y ^k (0 ≦ k ≦ m + 1). ) Is added to the unknown at the time of minimization.

DESCRIPTION OF SYMBOLS 1 Electric power system 2 Generator 3 Transmission line 4 Load 10, 11, 12, 13, 14, 15 Failure locus 20, 21, 22, 40, 44, 45 locus 30, 33 Critical locus 100 Information processing apparatus 101 CPU
102 Display device 103 Input device 104 Memory 105 Storage device 201, 401 Bus line 301, 302 Power transmission line

Claims (10)

A transient stability limit value calculation method for calculating a transient stability limit value of an auxiliary variable for specifying an operation state between a generator and a load in the power system before the power system fails,
A multidimensional state variable x ⁰ which is a function of the auxiliary variable;
A multidimensional state variable x ^{m + 1} (m is an integer) indicating the end point of a trajectory starting from the multidimensional state variable x ⁰ ;
A plurality of multidimensional state variables x ^k (1 ≦ k ≦ m: k is an integer) discretized between the multidimensional state variables x ⁰ and x ^{m + 1} ;
Euclidean distance ε between multidimensional state variables x ^k and x ^{k + 1} that are adjacent to each other among the multidimensional state variables x ^{0 to} x ^{m + 1} ,

Using the power system equation defined as

With an error vector defined as

The objective function is defined as, the multi-dimensional state variable x ⁰ is the critical failure time and the electric power system to be recoverable after the power system has failed is critical and time satisfying unrecoverable after failure A singular point including a removal time as a constant and the trajectory from the multidimensional state variable x ⁰ to the multidimensional state variable x ^{m + 1} is a dominant unstable equilibrium point of a predetermined function based on the power system equation Minimizing under the condition of passing
The multidimensional state based on variables x ^0, transient stability limit value calculation method characterized by calculating the transient stability limit of the auxiliary variable that minimizes the objective function. The multidimensional state variables x ⁰ is, r equation of the auxiliary variables (r ≧ 1: r is an integer) includes a variable vector which indicates a state of failure is removed in the critical state of the power system, it features the The transient stability limit value calculation method according to claim 1.   3. The transient according to claim 1, wherein the singular point is a singular point based on a synchronization force coefficient matrix K of a generator i (1 ≦ i ≦ n) linked to the power system. Stability limit calculation method. The synchronization force coefficient matrix K is
Using the acceleration power Pa _i and the phase angle δ _i of the generator i,

However,

The transient stability limit value calculation method according to claim 3, wherein the transient stability limit value calculation method is an n-dimensional square matrix defined as The synchronization force coefficient matrix K is
Using acceleration torque Ta _i and the phase angle [delta] _i of the generator i,

However,

The transient stability limit value calculation method according to claim 3, wherein the transient stability limit value calculation method is an n-dimensional square matrix defined as The singularity is
A first condition in which the synchronization force coefficient matrix K is an irregular matrix;
A second condition in which the direction of the n-dimensional eigenvector v having a multiplication result with the synchronization force coefficient matrix K equals the direction of the n-dimensional vector ω in which the angular frequencies ω _i are arranged;
The transient stability limit value calculating method according to claim 4 or 5, wherein the singular point where both of the conditions are satisfied is satisfied. The singularity is
As the first condition, using the synchronization force coefficient matrix K and a scalar q representing the magnitude of the n-dimensional eigenvector v,

Is established,
Using the scalar coefficient ks as the second condition,

The transient stability limit value calculation method according to claim 6, wherein the singular point is established. The multidimensional state variable x ^{m + 1} is a variable vector indicating the singular point;
Based on the first and second conditions

Using an error vector μ ^{m + 1} defined as and a square diagonal matrix W with positive diagonal elements

Using the square error function defined as

The transient stability limit value calculation method according to claim 7, wherein the multidimensional state variable x ⁰ that minimizes a function defined as is calculated. A transient stability limit value calculating device that calculates a transient stability limit value of an auxiliary variable that specifies an operation state between a generator and a load in the power system before the power system fails,
A multidimensional state variable x ⁰ which is a function of the auxiliary variable;
A multidimensional state variable x ^{m + 1} (m is an integer) indicating the end point of a trajectory starting from the multidimensional state variable x ⁰ ;
A plurality of multidimensional state variables x ^k (1 ≦ k ≦ m: k is an integer) discretized between the multidimensional state variables x ⁰ and x ^{m + 1} ;
Euclidean distance ε between multidimensional state variables x ^k and x ^{k + 1} that are adjacent to each other among the multidimensional state variables x ^{0 to} x ^{m + 1} ,

Using the power system equation defined as

With an error vector defined as

The objective function is defined as, the multi-dimensional state variable x ⁰ is the critical failure time and the electric power system to be recoverable after the power system has failed is critical and time satisfying unrecoverable after failure Under the condition that the removal time is included as a constant, and the trajectory from the multidimensional state variable x ⁰ to the multidimensional state variable x ^{m + 1} passes through a singular point of a predetermined function based on the power system equation. A first information processing unit minimized by
Based on the multi-dimensional state variable x ⁰ that minimizes the objective function, and the second information processing unit for calculating the transient stability limit of the parametric,
A transient stability limit value calculating device comprising: In the information processing device that calculates the transient stability limit value of the auxiliary variable that specifies the operation state between the generator and the load in the power system before the power system fails,
A multidimensional state variable x ⁰ which is a function of the auxiliary variable;
A multidimensional state variable x ^{m + 1} (m is an integer) indicating the end point of a trajectory starting from the multidimensional state variable x ⁰ ;
A plurality of multidimensional state variables x ^k (1 ≦ k ≦ m: k is an integer) discretized between the multidimensional state variables x ⁰ and x ^{m + 1} ;
Euclidean distance ε between multidimensional state variables x ^k and x ^{k + 1} that are adjacent to each other among the multidimensional state variables x ^{0 to} x ^{m + 1} ,

Using the power system equation defined as

With an error vector defined as

An objective function defined as: a critical fault in which the multidimensional state variable x ⁰ is critical between the time that the power system can recover after the power system fails and the time that the power system cannot recover after the power system fails Under the condition that the removal time is included as a constant, and the trajectory from the multidimensional state variable x ⁰ to the multidimensional state variable x ^{m + 1} passes through a singular point of a predetermined function based on the power system equation. A first procedure to minimize with
A second procedure for calculating a transient stability limit value of the auxiliary variable based on the multidimensional state variable x ⁰ that minimizes the objective function;
A program that executes

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