CN114156881A - Method for generating transient process sample of power system based on proxy model containing time sequence item - Google Patents

Method for generating transient process sample of power system based on proxy model containing time sequence item Download PDF

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CN114156881A
CN114156881A CN202111488746.8A CN202111488746A CN114156881A CN 114156881 A CN114156881 A CN 114156881A CN 202111488746 A CN202111488746 A CN 202111488746A CN 114156881 A CN114156881 A CN 114156881A
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佘东
马军
王晗
王运
严正
白鹭
蒙飞
徐潇源
李宏强
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Shanghai Jiaotong University
State Grid Corp of China SGCC
State Grid Ningxia Electric Power Co Ltd
Electric Power Research Institute of State Grid Ningxia Electric Power Co Ltd
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State Grid Corp of China SGCC
State Grid Ningxia Electric Power Co Ltd
Electric Power Research Institute of State Grid Ningxia Electric Power Co Ltd
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Abstract

The invention discloses a method for generating a transient process sample of a power system based on a proxy model containing a time sequence item, belonging to the technical field of transient analysis of the power system, and comprising the following steps: constructing a NARX data model Y (t, xi) which is used for calculating an output value of the power system at the time t under the influence of an input random variable xi; construction factor thetai(ξ) a PCE computation model; coefficient of thetaiSubstituting PCE calculation model of (xi) into NARX data model Y (t, xi) to obtain proxy model Y containing time sequence items(t, ξ); by including a time-series term proxy model Ys(T, ξ) and output values corresponding to historical times of the power system, and calculating output sample value Y of the power system at time Ts(T, ξ); by including a time-series term proxy model Ys(t, ξ), historical time of power systemCorresponding to output value and output sample value Y at time Ts(T, xi), gradually iterating and calculating the T + N of the power systemsOutput sample value Y of times(T+Nsξ), a transient process sample set of the power system is obtained.

Description

Method for generating transient process sample of power system based on proxy model containing time sequence item
Technical Field
The invention relates to the technical field of transient analysis of a power system, in particular to a method for generating a transient process sample of the power system based on a proxy model containing a time sequence item.
Background
With the increase of uncertainty factors in the power system, the traditional deterministic time domain simulation calculation gradually changes to time domain simulation uncertainty analysis so as to meet the requirement of transient analysis of the power system under the scene of strong randomness and volatility.
The transient analysis of the power system in an uncertain environment has the characteristics of time-varying property, nonlinearity and randomness, and the main research methods for the problem include Monte Carlo Simulation (MCS), random response surface method (SRSM), Polynomial Chaos Expansion (PCE), etc., or have the problems of low computational efficiency, or "dimensional disaster", etc., and a large number of samples need to be generated in the analysis process, so that the time cost is high.
Therefore, an efficient and practical sample generation method is needed for transient analysis of a power system under an uncertain environment.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a method for generating a transient process sample of a power system based on a proxy model containing a time sequence item, which can solve the problems of low calculation efficiency and large data volume generated in an analysis process and improve the generation calculation efficiency of the transient process sample of the power system in an uncertain environment.
The technical scheme adopted by the embodiment of the invention for solving the technical problem is as follows:
a method for generating transient process samples of a power system based on a proxy model containing time sequence items comprises the following steps:
constructing a NARX data model Y (t, xi) which is used for calculating an output value of the power system at the t moment under the influence of an input random variable xi,
Figure BDA0003398346900000021
wherein, the thetai(xi) is a coefficient containing the input random variable xi, the NgIs the basis function gi[u(t)]Number of terms of, said epsilont(t, xi) is the residual error of the NARX data model Y (t, xi) at the time t under the influence of uncertainty factors,
u(t)＝[X(t),…,X(t-tX),Y(t-1),…,Y(t-tY)]
the X (t) is external input excitation corresponding to the power system at the time t, and the X (t-t) isX) For the power system at historical time t-tXCorresponding outer partInputting excitation, wherein Y (t-1) is an output value corresponding to the power system at the historical moment t-1, and Y (t-t)Y) For the power system at historical time t-tYCorresponding output value, said tXFor maximum time delay of external input excitation, tYIs the maximum time delay of the output value at the historical moment;
constructing the coefficient thetai(ξ) a model of the PCE computation,
Figure BDA0003398346900000022
wherein, the alpha isi,jComputing coefficients of a model for the PCE, the
Figure BDA0003398346900000023
Is a multivariate orthogonal polynomial on the input random variable ξ; the above-mentioned
Figure BDA0003398346900000024
Is the number of terms of the multivariate orthogonal polynomial; the epsiloniCalculating a truncation error of the model for the PCE;
the coefficient theta is measurediSubstituting the PCE calculation model of (xi) into the NARX data model Y (t, xi) to obtain a proxy model Y containing a time sequence items(t,ξ)，
Figure BDA0003398346900000025
Wherein epsilon (t, xi) is the time sequence item-containing proxy model YsTotal error of (t, ξ);
through the time sequence item containing proxy model Ys(T, ξ) and output values corresponding to the historical time of the power system calculate output sample value Y of the power system at time Ts(T,ξ)；
Through the time sequence item containing proxy model Ys(T, ξ), an output value corresponding to the history time of the power system, and an output sample value Y at the time Ts(T, xi), gradually and iteratively calculating the T + N of the power systemsOutput sample value Y of times(T+Nsξ) of the power system, s being a sample volume value of the transient process sample set, N being a sample volume value of the transient process sample sets∈[0,s]。
Preferably, the constructing the NARX data model Y (t, ξ) includes:
constructing a NARX data model Y (t) representing the output value of the power system without the input random variable ξ at time t,
Figure BDA0003398346900000031
u(t)＝[X(t),…,X(t-tX),Y(t-1),…,Y(t-tY)]
wherein, the thetaiCoefficients of the NARX data model Y (t), NgIs the basis function gi[u(t)]Number of terms of, said epsilontIs the residual of said Y (t), εt～N(0,σ2(t))；
Selecting a maximum order m and said maximum time delay tYCreating a NARX model time sequence item candidate basis function set based on the NARX data model Y (t), wherein the number of time sequence item basis functions to be selected in the NARX model time sequence item candidate basis function set is (m + t)Y)！/(m！tY| A ) A plurality of;
establishing an input random variable sample set { xi ] with capacity of N(1),…,ξ(N)}；
Obtaining output variable samples Y (t, xi) corresponding to each time t through time domain simulation calculation of the power system(k)) Wherein k is [1, N ]]，t∈[1,TS]Said T isSFor discrete total time, TS>tY；
Dividing the sampling period into Ts-tYUnit sampling periods, each of which contains tYA time section;
when all the units in the sampling period are traversed for samplingAnd determining the key time sequence item basis function g 'corresponding to each unit sampling period'i[u(t)]To determine Ts-tYGrouping the key timing term basis functions;
calculating the total LOO error of all the unit sampling periods under each key time sequence term basis function;
selecting the key time sequence item basis function g 'with the minimum total LOO error'i[u(t)]As the basis function g for constructing the NARX data model Y (t, ξ)j[u(t)]；
Based on the input random variable sample set { ξ(1),…,ξ(N)And the basis function gj[u(t)]Through time-domain simulation calculation of the power system, tY～TSOutput variable samples of a time segment and the basis function gj[u(t)]The matrix expression of (a) is as follows,
Figure BDA0003398346900000041
wherein k is ∈ [1, N ∈ >]，t∈[1,TS]，TS>tYThe shorthand of the matrix expression is,
Yk＝ψkθ(ξ(k))+ε(ξ(k))
wherein the psikIs a time-series term basis function information matrix, the theta (xi)(k)) For the coefficient vector to be solved, the epsilon (xi)(k)) Is a residual vector, θ (ξ)(k)) And psikAnd epsilon (xi)(k)) Respectively satisfy the conditions that,
Figure BDA0003398346900000042
Figure BDA0003398346900000043
Figure BDA0003398346900000044
calculating the coefficient vector theta (xi) to be solved by using a least square method(k))，
Figure BDA0003398346900000051
According to the coefficient vector theta (xi) to be solved(k)) Deducing the basis function g under N random samplesj[u(t)]The corresponding coefficient theta (xi) to derive the NARX data model Y (t, xi).
Preferably, the unit sampling periods in the sampling periods are traversed, and the key time sequence item basis function g 'corresponding to each unit sampling period is determined'i[u(t)]To determine Ts-tYGrouping the key timing term basis functions includes:
defining a jth of said unit sampling periods [ j, …, j + t ] of said sampling periodsY-1]Defining j + t for historical timeYThe time is the current time, then at j + tYOutput variable vector Y of timeSIn order to realize the purpose,
YS＝[Y(j+tY,ξ(1)),…,Y(j+tY,ξ(N))]T,j∈[1,Ts-tY]
the j + tYThe time is the next time of the final time of the jth unit sampling period;
and said YSCorresponding time sequence item basic function information matrix psi to be selectedSIn order to realize the purpose,
Figure BDA0003398346900000052
wherein, M isgRepresenting the total number of the items of the base to be selected;
based on the YSAnd said psiSDetermining the jth unit sampling period [ j, …, j + t ] through LAR strategyY-1]Corresponding key timing term basis function g'i[u(t)]；
Let j be {1, …, Ts-tYSequentially traversing all the unit sampling time periods in the sampling time period to determine the Ts-tYGroup the key time sequence term basis function g'i[u(t)]。
Preferably, said base is based on said YSAnd said psiSDetermining the jth unit sampling period [ j, …, j + t ] through LAR strategyY-1]Corresponding key timing term basis function g'i[u(t)]The method comprises the following steps:
initializing a coefficient vector alpha1＝α2When … is 0, the residual error epsilon is obtained(0)LAR，ε(0)LAR＝YS-μ(0)＝YSWherein, mu(p)Predicting a vector for an output response obtained based on the current model in the p iteration process;
setting the iteration number as k, and calculating a correlation function ck＝ψT(YS-μ(k)) Obtaining the correlation between each time sequence base to be selected of the jth unit sampling period and the output response, and screening out the residual error term epsilon under the current iteration(k)LARThe most relevant time sequence item basic function, and the screened time sequence item basic function is moved into an active set A;
by iterative step size gamma(k)And the direction of iteration omega(k)Updating the coefficient vector to be solved, wherein alpha(k+1)i＝α(k)i+γ(k)ω(k)；
In the process of updating the coefficient vector, when the time sequence item basis function g to be selected in the NARX model time sequence item basis function set to be selectedφThe g is equal to the correlation of all the basis functions in the active set A to the output vector residualφMoving into the active set A;
calculating the LOO error under the kth iteration until the LOO error is smaller than a preset error value epsilonNARXOr the number of iterations k is equal to said Ng；
The active set a obtained after the iterative computation is the j-th unit sampling period [ j, …,j+tY-1]a corresponding set of timing term basis functions, wherein A ═ g'1,j[u(t)],g'2,j[u(t)],…,g'Nj,j[u(t)]}。
Preferably, said calculating the total LOO error for all said unit sampling periods under each said key timing term basis function comprises:
sampling time interval [ j, …, j + t ] of j unit of the ith group of key time sequence term basis function Y (t)Y-1]The output variable sample in the table is historical data, and j + t is used asYSolving the coefficient of the NARX data model Y (t) by using a least square method to construct the key time sequence term basis function g 'of the ith group'i[u(t)]The jth cell sampling period [ j, …, j + t [ ]Y-1]The corresponding NARX data model y (t);
calculating the i < th > group of key time sequence term basis functions g'i[u(t)]The jth cell sampling period [ j, …, j + t [ ]Y-1]LOO error L of the corresponding NARX data model Y (t)i,jWherein j is equal to [1, Ts-t ∈ [ ]Y]
Calculating a base function g 'based on the ith group of key time sequence items'i[u(t)]The built NARX model Y (t) has total LOO error L in all sampling time periodsi，
Figure BDA0003398346900000071
Let i be [1, Ts-t ]Y]Calculating each key time sequence term basis function g'i[u(t)]Corresponding total LOO error Li。
Preferably, said constructing a PCE computation model of said coefficient θ (ξ) comprises:
creating a PCE orthogonal polynomial basis function set to be selected, wherein the polynomial basis function to be selected in the PCE orthogonal polynomial basis function set to be selected comprises at least one of a Hermite basis function, a Legendre basis function and a Laguerre basis function;
selecting effective orthogonal polynomial basis functions from the PCE orthogonal polynomial basis function set to be selected;
substituting the input random variable sample ξ into the effective orthogonal polynomial basis function(k)(k＝1,…,N)；
Calculating coefficient vector alpha by using least square methodi,j，
Figure BDA0003398346900000072
Figure BDA0003398346900000073
The above-mentioned
Figure BDA0003398346900000074
Substituting the input random variable sample xi for the effective orthogonal polynomial basis function(k)The information matrix obtained later;
based on the significant orthogonal polynomial basis function and the coefficient vector alphai,jAnd obtaining a PCE model of the theta (xi).
Preferably, the simplified iterative formula of the output sample values is:
Figure BDA0003398346900000081
wherein, TS>tYThe xi is(k)Representing the kth sample of said input random variable.
Preferably, the value of the maximum order m is 3.
It can be known from the above technical solutions that, in the method for generating a transient process sample of an electric power system based on a proxy model including a time sequence item provided in the embodiments of the present invention, first, a NARX data model Y (t, ξ) for calculating an output value of the electric power system at time t under the influence of an input random variable ξ is constructed, and then, a coefficient θ in the NARX data model Y (t, ξ) is constructedi(ξ) a PCE computation model; coefficient of thetaiSubstituting the PCE calculation model of (xi) into the NARX data model Y (t, xi) to obtain a time sequence term-containing generationPhysical model Ys(t, xi) by proxy model Y with timing termss(T, ξ) and output values corresponding to historical times of the power system, and calculating output sample value Y of the power system at time Ts(T, xi), and then passing through a proxy model Y containing a time sequence items(T, ξ), output value corresponding to historical time of power system, and output sample value Y at time Ts(T, xi), gradually iterating and calculating the T + N of the power systemsOutput sample value Y of times(T+Nsξ) to obtain a transient process sample set of the power system, and provides a new sample generation model, which can solve the problems of low calculation efficiency and large data size generated in the analysis process and improve the calculation efficiency of the transient process sample generation of the power system in an uncertain environment.
Drawings
FIG. 1 is a flow chart of a method for generating a power system transient process sample based on a proxy model containing a timing term according to an embodiment of the invention.
FIG. 2 shows 10000 relative power angle differences delta generated in an IEEE9 node system by an embodiment of the method of the invention12A sample curve;
FIG. 3 shows an IEEE39 node for an output variable δ according to an embodiment of the present invention74The results of the analysis of (2) are compared with a graph.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, are within the scope of the present invention.
The method of the invention adopts a nonlinear autoregressive with exogenous input, NARX (network adaptive regression) model to represent the dynamic characteristics of the system, utilizes a Polynomial Chaotic Expansion (PCE) method to depict the influence of random factors in the system, and represents the output variable of the power system in uncertain environment as the polynomial chaotic sum of the historical moment of the output variable, thereby providing a quick and efficient method for generating samples in the transient process of the power system.
The technical scheme and the technical effect of the invention are further elaborated in the following by combining the drawings of the invention.
The invention provides a method for generating a transient process sample of a power system based on a proxy model containing a time sequence item, which comprises the following steps:
step S1, constructing a NARX data model Y (t, xi) which is used for calculating the output value of the power system at the time t under the influence of the input random variable xi,
Figure BDA0003398346900000091
wherein, thetai(xi) is a coefficient containing an input random variable xi, NgIs a basis function gi[u(t)]Number of terms of epsilont(t, xi) is the residual error of the NARX data model Y (t, xi) at the time t under the influence of uncertainty factors,
u(t)＝[X(t),…,X(t-tX),Y(t-1),…,Y(t-tY)] (2)
wherein X (t) is external input excitation corresponding to the power system at the time t, and X (t-t)X) For power system at historical time t-tXCorresponding external input excitation, Y (t-1) is an output value corresponding to the historical time t-1 of the power system, and Y (t-t)Y) For power system at historical time t-tYCorresponding output value, tXMaximum time delay of external input excitation, tYIs the maximum time delay of the output value at the historical moment;
step S2, constructing a coefficient thetai(ξ) a model of the PCE computation,
Figure BDA0003398346900000101
wherein alpha isi,jComputing a modulus for a PCEThe coefficient of the type(s) is,
Figure BDA0003398346900000102
is a multivariate orthogonal polynomial on the input random variable ξ;
Figure BDA0003398346900000103
is the number of terms of a multivariate orthogonal polynomial; epsiloniCalculating the truncation error of the model for the PCE;
step S3, the coefficient theta is adjustediSubstituting the PCE calculation model of (xi) into the NARX data model Y (t, xi) to obtain the proxy model Y containing the time sequence items(t,ξ)，
Figure BDA0003398346900000104
Wherein epsilon (t, xi) is a proxy model Y containing a time sequence itemsTotal error of (t, ξ);
step S4, proxy model Y containing time sequence items(T, ξ) and output values corresponding to historical times of the power system, and calculating output sample value Y of the power system at time Ts(T,ξ)；
Step S5, proxy model Y containing time sequence items(T, ξ), output value corresponding to historical time of power system, and output sample value Y at time Ts(T, xi), gradually iterating and calculating the T + N of the power systemsOutput sample value Y of times(T+Nsξ) of the power system, where s is the sample volume value of the transient process sample set and N is the sample volume value of the transient process sample sets∈[0,s]。
Specifically, step S1 is to set up a system model, select a random variable, determine a sample size N, and obtain an input random variable set ξS＝(ξ(1),…,ξ(N)) And constructing a basis function to be selected according to the principle of the NARX method, selecting an effective basis according to a Least Angle Regression (LAR) algorithm, and calculating a corresponding coefficient vector to further complete the construction of the NARX model. The specific process for constructing the NARX data model Y (t, ξ) comprises the following steps:
step S11, constructing a NARX data model Y (t) without considering the external input excitation signal, wherein the NARX data model Y (t) represents the output value of the power system without the input random variable xi at the time t,
Figure BDA0003398346900000111
u(t)＝[X(t),…,X(t-tX),Y(t-1),…,Y(t-tY)] (6)
wherein, thetaiCoefficient of NARX data model Y (t), NgIs a basis function gi[u(t)]Number of terms of epsilontIs the residual of Y (t), εt～N(0,σ2(t))；
Step S12, selecting the maximum order m and the maximum time delay t of the time sequence item basis function to be selected of the NARX moduleYCreating a time sequence item candidate basis function set of the NARX model based on the NARX data model Y (t), wherein the number of time sequence item basis functions to be selected in the time sequence item candidate basis function set of the NARX model is (m + t)Y)！/(m！tY| A ) And (4) respectively. Typically, the maximum order m is 3, where m is 3, tYFor example, 3, the number of the time sequence item basis functions to be selected in the created NARX model time sequence item basis function set is 19, and a distribution table of the time sequence item basis functions to be selected is shown in table 1:
Figure BDA0003398346900000112
TABLE 1
In table 1, each candidate basis function is a set of historical output values, u (t) is considered as a set of historical output values, and the candidate basis functions Y (t-1), Y (t-2) and Y (t-3) represent that g is multiplied by a polynomiali[u(t)]And gi[u(t)]The concrete expression of (1).
Step S13, establishing an input random variable sample set { ξ } with the capacity of N(1),…,ξ(N)}；
Step S14, obtaining each time t through the time domain simulation calculation of the power systemOutput variable sample Y (t, xi) corresponding to moment(k)) Wherein k is [1, N ]]，t∈[1,TS]，TSFor discrete total time, TS>tY；
Step S15, dividing the sampling period into Ts-tYUnit sampling periods, each unit sampling period containing tYA time section;
step S16, traversing all unit sampling periods in the sampling period, and determining the key time sequence item basis function g 'corresponding to each unit sampling period'i[u(t)]To determine Ts-tYGroup key timing term basis function g'i[u(t)]The specific implementation process is as follows:
defining a jth unit sample period [ j, …, j + t ] in the sample periodY-1]Defining j + t for historical timeYThe time is the current time, let j + tYThe time is the next time of the final time of the jth unit sampling period, then at j + tYOutput variable vector Y of timeSIn order to realize the purpose,
YS＝[Y(j+tY,ξ(1)),…,Y(j+tY,ξ(N))]T,j∈[1,Ts-tY] (7)
and YSThe corresponding time sequence item basic function information matrix to be selected is psiS，
Figure BDA0003398346900000121
Figure BDA0003398346900000122
Wherein M isgRepresenting the total number of the items of the base to be selected;
based on YSAnd psiSDetermining the jth unit sampling period [ j, …, j + t ] by LAR strategyY-1]Corresponding key timing term basis function g'i[u(t)]；
Let j be {1, …, Ts-tYSequentially traversing all unit sampling time periods in the sampling time period to determine Ts-tYWhen group is criticalSequence term basis function g'i[u(t)]。
Further, based on YSAnd psiSDetermining the jth unit sampling period [ j, …, j + t ] by LAR strategyY-1]Corresponding key timing term basis function g'i[u(t)]The specific implementation of (A) comprises:
a: initializing a coefficient vector alpha1＝α2When … is 0, the residual error epsilon is obtained(0)LAR，ε(0)LAR＝YS-μ(0)＝YSWherein, mu(p)In the p iteration process, the output response prediction vector is obtained based on the current model;
b: setting the iteration number as k, and calculating a correlation function ck＝ψT(YS-μ(k)) Obtaining the correlation between each time sequence base to be selected in the jth unit sampling period and the output response, and screening out the residual error term epsilon under the current iteration(k)LARThe most relevant time sequence item basic function, and the screened time sequence item basic function is moved into an active set A;
c: by iterative step size gamma(k)And the direction of iteration omega(k)Updating the coefficient vector, wherein(k+1)i＝α(k)i+γ(k)ω(k)；
d: in the process of updating the coefficient vector, when the time sequence item basis function g to be selected in the time sequence item basis function set of the NARX modelφAnd g is equal to the correlation of all the basis functions in the active set A to the residual error of the output vectorφMoving into the active set A;
e: calculating the LOO error under the kth iteration until the LOO error is smaller than a preset error value epsilonNARXOr the number of iterations k equals Ng；
f: the active set a obtained after the iterative computation is the jth unit sampling period [ j, …, j + tY-1]A corresponding set of timing term basis functions, wherein A ═ g'1,j[u(t)],g′2,j[u(t)],…,g′Nj,j[u(t)]}。
Step S17, calculating each key time sequence term basis function g'i[u(t)]Sampling at all unitsThe total LOO error of the time interval is implemented as follows:
sampling period [ j, …, j + t ] of j unit with ith group key time sequence item basis functionY-1]The output variable sample in the table is historical data, let j + tYThe moment is the current moment, the coefficient of an NARX data model Y (t) is solved by using a least square method, and a basis function g 'of the ith group of key time sequence terms is constructed'i[u(t)]The jth cell sampling period [ j, …, j + t [ ]Y-1]The corresponding NARX data model y (t);
calculating the jth unit sampling period [ j, …, j + t ] of the ith group key time sequence item basis function eY-1]LOO error L of the corresponding NARX data model Y (t)i,jWherein j is equal to [1, Ts-t ∈ [ ]Y]
Calculating the total LOO error L of the NARX data model Y (t) constructed based on the ith group of key time sequence term basis functions Y (t) in all sampling time periodsiAs shown in formula (9):
Figure BDA0003398346900000141
let i be [1, Ts-t ]Y]Calculating each key time sequence term basis function g'i[u(t)]Corresponding total LOO error Li。
Step S18, selecting a key time sequence item basis function g 'with the minimum total LOO error'i[u(t)]As a basis function g for constructing the NARX data model Y (t, ξ)j[u(t)]；
Step S19, based on input random variable sample set { ξ(1),…,ξ(N)And (c) calculating the time domain of the power system according to the selected key time sequence basis function and tY～TSThe relationship between the output variable samples of the time segment and the key timing basis function can be expressed as shown in formula (10), wherein k is equal to [1, N ∈]，t∈[1,TS]，TS>tY，
Figure BDA0003398346900000142
The abbreviation of equation (10) is equation (11):
Yk＝ψkθ(ξ(k))+ε(ξ(k)) (11)
wherein psikIs a time-series term basis function information matrix, theta (xi)(k)) For the coefficient vector, epsilon (xi) to be solved(k)) Is a residual vector, θ (ξ)(k)) And psikAnd epsilon (xi)(k)) Respectively satisfy formula (11), formula (12) and formula (13),
Figure BDA0003398346900000151
Figure BDA0003398346900000152
Figure BDA0003398346900000153
step S110, calculating the coefficient vector theta (xi) to be solved by using a least square method(k)) As shown in equation (15):
Figure BDA0003398346900000154
step S111, according to the coefficient vector theta (xi) to be solved(k)) Deducing N random sample lower sum basis functions gj[u(t)]The corresponding coefficient theta (xi) to derive NARX data model Y (t, xi). Specifically, a coefficient vector theta (xi) to be solved in front of a key time sequence basis function under the kth sample is solved through a formula (15)(k)) And repeating the operation for N times to obtain a coefficient theta (xi) under N random samples, and further constructing a PCE model between the coefficient and the orthogonal polynomial.
Further, the step of constructing a PCE computation model of the coefficient θ (ξ) in step S2 includes:
step S21, creating a PCE orthogonal polynomial basis function set to be selected, wherein the polynomial basis functions to be selected in the PCE orthogonal polynomial basis function set to be selected comprise at least one of a Hermite basis function, a Legendre basis function and a Laguerre basis function;
step S22, selecting effective orthogonal polynomial basis functions from the PCE orthogonal polynomial basis function set to be selected;
step S23, substituting input random variable sample xi into effective orthogonal polynomial basis function(k)(k＝1,…,N)；
Step S24, calculating coefficient vector α using the least square methodi,j，
Figure BDA0003398346900000161
Figure BDA0003398346900000162
Wherein the content of the first and second substances,
Figure BDA0003398346900000163
substituting input random variable samples xi for effective orthogonal polynomial basis functions(k)The information matrix obtained later;
step S25, based on the effective orthogonal polynomial basis function and the coefficient vector alphai,jThe PCE model for θ (ξ) is derived.
Based on the steps, the time sequence item-containing proxy model Y shown in formula (4) is finally obtaineds(t,ξ)，
Figure BDA0003398346900000164
Based on the formula (4), the output value corresponding to the historical time of the power system is input, and the output sample value Y of the power system at the current T time can be calculateds(T, xi), and further, the output sample value Y at time Ts(T, xi) is regarded as historical data at the time T +1, and an output sample value Y at the time T +1 is iterateds(T +1, ξ), whereby a stepwise iteration is possibleSubstitute for calculating the T + N of the power systemsOutput sample value Y of times(T+Nsξ) to obtain a transient process sample set with a sample capacity s of the power system, based on the above iteration mode, simplifying error values, and obtaining a simplified iteration formula of output sample values as shown in formula (18):
Figure BDA0003398346900000165
wherein, TS>tY，ξ(k)Representing the kth input random variable sample.
The method for generating the transient process sample of the power system based on the proxy model containing the time sequence item, provided by the embodiment of the invention, comprises the steps of firstly constructing an NARX data model Y (t, xi) for calculating the output value of the power system at the t moment under the influence of an input random variable xi, and then constructing a coefficient theta of the NARX data model Y (t, xi), wherein the coefficient theta isi(ξ) a PCE computation model; coefficient of thetaiSubstituting the PCE calculation model of (xi) into the NARX data model Y (t, xi) to obtain the proxy model Y containing the time sequence items(t, xi) by proxy model Y with timing termss(T, ξ) and output values corresponding to historical times of the power system, and calculating output sample value Y of the power system at time Ts(T, xi), and then passing through a proxy model Y containing a time sequence items(T, ξ), output value corresponding to historical time of power system, and output sample value Y at time Ts(T, xi), gradually iterating and calculating the T + N of the power systemsOutput sample value Y of times(T+Nsξ) to obtain a transient process sample set of the power system, thereby solving the problems of low calculation efficiency and large data volume generated in the analysis process.
The method has the advantages that the time sequence item contained in the constructed model can represent the dynamic characteristic of the system output variable changing along with time, the random parameter contained in the model can represent the random characteristic of the system influenced by uncertainty factors, and finally, the constructed model is beneficial to rapidly generating a large number of transient process simulation samples. The method provided by the embodiment of the invention is adopted to perform multiple transient process simulation, and the results are shown in fig. 2 and fig. 3. Compared with the traditional transient analysis method of the power system under the uncertain environment, the method provided by the embodiment of the invention is based on the time sequence item-containing proxy model of the NARX-PCE method, and can describe the dynamic characteristic of the system output variable changing along with time and simultaneously describe the random characteristic of the influence of uncertain factors; the method provided by the embodiment of the invention provides a novel way for generating the samples in the transient process of the power system through the constructed proxy model containing the time sequence item, so that the time for acquiring a large number of samples in an uncertain environment is greatly reduced, and the calculation efficiency of transient analysis of the power system in the uncertain environment is improved.
The above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the scope of the technical solutions of the embodiments of the present invention.

Claims (8)

1. A method for generating transient process samples of a power system based on a proxy model containing time sequence items is characterized by comprising the following steps:
constructing a NARX data model Y (t, xi) which is used for calculating an output value of the power system at the t moment under the influence of an input random variable xi,
Figure FDA0003398346890000011
wherein, the thetai(xi) is a coefficient containing the input random variable xi, the NgIs the basis function gi[u(t)]Number of terms of, said epsilont(t, xi) is the residual error of the NARX data model Y (t, xi) at the time t under the influence of uncertainty factors,
u(t)＝[X(t),…,X(t-tX),Y(t-1),…,Y(t-tY)]
the X (t) is external input excitation corresponding to the power system at the time t, and the X (t-t) isX) For the power system at historical time t-tXCorresponding external input excitation, wherein Y (t-1) is an output value corresponding to the power system at the historical moment t-1, and Y (t-t)Y) For the power system at historical time t-tYCorresponding output value, said tXFor maximum time delay of external input excitation, tYIs the maximum time delay of the output value at the historical moment; constructing the coefficient thetai(ξ) a model of the PCE computation,
Figure FDA0003398346890000012
wherein, the alpha isi,jComputing coefficients of a model for the PCE, the
Figure FDA0003398346890000013
Is a multivariate orthogonal polynomial on the input random variable ξ; the above-mentioned
Figure FDA0003398346890000015
Is the number of terms of the multivariate orthogonal polynomial; the epsiloniCalculating a truncation error of the model for the PCE;
the coefficient theta is measurediSubstituting the PCE calculation model of (xi) into the NARX data model Y (t, xi) to obtain a proxy model Y containing a time sequence items(t,ξ)，
Figure FDA0003398346890000014
Wherein epsilon (t, xi) is the time sequence item-containing proxy model YsTotal error of (t, ξ);
through the time sequence item containing proxy model Ys(T, ξ) and output values corresponding to the historical time of the power system calculate output sample value Y of the power system at time Ts(T,ξ)；
Through the time sequence item containing proxy model Ys(T, ξ), an output value corresponding to the history time of the power system, and an output sample value Y at the time Ts(T, xi), gradually and iteratively calculating the T + N of the power systemsOutput sample value Y of times(T+Nsξ) of the power system, s being a sample volume value of the transient process sample set, N being a sample volume value of the transient process sample sets∈[0,s]。
2. The method of generating power system transient process samples based on a proxy model with timing terms of claim 1, wherein the constructing a NARX data model Y (t, ξ) comprises:
constructing a NARX data model Y (t) representing the output value of the power system without the input random variable ξ at time t,
Figure FDA0003398346890000021
u(t)＝[X(t),…,X(t-tX),Y(t-1),…,Y(t-tY)]
wherein, the thetaiCoefficients of the NARX data model Y (t), NgIs the basis function gi[u(t)]Number of terms of, said epsilontIs the residual of said Y (t), εt～N(0,σ2(t))；
Selecting a maximum order m and said maximum time delay tYCreating a NARX model time sequence item candidate basis function set based on the NARX data model Y (t), wherein the number of time sequence item basis functions to be selected in the NARX model time sequence item candidate basis function set is (m + t)Y)！/(m！tY| A ) A plurality of;
establishing an input random variable sample set { xi ] with capacity of N(1),…,ξ(N)}；
Obtaining output variable samples Y (t, xi) corresponding to each time t through time domain simulation calculation of the power system(k)) Wherein k ∈[1,N]，t∈[1,TS]Said T isSFor discrete total time, TS>tY；
Dividing the sampling period into Ts-tYUnit sampling periods, each of which contains tYA time section;
traversing all the unit sampling periods in the sampling periods, and determining a key time sequence item basis function g 'corresponding to each unit sampling period'i[u(t)]To determine Ts-tYGrouping the key timing term basis functions; calculating the total LOO error of all the unit sampling periods under each key time sequence term basis function; selecting the key time sequence item basis function g 'with the minimum total LOO error'i[u(t)]As the basis function g for constructing the NARX data model Y (t, ξ)j[u(t)]；
Based on the input random variable sample set { ξ(1),…,ξ(N)And the basis function gj[u(t)]Through time-domain simulation calculation of the power system, tY～TSOutput variable samples of a time segment and the basis function gj[u(t)]The matrix expression of (a) is as follows,
Figure FDA0003398346890000031
wherein k is ∈ [1, N ∈ >]，t∈[1,TS]，TS>tYThe shorthand of the matrix expression is,
Yk＝ψkθ(ξ(k))+ε(ξ(k))
wherein the psikIs a time-series term basis function information matrix, the theta (xi)(k)) For the coefficient vector to be solved, the epsilon (xi)(k)) Is a residual vector, θ (ξ)(k)) And psikAnd epsilon (xi)(k)) Respectively satisfy the conditions that,
Figure FDA0003398346890000032
Figure FDA0003398346890000041
Figure FDA0003398346890000042
calculating the coefficient vector theta (xi) to be solved by using a least square method(k))，
Figure FDA0003398346890000043
According to the coefficient vector theta (xi) to be solved(k)) Deducing the basis function g under N random samplesj[u(t)]The corresponding coefficient theta (xi) to derive the NARX data model Y (t, xi).
3. The method of claim 2, wherein traversing all of the unit sampling periods in the sampling period determines a key timing term basis function g 'corresponding to each of the unit sampling periods'i[u(t)]To determine Ts-tYGrouping the key timing term basis functions includes:
defining a jth of said unit sampling periods [ j, …, j + t ] of said sampling periodsY-1]Defining j + t for historical timeYThe time is the current time, then at j + tYOutput variable vector Y of timeSIn order to realize the purpose,
YS＝[Y(j+tY,ξ(1)),…,Y(j+tY,ξ(N))]T,j∈[1,Ts-tY]
the j + tYThe time is the next time of the final time of the jth unit sampling period;
and said YSCorresponding time sequence item basic function information matrix psi to be selectedSIn order to realize the purpose,
Figure FDA0003398346890000051
wherein, M isgRepresenting the total number of the items of the base to be selected;
based on the YSAnd said psiSDetermining the jth unit sampling period [ j, …, j + t ] through LAR strategyY-1]Corresponding key timing term basis function g'i[u(t)]；
Let j be {1, …, Ts-tYSequentially traversing all the unit sampling time periods in the sampling time period to determine the Ts-tYGroup the key time sequence term basis function g'i[u(t)]。
4. The method of claim 3, wherein the basing on the YSAnd said psiSDetermining the jth unit sampling period [ j, …, j + t ] through LAR strategyY-1]Corresponding key timing term basis function g'i[u(t)]The method comprises the following steps:
initializing a coefficient vector alpha1＝α2When … is 0, the residual error epsilon is obtained(0)LAR，ε(0)LAR＝YS-μ(0)＝YSWherein, mu(p)Predicting a vector for an output response obtained based on the current model in the p iteration process;
setting the iteration number as k, and calculating a correlation function ck＝ψT(YS-μ(k)) Obtaining the correlation between each time sequence base to be selected of the jth unit sampling period and the output response, and screening out the residual error term epsilon under the current iteration(k)LARThe most relevant time sequence item basic function, and the screened time sequence item basic function is moved into an active set A; by iterative step size gamma(k)And the direction of iteration omega(k)Updating the coefficient vector, wherein(k+1)i＝α(k)i+γ(k)ω(k)；
In the process of updating the coefficient vector, when the NARX model timing itemBase function g of time sequence item to be selected in base function set to be selectedφThe g is equal to the correlation of all the basis functions in the active set A to the output vector residualφMoving into the active set A;
calculating the LOO error under the kth iteration until the LOO error is smaller than a preset error value epsilonNARXOr the number of iterations k is equal to said Ng；
The active set a obtained after the iterative computation is the jth unit sampling period [ j, …, j + tY-1]A corresponding set of timing term basis functions, wherein A ═ g'1,j[u(t)],g'2,j[u(t)],…,g'Nj,j[u(t)]}。
5. The method of claim 4, wherein said calculating the total LOO error for all of said unit sampling periods at each of said key timing term basis functions comprises:
sampling time interval [ j, …, j + t ] of j unit of the ith group of key time sequence term basis function Y (t)Y-1]The output variable sample in the table is historical data, and j + t is used asYSolving the coefficient of the NARX data model Y (t) by using a least square method to construct the key time sequence term basis function g 'of the ith group'i[u(t)]The jth cell sampling period [ j, …, j + t [ ]Y-1]The corresponding NARX data model y (t);
calculating the i < th > group of key time sequence term basis functions g'i[u(t)]The jth cell sampling period [ j, …, j + t [ ]Y-1]LOO error L of the corresponding NARX data model Y (t)i,jWherein j is equal to [1, Ts-t ∈ [ ]Y]Calculating a base function g 'based on the ith group of key time sequence items'i[u(t)]The built NARX model Y (t) has total LOO error L in all sampling time periodsi，
Figure FDA0003398346890000061
Let i be [1, Ts-t ]Y]Calculating each key time sequence term basis function g'i[u(t)]Corresponding total LOO error Li。
6. The method of claim 1, wherein constructing the PCE computation model for the coefficient θ (ξ) comprises:
creating a PCE orthogonal polynomial basis function set to be selected, wherein the polynomial basis function to be selected in the PCE orthogonal polynomial basis function set to be selected comprises at least one of a Hermite basis function, a Legendre basis function and a Laguerre basis function;
selecting effective orthogonal polynomial basis functions from the PCE orthogonal polynomial basis function set to be selected;
substituting the input random variable sample ξ into the effective orthogonal polynomial basis function(k)(k＝1,…,N)；
Calculating coefficient vector alpha by using least square methodi,j，
Figure FDA0003398346890000071
Figure FDA0003398346890000074
The above-mentioned
Figure FDA0003398346890000072
Substituting the input random variable sample xi for the effective orthogonal polynomial basis function(k)The information matrix obtained later;
based on the significant orthogonal polynomial basis function and the coefficient vector alphai,jAnd obtaining a PCE model of the theta (xi).
7. The method of claim 1, wherein the simplified iterative formula of the output sample values is:
Figure FDA0003398346890000073
wherein, TS>tYThe xi is(k)Representing the kth sample of said input random variable.
8. The method of claim 2, wherein the maximum order m is 3.
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