CN106548418B - Small interference stability evaluation method for power system - Google Patents

Small interference stability evaluation method for power system Download PDF

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CN106548418B
CN106548418B CN201611128492.8A CN201611128492A CN106548418B CN 106548418 B CN106548418 B CN 106548418B CN 201611128492 A CN201611128492 A CN 201611128492A CN 106548418 B CN106548418 B CN 106548418B
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周一辰
李永刚
于徳水
刘伟东
霍明霞
樊桢
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Abstract

The invention discloses a small interference stability assessment method for an electric power system, and relates to the field of operation analysis and control of the electric power system. The method comprises the following steps: establishing a photovoltaic system small interference analysis model; calculating a correlation coefficient matrix of random input variables; converting the random vector into a correlation coefficient matrix of a standard normal distribution random vector; generating a Hermite chaotic polynomial coordination point by adopting a linear independent coordination point method; converting the collocation points into corresponding illumination values and random load values by adopting Nataf conversion; obtaining a corresponding random output variable value; obtaining a coefficient expanded by a Hermite chaotic polynomial; and simulating the obtained Hermite chaotic polynomial expansion, obtaining the distribution characteristic of the random output variable by adopting nuclear density estimation, and evaluating the probability of small interference stability. The method can be used for processing the problem of double peaks of the illumination distribution, effectively estimating double peaks of PDF and bulges of CDF of the damping ratio of the key mode, and obtaining a better stable evaluation result.

Description

Small interference stability evaluation method for power system
Technical Field
The invention relates to the technical field of control methods of power systems, in particular to a small interference stability assessment method of a power system.
Background
Due to environmental constraints and sustainable energy supply requirements, clean and renewable new energy is vigorously developed, the permeability of the new energy in an electric power system is continuously improved, and photovoltaic power generation is a new energy power generation mode which occupies a larger proportion in the electric power system at present.
When clean energy is continuously injected into the system in photovoltaic power generation, the small interference stability characteristic of the power system is influenced, and uncertain factors of the power system are increased. Firstly, the influence of the internal dynamics of the photovoltaic power generation system on the small interference stability of the power system cannot be ignored, and sufficient modeling should be provided. The influence is shown in the following: 1) fluctuations in photovoltaic power output in Maximum Power Point Tracking (MPPT) mode can affect the power output of synchronous generators and the flow of critical lines. 2) Under the influence of zero inertia of the photovoltaic power generation system, the total inertia of the system is reduced after the traditional generator is replaced. 3) The damping torque may be reduced by control links and undesirable parameters of the photovoltaic power generation system. Secondly, the photovoltaic power generation system adds a new random variable to the small disturbance stability analysis of the power system. The photovoltaic power generation system is usually in a maximum power tracking operation mode, and due to the uncertainty of illumination and the influence of illumination fluctuation, the active power output of the photovoltaic power generation system is in a dynamic tracking process for a long time and shows strong randomness. Under certain illumination, a photovoltaic power generation system can cause negative damping of a power system, and threat to small disturbance stability. The photovoltaic power generation system then has a considerable dependency on the load, which should be taken into account.
The small interference stability evaluation of the power system is an effective means for processing random variables and providing the stable operation probability of the power system. Due to the nonlinearity and complexity of the power system, it is of great significance to find a method for rapidly and accurately evaluating the small interference stability. At present, the common methods include Monte Carlo simulation, point estimation method, semi-invariant method and the like. Monte carlo simulation is the most accurate method, but is time-consuming and requires a large number of repeated modal analyses, so that the method has more applications in detection accuracy. The point estimation method proposed by Rosenblueth is well-established, e.g. the widely used 2m +1 method, which is capable of accurately estimating the first four-step distance of the probability density function and may not have a solution when m is too large. The semi-invariant method can be used for any distributed input, linear processing is adopted, and local characteristics are reflected more. From the above analysis, it can be seen that the methods in the prior art all have certain limitations and poor use effect.
Disclosure of Invention
The invention aims to solve the technical problem of how to provide a small interference stability assessment method for an electric power system, which can process the double peaks of illumination distribution, effectively estimate the double peaks of PDF (Portable document Format) and the bulges of CDF (compact disc) of the damping ratio of a key mode and accurately control the double peaks.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: a method for evaluating the small interference stability of a power system is characterized by comprising the following steps:
establishing a photovoltaic system small interference analysis model;
taking illumination with correlation and random load as random input variables, and calculating a correlation coefficient matrix of the random input variables;
converting the correlation coefficient matrix of the random input variable into a correlation coefficient matrix of a random vector in standard normal distribution;
determining the order of a random response surface of a correlation coefficient matrix of a random vector of standard normal distribution, and generating a Hermite chaotic polynomial coordination point by adopting a linear independent coordination point method;
converting the collocation points into corresponding illumination values and random load values by adopting Nataf conversion;
performing modal analysis according to the obtained illumination value and the random load value, and taking the damping ratio of the key mode as a random output variable to obtain a corresponding random output variable value;
substituting the matched value and the random output variable value into a Hermite chaotic polynomial expansion to obtain a Hermite chaotic polynomial expansion coefficient;
and simulating the expansion of the Hermite chaotic polynomial by adopting a Monte Carlo method, solving the distribution characteristic of the random output variable by adopting nuclear density estimation, and evaluating the probability of small interference stability.
The further technical scheme is as follows: the method for establishing the photovoltaic system small interference analysis model comprises the following steps:
the structure of the photovoltaic system is a secondary photovoltaic power generation system comprising a photovoltaic array, a boost, an inverter, a filter and a phase-locked loop, wherein the output characteristic of a photovoltaic cell adopts an engineering calculation method, the temperature of the cell is a function of illumination, and random input variables of the whole photovoltaic system model are illumination and random load;
the control mode is maximum power point tracking control and fixed inverter direct-current voltage control, wherein all control links adopt PI control, the inverter operates in a unit power factor operation mode, the constant voltage of the direct-current side of the inverter is realized by an active power loop of the inverter, and the voltage of the low-voltage side of boost is determined by the maximum power point tracking control;
and (3) linearization is carried out at the operating point of the photovoltaic system, a photovoltaic system algebraic differential equation suitable for small interference stability analysis is established, and the photovoltaic system algebraic differential equation is combined with a generator algebraic differential equation and a network algebraic equation to obtain a power system linearization state equation containing the photovoltaic system.
The further technical scheme is as follows: the method for calculating the correlation coefficient matrix of the random input variables by taking the illumination with correlation and the random load as the random input variables comprises the following steps:
setting the illumination with correlation and the active power of random load as random input vectors:
X＝[RPV,PL]T
in the formula, RPV＝[r1,…,ri,…,rn]For illumination of photovoltaic systems, riFor the illumination of the ith photovoltaic system, n is the number of the photovoltaic systems, PL＝[p1,…,pj,…,pm]For a random load, pjIs jth random load, m is the number of random loads;
setting an analysis time interval and acquiring historical illumination data D of the photovoltaic system in the time intervalR＝[Dr1,…,Drn]And historical load data D of random loadL＝[DL1,…,DLm]And K is the number of the historical data, then DRDimension K × n, DLDimension is K × m;
calculating probability density function f of historical illumination data by adopting kernel density estimationri(ri) And cumulative distribution function Fri(ri) I represents the ith photovoltaic system; calculating probability density function f of historical load datapj(pj) And cumulative distribution function Fpj(pj) J represents the jth random load, and the correlation coefficient between random variables is calculated by the following formula:
Figure BDA0001175679340000031
in the formula, xiAnd xjIs any two random variables in the X, and the X is a random variable,
Figure BDA0001175679340000032
and
Figure BDA0001175679340000033
is xiAnd xjAverage value of (p)ijIs xiAnd xjThe correlation coefficient of (a); arranging the correlation coefficients of any two random variables in X according to the sequence of the random variables in X to obtain a correlation coefficient matrix of X
Figure BDA0001175679340000041
The further technical scheme is as follows: the method for converting the correlation coefficient matrix of the random input variable into the correlation coefficient matrix of the random vector of the standard normal distribution comprises the following steps:
solving the following implicit function, converting the correlation coefficient between random input variables into the correlation coefficient between standard normal random variables
Figure BDA0001175679340000042
In the formula, XiAnd XjIs any two random input variables in X, and conforms to Nataf distribution, YiAnd YjIs a reaction of with XiAnd XjThe corresponding standard normal random variable is used as the random variable,
Figure BDA0001175679340000046
and
Figure BDA0001175679340000047
is XiAnd XjIs a cumulative distribution function of a standard normal random variable, phi (·)2(Yi，Yj，ρ0ij) Is YiAnd YjOf a joint probability density function ofij0Is YiAnd YjThe correlation coefficient of (a);
will rhoij0According to XiAnd XjArranging in X order to obtain random vector correlation coefficient matrix C of random input variable in standard normal distributionY
Figure BDA0001175679340000043
The further technical scheme is as follows: the method for determining the order of the random response surface of the correlation coefficient matrix of the random vector of the standard normal distribution and generating the Hermite chaotic polynomial coordination point by adopting the linear independent coordination point method comprises the following steps:
determining the order l of Hermite chaotic polynomial expansion adopted by the random response surface according to needs, wherein the Hermite chaotic polynomial expansion of 2-5 orders is as follows:
Figure BDA0001175679340000044
Figure BDA0001175679340000045
Figure BDA0001175679340000051
Figure BDA0001175679340000052
in the formula, xi is a random output variable,
Figure BDA0001175679340000053
is an independent standard normal random vector and represents a random input variable, n is the number of elements in a vector U,
Figure BDA0001175679340000054
is an n-th order Hermite polynomial
Figure BDA0001175679340000055
When the degree of freedom of the random output variable is q, the number of terms expanded by the p-order Hermite chaotic polynomial is q
Figure BDA0001175679340000056
The undetermined coefficient of each term is set as a0，ai1，ai1i2，ai1i2i3，ai1i2,...,inDefinition of a ═ a0，ai1，ai1i2，ai1i2i3，ai1i2,...,in]；
The Hermite chaotic polynomial expansion is expressed as xi ═ Ha, and H is composed of a Hermite chaotic polynomial;
solving the distribution points expanded by the Hermite chaotic polynomial by adopting a linear independent probability distribution point method, wherein the number is
Figure BDA0001175679340000057
The further technical scheme is as follows: the method for solving the coordination point expanded by the Hermite chaotic polynomial by adopting the linear independent probability coordination point method comprises the following steps:
1) calculating the root of a Hermite polynomial of the order of p +1, and arranging and combining all possible coordination points;
2) forming a symmetrical distribution point group according to the symmetry of the distribution points about the origin;
3) according to the sequence of the descending probability of the symmetrical distribution point groups, sequentially checking each symmetrical distribution point group until all selected distribution points form an H array full rank, wherein the selected condition of the distribution point groups is as follows: the newly added symmetrical distribution point group is irrelevant to the linearity of an H array formed by all the existing distribution point groups;
4) if the symmetric matching point groups are completely detected and the H matrix is still not full of rank, sequentially checking each single matching point according to the descending order of the probability of matching points until the H matrix formed by all the selected matching points is full of rank, and selecting the single matching point under the condition that: the newly added single distribution point is irrelevant to the linearity of the H array formed by all the existing distribution points.
The further technical scheme is as follows: the method for transforming the collocation points into corresponding illumination values and random load values by adopting Nataf transformation is as follows:
calculating a combination of random input variables: dividing lighting data into high lighting DRHAnd low light DRLTwo groups:
DRH＝{Dr1H,…,DriH,…,DrnH},DRL＝{Dr1L,…,DriL,…,DrnL}
in the formula, DriHNumber of data of KiH，DRiLNumber of data of KiL(ii) a Using kernel density estimation to obtain high illumination DRHAnd low light DRLProbability density function f of each datariH(riH)、friL(riL) And a cumulative distribution function FriH(riH)、FriL(riL)；
According to DRH、DRLAnd DLList all possible data combinations of random input variables:
Dg＝{[DPV,DL]|DPV＝[DPV1,…,DPVi,…,DPVn]，DPVi∈{DriH,DriL}}
Dgin total 2nAn element;
for DgFor each element in the set, the Nataf transform is used to transform the collocation point into a corresponding illumination value and random load value, e.g. for element [ D ]PV1,…,DPVi,…,DPVn,DL]Wherein D isPV1＝Dr1H，{DPV2，…，DPVn}∈DRL：
1) For correlation coefficient matrix CYCholesky decomposition
CY＝BBT
In which B is CYObtained by Cholesky decompositionA lower triangular matrix;
2) matching point U of independent standard normal random variable spaceCConversion into a standard normal random vector Y with correlationC：YC＝BUC；
3) According to the principle of equal probability transformation, Y is divided intoCConverting into original random input vector;
Figure BDA0001175679340000061
in the formula (I), the compound is shown in the specification,
Figure BDA0001175679340000062
is [ D ]PV1,…,DPVi,…,DPVn,DL]The inverse of the edge cumulative distribution function of any element in (1), if the element is DPV1Then the edge cumulative distribution function is Fr1HIf the element is DPV2,…,DPVnThen the edge cumulative distribution function is Fr1L(r1L),…,FrnL(rnL)；XCiInputting the variable value for the original random value; repeating 1) to 3) to obtain DgOriginal random input vector of all elements in the vector, denoted as Xg＝[Xg1,…,Xgi,…,Xgw]In the formula 1<i<2n,w＝2n；
That is to say correspond to Dg＝{[DPV,DL]|DPV＝[DPV1,…,DPVi,…,DPVn]The illumination and random load value of (A) is Xg。
The further technical scheme is as follows: the method for carrying out modal analysis according to the obtained illumination value and random load value, taking the damping ratio of the key mode as a random output variable and obtaining the corresponding random output variable value comprises the following steps:
the obtained illumination value and random load value XgThe small interference analysis model of the photovoltaic system is brought in, modal analysis is carried out to obtain the corresponding random output variable value xig＝[ξg1,…,ξgi,…,ξgw]In the formula 1<i<2n,w＝2n；
The further technical scheme is as follows: the method for solving the coefficient expanded by the Hermite chaotic polynomial by substituting the coordinate value and the random output variable value into the Hermite chaotic polynomial expansion comprises the following steps:
the obtained illumination value and random load value XgAnd a random output variable value xigRespectively substituting into Hermite chaos polynomial to expand xigAs the output of the Hermite chaotic polynomial expansion, the coordinate Hermite chaotic polynomial expansion is taken as the input to obtain the coefficient of the Hermite chaotic polynomial expansion, which is marked as ag＝[ag1,…,agi,…,agw]。
The further technical scheme is as follows: the method for simulating the Hermite chaotic polynomial expansion by adopting the Monte Carlo method, obtaining the distribution characteristic of the random output variable by adopting the kernel density estimation and evaluating the probability of small interference stability comprises the following steps:
calculating the ratio of each random variable data combination for DgMiddle element [ D ]PV1,…,DPVi,…,DPVn,DL]Wherein D isPV1＝Dr1H，{DPV2，…，DPVn}∈DRLIn a ratio of
(K1H/K)×(K2L/K)×…×(KiL/K)×…×(KnL/K),2<i<n
The calculated ratio is calculated according to the element in XgThe corresponding sequences in (1) are arranged to obtain Tg＝[Tg1,…,Tgi,…,Tgw]In the formula 1<i<2n,w＝2n；
Generating M U vector values, distributing the U vector values to each random variable data combination according to the proportion, and obtaining M in each groupg＝MTgA U value; substituting the U value obtained by each group into the corresponding Hermite chaotic polynomial expansion of the group to obtain the value of the corresponding random output variable; the value of the random output variable obtained by combining all random variable data is recorded as xiMI.e. the damping ratio of the key modelHas a value of xiMThe number of the M is;
by using kernel density estimation, xi is solvedMCumulative distribution function FξM(ξM) F of the reaction mixtureξM(ξM) The function is the cumulative distribution function of the damping ratio of the key mode;
evaluating small interference stability by using small interference stability probability: the probability of small interference instability is FξM(0) The probability of small interference being stable is 1-FξM(0)。
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in: the double peak problem of the illumination distribution can be processed; the double peaks of the PDF of the key mode damping ratio can be well estimated; the projection of the CDF of the key mode damping ratio can be well estimated; the method has very accurate small interference stability evaluation results.
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The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.
FIG. 1 is a flow chart of a method according to an embodiment of the invention;
FIG. 2 is a functional block diagram of a photovoltaic power generation system according to an embodiment of the present invention;
FIG. 3 is a diagram of a model for establishing a small interference analysis of a photovoltaic system according to an embodiment of the present invention;
FIG. 4 is a flow chart of a linear independent probabilistic matching method in the method according to the embodiment of the invention;
FIG. 5 is a schematic block diagram of an IEEE10 machine 39 node power generation system with photovoltaic access in an embodiment of the present invention;
FIG. 6a is a graph of an illumination probability density function according to an embodiment of the present invention;
FIG. 6b is a graph of the cumulative distribution function of illumination according to an embodiment of the present invention;
FIG. 6c is a graph of a load probability density function according to an embodiment of the present invention;
FIG. 6d is a graph of the cumulative distribution function of the load according to the embodiment of the present invention;
FIG. 7a is a graph of a segmented illumination probability density function according to an embodiment of the present invention;
FIG. 7b is a graph of a function of the cumulative distribution of the segmented illumination according to an embodiment of the present invention;
FIG. 8a is a graph of the key mode damping ratio probability density function in an embodiment of the present invention;
FIG. 8b is a graph of the cumulative distribution function of the damping ratio of the critical mode in an embodiment of the present invention;
wherein: 1-39 represent No. 1 to No. 39 bus bars, respectively; 40. the system comprises an interconnection transmission line 41, a power transformation station transformer 42, an equivalent lumped parameter system 43, an equivalent step-up transformer 44 and a photovoltaic power station.
Detailed Description
The technical solutions in the embodiments of the present invention are 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 only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, but the present invention may be practiced in other ways than those specifically described and will be readily apparent to those of ordinary skill in the art without departing from the spirit of the present invention, and therefore the present invention is not limited to the specific embodiments disclosed below.
Theoretical analysis:
A. random input representation
1) Independent random input
The first step is to represent the input random variable vector X by the standard normally distributed random variables (SRVs)
Figure BDA0001175679340000091
In the formula
Figure BDA0001175679340000092
Is the inverse of the cumulative distribution function of X, and phi (-) is the cumulative distribution function of a standard normal distribution. U shapeiIs a standard normal variable with zero mean and unit variance,this is often selected from a standard normal random variable that is independent and identically distributed.
2) Coherent random input and Nataf transform
The ith independent input X in equation (1)iIs directly determined by the ith independent random input variable UiIs expressed as a function of (c). However, equation (1) does not necessarily hold in the power system, because the random input vectors X are usually related to each other. Thus, the related non-standard random variables in the vector X can be transformed into independent standard normal random variables U using a Nataf transform.
Assuming that the above-mentioned related random vector X has an edge probability density function CDFXi(Xi) (i-1, …, n) and a correlation coefficient matrix CX
Figure BDA0001175679340000093
Figure BDA0001175679340000094
Where ρ isijAnd cov (X)i,X j) Is a correlation coefficient XiAnd XjThe correlation coefficient and the covariance of (a),
Figure BDA0001175679340000095
and
Figure BDA0001175679340000096
are each XiAnd XjStandard deviation of (2). First, according to an equiprobable transition, XiAnd a standard normal variable YiCDF of (a) satisfies
Figure BDA0001175679340000101
I.e. the associated random variable XiCan be converted into a normalized normal random variable Y with zero mean and standard deviation per uniti
Figure BDA0001175679340000102
Arrangement YiWill form an AND matrix CYAnd the edge probability density function phin(Y,CY) Vector-related n-dimensional normal vector Y
Figure BDA0001175679340000103
Figure BDA0001175679340000104
ρ0ijIs YiAnd YjThe correlation coefficient of (2). Matrix CYThe medium element can be obtained by solving the following formula
Figure BDA0001175679340000105
Figure BDA0001175679340000106
And
Figure BDA0001175679340000107
is the mean value and the associated random variable XjStandard deviation of (2).
When in use
Figure BDA0001175679340000108
Edge probability distribution function and XiAnd XjCorrelation coefficient of (1) ("rho")ijWhen known, the equivalent correlation coefficient ρ can be determined by solving the nonlinear equation shown in equation (7)0ijWhen C is presentYKnown correlation coefficient matrix C for standard normal random variablesYCholesky decomposition
CY＝BBT (8)
B is CYAnd (5) decomposing the lower triangular matrix by Cholesky. The relevant standard normal random variable vector Y can be converted into an independent standard normal random vector U using B.
U＝B-1Y (9) thus far completes the forward transformation process of the Nataf transformation, and the original related non-standard random variable vector X can be represented by a vector U
Figure BDA0001175679340000111
B. Estimated output
By estimating the output performance by using Hermite chaotic polynomial expansion, the implicit functional relation of the complex system can be replaced by a simple explicit function. In this step, all random inputs are considered, since any random input will affect the random output. The output and input may be represented as functions of the same standard normal random variable.
The random output variable can be estimated using a Hermite random polynomial expansion as follows:
Figure BDA0001175679340000112
Figure BDA0001175679340000113
is a vector representing a standard normal random variable which is input randomly, and n is the number of the standard normal random variables;
Figure BDA0001175679340000114
is Hermite polynomial of the order of n, and the calculation formula is as follows:
Figure BDA0001175679340000115
when the degree of freedom of random output in a random space is N, the number of terms N of p-order Hermite random polynomial expansion is NaThe calculation formula of (2):
Figure BDA0001175679340000116
C. calculating coefficients of a random polynomial expansion expression
First, a probability distribution point method (PCM) is adopted to provide distribution points for a standard normal random vector U. Next, the original input random vector X corresponding to the collocation point is calculated using equation (10) or (1). Then, through modal analysis, the corresponding output Z ═ Z is obtained1,Z2,…,ZN]TThus, equation (11) can be replaced with:
Z＝Ha (14)
wherein H is NxNaAn independent spatial matrix is maintained, and the unknown coefficient a can be solved by the formula.
The principle of the probability matching method is as follows: 1) in general, the value of the input random variable is determined by the root of Hermite polynomial of order p +1, and the number of the optional nodes is (p +1)n. 2) The number of selected coordination points is generally more than the number of undetermined coefficients. 3) And preferentially selecting the collocation points in the high probability region. 4) The matching points are as symmetrical as possible about the origin. 5) Matrix H is full rank.
Based on these principles, the present invention provides a linearly independent probabilistic matching method (FIG. 4). The configuration points are selected one after the other. The fix point is selected only if the Hermite polynomial information matrix is linearly independent. First, a screen is selected from the matching points that are symmetric about the origin and have a high probability, so that the H-array is linearly uncorrelated. Secondly, when the symmetric matching points are not enough to enable the H array to be full, screening is carried out from the asymmetric matching points with high probability, so that the H array is not linearly related. Until the H-matrix reaches full rank.
D. Application of random response surface method
After the unknown coefficient in the formula (11) is determined, the p-order Hermite chaotic polynomial can be used for small interference stability evaluation. And (3) performing a large amount of simulation on the formula (11) by using a Monte Carlo Method (MCS), then adopting PDF and CDF for estimating random output (key mode damping ratio), and finally obtaining the probability that the key mode damping ratio is greater than 0, namely the probability that the small interference probability is stable by the CDF.
Through the above analysis, generally, as shown in fig. 1, the present invention discloses a method for evaluating the small interference stability of a power system, the method includes the following steps:
s101: and establishing a photovoltaic system small interference analysis model which considers the internal dynamic characteristics of the photovoltaic power generation system and adopts a maximum power point tracking operation mode.
S102: performing modal analysis, and taking a damping ratio of a key mode as a random output variable;
s103: taking illumination with correlation and random load as random input variables, acquiring historical data of the random input variables, obtaining the distribution characteristics of the random input variables by adopting kernel density estimation, and calculating a correlation coefficient matrix of the random input variables;
s104: converting the correlation coefficient matrix of the random input variable into a correlation coefficient matrix of a random vector in standard normal distribution;
s105: determining the order of a random response surface, and generating a Hermite chaotic polynomial coordination point by adopting a linear independent coordination point method;
s106: converting the configuration points into corresponding illumination values and random load values by adopting Nataf conversion;
s107: according to the obtained illumination value and the random load value, performing modal analysis to obtain a corresponding random output variable value;
s108: substituting the matched value and the random output variable value into a Hermite chaotic polynomial expansion to obtain a Hermite chaotic polynomial expansion coefficient;
s109: and simulating the expansion of the Hermite chaotic polynomial by adopting a Monte Carlo method, solving the distribution characteristic of the random output variable by adopting nuclear density estimation, and evaluating the probability of small interference stability.
Specifically, the step S101 specifically includes the following steps:
s1011, establishing a small interference analysis model of the photovoltaic system, as shown in FIGS. 2-3:
the structure of the photovoltaic power generation system is a secondary photovoltaic power generation system comprising a photovoltaic array, a boost, an inverter, a filter and a phase-locked loop, wherein the output characteristic of a photovoltaic cell adopts an engineering calculation method, the temperature of the cell is a function of illumination, and the random input variable of the whole photovoltaic system model is illumination;
the control mode is maximum power point tracking control and fixed inverter direct-current voltage control, wherein all control links adopt PI control, the inverter operates in a unit power factor operation mode, the constant voltage of the direct-current side of the inverter is realized by an active power loop of the inverter, and the voltage of the low-voltage side of boost is determined by the maximum power point tracking control;
and S1012, linearizing the operation point of the photovoltaic system, and establishing an algebraic differential equation of the photovoltaic system suitable for small interference stability analysis. The photovoltaic system algebraic differential equation is combined with the generator algebraic differential equation and the network algebraic equation to obtain a power system linear state equation containing the photovoltaic system.
The step S103 specifically includes the following steps:
and S1031, setting the illumination and random load active power with correlation as a random input vector:
X＝[RPV,PL]T (17)
in the formula, RPV＝[r1,…,ri,…,rn]For illumination of photovoltaic systems, riFor the illumination of the ith photovoltaic system, n is the number of the photovoltaic systems, PL＝[p1,…,pj,…,pm]For a random load, pjIs the jth random load, and m is the number of random loads.
S1032, setting an analysis period, such as 11:30-12:30 in 8 months, and acquiring historical illumination data D of the photovoltaic system in the periodR＝[Dr1,…,Drn]And historical load data D of random loadL＝[DL1,…,DLm]And K is the number of the historical data, then DRDimension K × n, DLDimension is K × m;
s1033: calculating probability density function f of historical illumination data by adopting kernel density estimationri(ri) And cumulative distribution function Fri(ri) And i denotes the ith photovoltaic system: calculating probability density function f of historical load datapj(pj) And cumulative distribution function Fpj(pj) And j denotes the jth random load. And calculating the correlation coefficient among random variables, wherein the formula is as follows:
Figure BDA0001175679340000141
in the formula, xiAnd xjIs any two random variables in the X, and the X is a random variable,
Figure BDA0001175679340000142
and
Figure BDA0001175679340000143
is xiAnd xjAverage value of (p)ijIs xiAnd xjThe correlation coefficient of (2). Arranging the correlation coefficients of any two random variables in X according to the sequence of the random variables in X to obtain a correlation coefficient matrix of X
Figure BDA0001175679340000144
The step S104 specifically includes the following steps:
s1041: solving the following implicit function, converting the correlation coefficient between random input variables into the correlation coefficient between standard normal random variables
Figure BDA0001175679340000145
In the formula, XiAnd XjAny two random input variables in bit X, according to Nataf distribution, YiAnd YjIs a reaction of with XiAnd XjThe corresponding standard normal random variable is used as the random variable,
Figure BDA0001175679340000146
and
Figure BDA0001175679340000147
is XiAnd XjIs a cumulative distribution function of a standard normal random variable, phi (·)2(Yi,Yj，ρ0ij) Is YiAnd YjOf a joint probability density function ofij0Is YiAnd YjThe correlation coefficient of (2).
S1042: let ρ ij0 be XiAnd XjThe sequences in X are arranged to obtain (a correlation coefficient matrix of random vectors of random input variables in a standard normal distribution) CY
Figure BDA0001175679340000151
The step S105 specifically includes the following steps:
and S1051, determining the order l of Hermite chaotic polynomial expansion adopted by the random response surface according to needs, wherein the higher the order is, the higher the accuracy is. Usually about 3 orders of magnitude can meet the precision requirement. The 2-5 order Hermite chaotic polynomial expansion is given as follows:
Figure BDA0001175679340000152
Figure BDA0001175679340000153
Figure BDA0001175679340000154
Figure BDA0001175679340000155
in the formula, xi is a random output variable,
Figure BDA0001175679340000156
is an independent standard normal random vector and represents a random input variable, n is the number of elements in a vector U,
Figure BDA0001175679340000157
is an n-th order Hermite polynomialFormula (II)
Figure BDA0001175679340000158
When the degree of freedom of the random output variable is q, the number of terms expanded by the p-order Hermite chaotic polynomial is q
Figure BDA0001175679340000159
The undetermined coefficient of each term is set as a0，ai1，ai1i2，ai1i2i3，ai1i2,...,inDefinition of a ═ a0，ai1，ai1i2，ai1i2i3，ai1i2,...,in]。
The Hermite chaotic polynomial expansion can be expressed as ξ ═ Ha, H consisting of a Hermite chaotic polynomial.
S10512, adopting a linear independent probability point matching method (as shown in figure 4) to obtain matching points expanded by Hermite chaotic polynomial, wherein the number of the matching points is
Figure BDA0001175679340000161
1) Calculating the root of a Hermite polynomial of the order of p +1, and arranging and combining all possible coordination points;
2) forming a symmetrical distribution point group according to the symmetry of the distribution points about the origin;
3) and sequentially checking each symmetrical distribution point group according to the descending order of the probability of the symmetrical distribution point groups until the H array full rank formed by all the selected distribution points is reached. The matching point group is selected as follows: the newly added symmetrical distribution point group is irrelevant to the linearity of an H array formed by all the existing distribution point groups;
4) and if the symmetric distribution point groups are completely detected and the H array is still not full of rank, sequentially checking each single distribution point according to the descending order of distribution point probability until the H array full of rank is formed by all the selected distribution points. The conditions for selecting a single point are: the newly added single distribution point is irrelevant to the linearity of the H array formed by all the existing distribution points. Thus, the coordination point of U in Hermite chaotic polynomial expansion is obtained.
The step S106 specifically includes the following steps:
s1061, calculating the combination of random variables: usually, the historical illumination data presents obvious bimodal distribution, and the illumination data is divided into high illumination DRHAnd low light DRLTwo groups:
DRH＝{Dr1H,…,DriH,…,DrnH},DRL＝{Dr1L,…,DriL,…,DrnL} (27)
in the formula, DriHNumber of data of KiH，DRiLNumber of data of KiL. Using kernel density estimation to obtain high illumination DRHAnd low light DRLProbability density function f of each datariH(riH)、friL(riL) And cumulative distribution function FriH(riH)、FriL(riL)；
S1062, because the illumination can be high illumination or low illumination, according to DRH、DRLAnd DLData combinations of all possible random variables are listed:
Dg＝{[DPV,DL]|DPV＝[DPV1,…,DPVi,…,DPVn]，DPVi∈{DriH，DriL}} (28)
it can be seen that DgIn total 2nAnd (4) each element.
S1063, centering with DgAnd each element adopts a Nataf transform to transform the collocation points into corresponding illumination values and random load values. E.g. for the element [ D ]PV1,…,DPVi,…,DPVn,DL]Wherein D isPV1＝Dr1H,{DPV2,…,DPVn}∈DRL：
1) For correlation coefficient matrix CYCholesky decomposition
CY＝BBT (29)
In which B is CYAnd (5) decomposing the lower triangular matrix by Cholesky.
2) Matching point (denoted as U) of independent standard normal random variable spaceC) Transformation to a standard normal response with correlationMachine vector YC：YC＝BUC.
3) According to the principle of equal probability transformation, Y is divided intoCConverting into original random input vector (illumination and random load);
Figure BDA0001175679340000171
in the formula (I), the compound is shown in the specification,
Figure BDA0001175679340000172
is step S1062 [ D ]PV1,…,DPVi,…,DPVn,DL]The inverse of the edge cumulative distribution function of any element in the list, if the element is DPV1Then the edge cumulative distribution function is Fr1HIf the element is DPV2,…,DPVnThen the edge cumulative distribution function is Fr1L(r1L),…,FrnL(rnL)；XCiThe original random input variable value is obtained. To this end, the result corresponds to DgMiddle element [ D ]PV1,…,DPVi,…,DPVn,DL]The original random input vector illumination and the value of the random load. Repeating 1) to 3) to obtain DgOriginal random input vector of all elements in the vector, denoted as Xg＝[Xg1,…,Xgi,…,Xgw]In the formula 1<i<2n,w＝2n。
Registered as corresponding to Dg＝{[DPV,DL]|DPV＝[DPV1,…,DPVi,…,DPVn]The illumination and random load value of (A) is Xg，
The step S107 specifically includes the step of comparing the illumination value and the random load value X obtained in the step S106gSubstituting the system model, and performing modal analysis to obtain corresponding random output variable value xig＝[ξg1,…,ξgi,…,ξgw]In the formula 1<i<2n,w＝2n。
The step S108 specifically includes the step of comparing the illumination value and the random load value obtained in the steps S106 and S107XgAnd a random output variable value xigRespectively substituting into Hermite chaos polynomial to expand xigAs the output of the Hermite chaotic polynomial expansion, the coordinate Hermite chaotic polynomial expansion is taken as the input to obtain the coefficient of the Hermite chaotic polynomial expansion, which is marked as ag＝[ag1,…,agi,…,agw]。
The step S109 includes the steps of:
s1091 computing the fraction of each random variable data combination, e.g., for DgMiddle element [ D ]PV1,…,DPVi,…,DPVn,DL]Wherein D isPV1＝Dr1H，{DPV2，…，DPVn}∈DRLIn a ratio of
(K1H/K)×(K2L/K)×…×(KiL/K)×…×(KnL/K),2<i<n (31)
The calculated ratio is calculated according to the element in XgThe corresponding sequences in (1) are arranged to obtain Tg＝[Tg1,…,Tgi,…,Tgw]In the formula 1<i<2n,w＝2n。
S1092, generating M U vector values, distributing the U vector values to each random variable data combination according to the proportion, and obtaining M in each groupg＝MTgAnd U value. And substituting the U value obtained by each group into the corresponding Hermite chaotic polynomial expansion of the group to obtain the value of the corresponding random output variable. The value of the random output variable (namely the key mode damping ratio) obtained by recording all random variable data combination is xiMThe number of the M pieces of the Chinese herbal medicine is M.
S1093: by using kernel density estimation, xi is solvedMCumulative distribution function FξM(ξM) F of the reaction mixtureξM(ξM) I.e. the cumulative distribution function of the damping ratio of the critical mode.
S1094: evaluating small interference stability by using small interference stability probability: the probability of small interference instability is FξM(0) The probability of small interference being stable is 1-FξM(0)。
The practice of the present invention will be further illustrated with reference to the accompanying drawings and examples, but the practice and inclusion of the invention is not limited thereto.
Taking a 10-machine 39-node power system as an example, as shown in fig. 5, the No. 19 bus is connected to a photovoltaic system. In FIG. 5, G1-G10 represent generators, A1-A2 and A4-A5 represent regions, BP1-BP4A bus bar is shown.
Determining a random input vector: illumination and random 16 th and 20 th random loads at the 16 th and 20 th bus bars. Determining a random output variable: when the light is illuminated, the standard 1000M/M is obtained2When the load at random takes the load flow calculation result, the corresponding key mode is 0.46Hz, the damping ratio is 0.002, and the damping ratio of the mode is taken as the random output variable. Determining the distribution and correlation coefficient of random input variables: the illumination data of the photovoltaic system is taken from historical illumination data of August 11:30-12:30 in the last 10 years in a certain place, the random load data is taken from random load data of adjacent areas, the illumination distribution and the load distribution are obtained by adopting nuclear density estimation and are shown in figures 6a-6d, and a correlation coefficient C is calculatedXConvert it to CY。
Determining that the expansion order of the Hermite chaotic polynomial is 3, and solving a coordination point U expanded by the Hermite chaotic polynomial by adopting a linear independent probability coordination point methodC. Determining a data packet: dividing the illumination into high illumination DrHAnd low light DrLTwo sets of data, then a random variable data combination Dg＝{Dg1，Dg2}，Dg1＝[DrH,DL16,DL20],Dg2＝[DrL,DL16,DL20]. The respective distribution characteristics were determined using kernel density estimation, as shown in FIGS. 7a-7 b. Respectively calculating the corresponding Dg of the coordinate points by adopting Nataf transformation1And Dg2Illumination and random load value Xg1And Xg2. Each group of illumination and random load value Xg1And Xg2Carrying out modal analysis to obtain damping ratio xi g of each group of key modes1And xig2(ii) a Carrying in Hermite chaotic polynomial expansion to obtain each group of coefficients ag1And ag2。
Probability density function and accumulation of key mode damping ratio by adopting Monte Carlo simulationIntegral distribution function: calculating Dg1And Dg2Ratio of (1) to (Tg)1And Tg2Generating 1000U values, then Mg1＝1000Tg1And Mg2＝1000Tg2Substituting U into each group of Hermite chaotic polynomial expansion to obtain the key mode damping ratio xi of all groupsM. By using kernel density estimation, xi is solvedMCumulative distribution function FξM(ξM) F of the reaction mixtureξM(ξM) I.e. the cumulative distribution function of the damping ratio of the critical mode. The results are shown in FIGS. 8a-8b, where Monte Carlo was performed using modal analysis to represent the true results. Therefore, the method provided by the invention can be used for treating the double peak problem of the illumination distribution; double peaks of PDF of the damping ratio of the key mode are well estimated; the projection of the CDF of the damping ratio of the key mode is well estimated; the method has very accurate small interference stability evaluation results. From fig. 8b, it can be estimated that the probability of the system small interference being stable is 89%. The small interference stability evaluation result provides effective information for operation scheduling personnel to monitor the stability of the power system.

Claims (2)

1. A method for evaluating the small interference stability of a power system is characterized by comprising the following steps:
establishing a photovoltaic system small interference analysis model, wherein the structure of the photovoltaic system small interference analysis model is a secondary photovoltaic power generation system comprising a photovoltaic array, a boost, an inverter, a filter and a phase-locked loop, the output characteristic of a photovoltaic cell adopts an engineering calculation method, the temperature of the cell is a function of illumination, and the random input variable of the whole photovoltaic system model is illumination; the control mode is maximum power point tracking control and fixed inverter direct-current voltage control, wherein all control links adopt PI control, the inverter operates in a unit power factor operation mode, the constant voltage of the direct-current side of the inverter is realized by an active power loop of the inverter, and the voltage of the low-voltage side of boost is determined by the maximum power point tracking control; linearization is carried out at the operating point of the photovoltaic system, a photovoltaic system algebraic differential equation suitable for small interference stability analysis is established, and the photovoltaic system algebraic differential equation is combined with a generator algebraic differential equation and a network algebraic equation to obtain a power system linearized state equation containing the photovoltaic system;
taking illumination with correlation and random load as random input variables, and calculating a correlation coefficient matrix of the random input variables, wherein the correlation coefficient matrix specifically comprises the following steps:
setting the illumination and random load active power with correlation as a random input vector,
X＝[RPV,PL]T
in the formula, RPV＝[r1,…,ri,…,rn]For illumination of photovoltaic systems, riFor the illumination of the ith photovoltaic system, n is the number of the photovoltaic systems, PL＝[p1,…,pj,…,pm]For a random load, pjIs jth random load, m is the number of random loads;
setting an analysis time interval and acquiring historical illumination data D of the photovoltaic system in the time intervalR＝[DR1,…,DRn]And historical load data D of random loadL＝[DL1,…,DLm]And K is the number of the historical data, then DRDimension K × n, DLDimension is K × m;
and calculating the correlation coefficient among random variables, wherein the formula is as follows:
Figure FDA0002745113090000011
in the formula, xiAnd xjIs any two random variables in the X, and the X is a random variable,
Figure FDA0002745113090000012
and
Figure FDA0002745113090000013
is xiAnd xjAverage value of (p)ijIs xiAnd xjThe correlation coefficient of (a); arranging the correlation coefficients of any two random variables in X according to the sequence of the random variables in X to obtain a correlation coefficient matrix of X:
Figure FDA0002745113090000021
converting the correlation coefficient matrix of the random input variable into a correlation coefficient matrix of a random vector in standard normal distribution, which specifically comprises the following steps:
solving the following implicit function, and converting the correlation coefficient among random input variables into the correlation coefficient among standard normal random variables:
Figure FDA0002745113090000022
in the formula, XiAnd XjIs any two random input variables in X, and conforms to Nataf distribution, YiAnd YjIs a reaction of with XiAnd XjThe corresponding standard normal random variable is used as the random variable,
Figure FDA0002745113090000023
are each XiThe average value and the standard deviation of (a),
Figure FDA0002745113090000024
Figure FDA0002745113090000025
are each XjIs a cumulative distribution function of a standard normal random variable, phi (·)2(Yi,Yj,ρ0ij) Is YiAnd YjOf a joint probability density function of0ijIs YiAnd YjThe correlation coefficient of (a);
will rho0ijAccording to XiAnd XjArranging in X order to obtain random vector correlation coefficient matrix C of random input variable in standard normal distributionY，
Figure FDA0002745113090000026
Determining the order of a random response surface of a correlation coefficient matrix of a random vector of standard normal distribution, and generating a Hermite chaotic polynomial coordination point by adopting a linear independent coordination point method, wherein the method specifically comprises the following steps:
determining the order l of Hermite chaotic polynomial expansion adopted by the random response surface according to needs, wherein the Hermite chaotic polynomial expansion of 2-5 orders is as follows:
Figure FDA0002745113090000027
Figure FDA0002745113090000028
Figure FDA0002745113090000031
Figure FDA0002745113090000032
in the formula, xi is a random output variable,
Figure FDA0002745113090000033
is an independent standard normal random vector, represents a random input variable,
Figure FDA0002745113090000034
is Hermite polynomial of order p
Figure FDA0002745113090000035
When the degree of freedom of the random output variable is q,the term number of the p-order Hermite chaotic polynomial expansion is
Figure FDA0002745113090000036
The undetermined coefficient of each term is set as
Figure FDA0002745113090000037
Definition of
Figure FDA0002745113090000038
Figure FDA0002745113090000039
The Hermite chaotic polynomial expansion is expressed as xi ═ Ha, and H is composed of a Hermite chaotic polynomial;
solving the distribution points expanded by the Hermite chaotic polynomial by adopting a linear independent probability distribution point method, wherein the number is
Figure FDA00027451130900000310
The method specifically comprises the following steps:
1) calculating the root of a Hermite polynomial of the order of p +1, and arranging and combining all possible coordination points;
2) forming a symmetrical distribution point group according to the symmetry of the distribution points about the origin;
3) according to the sequence of the descending probability of the symmetrical distribution point groups, sequentially checking each symmetrical distribution point group until all selected distribution points form an H array full rank, wherein the selected condition of the distribution point groups is as follows: the newly added symmetrical distribution point group is irrelevant to the linearity of an H array formed by all the existing distribution point groups;
4) if the symmetric matching point groups are completely detected and the H matrix is still not full of rank, sequentially checking each single matching point according to the descending order of the probability of matching points until the H matrix formed by all the selected matching points is full of rank, and selecting the single matching point under the condition that: adding a single coordination point which is unrelated to the linearity of the H array formed by all the existing coordination points, and thus obtaining a U coordination point in Hermite chaotic polynomial expansion;
and converting the collocation points into corresponding illumination values and random load values by adopting Nataf conversion, which specifically comprises the following steps:
calculating a combination of random variables: dividing lighting data into high lighting DRHAnd low light DRLTwo groups:
DRH＝{Dr1H,…,DriH,…,DrnH},DRL＝{Dr1L,…,DriL,…,DrnL} (27)
in the formula, DriHNumber of data of KiH，DRiLNumber of data of KiLUsing kernel density estimation to obtain high illumination DRHAnd low light DRLProbability density function f of each datariH(riH)、friL(riL) And cumulative distribution function FriH(riH)、FriL(riL)；
According to DRH、DRLAnd DLData combinations of all possible random variables are listed:
Dg＝{[DPV,DL]|DPV＝[DPV1,…,DPVi,…,DPVn]，DPVi∈{DriH,DriL}} (28)
it can be seen that DgIn total 2nAn element;
for DgFor each element in the set, the Nataf transform is used to transform the collocation point into a corresponding illumination value and random load value, e.g. for element [ D ]PV1,…,DPVi,…,DPVn,DL]Wherein D isPV1＝Dr1H,{DPV2,…,DPVn}∈DRL；
5) For correlation coefficient matrix CYCholesky decomposition
CY＝BBT (29)
In which B is CYThe lower triangular matrix obtained by Cholesky decomposition,
6) matching points of independent standard normal random variable space, and recording the matching points as UCTransformed into a standard normal random vector Y with correlationC：YC＝BUC，
7) According to the principle of equal probability transformation, Y is divided intoCConverting the random input vector into an original random input vector, wherein the original random input vector is illumination and random load;
Figure FDA0002745113090000041
in the formula (I), the compound is shown in the specification,
Figure FDA0002745113090000042
is { [ D ] in formula (28)PV,DL]|DPV＝[DPV1,…,DPVi,…,DPVn]，DPVi∈{DriH,DriL} if the element is D, the inverse of the edge-cumulative distribution function of any element in the latticePV1Then the edge cumulative distribution function is Fr1HIf the element is DPV2,…,DPVnThen the edge cumulative distribution function is Fr2L(r12L),…,FrnL(rnL)；XCiFor the original random input variable value found, up to this point, the value corresponding to D is obtainedgMiddle element [ D ]PV1,…,DPVi,…,DPVn,DL]Repeating the values of the original random input vector illumination and the random load by 5) -7) to obtain DgOriginal random input vector of all elements in the vector, denoted as Xg＝[Xg1,…,Xgi,…,Xgw]In the formula 1<i<2n，w＝2n，
I.e. obtained corresponds to { [ D ]PV,DL]|DPV＝[DPV1,…,DPVi,…,DPVn]，DPVi∈{DriH,DriLThe illumination and random load value of is Xg；
Performing modal analysis according to the obtained illumination value and the random load value, and taking the damping ratio of the key mode as a random output variable to obtain a corresponding random output variable value; specifically, the obtained illumination value and random load value XgBringing the small interference analysis model into a photovoltaic system, and carrying out modal analysis to obtain corresponding random outputDelta value xig＝[ξg1,…,ξgi,…,ξgw]In the formula 1<i<2n，w＝2n；
Substituting the matched value and the random output variable value into a Hermite chaotic polynomial expansion to obtain a Hermite chaotic polynomial expansion coefficient;
simulating the expansion of the Hermite chaotic polynomial by adopting a Monte Carlo method, obtaining the distribution characteristic of a random output variable by adopting nuclear density estimation, and evaluating the probability of small interference stability, wherein the method specifically comprises the following steps:
calculating the ratio of each random variable data combination for DgMiddle element { [ D ]PV,DL]|DPV＝[DPV1,…,DPVi,…,DPVn]，DPVi∈{DriH,DriL} in which D isPV1＝Dr1H,{DPV2,…,DPVn}∈DRLIn a ratio of
(K1H/K)×(K2L/K)×…×(KiL/K)×…×(KnL/K),2<i<n；
The calculated ratio is calculated according to the element in XgThe corresponding sequences in (1) are arranged to obtain Tg＝[Tg1,…,Tgi,…,Tgw]In the formula 1<i<2n，w＝2n；
Generating M U vector values, distributing the U vector values to each random variable data combination according to the proportion, and obtaining M in each groupg＝MTgA U value; substituting the U value obtained by each group into the corresponding Hermite chaotic polynomial expansion of the group to obtain the value of the corresponding random output variable; the value of the random output variable obtained by combining all random variable data is recorded as xiMI.e. the value of the key model damping ratio is xiMThe number of the M is;
by using kernel density estimation, xi is solvedMCumulative distribution function FξM(xi M), the FξM(ξM) The function is the cumulative distribution function of the damping ratio of the key mode;
evaluating small interference stability by using small interference stability probability: small interference instabilityHas a probability of FξM(0) The probability of small interference being stable is 1-FξM(0)。
2. The method for evaluating the small interference stability of the power system as claimed in claim 1, wherein the method for substituting the coordination value and the random output variable value into the Hermite chaotic polynomial expansion to obtain the coefficient of the Hermite chaotic polynomial expansion comprises the following steps:
the obtained illumination value and random load value XgAnd a random output variable value xigRespectively substituting into Hermite chaos polynomial to expand xigAs the known output value of Hermite chaotic polynomial expansion, the illumination value and the random load value X are usedgObtaining a coefficient of Hermite chaotic polynomial expansion as an input known value of the Hermite chaotic polynomial expansion, and marking the coefficient as ag＝[ag1,…,agi,…,agw]。
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