CN117313304B - Gaussian mixture model method for analyzing overall sensitivity of power flow of power distribution network - Google Patents

Gaussian mixture model method for analyzing overall sensitivity of power flow of power distribution network Download PDF

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CN117313304B
CN117313304B CN202310546389.9A CN202310546389A CN117313304B CN 117313304 B CN117313304 B CN 117313304B CN 202310546389 A CN202310546389 A CN 202310546389A CN 117313304 B CN117313304 B CN 117313304B
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高元海
徐潇源
严正
黄兴德
谢伟
方陈
王晗
平健
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    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
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    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
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Abstract

The invention discloses a Gaussian mixture model method for analyzing the overall sensitivity of power flow of a power distribution network, which relates to the field of power distribution networks, and comprises the following steps: step 1, establishing a Gaussian mixture model of a joint probability density function of a power flow input variable and an output variable of a power distribution network; step 2, obtaining an analytic type of probability density function of a power flow output variable of the power distribution network, and calculating variance of the analytic type; step 3, obtaining an analysis type of a conditional probability density function and a conditional variance of a power flow output variable of the power distribution network relative to an input variable, and calculating an expected value of the conditional variance; and step 4, calculating a power flow global sensitivity index of the power distribution network according to the results obtained in the step 2 and the step 3, and completing power flow global sensitivity analysis of the power distribution network. The method only needs less times of power flow calculation, consumes short time, avoids the time-consuming agent model construction and Monte Carlo simulation calculation process, and remarkably improves the calculation efficiency of power flow global sensitivity analysis of the power distribution network.

Description

Gaussian mixture model method for analyzing overall sensitivity of power flow of power distribution network
Technical Field
The invention relates to the field of power distribution networks, in particular to a Gaussian mixture model method for analyzing the overall sensitivity of power flow of a power distribution network.
Background
With the acceleration of the construction of the novel power distribution network, a large number of novel sources/charges with strong uncertainty are connected, so that the uncertainty of the running state of the power distribution network is outstanding. The global sensitivity analysis is a key technology for quantifying uncertainty propagation, and the importance quantification result of each uncertain source/load can be obtained through the power distribution network tide global sensitivity analysis, and the key source/load causing the uncertainty of the running state of the power distribution network is identified, so that the uncertainty of the key source/load is actively managed in a targeted manner. The uncertain source/load in the power distribution network has complex probability distribution characteristics and remarkable correlation, meanwhile, the power distribution network power flow model has remarkable nonlinear characteristics, and a Monte Carlo simulation method is generally adopted for power distribution network power flow global sensitivity analysis. The Monte Carlo simulation method has heavy calculation load, and the Monte Carlo simulation method based on the proxy model can reduce the time consumption of power flow calculation simulation of the power distribution network, but the total calculation time consumption is still longer.
Disclosure of Invention
The invention aims to provide a Gaussian mixture model method for analyzing the overall sensitivity of the power distribution network, which greatly improves the calculation efficiency of the overall sensitivity analysis of the power distribution network and solves the problems in the background technology.
In order to achieve the above purpose, the present invention provides the following technical solutions:
the invention discloses a Gaussian mixture model method for analyzing the overall sensitivity of power flow of a power distribution network, which comprises the following steps:
step 1: establishing a Gaussian mixture model of a joint probability density function of a power flow input variable and an output variable of the power distribution network;
step 2: obtaining an analytic type of probability density function of a power flow output variable of the power distribution network, and calculating variance of the analytic type;
step 3: obtaining an analytic expression of a conditional probability density function and a conditional variance of a power flow output variable of the power distribution network relative to an input variable, and calculating an expected value of the conditional variance;
step 4: and (3) calculating a power flow global sensitivity index of the power distribution network according to the results obtained in the step (2) and the step (3), and completing power flow global sensitivity analysis of the power distribution network.
As a further scheme of the present invention, the step 1 specifically includes:
firstly, establishing a Gaussian mixture model of a power flow input variable joint probability density function of a power distribution network;
the power flow input variable vector of the distribution network is X, and n historical samples of the power flow input variable vector are X 1 、X 2 、…、X n Establishing a joint probability density function of x by adopting a non-parameter kernel density estimation method:
wherein: n (·) represents a Gaussian kernel function, H k Represents the kth sample X k A corresponding bandwidth matrix, the probability density function based on non-parametric kernel density estimation being a basic Gaussian mixture model with Gaussian component number K b Weight coefficient equal to n, kth gaussian componentEqual to 1/n, mean vector->Equal to X k Covariance matrix->Equal to H k ；
And (3) adopting a layering expectation maximization algorithm of density retention to perform Gaussian component number reduction on the basic Gaussian mixture model shown in the formula (1), and obtaining a simplified Gaussian mixture model as follows:
wherein: omega k 、μ x,k 、Σ x,k Respectively representing a weight coefficient, a mean vector and a covariance matrix of the kth Gaussian component, wherein K represents the reduced Gaussian component number;
secondly, on the basis of the formula (2), establishing a piecewise linearization power flow model of the power distribution network;
the compact form of the nonlinear power distribution network power flow model is:
y＝Γ(x) (3)
wherein: Γ (·) represents a nonlinear power flow equation, and y represents an output variable vector of power flow of the power distribution network;
the kth Gaussian component N (x|mu) of the Gaussian mixture model given the input variable of formula (2) x,k ,Σ x,k ) For the nonlinear tide model shown in (3), the nonlinear tide model is shown in mu x,k Linearization is performed, and a kth linearization model is obtained:
y＝T k x+[Γ(μ x,k )-T k μ x,k ] (4)
wherein: t (T) k Mu for nonlinear tide equation x,k An inverse jacobian matrix at; each Gaussian component of the Gaussian mixture model of the input variable joint probability density function shown in the formula (2) corresponds to a linearized power flow model, namely, 1 piecewise linear power flow model comprising K linear segments is established;
then, on the basis of the formula (2) and the formula (4), establishing a joint probability density function of the power flow input variable and the power flow output variable of the power distribution network;
the kth Gaussian component N (x|mu) of the Gaussian mixture model for the input variable of the given formula (2) x,k ,Σ x,k ) The power flow input variable vector x of the power distribution network is expressed as:
x＝L k u+μ x,k (5)
wherein: l (L) k Is sigma-delta x,k The lower triangular matrix obtained by the George decomposition is a variable vector meeting standard normal distribution;
according to the formulas (4) and (5), for the kth gaussian component N (x|μ) of the gaussian mixture model given the input variable represented by the formula (2) x,k ,Σ x,k ) The combined variable vector of the power flow input variable and the power flow output variable of the power distribution network is as follows:
according to the formula (6) and the principle of the Georgi decomposition, the kth Gaussian component N (x|mu) of the Gaussian mixture model for the input variable represented by the given formula (2) x,k ,Σ x,k ) The joint probability density function of the power flow input variable and the power flow output variable of the power distribution network is as follows:
wherein: theta (theta) k The kth Gaussian component N (x|mu) of the Gaussian mixture model representing the input variable of the given formula (2) x,k ,Σ x,k ) With a probability P (Θ) k )＝ω k ；
According to the formula (7) and the full probability formula, the final obtaining of the joint probability density function of the power flow input variable and the power flow output variable of the power distribution network is as follows:
as a further scheme of the present invention, the step 2 specifically includes:
according to the analysis of the joint probability density function of the input variable and the output variable of the power flow of the power distribution network shown in the formula (8), the first output variable y of the power flow of the power distribution network l The probability distribution of (2) is the edge probability distribution of equation (8), and y is obtained l The probability density function of (2) is as follows:
wherein: mu (mu) l,k ＝μ y,k (l)、Σ ll,k ＝Σ y,k (l, l), (l, l) represent the indices of the vector and matrix elements, respectively;
according to formula (9), y l Is expressed as:
wherein: e [ N (y) l |μ l,k ,Σ ll,k ) M ]Shows a Gaussian distribution N (y) l |μ l,k ,Σ ll,k ) An M-order origin moment of (a);
according to formula (10), y l The variance of (c) is expressed as:
and calculating according to the formula (11) to obtain the variance of the power flow output variable of the power distribution network.
As a further scheme of the present invention, the step 3 specifically includes:
first, the power distribution network power flow input variable vector x is divided into complementary subsets x c And x d I.e. x= { x c ,x d }；
Next, the output variable y is calculated l With respect to input variable x c Is a desired value of conditional variance of (a);
will x c As input variable to be analyzed, x c And the first output variable y l The combined variable vector is represented as w, namely:
the probability distribution of w is the edge distribution of the formula (8), and the analytical formula of the joint probability density function of w is:
wherein: mu (mu) c,k ＝μ x,k (c)、Σ cc,k ＝Σ x,k (c,c)、Σ cl,k ＝Σ xy,k (c,l)、Σ lc,k ＝Σ yx,k (l, c), c represents x c Index in x, l denotes y l An index in y;
obtaining y according to formula (13) and Bayes theorem l Concerning x c The analytical formula of the conditional probability density function is:
wherein:
according to formulas (14) - (17), y is obtained l Concerning x c The conditional variance analysis formula of (2) is:
calculating y according to formula (18) l Concerning x c The expected value E [ Var (y) l |x c )]；
E[Var(y l |x c )]The calculation formula of (2) is as follows:
E[Var(y l |x c )]＝∫Var(y l |x c )f(x c )dx c (19)
wherein:
the numerical integration method is adopted to calculate the formula (19), and the method is specifically as follows:
wherein: x is X c,1 、X c,2 、…、X c,N For x generated according to formula (20) c Is a sample of N samples;
then, calculate the output variable y l With respect to input variable x d Is a desired value of conditional variance of (a);
will x d As input variable to be analyzed, x d And the first output variable y l The composed joint variable vector is denoted v, namely:
the probability distribution of v is the edge distribution of the formula (8), and the analytical formula of the joint probability density function for obtaining v is as follows:
wherein: mu (mu) d,k ＝μ x,k (d)、Σ dd,k ＝Σ x,k (d,d)、Σ dl,k ＝Σ xy,k (d,l)、Σ ld,k ＝Σ yx,k (l, d), d represents x d Index in x, l denotes y l An index in y;
obtaining y according to formula (23) and Bayes theorem l Concerning x d The analytical formula of the conditional probability density function is:
wherein:
according to formulae (24) - (27), y is obtained l Concerning x d The conditional variance analysis formula of (2) is:
calculating y according to formula (28) l Concerning x d The expected value E [ Var (y) l |x d )]；
E[Var(y l |x d )]The calculation formula of (2) is as follows:
E[Var(y l |x d )]＝∫Var(y l |x d )f(x d )dx d (29)
wherein:
the numerical integration method is adopted to calculate the formula (29), specifically:
wherein: x is X d,1 、X d,2 、…、X d,N For x generated according to formula (30) d Is a sample of N samples;
as a further scheme of the present invention, the step 4 specifically includes:
obtained according to step 2Var(y l ) And E [ Var (y) obtained in step 3 l |x c )]Calculating the output variable y l With respect to input variable x c Main effect global sensitivity index of (2):
var (y) obtained according to step 2 l ) And E [ Var (y) obtained in step 3 l |x d )]Calculating the output variable y l With respect to input variable x c Global sensitivity index of the total effect of (a):
and comprehensively, completing the overall sensitivity analysis of the power flow of the power distribution network.
Compared with the prior art, the invention has the beneficial effects that:
1) Based on the Gaussian mixture model and the piecewise linearized power flow model, the analytic type of the joint probability density function of the power flow input variable and the power flow output variable of the power distribution network can be obtained, the joint probability density function comprises the probability information of the complete input variable and the complete output variable and the piecewise linearized power flow model information of the power distribution network, and the construction process only needs less times of power flow calculation, so that the time-consuming proxy model construction process is avoided;
2) Based on the analytic expression of the joint probability density function of the input variable and the output variable of the power distribution network, the analytic expression of the variance of the output variable and the analytic expression of the conditional variance of the output variable relative to the input variable are obtained, and the time-consuming Monte Carlo simulation calculation process of the power distribution network is avoided;
by virtue of the characteristics, the analysis method for analyzing the overall power flow sensitivity of the power distribution network is realized, and the calculation efficiency of the overall power flow sensitivity analysis of the power distribution network is remarkably improved.
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FIG. 1 is a flow chart of the method of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely, and it is apparent that the described embodiments are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
In the embodiment of the invention, the Gaussian mixture model method for analyzing the overall sensitivity of the power flow of the power distribution network comprises the following steps:
step 1: and establishing a Gaussian mixture model of a joint probability density function of the power flow input variable and the power flow output variable of the power distribution network.
The detailed process is as follows:
firstly, a Gaussian mixture model of a power flow input variable joint probability density function of a power distribution network is established.
The power flow input variable vector of the distribution network is X, and n historical samples of the power flow input variable vector are X 1 、X 2 、…、X n Establishing a joint probability density function of x by adopting a non-parameter kernel density estimation method:
wherein: n (·) represents a Gaussian kernel function, H k Represents the kth sample X k A corresponding bandwidth matrix. The probability density function based on the non-parametric kernel density estimation is a basic Gaussian mixture model with Gaussian component number K b Weight coefficient equal to n, kth gaussian componentEqual to 1/n, mean vector->Equal to X k Covariance matrix->Equal to H k 。
And (3) adopting a layering expectation maximization algorithm of density retention to perform Gaussian component number reduction on the basic Gaussian mixture model shown in the formula (1), and obtaining a simplified Gaussian mixture model as follows:
wherein: omega k 、μ x,k 、Σ x,k The weight coefficient, the mean vector and the covariance matrix of the kth gaussian component are respectively represented, and K represents the reduced Gao Sicheng fraction.
And secondly, on the basis of the formula (2), establishing a piecewise linear power distribution network power flow model.
The compact form of the nonlinear power distribution network power flow model is:
y＝Γ(x) (3)
wherein: Γ (·) represents a nonlinear power flow equation, and y represents an output variable vector of the power flow of the power distribution network.
The kth Gaussian component N (x|mu) of the Gaussian mixture model given the input variable of formula (2) x,k ,Σ x,k ) For the nonlinear tide model shown in (3), the nonlinear tide model is shown in mu x,k Linearization is performed, and a kth linearization model is obtained:
y＝T k x+[Γ(μ x,k )-T k μ x,k ] (4)
wherein: t (T) k Mu for nonlinear tide equation x,k An inverse jacobian matrix at. Each gaussian component of the gaussian mixture model of the input variable joint probability density function shown in formula (2) corresponds to a linearized power flow model, i.e. 1 piecewise linear power flow model comprising K linear segments is established.
Then, on the basis of the formula (2) and the formula (4), a joint probability density function of the power flow input variable and the power flow output variable of the power distribution network is established.
The kth Gaussian component N (x|mu) of the Gaussian mixture model for the input variable of the given formula (2) x,k ,Σ x,k ) The power flow input variable vector x of the power distribution network is expressed as:
x＝L k u+μ x,k (5)
wherein: l (L) k Is sigma-delta x,k And (3) a lower triangular matrix obtained by the Georgi decomposition, wherein u is a variable vector meeting standard normal distribution.
According to the formulas (4) and (5), for the kth gaussian component N (x|μ) of the gaussian mixture model given the input variable represented by the formula (2) x,k ,Σ x,k ) The combined variable vector of the power flow input variable and the power flow output variable of the power distribution network is as follows:
according to the formula (6) and the principle of the Georgi decomposition, the kth Gaussian component N (x|mu) of the Gaussian mixture model for the input variable represented by the given formula (2) x,k ,Σ x,k ) The joint probability density function of the power flow input variable and the power flow output variable of the power distribution network is as follows:
wherein: theta (theta) k The kth Gaussian component N (x|mu) of the Gaussian mixture model representing the input variable of the given formula (2) x,k ,Σ x,k ) With a probability P (Θ) k )＝ω k 。
According to the formula (7) and the full probability formula, the final obtaining of the joint probability density function of the power flow input variable and the power flow output variable of the power distribution network is as follows:
step 2: and obtaining an analytic expression of a probability density function of the power flow output variable of the power distribution network, and calculating the variance of the analytic expression.
The detailed process is as follows:
according to the analysis of the joint probability density function of the input variable and the output variable of the power flow of the power distribution network shown in the formula (8), the first output variable y of the power flow of the power distribution network l The probability distribution of (2) is the edge probability distribution of equation (8), and y is obtained l The probability density function of (2) is as follows:
wherein: mu (mu) l,k ＝μ y,k (l)、Σ ll,k ＝Σ y,k (l, l), (l, l) represent the indices of the vector and matrix elements, respectively.
According to formula (9), y l Is expressed as:
wherein: e [ N (y) l |μ l,k ,Σ ll,k ) M ]Shows a Gaussian distribution N (y) l |μ l,k ,Σ ll,k ) Is the M-order origin moment of (c).
According to formula (10), y l The variance of (c) is expressed as:
and calculating according to the formula (11) to obtain the variance of the power flow output variable of the power distribution network.
Step 3: and obtaining an analytical formula of the conditional probability density function and the conditional variance of the power flow output variable of the power distribution network with respect to the input variable, and calculating an expected value of the conditional variance.
The detailed process is as follows:
first, the power distribution network power flow input variable vector x is divided into complementary subsets x c And x d I.e. x= { x c ,x d }。
Next, the output variable y is calculated l With respect to input variable x c Is a desired value of conditional variance of (a);
will x c As input variable to be analyzed, x c And the first output variable y l The combined variable vector is represented as w, namely:
the probability distribution of w is the edge distribution of the formula (8), and the analytical formula of the joint probability density function of w is:
wherein: mu (mu) c,k ＝μ x,k (c)、Σ cc,k ＝Σ x,k (c,c)、Σ cl,k ＝Σ xy,k (c,l)、Σ lc,k ＝Σ yx,k (l, c), c represents x c Index in x, l denotes y l Index in y.
Obtaining y according to formula (13) and Bayes theorem l Concerning x c The analytical formula of the conditional probability density function is:
wherein:
according to formulas (14) - (17), y is obtained l Concerning x c The conditional variance analysis formula of (2) is:
calculating y according to formula (18) l Concerning x c The expected value E [ Var (y) l |x c )]。
E[Var(y l |x c )]The calculation formula of (2) is as follows:
E[Var(y l |x c )]＝∫Var(y l |x c )f(x c )dx c (19)
wherein:
the numerical integration method is adopted to calculate the formula (19), and the method is specifically as follows:
wherein: x is X c,1 、X c,2 、…、X c,N For x generated according to formula (20) c Is a sample of the N samples.
Then, calculate the output variable y l With respect to input variable x d Is a desired value of conditional variance of (a);
will x d As input variable to be analyzed, x d And the first output variable y l The composed joint variable vector is denoted v, namely:
the probability distribution of v is the edge distribution of the formula (8), and the analytical formula of the joint probability density function for obtaining v is as follows:
wherein: mu (mu) d,k ＝μ x,k (d)、Σ dd,k ＝Σ x,k (d,d)、Σ dl,k ＝Σ xy,k (d,l)、Σ ld,k ＝Σ yx,k (l, d), d represents x d Index in x, l denotes y l Index in y.
Obtaining y according to formula (23) and Bayes theorem l With respect tox d The analytical formula of the conditional probability density function is:
wherein:
according to formulae (24) - (27), y is obtained l Concerning x d The conditional variance analysis formula of (2) is:
calculating y according to formula (28) l Concerning x d The expected value E [ Var (y) l |x d )]。
E[Var(y l |x d )]The calculation formula of (2) is as follows:
E[Var(y l |x d )]＝∫Var(y l |x d )f(x d )dx d (29)
wherein:
the numerical integration method is adopted to calculate the formula (29), specifically:
wherein: x is X d,1 、X d,2 、…、X d,N For x generated according to formula (30) d Is a sample of the N samples.
Step 4: and (3) calculating a power flow global sensitivity index of the power distribution network according to the results obtained in the step (2) and the step (3), and completing power flow global sensitivity analysis of the power distribution network.
The detailed process is as follows:
var (y) obtained according to step 2 l ) And E [ Var (y) obtained in step 3 l |x c )]Calculating the output variable y l With respect to input variable x c Main effect global sensitivity index of (2):
var (y) obtained according to step 2 l ) And E [ Var (y) obtained in step 3 l |x d )]Calculating the output variable y l With respect to input variable x c Global sensitivity index of the total effect of (a):
and comprehensively, completing the overall sensitivity analysis of the power flow of the power distribution network.
As a preferred embodiment of the present invention, the following is specific:
simulation case analysis
The scheme of the invention is verified by adopting an IEEE 33 node system containing 4 distributed photovoltaics, 1 distributed wind power and 1 electric vehicle charging station, the result obtained by the traditional Monte Carlo simulation method is taken as a benchmark, the scheme of the invention is compared with a Monte Carlo simulation method based on a Polynomial Chaotic Expansion (PCE) and Gaussian random process regression (GPR) proxy model, the Monte Carlo simulation method based on the proxy model is the closest prior art scheme to the scheme of the invention, and the PCE and the GPR are 2 commonly used proxy models. Each distributed photovoltaic has a rated active power of 0.8MW, distributed wind power has a rated active power of 1.2MW, and 20 charging piles (42 kW) are installed on the electric vehicle charging station. Wind, light distribution typeThe output data of the power supply are respectively obtained through conversion of measured wind speed and irradiation intensity, the load data of the electric vehicle charging station are obtained through collection of the actual electric charging station, the IEEE 33 node original load data are used as annual average values, the normalized load curve is obtained through a certain actual system, the conventional load in each hour window meets normal distribution, and the standard deviation is 10% of the mean value. The Gaussian component number of the scheme is 100, the number of samples of numerical integration is 1000, the number of training samples of proxy model modeling is 100, and the number of samples of Monte Carlo simulation is 10 6 . The simulation program was compiled and run on a computer configured as Intel i7-10510U 1.8GHz CPU and 16GB RAM.
1. Comparison of precision
Table 1 shows a partial global sensitivity index of the IEEE 33 node obtained by the scheme of the invention, and 4 representative output variables are selected: node 33 voltage amplitude V 33 Current amplitude I of branches 31-32 31-32 Active power P of branch 31-32 31-32 Total active loss P of network L The input variable examined is the active power P of the 4 th photovoltaic power generation PV4 。S KM (V 33 ,P PV4 )、S KM (I 31-32 ,P PV4 )、S KM (P 31-32 ,P PV4 )、S KM (P L ,P PV4 ) Respectively represent the 4 output variables V 33 、I 31-32 、P 31-32 、P L Regarding P PV4 Is a main effect global sensitivity index. The result shows that the scheme of the invention can obtain an accurate power flow global sensitivity index of the power distribution network, and the precision is obviously superior to that of the prior art scheme.
TABLE 1 global sensitivity index versus error contrast
2. Comparison of efficiency
The simulation case comprises 98 output variables (including node voltage, branch current, branch active power, total active power loss and total voltage deviation) and 38 input variables (32 conventional loads, 4 photovoltaics, 1 wind power and 1 electric vehicle charging station), the total sensitivity analysis needs to calculate 2×38×98 total sensitivity indexes, and table 2 shows comparison of calculation time consumption of different methods. The Monte Carlo simulation method based on the proxy model needs to consume a long time in the construction of the proxy model and the Monte Carlo simulation process, and although the proxy model method is remarkably lower than the traditional Monte Carlo simulation method in the calculation time consumption of the Monte Carlo simulation, the total time consumption is still long due to the large number of indexes to be calculated. Compared with the prior art, the method and the device have the advantages that the time-consuming agent model construction and Monte Carlo simulation process are avoided, and the method and the device belong to an analysis method, so that the calculation time is obviously lower than that of the prior art, and the calculation efficiency is improved by orders of magnitude.
TABLE 2 Global sensitivity index calculation time-consuming comparison
In summary, the scheme of the invention has the following advantages.
(1) The modeling process of the Gaussian mixture model of the joint probability density function of the power flow input variable and the power flow output variable of the power distribution network only needs less times of power flow calculation, the time consumption is short, and the time-consuming agent model construction process is avoided.
(2) The analysis formula of the power flow output variable variance of the power distribution network and the analysis formula of the condition variance of the power flow output variable of the power distribution network relative to the input variable are obtained, so that the calculation time of the global sensitivity index is short, and the time-consuming Monte Carlo simulation calculation process of the power flow of the power distribution network is avoided.
In conclusion, the scheme of the invention remarkably improves the calculation efficiency of the overall sensitivity analysis of the power flow of the power distribution network.
It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.
Furthermore, it should be understood that although the present disclosure describes embodiments, not every embodiment is provided with a separate embodiment, and that this description is provided for clarity only, and that the disclosure is not limited to the embodiments described in detail below, and that the embodiments described in the examples may be combined as appropriate to form other embodiments that will be apparent to those skilled in the art.

Claims (2)

1. The Gaussian mixture model method for analyzing the overall sensitivity of the power flow of the power distribution network is characterized by comprising the following steps of:
step 1: establishing a Gaussian mixture model of a joint probability density function of a power flow input variable and an output variable of the power distribution network;
step 2: obtaining an analytic type of probability density function of a power flow output variable of the power distribution network, and calculating variance of the analytic type;
step 3: obtaining an analytic expression of a conditional probability density function and a conditional variance of a power flow output variable of the power distribution network relative to an input variable, and calculating an expected value of the conditional variance;
step 4: calculating a power flow global sensitivity index of the power distribution network according to the results obtained in the step 2 and the step 3, and completing power flow global sensitivity analysis of the power distribution network;
the step 1 specifically comprises the following steps:
firstly, establishing a Gaussian mixture model of a power flow input variable joint probability density function of a power distribution network;
the power flow input variable vector of the distribution network is X, and n historical samples of the power flow input variable vector are X 1 、X 2 、…、X n Establishing a joint probability density function of x by adopting a non-parameter kernel density estimation method:
wherein: n (·) represents a Gaussian kernel function, H k Represents the kth sample X k A corresponding bandwidth matrix, the probability density function based on non-parametric kernel density estimation being a basic Gaussian mixture model with Gaussian component number K b Weight coefficient equal to n, kth gaussian componentEqual to 1/n, mean vector->Equal to X k Covariance matrix->Equal to H k ；
And (3) adopting a layering expectation maximization algorithm of density retention to perform Gaussian component number reduction on the basic Gaussian mixture model shown in the formula (1), and obtaining a simplified Gaussian mixture model as follows:
wherein: omega k 、μ x,k 、Σ x,k Respectively representing a weight coefficient, a mean vector and a covariance matrix of the kth Gaussian component, wherein K represents the reduced Gaussian component number;
secondly, on the basis of the formula (2), establishing a piecewise linearization power flow model of the power distribution network;
the compact form of the nonlinear power distribution network power flow model is:
y＝Γ(x) (3)
wherein: Γ (·) represents a nonlinear power flow equation, and y represents an output variable vector of power flow of the power distribution network;
the kth Gaussian component N (x|mu) of the Gaussian mixture model given the input variable of formula (2) x,k ,Σ x,k ) For the nonlinear tide model shown in (3), the nonlinear tide model is shown in mu x,k Linearization is performed, and a kth linearization model is obtained:
y＝T k x+pΓ(μ x,k )-T k μ x,k ] (4)
wherein: t (T) k Mu for nonlinear tide equation x,k An inverse jacobian matrix at; each Gaussian component of the Gaussian mixture model of the input variable joint probability density function shown in the formula (2) corresponds to a linearized power flow model, namely, 1 piecewise linear power flow model comprising K linear segments is established;
then, on the basis of the formula (2) and the formula (4), establishing a joint probability density function of the power flow input variable and the power flow output variable of the power distribution network;
the kth Gaussian component N (x|mu) of the Gaussian mixture model for the input variable of the given formula (2) x,k ,Σ x,k ) The power flow input variable vector x of the power distribution network is expressed as:
x＝L k u+μ x,k (5)
wherein: l (L) k Is sigma-delta x,k The lower triangular matrix obtained by the George decomposition is a variable vector meeting standard normal distribution;
according to the formulas (4) and (5), for the kth gaussian component N (x|μ) of the gaussian mixture model given the input variable represented by the formula (2) x,k ,Σ x,k ) The combined variable vector of the power flow input variable and the power flow output variable of the power distribution network is as follows:
according to the formula (6) and the principle of the Georgi decomposition, the kth Gaussian component N (x|mu) of the Gaussian mixture model for the input variable represented by the given formula (2) x,k ,Σ x,k ) The joint probability density function of the power flow input variable and the power flow output variable of the power distribution network is as follows:
wherein: theta (theta) k The kth Gaussian component N (x|mu) of the Gaussian mixture model representing the input variable of the given formula (2) x,k ,Σ x,k ) With a probability P (Θ) k )＝ω k ；
According to the formula (7) and the full probability formula, the final obtaining of the joint probability density function of the power flow input variable and the power flow output variable of the power distribution network is as follows:
the step 2 specifically comprises the following steps:
according to the analysis of the joint probability density function of the input variable and the output variable of the power flow of the power distribution network shown in the formula (8), the first output variable y of the power flow of the power distribution network l The probability distribution of (2) is the edge probability distribution of equation (8), and y is obtained l The probability density function of (2) is as follows:
wherein: mu (mu) l,k ＝μ y,k (l)、Σ ll,k ＝Σ y,k (l, l), (l, l) represent the indices of the vector and matrix elements, respectively;
according to formula (9), y l Is expressed as:
wherein: e [ N (y) l |μ l,k ,Σ ll,k ) M ]Shows a Gaussian distribution N (y) l |μ l,k ,Σ ll,k ) An M-order origin moment of (a);
according to formula (10), y l The variance of (c) is expressed as:
calculating to obtain the variance of the power flow output variable of the power distribution network according to the formula (11);
the step 3 specifically comprises the following steps:
first, the power distribution network power flow input variable vector x is divided into complementary subsets x c And x d I.e. x= { x c ,x d }；
Next, the output variable y is calculated l With respect to input variable x c Is a desired value of conditional variance of (a);
will x c As input variable to be analyzed, x c And the first output variable y l The combined variable vector is represented as w, namely:
the probability distribution of w is the edge distribution of the formula (8), and the analytical formula of the joint probability density function of w is:
wherein: mu (mu) c,k ＝μ x,k (c)、Σ cc,k ＝Σ x,k (c,c)、Σ cl,k ＝Σ xy,k (c,l)、Σ lc,k ＝Σ yx,k (l, c), c represents x c Index in x, l denotes y l An index in y;
obtaining y according to formula (13) and Bayes theorem l In relation to x c The analytical formula of the conditional probability density function is:
wherein:
according to formulas (14) - (17), y is obtained l Concerning x c The conditional variance analysis formula of (2) is:
calculating y according to formula (18) l Concerning x c The expected value E [ Var (y) l |x c )]；
E[Var(y l |x c )]The calculation formula of (2) is as follows:
E[Var(y l |x c )]＝∫Var(y l |x c )f(x c )dx c (19)
wherein:
the numerical integration method is adopted to calculate the formula (19), and the method is specifically as follows:
wherein: x is X c,1 、X c,2 、…、X c,N For x generated according to formula (20) c Is a sample of N samples;
then, calculate the output variable y l With respect to input variable x d Is a desired value of conditional variance of (a);
will x d As input variable to be analyzed, x d And the first output variable y l The composed joint variable vector is denoted v, namely:
the probability distribution of v is the edge distribution of the formula (8), and the analytical formula of the joint probability density function for obtaining v is as follows:
wherein: mu (mu) d,k ＝μ x,k (d)、Σ dd,k ＝Σ x,k (d,d)、Σ dl,k ＝Σ xy,k (d,l)、Σ ld,k ＝Σ yx,k (l, d), d represents x d Index in x, l denotes y l An index in y;
obtaining y according to formula (23) and Bayes theorem l Concerning x d The analytical formula of the conditional probability density function is:
wherein:
according to formulae (24) - (27), y is obtained l Concerning x d The conditional variance analysis formula of (2) is:
calculating y according to formula (28) l Concerning x d The expected value E [ Var (y) l |x d )]；
E[Var(y l |x d )]The calculation formula of (2) is as follows:
E[Var(y l |x d )]＝∫Var(y l |x d )f(x d )dx d (29)
wherein:
the numerical integration method is adopted to calculate the formula (29), specifically:
wherein: x is X d,1 、X d,2 、…、X d,N For x generated according to formula (30) d Is a sample of the N samples.
2. The gaussian mixture model method for power distribution network power flow global sensitivity analysis according to claim 1, wherein said step 4 is specifically:
var (y) obtained according to step 2 l ) And E [ Var (y) obtained in step 3 l |x c )]Calculating the output variable y l With respect to input variable x c Main effect global sensitivity index of (2):
var (y) obtained according to step 2 l ) And E [ Var (y) obtained in step 3 l |x d )]Calculating the output variable y l With respect to input variable x c Global sensitivity index of the total effect of (a):
and comprehensively, completing the overall sensitivity analysis of the power flow of the power distribution network.
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