CN115017735A - Multi-agent probability voltage stability calculation method for high-dimensional system - Google Patents
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Abstract
The invention discloses a multi-agent probability voltage stability calculation method for a high-dimensional system, which comprises the following steps of S1: collecting historical data of random variables in the power system, and estimating probability density functions of the random variables; s2: determining an operation scene of the power system, and constructing a plurality of different probability voltage stability evaluation models based on different operation scenes of the power system in an operation period; s3: establishing an agent model of a deterministic voltage stability assessment model corresponding to an operation scene based on regularization low-rank approximation; s4: sampling and inputting an agent model on a random variable probability density function based on a Monte Carlo simulation method to perform probability voltage stability evaluation; s5: and defining a risk calculation formula of the probability voltage stability evaluation index, and evaluating the voltage stability risk in different scenes of the power system. The invention can give consideration to the contradiction between the calculation speed and the calculation precision in the power system probability voltage stability analysis.
Description
Technical Field
The invention belongs to the technical field of power systems, and particularly relates to a multi-agent probability voltage stability calculation method for a high-dimensional system.
Background
In recent years, with the continuous expansion of the scale of a power grid and the large-scale access of renewable energy sources, the number of uncertain sources (such as wind power plants and photovoltaic power plants) in a power system rapidly rises. Traditional power systems are evolving into new energy power systems containing high dimensional sources of uncertainty. Under the background of high-dimensional uncertain source access, the potential hidden danger that the power system is difficult to predict due to increased running risk and complicated running conditions of a power grid needs to be concerned urgently while the running economy and flexibility of the power system are concerned. Therefore, studying the stability of the power system in this context helps to reveal grid risks.
Static voltage stabilization is one of the key contents of power system stability, and the voltage stabilization level of the current operation state is drawn at the moment by continuously increasing the system load to test the distance between the current operation state and the voltage stabilization breakdown state. Meanwhile, renewable energy sources represented by wind power and photovoltaic have volatility and uncertainty, and the large-scale access of the renewable energy sources causes random change of the operation mode of a power system, so that the uncertainty of the static voltage stability of each node is aggravated. Conventional static voltage stability research is based on a determination model, randomness of wind power and uncertainty of load are ignored, the operation situation of a power system cannot be comprehensively and accurately evaluated at present when renewable energy sources are increased day by day and novel load is increased continuously, and a probabilistic static voltage stability evaluation algorithm containing a high-dimensional uncertainty source needs to be researched urgently. Meanwhile, in the actual operation of the power grid, factors such as equipment maintenance and element faults cause frequent changes of the power grid topology, and when the probability voltage stability evaluation is performed on the power grid, the changes of the power system topology need to be considered, so that the research of a high-dimensional probability voltage stability evaluation algorithm considering multiple operation scenes has practical significance.
Disclosure of Invention
In view of the problems in the prior art, the present invention aims to provide a multi-agent probability voltage stability calculation method for a high-dimensional system.
In order to achieve the purpose, the invention provides the following technical scheme: a multi-agent probability voltage stability calculation method for a high-dimensional system comprises the following steps:
step S1: collecting historical data of random variables in the power system, and estimating probability density functions of the random variables;
step S2: determining an electric power system operation scene needing to check voltage stability in a specific operation period according to actual electric power system operation experience, and constructing a plurality of different probability voltage stability evaluation models based on different operation scenes of the electric power system in the operation period;
step S3: according to the operation scene of the power system, establishing an agent model of a deterministic voltage stability evaluation model corresponding to the operation scene on the basis of regularization low-rank approximation;
step S4: sampling and inputting an agent model on a random variable probability density function based on a Monte Carlo simulation method to perform probability voltage stability evaluation;
step S5: calculating probability density function of probability voltage stability index by combining probability voltage stability calculation result based on agent model
And defining a risk calculation formula of the probability voltage stability evaluation index, and evaluating the voltage stability risk in different scenes of the power system.
Further, the random variables in the power system of step S1 include wind speed, light intensity and load, which may be represented as X ═ X 1 ,x 2 ,x 3 ,…,x n_random ]Wherein n _ random represents the dimension of the random variable and the probability density function of the ith random variable is represented as f i (x i )(i＝1,2,3,…,n_random)。
Further, step S2 specifically includes the following steps:
s21: voltage stability conditions of m operation scenes with voltage stability need to be checked in an operation period, and the operation scene of the nth _ m power system in the m operation scenes is assumed to be L n_m Expressed as:
wherein
The line state which represents the n _ m power system operation scene number is 1, the line fault value is 0, the line normal value is 1,
representing the state of a conventional generator with the n _ m-th power system operation scene number of 1, wherein the fault value of the generator is 0, and the normal operation value of the generator is 1;
n line representing the number of lines in the power system, n Gen Representing the number of conventional generators in the power system. Note that the state of the lines and conventional generators in the operating scenario may be determined empirically by grid operations and planners.
S22: constructing m voltage stability evaluation models based on m operation scene lines and the state of a conventional generator, wherein the nth _ m voltage stability evaluation models can be expressed as:
wherein X sample Set of samples, ε, representing random variables in an electric power system n_m Represents the nth _ m power system voltage stability index set,
representing the (n _ m) th deterministic voltage stability evaluation model; if the uncertainty of wind speed, illumination intensity and load in the power system is considered, random variable samples are input into a deterministic voltage stability evaluation model group by group, and the model is evolved into a probabilistic voltage stability analysis model.
Further, step S3 specifically includes the following steps:
s31: transforming random variable in electric power system to uniform distribution space U ═ U 1 ,u 2 ,…,u n _ random ]The transformation formula of the ith random variable is as follows:
u i ＝∫f i (x i )dx i (2)
s32: constructing a proxy model of m deterministic voltage stabilization models by using low-rank approximation; taking the proxy model solution of the deterministic voltage stabilization model of the nth _ m scene as an example, the steps and the idea of solving the proxy model are described, and the proxy model solution methods of the deterministic voltage stabilization models of other scenes are similar.
S321: the voltage stability indicator for the nth _ m power system operating scenario may be represented by the agent:
wherein, b l Is a normalized weight coefficient; omega l As a function of the rank of U, it can be expressed as:
in the formula (I), the compound is shown in the specification,
representing a univariate function in the ith dimension of the ith rank-one function, equations (3), (4) represent a regular low rank approximation,
is approximately:
wherein, R represents the number of rank one functions;
a univariate polynomial base of the kth order for the ith input variable; p is a radical of i Is composed of
The highest order of (a);
as a function of the l-th rank
Rank coefficient of (d);
s322: determination of proxy model parameters:
(1) determining univariate orthogonal polynomial basis
Because the input random variable U ═ U 1 ,u 2 ,…,u n_random ]Following a uniform distribution, univariate orthogonal polynomial bases can be determined
Hilbert basis for Legendre polynomials
k is the order;
(2) obtaining a set of test sample points
Uniformly distributed sample point set U is selected on uniformly distributed U based on Latin hypercube sampling algorithm C Set the sample points U C Sequentially inputting the inverse function of the formula (2) to obtain a sample point X on the original distribution C (ii) a Sample point X on the original distribution C Inputting the (n _ m) th electric power system operation scene deterministic voltage stability analysis model to obtain a voltage stability analysis index
Forming an n _ m power system operation scene test sample point set
(3) Selection of polynomial order p
The implementation of the regularized low rank approximation takes into account a common polynomial order in all dimensions, i.e., p in equation (5) 1 ＝…＝p n_random P; according to a large amount of analog calculation and power system analysis experience, the order p is 5, so that p 1 ＝…＝p n_random ＝p＝5；
(4) L1 norm regularization least square algorithm based on sparsity induction for calculating rank coefficient z and weight coefficient b
Introducing a regularization method, changing the solving parameters into a least square problem with sparsity induced L1 regularization, and meanwhile, sparsity low-rank function of a matrix, wherein the method comprises the following steps:
the overall idea of solving rank coefficient z and weight coefficient b based on L1 norm "correction-update" is as follows:
1) initialization:
let r be 1 and r be equal to 1,
2) correction step based on L1 norm regularization:
the r-th step is to find a new rank-tensor ω r This can be obtained by solving the following minimization problem:
where W represents the space of a rank one tensor, subscript
Represents the minimization of the experimental design; equation (6) is solved by an alternating least squares approach involving minimization of the sequence along each dimension i-1, …, n random, while "freezing" the coefficients in all remaining dimensions;
sequentially calculating the optimization problem for each j dimension to obtain corresponding rank coefficient
Meanwhile, a regularization method is introduced, a penalty term is added in an original objective function, and a model with high complexity is punished:
optimization equation (7) can be solved based on a minimum angle regression method;
3) updating step based on L1 norm regularization:
after the correction of the step r is completed, the algorithm enters the step r for updating, and a newly solved first-order function omega is determined r Weight factor b of (X) r At the same time, the existing weight coefficient (b) is updated 1 ,b 2 ,···,b r-1 ) (ii) a Based on the regularization approach, the update step can be implemented by solving the following minimization problem:
4) convergence criterion:
is realized by a series of steps of 'correcting and updating' based on L1 norm, and in the step of r correction, the method establishesNew rank-one function omega r And in the update of the r step, a set of weighting factors (b) is determined 1 ,b 2 ,···,b r )；
Convergence criterion is the number of iterations I r And decrease of error measurement
Combined in two successive iterations; the error measure used is a relative empirical error, as follows:
in the above-mentioned formula, the compound of formula,
empirical variance of a set consisting of model responses when designing an experiment; therefore, if I r To a maximum allowable value I max Or is or
Less than a prescribed threshold value
The algorithm converges and the procedure terminates; otherwise, r ═ r +1, and return to step 2); UQLAB uses this convergence criterion and defaults to I max 100 and
(5) selection of optimal rank R
1) Initialization:
let r be 1;
3) and (3) error calculation:
to determine the optimal rank R e { R ═ 1, …, R max The relative generalization error can be measured by adopting the size of the relative generalization error, and the relative generalization error is calculated by adopting a 3-time cross validation method based on an experimental design sample; the process first requires partitioning the experimental design into 3 approximately equal sized subsets, with increasing rankThe low rank of (2) approximately constitutes a training set TR in 2 subsets; while the remaining subset is used as the test set TS to evaluate the error of the low rank approximation constructed with the training set:
wherein
An LRA meta-model built for the training set,
evaluating the empirical variance of the formed set for the model on the test set;
3) convergence criterion:
with alternating 3 sets, 3 metamodels are obtained in this way; their average error provides an estimate of the meta-model generalization error for controlling the early-stop option and selecting the optimal rank R; when r reaches the maximum allowable value r stop Or is or
Less than a prescribed threshold value
The algorithm converges and the procedure terminates; otherwise, r ═ r +1, and return to step 2);
after determining the surrogate model parameters of the n _ m power system operation scenes, sequentially obtaining the surrogate model parameters in the m operation scenes;
s33: and inputting the obtained proxy model parameters in the m operation scenes into a formula (5) to form m proxy models.
Further, step S4 specifically includes the following steps:
s41: large-scale sampling is carried out on a random variable probability density function based on a Monte Carlo simulation method to obtain a sample point set X S ；
S42: by the formula (2) to convert X S Equal probability changeIs changed into U S (ii) a Will U S Inputting the (n _ m) th power system operation scene into the proxy model shown in the formula (3), and obtaining the voltage stability analysis index based on the proxy model instead of the original model
According to the method, sample sets of output responses in m operation scenes can be obtained in sequence.
Further, step S5 defines a risk calculation formula of the probabilistic voltage stability assessment indicator as follows:
Compared with the prior art, the invention has the beneficial effects that:
1. and introducing a regularization method to solve the model parameters in the parameter solving process of the proxy model. Specifically, a penalty term is added in an original objective function, a model with high complexity is punished, and a solving parameter is changed into a least square problem with sparsity-induced L1 norm regularization. The advantages of the proposed solution are: 1) the proxy model can be prevented from being over-fitted, the fitting precision of the proxy model is improved, and the solving precision of the probability voltage stability is further improved; 2) based on the L1 norm regularization technology, the sparsity of the matrix can be utilized, the complexity of the model is greatly reduced, the parameter solving efficiency is improved, and the probability analysis efficiency is further improved.
2. A proxy model order empirical calculation method suitable for voltage stability evaluation of a power system is provided. According to a large amount of simulation calculation and power system analysis experience, the order of the proxy model is determined, and the calculation precision of the high-dimensional probability voltage stability evaluation problem can be greatly improved.
3. An optimal rank determination method of an adaptive agent model is provided. The method can adaptively determine the optimal rank of the agent model according to different power system operation scenes, further determine the most matched agent model, and consider the contradiction between the calculation speed and the calculation precision in the power system probability voltage stability analysis.
Drawings
FIG. 1 is a flow chart of a multi-agent probabilistic voltage stabilization calculation method for a high dimensional system of the present invention;
FIG. 2 is a probability distribution diagram of a system maximum load margin for a normal operating scenario;
FIG. 3 is a probability distribution diagram of a line 4-11 disconnection scenario system maximum load margin;
FIG. 4 is a probability distribution plot of line 12-14 disconnection scenario system maximum load margin;
FIG. 5 is a probability distribution diagram of a line 20-21 disconnection scenario system maximum load margin;
FIG. 6 is a probability distribution diagram of line 60-61 disconnection scenario system maximum load margin.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention. It is to be understood that the described embodiments are merely a few embodiments of the invention, and not all embodiments. All other embodiments, which can be obtained by a person skilled in the art without any inventive step based on the embodiments of the present invention, belong to the scope of the present invention.
A multi-agent probability voltage stability calculation method for a high dimensional system, as shown in fig. 1, includes the following steps:
step S1: collecting historical data of random variables (mainly comprising wind speed, illumination intensity and load data) in the power system, and estimating a probability density function of the random variables;
the random variables in the power system are typically wind speed, illumination intensity and load, which can be expressed as X ═ X 1 ,x 2 ,x 3 ,…,x n_random ](where n _ random represents the dimension of the random variable). Wherein the probability density function of the ith random variable is represented as f i (x i )(i＝1,2,3,…,n_random)。
Step S2: determining an electric power system operation scene needing to be checked for voltage stability in a specific operation period (the operation period can be one day, one week, one month and the like, and is determined according to the needs of power grid operators) according to the actual electric power system operation experience, and constructing a plurality of different probability voltage stability evaluation models based on different operation scenes of the electric power system in the operation period;
s21: and in the operation period, the voltage stability conditions of m operation scenes with stable voltage need to be checked. Assuming that the operation scene of the n _ m power system in the m operation scenes is L n_m Expressed as:
wherein
The line state (the line fault numerical value is 0, the line normal numerical value is 1) of the n _ m power system operation scene number is represented,
the normal generator state (the generator fault value is 0, and the generator normal operation value is 1) with the number of the n _ m power system operation scene being 1 is represented;
n line representing the number of lines in the power system, n Gen Representing the number of conventional generators in the power system. Note that the state of the lines and conventional generators in the operating scenario can be determined empirically by the grid operating and planning personnel.
S22: constructing m voltage stability evaluation models based on m operation scene lines and the state of a conventional generator, wherein the nth _ m voltage stability evaluation models can be expressed as:
wherein X sample Set of samples, ε, representing random variables in an electric power system n_m Represents the nth _ m power system voltage stability index set,
represents the nth _ m deterministic voltage stability evaluation model. If the uncertainty of wind speed, illumination intensity and load in the power system is considered, random variable samples are input into a deterministic voltage stability evaluation model group by group, and the model is evolved into a probabilistic voltage stability analysis model.
Step S3: establishing an agent model of a deterministic voltage stability assessment model corresponding to an operation scene on the basis of regularization low-rank approximation according to the operation scene of the power system, wherein the number of the agent models is the same as that of the operation scene and the agent models are in one-to-one correspondence;
s31: transforming random variable in electric power system to uniform distribution space U ═ U 1 ,u 2 ,…,u n _ random ]The transformation formula of the ith random variable is as follows:
u i ＝∫f i (x i )dx i (2)
s32: and constructing the proxy models of the m deterministic voltage stabilization models by using low-rank approximation. In this embodiment, the step and the idea of solving the proxy model of the deterministic voltage stabilization model of the nth _ m scenario are described as an example, and the proxy model solving methods of the deterministic voltage stabilization models of other scenarios are similar.
S321: the voltage stability indicator for the nth _ m power system operating scenario may be represented by the agent:
wherein, b l Is a normalized weight coefficient; omega l As a function of the rank of U, it can be expressed as:
in the formula (I), the compound is shown in the specification,
representing a univariate function in the ith dimension of the ith rank-one function. Usually the value of R is small, so equations (3), (4) represent a regular low rank approximation. Thus, it is possible to provide
Is approximately:
wherein, R represents the number of rank one functions;
a univariate polynomial base of the kth order for the ith input variable; p is a radical of i Is composed of
The highest order of (a);
as a function of the l-th rank
Is calculated.
S322: determination of proxy model parameters:
(1) determining univariate orthogonal polynomial basis
Because the input random variable U is [ U ═ U 1 ,u 2 ,…,u n_random ]Following a uniform distribution, univariate orthogonal polynomial bases can be determined
Hilbert basis for Legendre polynomials
(k is the order).
(2) Obtaining a set of test sample points
Uniformly distributed sample point set U is selected on uniformly distributed U based on Latin hypercube sampling algorithm C Set the sample points U C Sequentially inputting the inverse function of the formula (2) to obtain a sample point X on the original distribution C . Sample point X on the original distribution C Inputting the (n _ m) th electric power system operation scene deterministic voltage stability analysis model to obtain a voltage stability analysis index
Forming an n _ m power system operation scene test sample point set
(3) Selection of polynomial order p
The implementation of the regularized low rank approximation takes into account a common polynomial order in all dimensions, i.e., p in equation (5) 1 ＝…＝p n_random P. According to a large amount of analog calculation and power system analysis experience, 5-order fitting effect selected from order p is good, so that p in the text is good 1 ＝…＝p n_random ＝p＝5。
(4) L1 norm regularization least square algorithm based on sparsity induction for calculating rank coefficient z and weight coefficient b
A regularization method is introduced, and solution parameters are changed into a least square problem with sparsity-induced L1 regularization. The overfitting of the proxy model can be effectively prevented, and the model construction precision is improved. Meanwhile, the sparsity of the matrix is utilized to sparsify the low-rank function, so that the complexity of the model is reduced, and the parameter solving efficiency is improved. The detailed steps are as follows:
the overall idea of solving rank coefficient z and weight coefficient b based on L1 norm 'correction-update' is as follows:
1) initialization:
let r be 1 and r be equal to 1,
2) correction step based on L1 norm regularization:
the r-th step is to find a new rank-tensor ω r This can be obtained by solving the following minimization problem:
where W represents the space of rank one tensor and the subscript χ represents the minimization of the design of the experiment. Equation (6) is solved by an alternating least squares approach that involves minimizing the sequence along each dimension i-1, …, n random, while "freezing" the coefficients in all remaining dimensions.
The optimization problem is respectively calculated for each j dimension in turn, and then the corresponding rank coefficient can be obtained
Meanwhile, a regularization method is introduced, a penalty term is added in an original objective function, and a model with high complexity is punished:
the optimization equation (7) can be solved based on a minimum angle regression method.
3) Updating step based on L1 norm regularization:
after the correction of the step r is completed, the algorithm enters the step r for updating, and a newly solved first-order function omega is determined r Weight factor b of (X) r At the same time, the existing weight coefficient (b) is updated 1 ,b 2 ,···,b r-1 ). Based on the regularization approach, the update step can be implemented by solving the following minimization problem:
4) convergence criterion:
the method is realized by a series of steps of 'correction-update' based on L1 norm, and in the step of r correction, a new rank-one function omega is established r And in the update of the r step, a set of weighting factors (b) is determined 1 ,b 2 ,···,b r )。
The convergence criterion of the invention is the number of iterations I r And decrease of error measurement
Combined in two successive iterations. The error measure used is a relative empirical error, as follows:
in the above-mentioned formula, the compound of formula,
empirical variance of the set consisting of model responses when designed for experiments. Therefore, if I r To a maximum allowable value I max Or is or
Less than a prescribed threshold value
The algorithm converges and the procedure terminates; otherwise, r ═ r +1, and return to step 2). UQLAB uses this convergence criterion and defaults to I max 100 and
(5) selection of optimal rank R
1) Initialization:
let r equal to 1;
4) and (3) error calculation:
to determine the optimal rank R e { R ═ 1, …, R max Relative generalization error can be adoptedAnd measuring the difference, and calculating the relative generalization error by adopting a 3-time cross validation method based on the experimental design sample. The process first requires partitioning the experimental design into 3 approximately equal sized subsets, with the low rank of increasing rank forming the training set TR in approximately 2 subsets. While the remaining subset is used as the test set TS to evaluate the error of the low rank approximation constructed with the training set:
wherein
An LRA meta-model built for the training set,
the empirical variance of the composed set is evaluated for the model on the test set.
3) Convergence criterion:
with alternating 3 sets, 3 metamodels are obtained in this way; their average error provides an estimate of the meta-model generalization error for controlling the early stop option and selecting the optimal rank R. When r reaches the maximum allowable value r stop Or is or
Less than a prescribed threshold value
The algorithm converges and the procedure terminates; otherwise, r ═ r +1, and return to step 2).
After the surrogate model parameters of the n _ m-th power system operation scene are determined, the surrogate model parameters in the m operation scenes can be sequentially obtained according to the method thought.
S33: and inputting the obtained proxy model parameters in the m operation scenes into a formula (5) to form m proxy models.
Step S4: sampling and inputting an agent model on a random variable probability density function based on a Monte Carlo simulation method to perform probability voltage stability evaluation;
s41: large-scale sampling is carried out on a random variable probability density function based on a Monte Carlo simulation method to obtain a sample point set X S 。
S42: by the formula (2) to convert X S Equiprobable transformation to U S . Will U S Inputting the (n _ m) th power system operation scene into the proxy model shown in the formula (3), and obtaining the voltage stability analysis index based on the proxy model instead of the original model
Thereby greatly improving the probability analysis efficiency.
According to the method, sample sets of output responses in m operation scenes can be obtained in sequence.
Step S5: calculating probability density function of probability voltage stability index by combining probability voltage stability calculation result based on agent model
Defining a risk calculation formula of a probability voltage stability evaluation index, and evaluating voltage stability risks in different scenes of the power system:
Experimental case
To evaluate the effectiveness of the proposed algorithm, a probabilistic static voltage stabilization calculation was performed using the IEEE118 node system. Wind farms WFD1, WFD2 and pv plants PVD1 are connected to the IEEE118 node system bus 1, bus 4 and bus 38, and wind farms WFA1, WFA2 and pv plants PVA1 are connected to the bus 72, bus 79 and bus 99, respectively. Wind speed and historical illumination data are both from a provincial power grid in the south of China. In this embodiment, the PVSE calculation is performed based on a Matlab simulation platform, and the computer hardware conditions are Inter Core i52.40-GHz CPU and 8GB RAM. Active load in the system is greatly influenced by power utilization behaviors of users, strong uncertainty exists, and simulation is carried out by utilizing Gaussian distribution. The number of dimensions of the input random variables in the system is up to 105.
Setting an operation scene: the method comprises the steps of conducting line N-1 scanning on an IEEE118 node system, namely, conducting line-by-line disconnection on the IEEE118 node system to form an operation scene, and then conducting probabilistic voltage stability evaluation to evaluate which lines are disconnected and have the most serious influence on the voltage stability level of the system. According to the operation scenario setting principle, the total number of the operation scenarios is 55, including 1 normal operation scenario and 54 different single line element fault scenarios.
In order to evaluate the correctness of the performance of the proposed algorithm, the result output by using the monte carlo simulation method (referred to as "reference value" for short) is used as a reference result. In order to evaluate the superiority of the algorithm of the present invention, the algorithm of the present invention is compared with the following algorithm:
(1) according to a document [ Pan faithful, Liu Jian, Wu Jian, Pai Huan, same time onward ] three-phase probability power flow calculation of a droop control island microgrid based on a low-rank approximation method [ J ]. China Motor engineering reports, 2020,40(20):6506 + 6516 ], a probability power flow calculation method based on an optimal multiplier cow-drawn method and a low-rank approximation method is provided. The method is applied to the field of probability voltage stability evaluation, and is called a comparison method 1 for short.
(2) The literature [ Sun Xin, available transmission capacity calculation method considering wind power uncertainty research [ D ]. university of science and technology in Huazhong, 2019 ] provides a probability available transmission capacity calculation method based on low-rank approximation. The method is applied to the field of probability voltage stability evaluation, and is called a comparison method 2 for short.
TABLE 1 mean of maximum load margin, standard deviation, third order moment and fourth order moment relative error
Table 1 shows the mean, standard deviation, third moment and fourth moment relative errors of the proposed algorithm, comparison algorithm 1 and comparison algorithm 2 over all operating scenarios. The error of the proposed algorithm is minimal, with a relative error of the fourth moment of only 5.89%. The relative error averages of comparative algorithm 1 and comparative algorithm 2 are much higher than those of the proposed algorithm, and the relative error averages of the fourth moment are 15.66% and 19.33%, respectively. The test results verify the effectiveness and high precision of the proposed algorithm.
The relative error data in table 1 and the probability distribution maps in fig. 2 to fig. 6 are combined to find that the calculation precision of the method is the highest, the calculation precision of the comparison method is not ideal enough, and the calculation precision of the probability voltage stability analysis can be greatly improved. The calculation accuracy of a droop control island microgrid three-phase probability power flow calculation based on a low rank approximation method [ J ] China Motor engineering report, 2020,40(20):6506 + 6516 ] and a document [ Sunxin ] research on available transmission capacity calculation method considering wind power uncertainty [ D ] Huazhong university of science and technology, 2019 ] algorithm (comparison method) is not ideal, and the main reasons are as follows: 1) the above papers all use conventional least squares techniques to solve the coefficients of a low rank approximation model (proxy model). When the conventional least square method technology is used for solving the high-dimensional low-rank approximate agent model coefficient, overfitting of the model can be caused due to overhigh dimension of the input variable, the precision of the agent model is further influenced, and the stability evaluation precision of the probability voltage is unsatisfactory. 2) The calculation method based on the polynomial order of the low-rank approximate agent model is unclear and has unclear rules, and the problem of high-dimensional probability voltage stability analysis is faced, so that the accuracy of the agent model is reduced due to inaccurate order determination; 3) the optimal rank calculation method based on the low-rank approximate agent model depends on artificial experience, and the accuracy of the agent model is possibly insufficient in the face of the problem of high-dimensional voltage stability probability analysis, so that the accuracy of a probability voltage stability result is influenced.
TABLE 2 calculation of time contrast(s)
In table 2, the total calculation time for the reference algorithm, the proposed method, the comparative method 1 and the comparative method 2 was 617925s, 1174.25s, 1592.25s and 1619.75s, respectively. Compared with a reference method and a comparison method, the method can greatly improve the calculation efficiency, mainly because the algorithm can be based on an L1 norm regularization technology, the sparsity of the matrix can be utilized, the complexity of the model is greatly reduced, the parameter solving efficiency is improved, and the probability analysis efficiency is further improved.
Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that changes, modifications, substitutions and alterations can be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.
Claims (6)
1. A multi-agent probability voltage stability calculation method for a high-dimensional system is characterized by comprising the following steps: the method comprises the following steps:
step S1: collecting historical data of random variables in the power system, and estimating probability density functions of the random variables;
step S2: determining an electric power system operation scene needing to check voltage stability in a specific operation period according to actual electric power system operation experience, and constructing a plurality of different probability voltage stability evaluation models based on different operation scenes of the electric power system in the operation period;
step S3: according to the operation scene of the power system, establishing an agent model of a deterministic voltage stability evaluation model corresponding to the operation scene on the basis of regularization low-rank approximation;
step S4: sampling and inputting an agent model on a random variable probability density function based on a Monte Carlo simulation method to perform probability voltage stability evaluation;
step S5: calculating probability density function of probability voltage stability index by combining probability voltage stability calculation result based on agent model
And defining a risk calculation formula of the probability voltage stability evaluation index, and evaluating the voltage stability risk in different scenes of the power system.
2. The method according to claim 1, wherein the method comprises the following steps: the random variables in the power system of step S1 include wind speed, light intensity and load, which may be represented as X ═ X 1 ,x 2 ,x 3 ,…,x n_random ]Wherein n _ random represents the dimension of the random variable and the probability density function of the ith random variable is represented as f i (x i )(i＝1,2,3,…,n_random)。
3. The multi-agent probabilistic voltage stabilization method for high dimensional systems according to claim 2, wherein: step S2 specifically includes the following steps:
s21: voltage stability conditions of m operation scenes with voltage stability need to be checked in an operation period, and the operation scene of the nth _ m power system in the m operation scenes is assumed to be L n_m Expressed as:
wherein
The line state which represents the n _ m power system operation scene number is 1, the line fault value is 0, the line normal value is 1,
representing the state of a conventional generator with the n _ m-th power system operation scene number of 1, wherein the fault value of the generator is 0, and the normal operation value of the generator is 1;
n line representing the number of lines in the power system, n Gen Representing the number of conventional generators in the power system, lines and conventional generators in the operating scenarioThe state of the motor can be determined empirically by grid operation and planners;
s22: constructing m voltage stability evaluation models based on m operation scene lines and the state of a conventional generator, wherein the nth _ m voltage stability evaluation models can be expressed as:
wherein X sample Set of samples, ε, representing random variables in an electric power system n_m Represents the nth _ m power system voltage stability index set,
representing the (n _ m) th deterministic voltage stability evaluation model; if the uncertainty of wind speed, illumination intensity and load in the power system is considered, random variable samples are input into a deterministic voltage stability evaluation model group by group, and the model is evolved into a probabilistic voltage stability analysis model.
4. The multi-agent probabilistic voltage stabilization calculation method for high dimensional systems according to claim 3, wherein: step S3 specifically includes the following steps:
s31: transforming random variables in power system to uniform distribution space U ═ U 1 ,u 2 ,…,u n_random ]The transformation formula of the ith random variable is as follows:
u i ＝∫f i (x i )dx i (2)
s32: constructing a proxy model of m deterministic voltage stabilization models by using low-rank approximation; taking the proxy model solution of the deterministic voltage stabilization model of the (n _ m) th scene as an example;
s321: the voltage stability indicator for the nth _ m power system operating scenario may be represented by the agent:
wherein, b l Is a normalized weight coefficient; omega l As a function of the rank of U, it can be expressed as:
in the formula (I), the compound is shown in the specification,
representing a univariate function in the ith dimension of the ith rank-one function, equations (3), (4) represent a regular low rank approximation,
is approximately:
wherein, R represents the number of rank one functions;
a univariate polynomial base of the kth order for the ith input variable; p is a radical of i Is composed of
The highest order of (a);
as a function of the l-th rank
Rank coefficient of (d);
s322: determination of proxy model parameters:
(1) determining univariate orthogonal polynomial basis
Due to input of random variablesU＝[u 1 ,u 2 ,…,u n_random ]Following a uniform distribution, the univariate orthogonal polynomial basis can be determined
Hilbert basis for Legendre polynomials
k is the order;
(2) obtaining a set of test sample points
Uniformly distributed sample point set U is selected on uniformly distributed U based on Latin hypercube sampling algorithm C Set the sample points U C Sequentially inputting the inverse function of the formula (2) to obtain a sample point X on the original distribution C (ii) a Sample point X on the original distribution C Inputting the (n _ m) th electric power system operation scene deterministic voltage stability analysis model to obtain a voltage stability analysis index
Forming an n _ m power system operation scene test sample point set
(3) Selection of polynomial order p
The implementation of the regularized low rank approximation takes into account a common polynomial order in all dimensions, i.e., p in equation (5) 1 ＝…＝p n_random P; according to a large amount of analog calculation and power system analysis experience, the order p is 5, so that p 1 ＝…＝p n_random ＝p＝5；
(4) L1 norm regularization least square algorithm based on sparsity induction for calculating rank coefficient z and weight coefficient b
Introducing a regularization method, changing the solving parameters into a least square problem with sparsity induced L1 regularization, and meanwhile, sparsity low-rank function of a matrix, wherein the method comprises the following steps:
the overall idea of solving rank coefficient z and weight coefficient b based on L1 norm "correction-update" is as follows:
1) initialization:
let r be 1 and r be equal to 1,
2) correction step based on L1 norm regularization:
the r-th step is to find a new rank-tensor ω r This can be obtained by solving the following minimization problem:
wherein W represents the space of a rank one tensor, and subscript χ represents the minimization of the design of the experiment; equation (6) is solved by an alternating least squares approach involving minimization of the sequence along each dimension i-1, …, n random, while "freezing" the coefficients in all remaining dimensions;
respectively calculating the optimization problem for each j dimension in turn to obtain corresponding rank coefficient
Meanwhile, a regularization method is introduced, a penalty term is added in an original objective function, and a model with high complexity is punished:
optimization equation (7) can be solved based on a minimum angle regression method;
3) updating step based on L1 norm regularization:
after the correction of the step r is completed, the algorithm enters the step r for updating, and a newly solved first-order function omega is determined r Weight factor b of (X) r At the same time, the existing weight coefficient (b) is updated 1 ,b 2 ,…,b r-1 ) (ii) a Base ofIn the regularization method, the updating step may be implemented by solving the following minimization problem:
4) convergence criterion:
is realized by a series of steps of 'correcting and updating' based on L1 norm, and in the step of r correction, a new rank-one function omega is established r And in the update of the r step, a set of weighting factors (b) is determined 1 ,b 2 ,…,b r )；
Convergence criterion is the number of iterations I r And decrease of error measurement
Combined in two successive iterations; the error measure used is a relative empirical error, as follows:
in the above-mentioned formula, the compound of formula,
empirical variance of a set consisting of model responses when designing an experiment; therefore, if I r To a maximum allowable value I max Or is or
Less than a prescribed threshold value
The algorithm converges and the procedure terminates; otherwise, r ═ r +1, and return to step 2); default I max 100 and
(5) selection of optimal rank R
1) Initialization:
let r be 1;
2) and (3) error calculation:
to determine the optimal rank R e { R ═ 1, …, R max The relative generalization error can be measured by adopting the size of the relative generalization error, and the relative generalization error is calculated by adopting a 3-time cross validation method based on an experimental design sample; the process firstly needs to divide an experimental design into 3 subsets with approximately equal sizes, and a low-rank approximation with increasing rank forms a training set TR in 2 subsets; while the remaining subset is used as the test set TS to evaluate the error of the low rank approximation constructed with the training set:
wherein
An LRA meta-model built for the training set,
evaluating the empirical variance of the formed set for the model on the test set;
3) convergence criterion:
with alternating 3 sets, 3 metamodels are obtained in this way; their average error provides an estimate of the proxy model generalization error for controlling the early stop option and selecting the optimal rank R; when r reaches the maximum allowable value r stop Or are each
Less than a prescribed threshold value
The algorithm converges and the procedure terminates; otherwise, r ═ r +1, and return to step 2);
after determining the surrogate model parameters of the n _ m power system operation scenes, sequentially obtaining the surrogate model parameters in the m operation scenes;
s33: and inputting the obtained proxy model parameters in the m operation scenes into a formula (5) to form m proxy models.
5. The multi-agent probabilistic voltage stabilization method for high dimensional systems according to claim 4, wherein: step S4 specifically includes the following steps:
s41: large-scale sampling is carried out on a random variable probability density function based on a Monte Carlo simulation method to obtain a sample point set X S ；
S42: by the formula (2) to convert X S Equiprobable transformation to U S (ii) a Will U S Inputting the (n _ m) th power system operation scene into the proxy model shown in the formula (3), and obtaining the voltage stability analysis index based on the proxy model instead of the original model
According to the method, sample sets of output responses in m operation scenes can be obtained in sequence.
6. The multi-agent probabilistic voltage stabilization method for high dimensional systems according to claim 4, wherein: step S5 defines a risk calculation formula of the probabilistic voltage stability assessment indicator as follows:
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