CN115173421B - Probability optimal power flow calculation method based on progressive Latin hypercube sampling - Google Patents
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Abstract
The invention discloses a probability optimal power flow calculation method based on gradual Latin hypercube sampling, which comprises the steps of 1) collecting topology parameters of a power system and establishing a deterministic optimal power flow model; 2) Converting into uniform distribution, and establishing a function mapping relation; 3) Generating an initial sample set from uniform distribution by using a traditional Latin hypercube sampling method; 4) Equally dividing into T different slices, and selecting the first slice as a first group of sample point subsets; 5) Transforming the newly added sample point subset to original distribution, inputting a deterministic optimal power flow model, and carrying out probability optimal power flow calculation; 6) The existing probability optimal power flow results are tided, and whether the probability optimal power flow results are converged or not is judged; 7) And (5) determining a slice with optimal space filling characteristics with the used slice based on a greedy heuristic algorithm to form a newly added sample point subset, and executing the step (5). The invention improves the convergence of the probability optimal power flow calculation and the robustness to the variability of the sampling result.
Description
Technical Field
The invention relates to the technical field of power systems, in particular to a probability optimal power flow calculation method based on progressive Latin hypercube sampling.
Background
The optimal power flow (Optimal Power Flow, OPF) as an effective tool for analysing the operation of the power system can find a power flow profile that satisfies the operation constraints and can reduce the operation costs to a lower level. Along with the development of a power system and the uncertainty caused by new energy access, the tide value in the power grid also changes with the uncertainty. The traditional deterministic optimal power flow (Deterministic Optimal Power Flow, DOPF) is difficult to adapt to the current power grid uncertainty scene, a probabilistic power flow (Probabilistic Power Flow, PPF) concept is proposed in 1974 in combination with the Borkowska, the probabilistic optimal power flow (Probabilistic Optimal Power Flow, POPF) can evaluate various uncertain factors in a power system, and the statistical information of a solution (output variable) of the optimal power flow is obtained based on the statistical information of random variables (input variables) in a model, so that potential weak links can be identified in advance, and the power grid operation risk is reduced.
Currently, the POPF can be used for evaluating potential risks and weak points of operation of the system, and provides information with reference value for planning, reliability analysis, system safety analysis and the like of the power system. POPF is a method for establishing a multi-element nonlinear equation by taking uncertainty factors into consideration in power flow calculation and calculating by using different methods by utilizing knowledge of probability theory, and the existing research methods can be roughly divided into three types of simulation methods, analysis methods and approximation methods.
The analysis method is to take the optimization process in POPF calculation as a probability mapping, and calculate the statistical information of the output quantity through a linearization load flow operation model. However, the method has two disadvantages, namely, due to the existence of inequality constraint in POPF calculation, the nonlinearity of the function relation between the output quantity and the input quantity is large, and the error introduced by linearizing the function relation is not negligible; secondly, the analysis method needs complex mathematical calculation and guarantee in the correlation processing, input variables are independent of each other, and the correlation among the output of the wind power plant units is considered.
The approximation method is to select a small number of sample points to perform deterministic calculation, so that the estimation of the mean value and variance of the output variable can be completed, but the error caused by the reduction of the calculated amount is also quite large, and the high-order statistical moment of the output quantity cannot be obtained. In particular, in the POPF calculation, due to the limitation of inequality constraint, even if the input random variable is normally distributed, the output variable is not normally, the probability distribution of many output variables is truncated, and the accuracy of the method is even less than sufficient.
In the three methods, the simulation method has the highest applicability, only the uncertain factors are used as random variables to establish a probability model, then samples of probability distribution are randomly extracted to perform optimal power flow calculation, and further the statistical moment, the probability density function and the cumulative distribution function of the output variables can be accurately obtained. Simulation is generally referred to as monte carlo simulation, which is extremely accurate in the case of sufficient samples, but requires a large amount of computation and high cost as a premise, mckay et al proposed in 1979 a monte carlo method based on latin hypercube sampling (Latin Hypercube Sampling, LHS) in order to effectively estimate the mean value of the output variables with a small number of samples. LHS is a hierarchical sampling method, which consists of interval sampling and correlation control steps, and can effectively reflect the overall distribution of random variables by using sampling values. The original LHS ensures that the generated samples have one-dimensional projection characteristics, indicating that the projection of sample points in two-or high-dimensional space in any dimension will follow a uniform distribution (or any other distribution of interest). Thus, the sample set is considered to be a "latin hypercube" if and only if it has one-dimensional projection properties.
The problems of the traditional Latin hypercube sampling technology are as follows: (1) The traditional Latin hypercube sample only can ensure one-dimensional projection characteristics, but the space filling characteristics in a multidimensional space are not considered enough; (2) They generate all the required sample point sets once in advance, which requires the user to specify the sample point set size before performing the probability optimal power flow analysis based on Latin hypercube sampling, so the user tends to use a larger sample size, thereby avoiding the situation that the sample size is too small to meet the POPF convergence criterion, but this brings about unnecessarily larger calculation requirements and high calculation burden; (3) If the convergence criterion is not met, the sample size needs to be enlarged, and the user faces the dilemma of using a new sample: or generating a new sample meeting the convergence condition by the LHS and adding it to the previously generated sample, the trade-off between the two samples would not be a latin hypercube; or wasting computational cost discarding previous samples and generating a larger new sample to meet the convergence criterion.
Disclosure of Invention
Aiming at the problems existing in the prior art, the invention aims to provide a probability optimal power flow calculation method based on progressive Latin hypercube sampling, and provides a new and efficient LHS (luted solution) called progressive Latin hypercube sampling (Progressive Latin Hypercube Sampling, PLHS). Unlike the traditional LHS method, which generates the entire sample set in one stage, PLHS first generates a series of smaller subsets, while: (1) the first subset is a latin hypercube; (2) The stepwise increasing subset is still a latin hypercube, i.e. meets the one-dimensional projection characteristics and meets the good space filling characteristics; (3) the entire sample set is a Latin hypercube. As the sample size gradually increases during analysis, the PLHS will maintain the desired distribution characteristics.
The invention is realized by the following technical scheme:
a probability optimal power flow calculation method based on progressive Latin hypercube sampling comprises the following steps:
step S1: collecting topology parameters of a power system, and establishing a deterministic optimal power flow model;
step S2: based on a probability density function of the random variable X, establishing a function mapping relation of the random variable in the power system from original distribution to uniform distribution, and further establishing a function mapping relation of the random variable from uniform distribution to original distribution;
step (a)S3: generating an initial sample set U from a uniform distribution using a conventional Latin hypercube sampling method sample ＝[u 1 ,u 2 ,…,u p ]Wherein p represents the dimension of the random variable, and the number of samples of the random variable in each dimension is n;
step S4: the initial sample set U obtained according to step S3 sample Equally dividing into T different slices, and selecting the first slice as a first group of sample point subsets;
step S5: transforming the newly added sample point subset to original distribution, inputting a deterministic optimal power flow model, and carrying out probability optimal power flow calculation;
step S6: the existing probability optimal power flow results are tided, whether the probability optimal power flow results are converged or not is judged, if yes, the program is ended, and the probability optimal power flow results are output; if not, executing step S7;
step S7: and determining a slice with optimal space filling characteristics with the used slice based on a greedy heuristic algorithm, forming a newly added sample point subset, and executing step S5.
Further, the specific process of step S1 is as follows:
the random variables in the power system refer to wind speed, illumination intensity and load data, which can be expressed as x= [ X ] 1 ,x 2 ,x 3 ,…,x p ]Where p represents the dimension of the random variable, and the probability density function of the ith random variable is estimated to be f i (x i )(i＝1,2,3,…,p)；
The deterministic optimal power flow model is as follows:
Z＝g(W) (1)
w refers to node injection power, Z refers to node voltage and branch current;
establishing a slave random variable X= [ X ] 1 ,x 2 ,x 3 ,…,x p ]Injecting power parameters W= [ W ] to nodes needed by tide calculation 1 ,w 2 ,w 3 ,…,w p ]The one-to-one mapping relation is obtained according to the formula (1), namely, the mapping relation of the random variable X, the output voltage, the phase angle and the line power flow Z is obtained:
Z＝G(X) (2)
x refers to the random variables of the power system, and Z refers to the node voltage and branch current.
Further, the specific process of step S2 is as follows:
s21, a probability density function f of a random variable X of the power system i (x i ) (i=1, 2, …, p) to obtain a cumulative distribution function U i ＝F i (x i )，x i Is x 1 ，x 2 ，…，x p Any random variable in (2), the cumulative distribution function U i ＝F i (x i ) Obeys [0,1 ]]The space is uniformly distributed;
s22, the original distribution refers to the obeyed probability distribution of the random variable X, in particular to the X of the wind speed, the illumination intensity and the load data i The method comprises the steps of carrying out a first treatment on the surface of the Uniformly distributed means U i The obeyed probability distribution is specifically probability distribution of a cumulative distribution function obtained by a probability density function;
s23, inverse function F of cumulative distribution function i -1 (U i ) And further establishing a function mapping relation for transforming the uniform distribution into the original distribution.
Further, the specific process of step S3 is as follows:
s31, integrating the distribution function U i The value range [0,1 ]]Equally divided into n subintervals, i.e. into n disjoint intervals [0,1/n ], [1/n,2/n ], …, [ (n-1)/n, 1]These intervals are represented by q (q=1, 2, …, n), a random number is extracted in each interval in a uniform distribution, and the extraction is not repeated for each sub-interval, thereby obtaining an n×p-order sample matrix U (n, p), and the extracted sample points are represented by x i,j ∈[0,1]Represents, wherein i=1, 2, …, n; j=1, 2, …, p;
s32, defining a new auxiliary variable y q,j I.e.
So that the sample conforms to
I.e. the sample matrix U (n, p) satisfies the one-dimensional projection characteristic of Latin hypercube;
s33, outputting an initial sample set U with the dimension of p and the sample size of n sample ＝[u 1 ,u 2 ,…,u p ]。
Further, the specific process of step S4 is as follows:
s41, the initial sample set U obtained in the step S3 sample Equally dividing into T equal parts to obtain T samples with the size of m=n/T;
s42, set S t Is a collective sample formed by the union of the subsamples, i.eWherein t=1, 2, …, T represents the number of slices, defining a set of auxiliary binary variables +.>Corresponding to the sample matrix->Where t=1, 2, …, T, each of the readily available subsamples satisfies the latin hypercube property and satisfies;
i.e. the progressive union of the samples satisfies the Latin hypercube property;
s43, forming a set of slice sample matrices S (n, p, T) from S42, wherein S t (m, p) for t=1, 2, …, T is a set of Latin hypercube sample matrices, for S t (m, p) each slice is numbered pi (1), pi (2), …, pi (T), and the first slice pi (1) is selected as the initial Latin hypercube subset S 1 (m,p)。
Further, the specific process of step S5 is as follows:
s51, combining the newly added sample point subset S t (m, p) (t=1, 2, …, T), by U i From the inverse F of the cumulative distribution function i -1 (U i ) Obtaining X i The newly added slice pi (T) (t=1, 2, …, T) is thus transformed to the original distribution;
s52, then combining equations (1) and (2) to obtain W (T), and then obtaining output data Z (T) through a deterministic optimal power flow model, where t=1, 2, …, T.
Further, the specific process of step S6 is as follows:
s61, since the number of each output random variable is more than one in the probability optimal power flow calculation, voltage amplitude can be selected as a reference, and the average value and variance of the output voltages are represented by mu (Z) and sigma (Z) respectively on the assumption that the output voltages are v (1), v (2), … and v (a), wherein
Wherein a refers to the total number of output voltages;
s62, kernel density estimation is a non-parametric method for estimating a probability density function, output voltages v (1), v (2), … and v (a) are a sample points which are independently distributed, the probability density function is set as f, and the kernel density estimation is as follows:
wherein K (.cndot.) is a suitable kernel function, K (x). Gtoreq.0,h a the smoothness parameter refers to window width or bandwidth;
s63, defining a convergence coefficient β=0.01, if the variance coefficient satisfies
Judging that the probability optimal power flow result is converged, ending the program, and outputting the probability optimal power flow result; if not, step S7 is performed.
Further, the specific process of step S7 is as follows:
s71, defining the maximum distance criterion is a measure of the space filling, the purpose of which is to maximize the minimum distance between each pair of points in the sample, namely:
wherein X is i And X j Is any two different sample points, n is the sample size, d (-) is the distance metric, two points X in p-dimensional space i And X j The distance d is expressed as:
and->Refers to two points X i And X j Coordinates in delta dimension space;
s72, assuming pi is a group of randomly arranged integers {1,2, …, T }, in order to find the optimal arrangement of the slices, pi (1) is selected as a first block slice, and in order to construct the optimal arrangement sequence of the slices, pi (k) of the next slice is selected to maximize the F (d) value of the formula (9) at each stage, namely, each iteration is maximized, so that the arrangement of the slices is obtained to optimize the progressive and space filling characteristics of the samples;
s73, S72 uses a greedy search algorithm to find the optimal solution of the optimization problem, and the algorithm steps are as follows:
(1) Setting the sample size n, the dimension p and the slice number T;
(2) Generating a set of Latin hypercubesSlicing is carried out, the slice is cut,
(3) The slice sequences are arranged arbitrarily, { pi (1), pi (2), …, pi (T) };
(4) The integers r=1, 2, …, T are arbitrarily set, let k=1, pi (1) =r, s=lhs π(1) (m,p)；
(5) Marking the 1 st slice as browsed and T-1 slices as unviewed;
and (3) circulation: when k is less than T, let F best ＝0,k＝k+1；
Let j = all unviewed;
constructing a new set S π(j) ＝S∪LHS π(j) (m,p)；
Evaluating the objective function value F (S) by equation (9) π(j) )；
If F (S) π(j) )＞F best ；
F best ＝F(S π(j) ),π(k)＝π(j)；
(6) Output slice sample arrangement
S74, finding out the space filling characteristic of optimizing the slice arrangement sequence through a greedy search algorithm, selecting a newly added slice to form a newly added sample point subset, and executing the step S5.
The beneficial effects of the invention are as follows:
the invention improves the convergence of the probability optimal power flow calculation and the robustness to the variability of the sampling result, and is characterized in that: (1) The correlation of POPF random variables in the case of small-scale sampling can be accurately processed; (2) Compared with the existing algorithm, the output result is more accurate at the same sampling scale. The probability optimal power flow calculation method based on the gradual Latin hypercube sampling is seen in effect, a large number of complicated calculation processes are reduced, the method has the characteristics of high precision and high timeliness, particularly has extremely high stability and accuracy in the aspect of solving the expected value of the random variable, and has important significance for the safe operation of the power system.
Drawings
FIG. 1 is a flow chart of a probability optimal power flow calculation method based on progressive Latin hypercube sampling;
FIG. 2 is a probability density distribution of branch active power;
FIG. 3 is the second and third moments of the active power of branches 39-42;
FIG. 4 is the cumulative calculation time for MVC-CLHS and methods herein.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. It will be apparent that the described embodiments are only some, but not all, embodiments of the invention. 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 fall within the scope of the invention.
A probability optimal power flow calculation method based on progressive Latin hypercube sampling is shown in figure 1, and comprises the following steps:
step S1: and collecting topological parameters of the power system, and establishing a deterministic optimal power flow model. The method comprises the following steps:
random variables in an electrical power system are generally referred to as wind speed, illumination intensity, and load data, which may be represented as x= [ X ] 1 ,x 2 ,x 3 ,…,x p ](where p represents the dimension of the random variable), the probability density function of the ith random variable is estimated to be f i (x i )(i＝1,2,3,…,p)；
The deterministic optimal power flow model is as follows:
Z＝g(W) (1)
w refers to node injection power, Z refers to node voltage (including voltage amplitude and phase angle) and branch current (including active and reactive);
establishing a slave random variable X= [ X ] 1 ,x 2 ,x 3 ,…,x p ]Injecting power parameters W= [ W ] to nodes needed by tide calculation 1 ,w 2 ,w 3 ,…,w p ]The one-to-one mapping relation is obtained according to the formula (1), namely, the mapping relation of the random variable X, the output voltage, the phase angle and the line power flow Z is obtained:
Z＝G(X) (2)
x refers to the random variables of the power system, and Z refers to the node voltage and branch current.
Step S2: based on the probability density function of the random variable X, a function mapping relation of the random variable in the power system is established, wherein the function mapping relation is converted from the original distribution to the uniform distribution, and the function mapping relation is further established.
The specific process of step S2 is as follows:
s21, a probability density function f of a random variable X of the power system i (x i ) (i=1, 2, …, p) to obtain a cumulative distribution function U i ＝F i (x i )，x i Is x 1 ，x 2 ，…，x p Any random variable in (2), the cumulative distribution function U i ＝F i (x i ) Obeys [0,1 ]]The space is uniformly distributed;
s22, the original distribution refers to probability distribution obeyed by random variable X, in particular to X of wind speed, illumination intensity and load data i The method comprises the steps of carrying out a first treatment on the surface of the Uniformly distributed means U i The obeyed probability distribution is specifically probability distribution of a cumulative distribution function obtained by a probability density function;
s23, inverse function F of cumulative distribution function i -1 (U i ) And further establishing a function mapping relation for transforming the uniform distribution into the original distribution.
Step S3: generating an initial sample set U from a uniform distribution using a conventional Latin hypercube sampling method sample ＝[u 1 ,u 2 ,…,u p ]Where p represents the dimension of the random variable, and the number of samples per dimension of the random variable is n. The specific process of step S3 is as follows:
s31, integrating the distribution function U i The value range [0,1 ]]Equally divided into n subintervals, i.e. into n disjoint intervals [0,1/n ], [1/n,2/n ], …, [ (n-1)/n, 1]This is denoted by q (q=1, 2, …, n)Extracting a random number in each interval according to uniform distribution, and not repeatedly extracting each subinterval to obtain n multiplied by p order sample matrix U (n, p), wherein the extracted sample points are x i,j ∈[0,1]Represents, wherein i=1, 2, …, n; j=1, 2, …, p;
s32, defining a new auxiliary variable y q,j I.e.
So that the sample conforms to
I.e. the sample matrix U (n, p) satisfies the one-dimensional projection characteristic of Latin hypercube;
s33, outputting an initial sample set U with the dimension of p and the sample size of n sample ＝[u 1 ,u 2 ,…,u p ]。
Step S4: the initial sample set U obtained according to step S3 sample Aliquoted into T different slices, with the first slice selected as the first set of sample point subsets. The specific process of step S4 is as follows:
s41, the initial sample set U obtained in the step S3 sample Equally dividing into T equal parts to obtain T samples with the size of m=n/T;
s42, set S t Is a collective sample formed by the union of the subsamples, i.eWherein t=1, 2, …, T represents the number of slices, defining a set of auxiliary binary variables +.>Corresponding to the sample matrix->Wherein t=1, 2, …, T, facileEach sub-sample is satisfied with Latin hypercube characteristics and satisfied;
i.e. the progressive union of the samples satisfies the Latin hypercube property;
s43, forming a set of slice sample matrices S (n, p, T) from S42, wherein S t (m, p) for t=1, 2, …, T is a set of Latin hypercube sample matrices, for S t (m, p) numbering each slice (subsamples) pi (1), pi (2), …, pi (T), selecting the first slice pi (1) as the initial Latin hypercube subset S 1 (m,p)。
Step S5: and transforming the newly added sample point subset to original distribution, inputting a deterministic optimal power flow model, and carrying out probability optimal power flow calculation. The specific process of step S5 is as follows:
s51, combining the newly added sample point subset S t (m, p) (t=1, 2, …, T), by U i From the inverse F of the cumulative distribution function i -1 (U i ) Obtaining X i The newly added slice pi (T) (t=1, 2, …, T) is thus transformed to the original distribution;
s52, then combining equations (1) and (2) to obtain W (T), and then obtaining output data Z (T) through a deterministic optimal power flow model, where t=1, 2, …, T.
Step S6: the existing probability optimal power flow results are tided, whether the probability optimal power flow results are converged or not is judged, if yes, the program is ended, and the probability optimal power flow results are output; if not, step S7 is performed. The specific process of step S6 is as follows:
s61, since the number of each output random variable is more than one in the probability optimal power flow calculation, voltage amplitude (same voltage phase angle and power) can be generally selected as reference, and the average value and variance of the output voltages are represented by mu (Z) and sigma (Z) respectively under the assumption that the output voltages are v (1), v (2), … and v (a)
Wherein a refers to the total number of output voltages;
s62, kernel density estimation is a non-parametric method for estimating a probability density function, output voltages v (1), v (2), … and v (a) are a sample points which are independently distributed, the probability density function is set as f, and the kernel density estimation is as follows:
wherein K (.cndot.) is a suitable kernel function, K (x). Gtoreq.0,h a a smoothness parameter, typically window width or bandwidth;
s63, defining a convergence coefficient β=0.01, if the variance coefficient satisfies
Judging that the probability optimal power flow result is converged, ending the program, and outputting the probability optimal power flow result; if not, step S7 is performed.
Step S7: and determining a slice with optimal space filling characteristics with the used slice based on a greedy heuristic algorithm, forming a newly added sample point subset, and executing step S5. The specific process of step S7 is as follows:
s71, defining the maximum distance criterion is a measure of the space filling, the purpose of which is to maximize the minimum distance between each pair of points in the sample, namely:
wherein X is i And X j Is any two different sample points, n is the sample size, d (-) is the distance metric (here the euclidean metric), two points X in p-dimensional space i And X j The distance d is expressed as:
and->Refers to two points X i And X j Coordinates in delta dimension space;
s72, assuming pi is a group of randomly arranged integers {1,2, …, T }, in order to find the optimal arrangement of the slices, pi (1) is selected as a first block slice, and in order to construct the optimal arrangement sequence of the slices, pi (k) of the next slice is selected to maximize the F (d) value of the formula (9) at each stage, namely, each iteration is maximized, so that the arrangement of the slices is obtained to optimize the progressive and space filling characteristics of the samples;
s73, S72 uses a greedy search algorithm to find the optimal solution of the optimization problem, and the algorithm steps are as follows:
(1) Setting a sample size n (=10000), a dimension p and the number of slices T;
(2) A set of latin hypercube slices is generated,
(3) The slice sequences are arranged arbitrarily, { pi (1), pi (2), …, pi (T) };
(4) The integers r=1, 2, …, T are arbitrarily set, let k=1, pi (1) =r, s=lhs π(1) (m,p)；
(5) Marking the 1 st slice as browsed and T-1 slices as unviewed;
and (3) circulation: when k is less than T, let F best ＝0,k＝k+1；
Let j = all unviewed;
constructing a new set S π(j) ＝S∪LHS π(j) (m,p)；
Evaluating the objective function value F (S) by equation (9) π(j) )；
If F (S) π(j) )＞F best ；
F best ＝F(S π(j) ),π(k)＝π(j)；
(6) Output slice sample arrangement
S74, finding out the space filling characteristic of optimizing the slice arrangement sequence through a greedy search algorithm, selecting a newly added slice to form a newly added sample point subset, and executing the step S5.
Description of the preferred embodiments
To evaluate the effectiveness of the proposed algorithm, a probabilistic optimal power flow calculation is performed using the IEEE118 node system. Wind farm WFD1, wind farm WFD2 and photovoltaic power plant PVD1 are connected to bus 1, bus 4 and bus 38 of the IEEE118 node system, and wind farm WFA1, wind farm WFA2 and photovoltaic power plant PVA1 are connected to bus 72, bus 79 and bus 99 respectively.
Calculation example wind power plant and photovoltaic power plant output models and parameters are derived from actual wind power plant and photovoltaic power plant parameters. Wind speed and illumination historical data are both from a certain provincial power grid in the south of China, and a probability density distribution function of an input random variable is constructed based on actual wind-light data. Meanwhile, the deterministic optimal power flow model is sampled and input on the basis of the sampling algorithm on the basis of the input probability distribution so as to realize probability optimal power flow calculation. PVSE calculation is carried out based on a Matlab simulation platform, and the computer hardware conditions are an InterCore i52.40-GHz CPU and an 8GB RAM. The algorithm proposed in the present invention is simply referred to as "the method herein".
(1) Method herein validity assessment
In order to evaluate the effectiveness of the method, 20000 groups of historical data are input into a deterministic optimal power flow module by using an MCS algorithm to calculate a probability optimal power flow, and the obtained result is used as a reference value, wherein the method is called as an MCS for short.
Fig. 2 shows probability density profiles of branch active power obtained by the methods herein and MCS method. The branch active power probability density profile obtained by the method herein is highly fitting to the MCS (reference algorithm) because: the probability information can be effectively transmitted in the voltage stability analysis module by using the method. This fully demonstrates the effectiveness of the methods herein. In addition, the method can output the probability density distribution diagram of the key indexes of the power system, and can comprehensively display the power grid operation information to reference power system operators.
(2) Superiority of the method herein
The probability optimal power flow needs to be calculated in a large scale to realize the transmission of probability information, and the calculation process is extremely time-consuming. Therefore, the calculation efficiency of the probability optimal power flow is improved.
To demonstrate the superiority of the method herein, the method herein will be compared to a conventional Latin hypercube algorithm, the comparison method being abbreviated as "MVC-CLHS".
FIG. 3 illustrates the maximum load margin second and third moment information obtained by the MVC-CLHS and methods herein. From a computational process perspective, both MVC-CLHS and the methods herein exhibit a fluctuating convergence characteristic as the sample size increases, and the values of the output second and third moments are substantially identical. When the number of sample points is 3500 and 4000, respectively, the third order moments obtained by the MVC-CLHS and the method herein are 3.0821 and 3.0911 (MVC-CLHS), 3.1021 and 3.1024 (method herein), respectively. The results of the two calculations are very similar, which further verifies the effectiveness and high accuracy of the methods herein.
The cumulative calculation time for MVC-CLHS and the methods herein is shown in FIG. 4. Taking the calculation as an example of a 500 sample point step size, the cumulative calculation time for MVC-CLHS convergence is as high as 4160.23s, with the method herein requiring only 924.18s. In the case of a specific probability optimal power flow analysis calculation scene, the MVC-CLHS is difficult to determine the total amount of samples required at one time, and multiple attempts of calculation (probability analysis) are needed in the calculation process to determine whether the selected sample amount meets the calculation accuracy requirement; on the other hand, the MVC-CLHS algorithm cannot reuse the already obtained calculation results, resulting in extremely long cumulative calculation time. The method can overcome the problem, because the method can inherit the used sample points and the calculation result, the algorithm only needs to calculate the newly added sample points when iterating, the repeated calculation amount is avoided, and the calculation efficiency of the probability optimal power flow is greatly improved.
Although embodiments of the present invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made therein without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.
Claims (8)
1. The probability optimal power flow calculation method based on progressive Latin hypercube sampling is characterized by comprising the following steps of:
step S1: collecting topology parameters of a power system, and establishing a deterministic optimal power flow model;
step S2: based on a probability density function of the random variable X, establishing a function mapping relation of the random variable in the power system from original distribution to uniform distribution, and further establishing a function mapping relation of the random variable from uniform distribution to original distribution;
step S3: generating an initial sample set U from a uniform distribution using a conventional Latin hypercube sampling method sample ＝[u 1 ,u 2 ,…,u p ]Wherein p represents the dimension of the random variable, and the number of samples of the random variable in each dimension is n;
step S4: the initial sample set U obtained in the step S3 sample Equally dividing into T different slices, and selecting the first slice as a first group of sample point subsets;
step S5: transforming the newly added sample point subset to original distribution, inputting a deterministic optimal power flow model, and carrying out probability optimal power flow calculation;
step S6: the existing probability optimal power flow results are tided, whether the probability optimal power flow results are converged or not is judged, if yes, the program is ended, and the probability optimal power flow results are output; if not, executing step S7;
step S7: and determining a slice with optimal space filling characteristics with the used slice based on a greedy heuristic algorithm, forming a newly added sample point subset, and executing step S5.
2. The method for calculating the probability optimal power flow based on progressive Latin hypercube sampling as set forth in claim 1, wherein the specific process of step S1 is as follows:
the random variables in the power system refer to wind speed, illumination intensity and load data, which can be expressed as x= [ X ] 1 ,x 2 ,x 3 ,…,x p ]Where p represents the dimension of the random variable, and the probability density function of the ith random variable is estimated to be f i (x i )(i＝1,2,3,…,p)；
The deterministic optimal power flow model is as follows:
Z＝g(W) (1)
w refers to node injection power, Z refers to node voltage and branch current;
establishing a slave random variable X= [ X ] 1 ,x 2 ,x 3 ,…,x p ]Injecting power parameters W= [ W ] to nodes needed by tide calculation 1 ,w 2 ,w 3 ,…,w p ]The one-to-one mapping relation is obtained according to the formula (1), namely, the mapping relation of the random variable X, the output voltage, the phase angle and the line power flow Z is obtained:
Z＝G(X) (2)
x refers to the random variables of the power system, and Z refers to the node voltage and branch current.
3. The method for calculating the probability optimal power flow based on progressive Latin hypercube sampling as set forth in claim 2, wherein the specific process of step S2 is as follows:
s21, a probability density function f of a random variable X of the power system i (x i ) (i=1, 2, …, p) to obtain a cumulative distribution function U i ＝F i (x i )，x i Is x 1 ，x 2 ，…，x p Any random variable in (2), the cumulative distribution function U i ＝F i (x i ) Subject to the compliance of [0 ],1]the space is uniformly distributed;
s22, the original distribution refers to probability distribution obeyed by random variable X, in particular to X of wind speed, illumination intensity and load data i The method comprises the steps of carrying out a first treatment on the surface of the Uniformly distributed means U i The obeyed probability distribution is specifically probability distribution of a cumulative distribution function obtained by a probability density function;
s23, inverse function F of cumulative distribution function i -1 (U i ) And further establishing a function mapping relation for transforming the uniform distribution into the original distribution.
4. The method for calculating the probability optimal power flow based on progressive Latin hypercube sampling as set forth in claim 3, wherein the specific process of step S3 is as follows:
s31, integrating the distribution function U i The value range [0,1 ]]Equally divided into n subintervals, i.e. into n disjoint intervals [0,1/n ], [1/n,2/n ], …, [ (n-1)/n, 1]These intervals are represented by q (q=1, 2, …, n), a random number is extracted in each interval in a uniform distribution, and the extraction is not repeated for each sub-interval, thereby obtaining an n×p-order sample matrix U (n, p), and the extracted sample points are represented by x i,j ∈[0,1]Represents, wherein i=1, 2, …, n; j=1, 2, …, p;
s32, defining a new auxiliary variable y q,j I.e.
So that the sample conforms to
I.e. the sample matrix U (n, p) satisfies the one-dimensional projection characteristic of Latin hypercube;
s33, outputting an initial sample set U with the dimension of p and the sample size of n sample ＝[u 1 ,u 2 ,…,u p ]。
5. The method for calculating the probability optimal power flow based on progressive Latin hypercube sampling as set forth in claim 4, wherein the specific process of step S4 is as follows:
s41, the initial sample set U obtained in the step S3 sample Equally dividing into T equal parts to obtain T samples with the size of m=n/T;
s42, set S t Is a collective sample formed by the union of the subsamples, i.eWherein t=1, 2, …, T represents the number of slices, defining a set of auxiliary binary variables +.>Corresponding to the sample matrix->Where t=1, 2, …, T, each of the readily available subsamples satisfies the latin hypercube property and satisfies;
i.e. the progressive union of the samples satisfies the Latin hypercube property;
s43, forming a set of slice sample matrices S (n, p, T) from S42, wherein S t (m, p) for t=1, 2, …, T is a set of Latin hypercube sample matrices, for S t (m, p) each slice is numbered pi (1), pi (2), …, pi (T), and the first slice pi (1) is selected as the initial Latin hypercube subset S 1 (m,p)。
6. The method for calculating the probability optimal power flow based on progressive Latin hypercube sampling as set forth in claim 5, wherein the specific process of step S5 is as follows:
s51, combining the newly added samplesPoint subset S t (m, p) (t=1, 2, …, T), by U i From the inverse F of the cumulative distribution function i -1 (U i ) Obtaining X i Transforming the newly added slice pi (T) (t=1, 2, …, T) to the original distribution;
s52, then combining equations (1) and (2) to obtain W (T), and then obtaining output data Z (T) through a deterministic optimal power flow model, where t=1, 2, …, T.
7. The method for calculating the probability optimal power flow based on progressive Latin hypercube sampling as set forth in claim 6, wherein the specific process of step S6 is as follows:
s61, since the number of each output random variable is more than one in the probability optimal power flow calculation, voltage amplitude can be selected as a reference, and the output voltages are assumed to be v (1), v (2), … and v (a), and the mean value and variance of the output voltages are represented by mu (Z) and sigma (Z), namely:
wherein a refers to the total number of output voltages;
s62, kernel density estimation is a non-parametric method for estimating a probability density function, output voltages v (1), v (2), … and v (a) are a sample points which are independently distributed, the probability density function is set as f, and the kernel density estimation is as follows:
wherein K (.cndot.) is a suitable kernel function, K (x). Gtoreq.0,h a the smoothness parameter refers to window width or bandwidth;
s63, defining a convergence coefficient β=0.01, if the variance coefficient satisfies
Judging that the probability optimal power flow result is converged, ending the program, and outputting the probability optimal power flow result; if not, step S7 is performed.
8. The method for calculating the probability optimal power flow based on progressive Latin hypercube sampling as set forth in claim 7, wherein the specific process of step S7 is as follows:
s71, defining a measure of the maximum distance criterion characterizing the space filling, the purpose of which is to maximize the minimum distance between each pair of points in the sample, namely:
wherein X is i And X j Is any two different sample points, n is the sample size, d (-) is the distance metric, two points X in p-dimensional space i And X j The distance d is expressed as:
and->Refers to two points X i And X j Coordinates in delta dimension space;
s72, assuming pi is a set of randomly arranged integers {1,2, …, T }, to find the optimal arrangement of slices, pi (1) is selected as the first block slice, and to construct the optimal arrangement sequence of slices, the next slice pi (k) is selected to maximize the F (d) value of formula (9) at each stage, i.e., each iteration is maximized, and the arrangement of slices is obtained to optimize the progressive union space filling characteristics of the samples;
s73, S72 uses a greedy search algorithm to find the optimal solution of the optimization problem, and the algorithm steps are as follows:
(1) Setting the sample size n, the dimension p and the slice number T;
(2) A set of latin hypercube slices is generated,
(3) The slices are numbered, { pi (1), pi (2), …, pi (T) };
(4) The integers r=1, 2, …, T are arbitrarily set, let k=1, pi (1) =r, s=lhs π(1) (m,p)；
(5) Marking the 1 st slice as browsed and T-1 slices as unviewed;
and (3) circulation: when k is less than T, let F best ＝0,k＝k+1；
Let j = all unviewed;
constructing a new set S π(j) ＝S∪LHS π(j) (m,p)；
Evaluating the objective function value F (S) by equation (9) π(j) )；
If F (S) π(j) )＞F best ；
F best ＝F(S π(j) ),π(k)＝π(j)；
(6) Output slice sample arrangement
S74, finding out the space filling characteristic of optimizing the slice arrangement sequence through a greedy search algorithm, selecting a newly added slice to form a newly added sample point subset, and executing the step S5.
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