Volume Previous issueNext issue. 10th International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE ). results in a probability density function or PDF for short. density estimation [Cra46] in the form of Edgeworth and Gram-Charlier.
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Indeed, even for a strongly mixing normal process, the distribution is unknown. Here, we do not assume any other assumption than a sufficiently fast decrease of the underlying distribution to make the Edgeworth expansion convergent. This results can obviously apply to strongly mixing normal process and provide an alternative to the work of Moschopoulos and Mathai Eric Benhamou.
Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday. Let X 1 , … , X n be a random sample and define the sample variance statistic as:. We are interested in the distribution of the sample variance under very weak conditions, namely that it admits a valid Edgeworth expansion.
It is insightful to notice that even with the additional constraint of a multi dimensional Gaussian distribution. In this particular setting, the sample variance is the squared norm of a multi dimensional Gaussian and can be seen as the linear combination of independent but not Homoscedastic variables. Standard theory states that the sample variance of a collection of independent and identically distributed normal variables follows a chi-squared distribution.
But in this particular case, the different variables. In other cases, the sample variance is a linear combination of Gamma distribution , and one has to rely on approximations as explained in. The series are the same but, they differ in the ordering of their terms. Hence the truncated series are different, as well as the accuracy of truncating the series. The key idea in these two series is to expand the characteristic function in terms of the characteristic function of a known distribution with suitable properties, and to recover the concerned distribution through the inverse Fourier transform.
In our case, a natural candidate to expand around is the normal distribution as the central limit theorem and its different extensions to non independent and non identically distributed variable state that the resulting distribution is a normal distribution or in the most general case to truncated symmetrical and. Cumulants definition state. By expanding the exponential and collecting terms according to the order of the derivatives, we arrive at the Gram—Charlier A series.
Such an expansion can be written compactly in terms of Bell polynomials as. If we include only the first two correction terms to the normal distribution, we obtain. If in the above expression, the cumulant are function of a parameter 1 n , we can rearrange terms by power of 1 n and find the Edgeworth expansion. The PDFs of active load power and wind power at bus 7 are depicted by histograms as shown in Figs.
According to 12 , the relationship between WAR and the number of clusters can be obtained, as shown in Fig. It can be observed that the WAR declines slowly after the number of clusters is more than This implies that the clustering performance will not significantly improve when the number of clusters is above Therefore, the K value is suggested to be Table 3 shows the clustering results of the K -means algorithm.
After clustering, input random variable samples are grouped into 40 clusters. The variance can reflect the fluctuation of one random variable. For each cluster, the variance of the random variable is calculated. As a result, 40 variance values corresponding to 40 clusters are obtained.
Among these 40 values, we choose the minimum, the maximum and the mean value to present the fluctuation level of each input random variable in each cluster. The chosen values are labelled as S min , S max and S mean , respectively. The column labelled S is the variance of a specific input random variable for the original total samples. It can be observed that the variances after the improved K -means clustering are much smaller than those for the original whole samples.
Table 4 lists the results of different methods used to solve PLF problems for this test system. The results are aggregated into: VA which stands for voltage angles, VM which stands for voltage magnitudes, PL which stands for line active power flows, and QL which stands for line reactive power flows, since it is difficult to present all output variables individually.
The mean and maximum values of the APE values are shown to demonstrate the scope of APE values for a class of variables. This variable may mislead the comparison on APE and should not be applied to assess the performance of different methods. There are two points to be pointed out about the CCM. The second, third and fourth cumulants can reflect the variance, skewness and kurtosis of an output random variable, respectively. Figures 7 and 8 show the PDFs of the VM at bus 9 and the active power flow in line 7—8, respectively.
From the comparison in Figs. It should be pointed out that in Fig. However, the actual ARMS values are only 0. It can be seen that the CCM perform worse on reactive quantities than active quantities. This characteristic can also be observed from the results of cumulants and PDFs.
The reason is that reactive quantities generally have higher degree of non-linearity than active quantities. Moreover, the computation time of each method consumes and the number of deterministic power flow DPF calculations conducted by each method are shown in Table 5. An additional experiment is conducted to examine the performance of the proposed method with more clusters, where the proposed method with 60 clusters is implemented on this test system.
The results indicate that the proposed method with 60 clusters is more accurate than 40 clusters. These values obtained using 60 clusters are 1. It can be seen that more clusters will produce more accurate results. Obviously, more clusters will require more computation time. In addition, more clusters can achieve higher accuracy at the expense of efficiency. The modified IEEE bus test system is used to examine the feasibility of the proposed method for a large system with multiple wind farms.
Table 6 lists the shape and scale parameters of wind speed distributions [ 36 ]. The correlations of wind speeds at buses 17 and 30, buses 59 and 80, and buses 92 and are set to be 0. All loads have constant power factors. The relationship between WAR and the number of clusters for case 2 can be obtained using The number of clusters is suggested to be In this test system, there are input random variables, including 6 wind power outputs and 99 load demands.
Therefore, the dimensionality reduction based on SVD is applied in the K -means process, where the first six singular values are selected and their sum is equal to It can be seen that the K -means algorithm achieves an efficiency improvement through the dimensionality reduction based on SVD. Table 7 presents the results of cumulants.
The reason is that the Cholesky decomposition algorithm used to handle correlations has some errors for high-order cumulants when input random variables are non-normal distributions. The proposed method is based on the clustering algorithm. Theoretically, the result of clustering is the local optimal solution, which is influenced by the initial cluster centers. In order to examine the stability of the proposed method, probabilistic power flow for the modified IEEE bus test system is conducted times with random initial cluster centers.
The APE values of the first four cumulants of each type of variables obtained in each simulation are summed and averaged. It can be seen that the errors in Table 9 are approximately equal to the corresponding values of the proposed method in Table 7 , which demonstrates that the proposed method can achieve stable and accurate results.
A novel PLF method considering large-scale wind power integration is proposed in this paper. In the process of the proposed method, an improved K -means algorithm is used to cluster the samples of input random variables, and the law of total probability is applied to combine the results obtained in each cluster. From the case studies on modified IEEE 9-bus and bus test systems, some conclusions are drawn as follows:. In other words, the proposed method can achieve a better performance with consideration of both computational efficiency and accuracy.
More clusters will produce more accurate results at the expense of time. The suggested number of clusters should be determined in advance. The traditional CM considering the correlation of input random variables generally has significant errors for reactive quantities. In conclusion, as the proposed method has been tested on the small and large test systems, it can provide an accurate and efficient tool for power system planning and operation with large-scale wind power.
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Rebew Sust Energ Rev — Download references. You can also search for this author in PubMed Google Scholar. Reprints and Permissions. DENG, X. Probabilistic load flow method considering large-scale wind power integration. Power Syst. Clean Energy 7, — Download citation. Received : 04 July Accepted : 06 December Published : 28 February Issue Date : 13 July Anyone you share the following link with will be able to read this content:. Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative. Skip to main content. Search SpringerLink Search. Download PDF. Abstract The increasing penetration of wind power brings great uncertainties into power systems, which poses challenges to system planning and operation. Introduction Load flow study is a vital tool for power system planning and operation. Proposed method The fundamental reason why the traditional CM has high errors for solving PLF of power systems containing large-scale wind power is that the wind power output can change significantly over time due to fluctuations of wind speeds.
Samples of input variables. Full size image. Overall procedure of improved K -means algorithm. Flow chart of proposed method. Case study The proposed method, namely the improved K -means based cumulant method IKCM , is tested on modified IEEE 9-bus and bus test systems [ 35 ], which are integrated with additional wind farms. Table 1 Particulars of wind farms Full size table. Table 2 Parameters of wind speeds case 1 Full size table. PDF of active load power at bus 7. PDF of wind power at bus 7.
Relationship between WAR and number of clusters. Table 3 Comparison of variances of input variables Full size table. Table 4 Comparison of first four cumulants case 1 Full size table. PDFs of VM at bus 9. PDFs of PL in line Table 5 Comparison of computation time case 1 Full size table. Table 6 Parameters of wind speeds case 2 Full size table.
Table 7 Comparison of first four cumulants case 2 Full size table. PDFs of QL in line Table 8 Comparison of computation time case 2 Full size table.
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