A Nataf Transformation Based on Extended Quasi Monte Carlo Simulation Method for Solving Probabilistic Load Flow Problems with Correlated Random Variables
Fang Sidun1, Cheng Haozhong1, Xu Guodong1, Yao Liangzhong2, Zeng Pingliang2
1. School of Electronic Information and Electrical Engineering Shanghai Jiao Tong University Shanghai 200240 China; 2. China Electric Power Research Institute Beijing 100192 China
Abstract：For probabilistic load flow (PLF) analysis, quasi Monte Carlo (QMC) simulation method has higher efficiency than Latin Hypercube Sampling, and its extension is more flexible. Therefore, a Nataf transformation based extended QMC (NEQM) method is proposed to solve the PLF problems. During the process, the extended procedure enlarges the sample size with arbitrary step, and then the Nataf transformation is performed to reconstruct the probability distributions of input variables. With the increase of sample size, this method retains the load flow results already obtained. Furthermore, singular value decomposition method is employed to control the correlation matrix before and after the extension in place of Cholesky decomposition, which is able to handle the non-positive definite correlation matrixes, simultaneously. Through IEEE 30 and IEEE 118 bus systems, the proposed method shows its efficiency and accuracy in comparison with Simple Random Sampling and Nataf Transformation based Extended LHS. The simulation results suggest that, NEQM enhance the accuracy of obtained variables especially the standard deviations with shorter consuming time.
方斯顿, 程浩忠, 徐国栋, 姚良忠, 曾平良. 基于Nataf变换含相关性的扩展准蒙特卡洛随机潮流方法[J]. 电工技术学报, 2017, 32(2): 255-263.
Fang Sidun, Cheng Haozhong, Xu Guodong, Yao Liangzhong, Zeng Pingliang. A Nataf Transformation Based on Extended Quasi Monte Carlo Simulation Method for Solving Probabilistic Load Flow Problems with Correlated Random Variables. Transactions of China Electrotechnical Society, 2017, 32(2): 255-263.
[1] Ackermann T. Wind power in power systems[M]. New York: Wiley, 2012. [2] Borkowska B. Probabilistic load flow[J]. IEEE Transactions on Power Apparatus and Systems, 1974, 27(3): 752-759. [3] Fang S D, Cheng H Z, Song Y. Stochastic optimal reactive power dispatch method based on point estimation considering load margin[C]//IEEE Pro- ceedings of Power Engineering Society Meeting, Washington DC, USA, 2014: 1-5. [4] 茆美琴, 周松林, 苏建徽. 基于风光联合概率分布的微电网概率潮流预测[J]. 电工技术学报, 2014, 29(2): 55-63. Mao Meiqin, Zhou Songlin, Su Jianhui. Probabilistic power flow forecasting of microgrid based on joint probability distribution about wind and irradiance[J]. Transactions of China Electrotechnical Society, 2014, 29(2): 55-63. [5] 刘小团, 赵晋泉, 罗卫华, 等. 基于 TPNT 和半不变量法的考虑输入量相关性概率潮流算法[J]. 电力系统保护与控制, 2013, 41(22): 13-18. Liu Xiaotuan, Zhao Jinquan, Luo Weihua, et al. A TPNT and cumulants based probabilistic load flow approach considering the correlation variables[J]. Power System Protection and Control, 2013, 41(22): 13-18. [6] 苏胜皓, 宋锐, 陈庆芳, 等. 基于序列运算的含分布式发电配电网潮流分析[J]. 电力系统保护与控制, 2014, 42(24): 12-17. Su Shenghao, Song Rui, Chen Qingfang, et al. Power flow analysis of distribution network containing distributed generation based on sequence operation[J]. Power System Protection and Control, 2014, 42(24): 12-17. [7] 宣锐峰, 王亚楠, 万要军, 等. 基于Faure序列的电力系统概率潮流计算[J]. 电力系统保护与控制, 2015, 43(20): 15-20. Xuan Ruifeng, Wang Yanan, Wang Yaojun, et al. Probabilistic power flow calculation based on faure sequence with wind farms[J]. Power System Protection and Control, 2015, 43(20): 15-20. [8] 任洲洋, 颜伟, 项波, 等. 考虑光伏和负荷相关性的概率潮流计算[J]. 电工技术学报, 2015, 30(24): 181-187. Ren Zhouyang, Yan Wei, Xiang Bo, et al. Probabilistic power flow analysis incorporating the correlations between PV power outputs and loads[J]. Transactions of China Electrotechnical Society, 2015, 30(24): 181-187. [9] 吴巍, 汪可友, 韩蓓, 等. 基于Pair Copula的随机潮流三点估计法[J]. 电工技术学报, 2015, 30(9): 121-128. Wu Wei, Wang Keyou, Han Bei, et al. Pair Coupla based three-point estimate method for probabilistic load flow calculation[J]. Transactions of China Electrotechnical Society, 2015, 30(9): 121-128. [10] 蔡德福, 石东源, 陈金富. 基于多项式正态变换和拉丁超立方采样的概率潮流计算方法[J]. 中国电机工程学报, 2013, 33(13): 92-101. Cai Defu, Shi Dongyuan, Chen Jinfu. Probabilistic load flow calculation method based on polynomial normal transformation and Latin Hypercube sampling[J]. Proceedings of the CSEE, 2013, 33(13): 92-101. [11] 李俊芳, 张步涵. 基于进化算法改进拉丁超立方抽样的概率潮流计算[J]. 中国电机工程学报, 2011, 31(25): 90-96. Li Junfang, Zhang Buhan. Probabilistic load flow based on improved Latin Hypercube sampling with evolutionary algorithm[J]. Proceedings of the CSEE, 2011, 31(25): 90-96. [12] 丁明, 王京景, 李生虎. 基于扩展拉丁超立方的采样的电力系统概率潮流计算[J]. 中国电机工程学报, 2013, 33(4): 163-170. Ding Ming, Wang Jingjing, Li Shenghu. Pro- babilistic load flow evaluation with extended Latin Hyperbube sampling[J]. Proceedings of the CSEE, 2013, 33(4): 163-170. [13] Niederreiter H. Random number generation and quasi-Monte Carlo methods[M]. Philadelphia, PA: SIAM, 1992. [14] Owen A B. Controlling correlations in Latin Hypercube sampling[J]. Journal of the American Statistical Association, 1994, 89(428): 1517-1522. [15] Singhee A, Rutenbar R A. Why quasi-Monte Carlo is better than Monte Carlo or Latin hypercube sampling for statistical circuit analysis[J]. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2010, 29(11): 1763-1776. [16] Zou Bin, Xiao Qing. Solving probabilistic optimal power flow problem using quasi Monte Carlo method and ninth-order polynomial normal transformation[J]. IEEE Transactions on Power Systems, 2014, 29(1): 300-306. [17] Breipohl A, Lee F N, Huang J, et al. Sample size reduction in stochastic production simulation[J]. IEEE Transactions on Power Systems, 1990, 5(3): 984-992. [18] Jirutitijaroen P, Singh C. Comparison of simulation methods for power system reliability indexes and their distributions[J]. IEEE Transactions on Power Systems, 2008, 23(2): 486-493. [19] Iman R L, Conover W J. A distribution-free approach to inducing rank correlation among input variables[J]. Communications in Statistics-Simulation and Com- putation, 1982, 11(3): 311-334. [20] Vořechovský M, Novák D. Correlation control in small-sample Monte Carlo type simulations I: a simu- lated annealing approach[J]. Probabilistic Engin- eering Mechanics, 2009, 24(3): 452-462. [21] Liefvendahl M, Stocki R. A study on algorithms for optimization of Latin hypercubes[J]. Journal of Statistical Planning and Inference, 2006, 136(9): 3231-3247. [22] Tong C. Refinement strategies for stratified sampling methods[J]. Reliability Engineering and System Safety, 2006, 91(S10-11): 1257-1265. [23] Sallaberry C J, Helton J C, Hora S C. Extension of Latin hypercube samples with correlated variables[J]. Reliability Engineering and System Safety, 2008, 93(7): 1047-1059. [24] Chen X, Tung Y K. Investigation of polynomial normal transform[J]. Structural Safety, 2003, 25(4): 423-445. [25] Chen Yan, Wen Jinyu, Cheng Shijie. Probabilistic load flow method based on Nataf transformation and Latin Hypercube sampling[J]. IEEE Transactions on Power Systems, 2013, 4(2): 294-301. [26] Kiureghian A D, Asce M, Liu P L. Structural reliability under incomplete probability infor- mation[J]. Joumal of Engineering Mechanics, 1986, 112(1): 721-740. [27] Liu P L, Kiureghian A D. Multivariate distribution models with prescribed marginals and covariances[J] Probabilistic Engineering Mechanics, 1986, 1(2): 105-112. [28] Shuang L H, Zhou L Z, Kai Y X. Nataf trans- formation based point estimate method[J]. Chinese Science Bulletin, 2008, 53(17): 2586-2592. [29] Power system engineering research matpower4.1[Z]. http://www.pserc.cornell.edu/matpower/.
2017, Vol. 32 
