基于复杂网络理论的互联电网Braess悖论现象分析

doi:10.19596/j.cnki.1001-246x.8387

摘要/Abstract

摘要：

为了探究互联电力网络中的Braess悖论现象, 采用二阶类Kuramoto相振子模型对电网进行动力学建模, 将两个子网通过大度节点相连构建互联电网。当两个子网间有功率传输时, 分别在两个子网内部新增传输线路探究互联电网发生Braess悖论现象的概率并分析其原因。研究发现: 当互联电网中两个子网间的功率传输达到某一临界值时, 受电子网的同步能力远优于供电子网的同步能力, 供电子网新增传输线路引起互联电网发生Braess悖论的概率远高于受电子网新增传输线路引发的Braess悖论概率。通过定义子网序参数对上述现象的产生进行深入分析。本研究对互联电网的拓扑优化具有重要意义。

关键词: 复杂网络, 互联电网, 同步, Braess悖论

Abstract:

To explore Braess paradox phenomenon in an interconnected power grid, a second order Kuramoto-like phase oscillator model is applied to model dynamics of the power grid. The two subnets are connected by the largest degree nodes to build an interconnected power grid. As power transmission between two subnets occurs, new transmission lines are added in the two subnets repectively to explore the probability of Braess paradox phenomenon and analyze the reasons. It is found that as the power transmission between the two subnets reaches a critical value, synchronizability of the power receiving subnet is much better than that of the power supply subnet. The probability of Braess paradox caused by adding a new transmission line in the power supply subnet is much higher than that caused by adding a new transmission line in the power receiving subnet. The phenomena are analyzed in depth by defining subnet order parameter. This study is important for the topology optimization of a interconnected power grid.

Key words: complex network, an interconnected power grid, synchronization, Braess paradox

张少泽, 邹艳丽, 谭秫毅, 李浩乾, 刘欣妍. 基于复杂网络理论的互联电网Braess悖论现象分析[J]. 计算物理, 2022, 39(2): 233-243.

Shaoze ZHANG, Yanli ZOU, Shuyi TAN, Haoqian LI, Xinyan LIU. Analysis of Braess Paradox in an Interconnected Power Grid Based on Complex Network Theory[J]. Chinese Journal of Computational Physics, 2022, 39(2): 233-243.

图/表 9

图1 IEEE14系统原网络及新增传输线路后网络频偏随时间的演化(a)原网络；(b)新增(1, 11)传输线路后网络

Fig.1 Frequency offset evolution of the original network of IEEE14 system and the network adding a new transmission line (a) original network; (b) network adding (1, 11) transmission line

表1 IEEE14系统原网络及新增(1, 11)传输线路后网络的稳态频偏

Table 1 Steady state frequency offset of the original network and the network adding (1, 11) transmission line in IEEE14 system

网络	稳态频偏
原网络	0.00
新增(1, 11)传输线路	2.57

图2 互联电网拓扑图(a)IEEE14-14互联电网；(b)IEEE14-30互联电网

Fig.2 Topology of interconnected power grids (a) IEEE14-14 interconnected power grid; (b) IEEE14-14 interconnected power grid

图3 互联电网中子网新增传输线路发生Braess悖论现象的概率随传输功率变化(a)IEEE14-14子网A；(b)IEEE14-14子网B；(c)IEEE14-30子网A；(d)IEEE14-30子网B

Fig.3 Probability of Braess paradox phenomenon as adding a transmission line in a subnet in an interconneted power grid as functions of transmission power (a)IEEE14-14 subnet A; (b)IEEE14-14 subnet B; (c)IEEE14-30 subnet A; (d)IEEE14-30 subnet B

图4 IEEE14-14互联电网稳态序参数以及供电、受电子网的稳态序参数随耦合强度的变化

Fig.4 Steady state order parameters of IEEE14-14 interconnected power grid, the power supply subnet and the power receiving subnet as functions of the coupling strength

图5 分别在供电子网和受电子网新增传输线路，IEEE14-14互联电网同步情况((a)、(b)、(c)在供电子网新增传输线路，(d)、(e)、(f)在受电子网新增传输线路。) (a)IEEE14-14互联电网；(b)IEEE14-14供电子网；(c)IEEE14-14受电子网；(d)IEEE14-14互联电网；(e)IEEE14-14供电子网；(f)IEEE14-14受电子网

Fig.5 Synchronizability of IEEE14-14 interconnected power grid with a new transmission line in the power supply subnet and in the power receiving subnet, respectivlely (In (a), (b) and (c) the new transmission line is added in the power supply subnet, and in (d), (e) and (f) the new transmission line is added in the power receiving subnet.) (a) IEEE14-14 interconnected power grid; (b) Power supply subnet of IEEE14-14; (c) Power receiving subnet of IEEE14-14; (d) IEEE14-14 interconnected power grid; (e) Power supply subnet of IEEE14-14; (f) Power receiving subnet of IEEE14-14

图6 IEEE14-30互联电网稳态序参数以及供电、受电子网的稳态子网序参数随耦合强度的变化

Fig.6 Steady state order parameter of IEEE14-30 interconnected power grid, the power supply subnet and the power receiving subnet as functions of the coupling strength

图7 分别在供电子网和受电子网新增传输线路，IEEE14-30互联电网同步情况((a)、(b)、(c)在供电子网新增传输线路，(d)、(e)、(f)在受电子网新增传输线路。) (a)IEEE14-30互联电网；(b)IEEE14-30供电子网；(c)IEEE14-30受电子网；(d)IEEE14-30互联电网；(e)IEEE14-30供电子网；(f)IEEE14-30受电子网

Fig.7 Synchronizability of IEEE14-30 interconnected power grid with a new transmission line in the power supply subnet and in the power receiving subnet, respectivlely (In (a), (b) and (c) the new transmission line is addded in the power supply subnet, and in (d), (e) and (f) the new transmission line is added in the power receiving subnet.) (a) IEEE14-30 interconnected power grid; (b) Power supply subnet of IEEE14-30; (c) Power receiving subnet of IEEE14-30, (d) IEEE14-30 interconnected power grid; (e) Power supply subnet of IEEE14-30; (f) Power receiving subnet of IEEE14-30

图8 互联电网中子网新增传输线路发生Braess悖论现象的概率随传输功率变化 (a)IEEE57-57子网A；(b)IEEE57-57子网B

Fig.8 Probability of the Braess paradox phenomenon with a transmission line in a subnent in an interconneted power grid as functions of the transmission power (a) IEEE57-57 subnet A; (b) IEEE57-57 subnet B

参考文献 27

1	BRAESS D . Über ein paradoxon aus der verkehrsplanung[J]. Mathematical Methods of Operational Research, 1968, 12 (1): 258- 268. DOI
2	BRAESS D , NAGURNEY A , WAKOLBINGER T . On a paradox of traffic planning[J]. Transportation Science, 2005, 39 (4): 446- 450. DOI
3	鲁丛林, 蔡宁. Braess's paradox的博弈论分析[J]. 交通科学与工程, 2003, 19 (3): 69- 72. DOI
4	COLOMBO R M , HOLDEN H . On the Braess paradox with nonlinear dynamics and control theory[J]. Journal of Optimization Theory and Applications, 2016, 168 (1): 216- 230. DOI
5	XUE J, GEHLOT H, UKKUSURI S V. Braess's paradox in scale-free networks[C]. The 8th International Symposium on Dynamic Traffic Assignment, 2020.
6	MA C , CAI Q , ALAM S , et al. Airway network management using Braess's paradox[J]. Transportation Research: Part C, 2019, 105 (1): 565- 579.
7	张国宝. "西电东送"工程: 中国电力史上的重要篇章[J]. 中国经济周刊, 2019, 1 (12): 48- 52.
8	李飏, 周永章. 珠江流域"西电东送"能源空间配置的综合效益分析[J]. 中国人口·资源与环境, 2008, (3): 147- 151. DOI
9	朱法华, 王圣, 刘思湄. 中国电力规划环评的发展与建议[J]. 电力科技与环保, 2007, 23 (5): 9- 13. DOI
10	WITTHAUT D , TIMME M . Nonlocal failures in complex supply networks by single link additions[J]. The European Physical Journal B, 2013, 86 (9): 1- 12.
11	WITTHAUT D , TIMME M . Braess's paradox in oscillator networks, desynchronization and power outage[J]. New Journal of Physics, 2012, 14 (1): 83036- 83051.
12	COLETTA T , JACQUOD P . Linear stability and the Braess paradox in coupled-oscillator networks and electric power grids[J]. Physical Review E, 2016, 93 (3): 032222. DOI
13	BAILLIEUL J, ZHANG B, WANG S. The Kirchhoff-Braess paradox and its implications for smart microgrids[C]. 2015 54th IEEE Conference on Decision and Control, 2015: 6556-6563.
14	TCHAWOU T E B , GOMILA DAMIÀ , PERE C , et al. Curing Braess' paradox by secondary control in power grids[J]. New Journal of Physics, 2018, 20 (8): 083005. DOI
15	WANG X, KOC Y, KOOIJ R E, et al. A network approach for power grid robustness against cascading failures[J]. Physics and Society, 2015, arXiv: 1505.06312.
16	FAZLYAB M, DORFLER F, PRECIADO V M, et al. Optimal network design for synchronization of Kuramoto oscillators[J]. Optimization and Control, 2015, arXiv: 1503.07154.
17	STROGATZ S H . Exploring complex networks[J]. Nature, 2001, 410 (6825): 268- 276. DOI
18	BOCCALETTI S , LATORA V , MORENO Y , et al. Complex networks: Structure and dynamics[J]. Physics Reports, 2006, 175- 308.
19	汪小帆, 李翔, 陈关荣. 复杂网络理论及其应用[M]. 北京: 清华大学出版社, 2006.
20	方锦清. 非线性网络的动力学复杂性研究的若干进展[J]. 自然科学进展, 2007, 17 (7): 841- 857. DOI
21	ZHOU J , YU X , LU J A . Node importance in controlled complex networks[J]. IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 2019, 66 (3): 1- 1. DOI
22	FILATRELLA G , NIELSEN A H , PEDERSEN N F . Analysis of a power grid using a Kuramoto-like model[J]. European Physical Journal B: Condensed Matter, 2008, 61 (4): 485- 491.
23	ROHDEN M , SORGE A , WITTHAUT D , et al. Impact of network topology on synchrony of oscillatory power grids[J]. Chaos, 2014, 24 (1): 013123(1-8).
24	ZOU Y , WANG R , WU L , et al. Optimization of synchronization performance in power grids with local order parameters[J]. Chinese Journal of Computational Physics, 2019, 36 (4): 498- 504.
25	ZOU Y , WANG R , GAO Z . Improve synchronizability of a power grid through power allocation and topology adjustment[J]. Physica A: Statistical Mechanics and Its Applications, 2020, 548 (1): 122956.
26	ZOU Y , GAO Z , LIANG M , et al. Optimization of synchronization performance and robustness analysis in power grids based on power tracing[J]. Chinese Journal of Computational Physics, 2020, 37 (5): 623- 630.
27	WANG Y , ZOU Y , HUANG L , et al. Key nodes identification of power grid considering local and global characteristics[J]. Chinese Journal of Computational Physics, 2018, 35 (1): 119- 126.

[1]	高正, 邹艳丽, 胡均万, 姚高华, 刘唐慧美. 基于复杂网络理论的电网耦合强度分配策略[J]. 计算物理, 2022, 39(5): 579-588.
[2]	刘凯歌, 韦笃取. 基于WOA-ESN的电机系统混沌振荡预测[J]. 计算物理, 2022, 39(4): 498-504.
[3]	付韬, 邬龙, 李晨光. 基于生成函数方法的现实网络座键渗流建模[J]. 计算物理, 2022, 39(2): 212-222.
[4]	李浩乾, 邹艳丽, 张少泽, 刘欣妍, 谭秫毅. 基于复杂网络的电网结构健壮性及Braess悖论现象研究[J]. 计算物理, 2021, 38(4): 470-478.
[5]	邹艳丽, 高正, 梁明月, 李志慧, 何铭. 基于潮流追踪的电网同步性能优化及鲁棒性分析[J]. 计算物理, 2020, 37(5): 623-630.
[6]	石建平, 李培生, 刘国平. 基于改进克隆选择算法的统一混沌系统同步控制与参数辨识[J]. 计算物理, 2020, 37(2): 240-252.
[7]	钟国翔, 韦笃取, 张波. 分布式发电系统感性负载的远程混沌同步[J]. 计算物理, 2019, 36(6): 719-725.
[8]	邹艳丽, 王瑞瑞, 吴凌杰, 姚飞, 汪洋. 基于局部序参数的电网同步性能优化[J]. 计算物理, 2019, 36(4): 498-504.
[9]	石建平, 杨兰天, 刘丹. 基于量子粒子群算法的混沌系统同步控制及参数辨识[J]. 计算物理, 2019, 36(2): 236-244.
[10]	查进道, 李春彪. 基于引力场理论的混沌同步[J]. 计算物理, 2018, 35(6): 737-749.
[11]	刘利花, 韦笃取, 张波. 基于动力中继的小世界复杂电机网络混沌同步控制[J]. 计算物理, 2018, 35(6): 750-756.
[12]	王意, 邹艳丽, 黄李, 李可. 综合考虑局部和全局特性的电网关键节点识别[J]. 计算物理, 2018, 35(1): 119-126.
[13]	钱慧, 于洪洁. SC混沌比例投影同步方法在保密通信中的应用[J]. 计算物理, 2016, 33(1): 117-126.
[14]	安海岗. 基于复杂网络的时间序列双变量联动波动[J]. 计算物理, 2014, 31(6): 742-750.
[15]	范文礼, 刘志刚. 一种基于效率矩阵的网络节点重要度评价算法[J]. 计算物理, 2013, 30(5): 714-719.