Analysis of Braess Paradox in an Interconnected Power Grid Based on Complex Network Theory

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

Abstract

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

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.

Figures/Tables 9

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

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

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

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

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

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

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

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

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

References 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.

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