基于Kuramoto耦合振子理论的孤岛微网系统建模

doi:10.11930/j.issn.1004-9649.202404032

中国电力 ›› 2025, Vol. 58 ›› Issue (11): 193-204.DOI: 10.11930/j.issn.1004-9649.202404032

基于Kuramoto耦合振子理论的孤岛微网系统建模

李成¹(), 张海昱²(), 宋蕙慧²(), 曲延滨², 张新³, 李盈盈³

1. 国家电投集团江苏电力有限公司，江苏南京　210008
2. 哈尔滨工业大学（威海）新能源学院，山东威海　264209
3. 国家电投集团科学技术研究院，北京　102200

收稿日期:2024-04-07 修回日期:2024-12-09 发布日期:2025-12-01 出版日期:2025-11-28
作者简介:
李成（1966），男，高级工程师，从事电力系统电气一二次设备、新能源光伏风电运行维护研究，E-mail：ept_lc@163.com
张海昱（2000），女，硕士研究生，从事多能互补系统和配电网优化调度研究，E-mail：zhanghaiyu17@163.com
宋蕙慧（1983），女，通信作者，副教授，从事新能源发电、储能系统的非线性控制研究，E-mail：songhh@hitwh.edu.cn
基金资助:
国家自然科学基金资助项目（微网系统Kuramoto建模与基于能量成型的无缝切换调频控制方法研究，61773137）。

Islanded Microgrid System Modeling Based on Kuramoto Coupled Oscillator Theory

LI Cheng¹(), ZHANG Haiyu²(), SONG Huihui²(), QU Yanbin², ZHANG Xin³, LI Yingying³

1. State Power Investment Group Jiangsu Electric Power Co., Ltd., Nanjing 210008, China
2. School of New Energy, Harbin Institute of Technology at Weihai, Weihai 264209, China
3. New Energy Technology Research Institute, State Power Investment Corporation Science and Technology Research Institute, Beijing 102200, China

Received:2024-04-07 Revised:2024-12-09 Online:2025-12-01 Published:2025-11-28
Supported by:
This work is supported by National Natural Science Foundation of China (Kuramoto Modeling and Energy-Shaping Seamless Frequency Control for Microgrid, No.61773137)

摘要/Abstract

摘要：

为深入揭示同步现象下多种复杂系统隐含的集体动力学规律，引入Kuramoto模型以描述具有自然频率分布的全局正弦耦合相振子系统的演化过程。本文从振子同步的角度提出了孤岛微网系统的Kuramoto建模方法，阐明了微网各单元频率同步的暂态特性。首先，深入分析微网各单元运行特性和Kuramoto振子动态特性的相似性，根据运行特性差异建立其Kuramoto模型。然后，进一步根据其惯性特性推导出按惯性分类的Kuramoto振子动态方程，并构建一、二阶混合非均匀Kuramoto模型对系统整体进行表征。最后，基于Simulink仿真平台搭建3机5节点的微网系统，验证了Kuramoto同步理论在微网建模中的适用性，揭示了Kuramoto模型频率同步与微网由暂态到稳态过程的相似性，为暂态稳定性的定量分析以及暂态能量函数的构造奠定理论基础。

关键词: 孤岛微网, 耦合振子, 暂态稳定性, 同步, Kuramoto模型

Abstract:

To further explore the implicit collective dynamics within the synchronization phenomenon of various complex systems, the Kuramoto model is introduced to describe the evolution of globally sinusoidal coupled phase oscillator systems with natural frequency distributions. This paper proposes a Kuramoto modeling method for island microgrid systems from the perspective of oscillator synchronization, clarifying the transient characteristics of frequency synchronization among different units in the microgrid. Firstly, a deep analysis is conducted to explore the similarities between the operational characteristics of each unit in the microgrid and the dynamic characteristics of Kuramoto oscillators, and a Kuramoto model is established based on the differences in operational characteristics. Then, the dynamic equations of Kuramoto oscillators classified by inertia are derived according to their inertial properties, and a first- and second-order mixed inhomogeneous Kuramoto model is constructed to represent the entire system. Finally, a 3-machine 5-node microgrid system is built on the Simulink simulation platform to verify the applicability of the Kuramoto synchronization theory in microgrid modeling. The similarity between the frequency synchronization of the Kuramoto model and the transient-to-steady-state process of the microgrid is revealed, laying a theoretical foundation for quantitative analysis of transient stability and the construction of transient energy functions.

Key words: islanded microgrid, coupled oscillators, transient stability, synchronization, kuramoto model

李成, 张海昱, 宋蕙慧, 曲延滨, 张新, 李盈盈. 基于Kuramoto耦合振子理论的孤岛微网系统建模[J]. 中国电力, 2025, 58(11): 193-204.

LI Cheng, ZHANG Haiyu, SONG Huihui, QU Yanbin, ZHANG Xin, LI Yingying. Islanded Microgrid System Modeling Based on Kuramoto Coupled Oscillator Theory[J]. Electric Power, 2025, 58(11): 193-204.

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链接本文: https://www.electricpower.com.cn/CN/10.11930/j.issn.1004-9649.202404032

https://www.electricpower.com.cn/CN/Y2025/V58/I11/193

图/表 22

图 1 机械模拟Kuramoto弹簧振子模型

Fig.1 Mechanical simulation of Kuramoto spring oscillator model

图 2 同步发电机静态频率特性

Fig.2 Static frequency characteristics of synchronous generator

图 3 $ P - {\omega ^{\text{r}}} $下垂特性曲线

Fig.3 $ \boldsymbol{P}-\boldsymbol{\omega} $r droop characteristic curve

图 4 综合负荷的静态频率特性

Fig.4 Static frequency characteristics of common load

表 1 微网系统单元组件划分

Table 1 Element component partition in microgrid

有无惯性	发电单元	负荷单元	馈线单元	储能单元
有	小型柴油发电机；双轴结构的微型燃气轮机	电机型负荷	—	—
无	变速风力发电；光伏发电；单轴结构的微型燃气轮机	频率型负荷；恒功率型负荷；恒导纳型负荷	内部馈线 PCC开关	飞轮储能；蓄电池储能；超级电容；燃料电池

图 5 惯性单元振子模型

Fig.5 Oscillator model of the inertial unit

图 6 无惯性单元振子模型

Fig.6 Kuramoto oscillator model without inertia

表 2 典型无惯性单元Kuramoto振子模型

Table 2 Special Kuramoto oscillator models without inertia

典型无惯性单元		模型
微网PCC开关节点		微网并网模式： $0=P_{i}-\displaystyle\sum_{i=1}^{N} E_{i} E_{j}\left\|Y_{ij}\right\| \sin (\theta_{i}-\theta_{j}+\varphi_{ij}) $ 恒功率DER单元
微网PCC开关节点		微网孤岛模式： $ 0=\displaystyle\sum_{i=1}^{N} E_{i} E_{j}\left\|Y_{ij}\right\| \sin (\theta_{i}-\theta_{j}+\varphi_{ij}) $ 馈线单元
恒功率型负荷模型		$0=-P_{{\rm L}, i}-\displaystyle\sum_{j=1}^{N} E_{i} E_{j}\left\|Y_{ij}\right\| \sin (\theta_{i}-\theta_{j}+\varphi_{ij}) $
恒电纳型负荷模型		Kron化简

表 4 各节点参数

Table 4 Parameters of each node

节点	节点特性	M_i（×10⁴）	D_i（×10⁵）	P_M,i（×10⁴）	E_i
2	风力发电单元	0	0.56	1	310
3	柴油发电机	0.61	0.83	2	310
4	储能单元	0	1.60	3	310
5	综合负荷单元	1.51	0.60	–6	310

图 7 微网系统网络拓扑

Fig.7 Topology of microgrid system

表 3 线路阻抗参数

Table 3 Line impedance parameters

线路	节点	单位阻抗大小/(Ω·km^–1)	长度/km
1	1-2	0.0641+j2.68×10^–3	100
2	2-3	0.0641+j2.68×10^–3	2
3	2-4	0.0641+j2.68×10^–3	2
4	2-5	0.0641+j2.68×10^–3	1
5	3-4	0.0641+j2.68×10^–3	2
6	3-5	0.0641+j2.68×10^–3	1
7	4-5	0.0641+j2.68×10^–3	1

表 5 各节点之间的参数（$ {\varphi _{ij}} = \arctan ({G_{ij}}/{B_{ij}}) $）

Table 5 Parameters between nodes

节点号	R + jX	a_ij（×10⁴）	φ_ij
2-3	1.282+j5.36×10^–3	3.77	–0.3
2-4	1.282+j5.36×10^–3	3.77	–0.4
2-5	0.0641+j2.68×10^–3	7.54	0.3
3-4	1.282+j5.36×10^–3	3.77	–0.1
3-5	0.0641+j2.68×10^–3	7.54	0.6

图 8 微网系统频率同步过程对比

Fig.8 Comparison of frequency synchronization process in microgrid systems

图 9 微网各节点Kuramoto模型频率同步

Fig.9 Frequency synchronization of microgrid Kuramoto models

图 10 微网暂态稳定情况下Kuramoto模型相轨迹

Fig.10 Phase trajectories of microgrid Kuramoto model under transient stability

图 11 3机微网结构

Fig.11 The structure of microgrid with three DGs

图 12 负荷变化时系统频率仿真对比

Fig.12 System Frequency Comparison Simulation under Load Variations

图 13 负荷变化时各DG单元频率仿真

Fig.13 Frequency Simulation of DG Units with Load Variations

图 14 微网频率在扰动下失稳

Fig.14 Frequency instability of microgrid under pulse disturbance

图 15 微网各单元Kuramoto振子频率失同步

Fig.15 Frequency desynchronization of microgrid Kuramoto models

图 16 风电机组脱机产生的功率扰动

Fig.16 Power disturbance caused by off-line wind turbine

图 17 微网失稳时Kuramoto模型相轨迹

Fig.17 Phase trajectories of microgrid Kuramoto model in the case of instability

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基于Kuramoto耦合振子理论的孤岛微网系统建模

Islanded Microgrid System Modeling Based on Kuramoto Coupled Oscillator Theory

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基于Kuramoto耦合振子理论的孤岛微网系统建模

Islanded Microgrid System Modeling Based on Kuramoto Coupled Oscillator Theory

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