New Integral Inequality Approach on Stability Criteria for Delayed Load Frequency Control Systems
Guo Jianfeng1, Qian Wei1, Wang Nan1, Fei Shumin2
1. School of Electrical Engineering and Automation Henan Polytechnic University Jiaozuo 454000 China; 2. School of Automation Southeast University Nanjing 210096 China
Abstract：In wide-area interconnected power systems, load frequency control (LFC) has a pivotal role in addressing the issue of upholding the variations in system frequency and power exchange between different control areas at desired scheduled values. With the development of smart grid technologies and the emergence of numerous private networks, open communication networks are widely used, which inevitably leads to communication delay. The communication delay is an important factor affecting the stability and performance of LFC system. At present, Lyapunov-Krasovskii (L-K) functional method is one of the main methods to study the stability of LFC system. To this issue, how to construct appropriate L-K functional and estimate the functional derivatives accurately to reduce the conservatism of conclusions is the core problem, and the sustained efforts have been made, but it is far from enough in how to coordinate functional construction with estimating techniques efficiently. In this paper, the stability problem of LFC system with delay influence and load disturbance is further studied, by proposing some new methods, the less conservative stability criteria are obtained, and the influence of controller parameters on the delay margin is analyzed. Firstly, by considering time delay and load disturbance, the LFC system model is established. Secondly, a new L-K functional with augmented vector and multiple integral terms is constructed. The single integral terms and the double integral terms are introduced, which builds more relations among different vectors. Moreover, state vector and its derivative are augmented to deepen the relationships between L-K functional and the system. Besides, integral functionals with double and triple forms are also created to get a better exploitation of delay information, all of which conduce to the stability criteria with less conservatism. Then, in order to cooperate with the constructed functional effectively, two inequalities named delay-dependent-matrix-based and free-matrix-based integral inequality (DDMB and FMB integral inequality) and extended delay-dependent-matrix-based reciprocally convex inequality (EDDMB reciprocally convex inequality) are proposed to estimate the functional derivatives more accurately. Compared with the existing estimation approaches with constant matrices, DDMB and FMB integral inequality employs delay-dependent matrices and utilizes more information of time delay and its derivative, which provides more freedom in reducing the conservatism of the main results. Compared with the existing DDMB reciprocally convex inequality, EDDMB reciprocally convex inequality is more general because the value of delay lower bound is relaxed, which expands its scope of application. In addition, auxiliary-function-based integral inequalities (AFBII) together with relaxed integral inequality are also used, which helps to get less conservative stability conditions. The simulation of typical second-order delay system and single-region delay LFC system are given, and the influence of load disturbance and controller parameters on the upper bound of single-region delayed LFC system under different conditions is analyzed. The following conclusions can be drawn from the simulation analysis: (1) Compared with the research results of some existed literatures, the upper bound of time delay obtained by the proposed method is significantly larger, which indicates that the proposed method can effectively improve the stability margin of time delay and reduce the conservatism of the conclusion. (2) The influence of load disturbance and controller parameters on the upper bound of system delay is obvious, the larger the load disturbance is, the smaller the upper bound of system delay is. When the PI control is applied, the upper bound of time delay decreases with the increase of integral gain, and this trend is more obvious when the proportional gain is smaller. The relation between the upper bound of time delay and the proportional gain is more complicated, the upper bound of time delay increases first and then decreases with the increase of the proportional gain.
郭建锋, 钱伟, 王楠, 费树岷. 基于新积分不等式的时滞负荷频率控制系统稳定性分析[J]. 电工技术学报, 2023, 38(15): 4147-4161.
Guo Jianfeng, Qian Wei, Wang Nan, Fei Shumin. New Integral Inequality Approach on Stability Criteria for Delayed Load Frequency Control Systems. Transactions of China Electrotechnical Society, 2023, 38(15): 4147-4161.
[1] 陈宗遥, 卜旭辉, 郭金丽. 基于神经网络的数据驱动互联电力系统负荷频率控制[J]. 电工技术学报, 2022, 37(21): 5451-5461. Chen Zongyao, Bu Xuhui, Guo Jinli.Neural network based data-driven load frequency control for interconnected power systems[J]. Transactions of China Electrotechnical Society, 2022, 37(21): 5451-5461. [2] 常烨骙, 李卫东, 巴宇, 等. 基于运行安全的频率控制性能评价新方法[J]. 电工技术学报, 2019, 34(6): 1218-1229. Chang Yekui, Li Weidong, Ba Yu, et al.A new method for frequency control performance assessment on operation security[J]. Transactions of China Electrotechnical Society, 2019, 34(6): 1218-1229. [3] 梁煜东, 陈峦, 张国洲, 等. 基于深度强化学习的多能互补发电系统负荷频率控制策略[J]. 电工技术学报, 2022, 37(7): 1768-1779. Liang Yudong, Chen Luan, Zhang Guozhou, et al.Load frequency control strategy of hybrid power generation system: a deep reinforcement learning-based approach[J]. Transactions of China Electrotechnical Society, 2022, 37(7): 1768-1779. [4] 吕永青, 窦晓波, 杨冬梅, 等. 含荷电状态修正和通信延迟的储能电站负荷频率鲁棒控制[J]. 电力系统自动化, 2021, 45(10): 59-67. Lü Yongqing, Dou Xiaobo, Yang Dongmei, et al.Load-frequency robust control for energy storage power station considering correction of state of charge and communication delay[J]. Automation of Electric Power Systems, 2021, 45(10): 59-67. [5] Yang Feisheng, He Jing, Pan Quan.Further improvement on delay-dependent load frequency control of power systems via truncated B-L inequality[J]. IEEE Transactions on Power Systems, 2018, 33(5): 5062-5071. [6] Hua Changchun, Wang Yibo, Wu Shuangshuang.Stability analysis of micro-grid frequency control system with two additive time-varying delay[J]. Journal of the Franklin Institute, 2020, 357(8): 4949-4963. [7] 马燕峰, 霍亚欣, 李鑫, 等. 考虑时滞影响的双馈风电场广域附加阻尼控制器设计[J]. 电工技术学报, 2020, 35(1): 158-166. Ma Yanfeng, Huo Yaxin, Li Xin, et al.Design of wide area additional damping controller for doubly fed wind farms considering time delays[J]. Transactions of China Electrotechnical Society, 2020, 35(1): 158-166. [8] 钱伟, 王晨晨, 费树岷. 区间变时滞广域电力系统稳定性分析与控制器设计[J]. 电工技术学报, 2019, 34(17): 3640-3650. Qian Wei, Wang Chenchen, Fei Shumin.Stability analysis and controller design of wide-area power system with interval time-varying delay[J]. Transactions of China Electrotechnical Society, 2019, 34(17): 3640-3650. [9] Luo Haocheng, Hiskens I A, Hu Zechun.Stability analysis of load frequency control systems with sampling and transmission delay[J]. IEEE Transactions on Power Systems, 2020, 35(5): 3603-3615. [10] Shen Hao, Chen Mengshen, Wu Zhengguang, et al.Reliable event-triggered asynchronous extended passive control for semi-markov jump fuzzy systems and its application[J]. IEEE Transactions on Fuzzy Systems, 2020, 28(8): 1708-1722. [11] Qian Wei, Yuan Manman, Wang Lei, et al.Stabilization of systems with interval time-varying delay based on delay decomposing approach[J]. ISA Transactions, 2017, 70: 1-6. [12] Kwon O M, Lee S H, Park M J, et al.Augmented zero equality approach to stability for linear systems with time-varying delay[J]. Applied Mathematics and Computation, 2020, 381: 125329. [13] Qian Wei, Xing Weiwei, Fei Shumin.H(∞) state estimation for neural networks with general activation function and mixed time-varying delays[J]. IEEE Transactions on Neural Networks and Learning Systems, 2021, 32(9): 3909-3918. [14] Xu Haotian, Zhang Chuanke, Jiang Lin, et al.Stability analysis of linear systems with two additive time-varying delays via delay-product-type Lyapunov functional[J]. Applied Mathematical Modelling, 2017, 45: 955-964. [15] Zhang Zhiming, He Yong, Wu Min, et al.Exponential synchronization of neural networks with time-varying delays via dynamic intermittent output feedback control[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019, 49(3): 612-622. [16] Zeng Hongbing, He Yong, Wu Min, et al.Free-matrix-based integral inequality for stability analysis of systems with time-varying delay[J]. IEEE Transactions on Automatic Control, 2015, 60(10): 2768-2772. [17] Qian Wei, Gao Yanshan, Chen Yonggang, et al.The stability analysis of time-varying delayed systems based on new augmented vector method[J]. Journal of the Franklin Institute, 2019, 356(3): 1268-1286. [18] Yang Feisheng, He Jing, Wang Jing, et al.Auxiliary-function-based double integral inequality approach to stability analysis of load frequency control systems with interval time-varying delay[J]. IET Control Theory & Applications, 2018, 12(5): 601-612. [19] Seuret A, Gouaisbaut F.Stability of linear systems with time-varying delays using Bessel-Legendre inequalities[J]. IEEE Transactions on Automatic Control, 2018, 63(1): 225-232. [20] Zhang Ruimei, Zeng Deqiang, Park J H, et al.New approaches to stability analysis for time-varying delay systems[J]. Journal of the Franklin Institute, 2019, 356(7): 4174-4189. [21] Yang Feisheng, He Jing, Wang Dianhui.New stability criteria of delayed load frequency control systems via infinite-series-based inequality[J]. IEEE Transactions on Industrial Informatics, 2018, 14(1): 231-240. [22] Jin Li, Zhang Chuanke, He Yong, et al.Delay-dependent stability analysis of multi-area load frequency control with enhanced accuracy and computation efficiency[J]. IEEE Transactions on Power Systems, 2019, 34(5): 3687-3696. [23] van Hien L, Trinh H. Refined Jensen-based inequality approach to stability analysis of time-delay systems[J]. IET Control Theory & Applications, 2015, 9(14): 2188-2194. [24] Liu Yajuan, Park J H, Guo Baozhu.Results on stability of linear systems with time varying delay[J]. IET Control Theory & Applications, 2017, 11(1): 129-134. [25] Kwon O M, Park M J, Park J H, et al.Enhancement on stability criteria for linear systems with interval time-varying delays[J]. International Journal of Control, Automation and Systems, 2016, 14(1): 12-20. [26] Abolpour R, Dehghani M, Talebi H A.Stability analysis of systems with time-varying delays using overlapped switching Lyapunov Krasovskii functional[J]. Journal of the Franklin Institute, 2020, 357(15): 10844-10860. [27] Zhang Chuanke, He Yong, Jiang Lin, et al.Notes on stability of time-delay systems: bounding inequalities and augmented Lyapunov-Krasovskii functionals[J]. IEEE Transactions on Automatic Control, 2017, 62(10): 5331-5336. [28] Gyurkovics É, Takács T.Comparison of some bounding inequalities applied in stability analysis of time-delay systems[J]. Systems & Control Letters, 2019, 123: 40-46. [29] Ramakrishnan K, Ray G.Stability criteria for nonlinearly perturbed load frequency systems with time-delay[J]. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2015, 5(3): 383-392. [30] Park M J, Kwon O M, Park J H, et al.Stability of time-delay systems via Wirtinger-based double integral inequality[J]. Automatica, 2015, 55: 204-208. [31] Qian Wei, Xing Weiwei, Wang Lei, et al.New optimal analysis method to stability and H(∞) performance of varying delayed systems[J]. ISA Transactions, 2019, 93: 137-144.
2023, Vol. 38 
