H_∞ performance for load frequency control systems with random delays

Figures & data

Figure 1. Schematic diagram of one-area LFC scheme.

Table 1. Admissible upper bounds h for given μ.

μ	0.1	0.2	0.5	0.8
Seuret and Gouaisbaut (2018) (N = 1)	4.80	3.99	2.79	2.42
T. H. Lee and Park (2017)	4.8257	4.1274	3.1131	2.6763
W. I. Lee et al. (2018)	4.908	4.199	3.166	2.735
Zeng et al. (2019)	4.921	4.218	3.221	2.792
Li et al. (2019)	4.996	4.308	3.251	2.867
Chen and Chen (2019)	4.943	4.270	3.322	2.899
M. J. Park et al. (2018)	4.9445	4.2747	3.3058	2.8505
Seuret and Gouaisbaut (2018) (N = 4)	5.01	4.29	3.19	2.70
Corollary 3.2	5.237	4.602	3.475	3.361

Table 2. Maximum delay upper bounds $h\propto \left(K_P,K_I\right)$ with $\mu =0.9$.

h	$K_I$
$K_P$	0.05	0.1	0.15	0.2	0.4	0.6	1.0
0	38.735	24.475	17.833	13.395	6.654	4.518	3.169
0.05	32.241	20.179	15.995	13.144	6.493	4.484	3.049
0.10	33.785	20.414	15.422	9.979	6.500	4.421	3.119
0.20	20.180	11.686	9.520	8.807	5.350	4.048	3.035
0.40	3.568	3.499	3.489	3.482	3.465	3.128	2.696
0.60	2.685	2.650	2.612	2.434	2.423	2.299	2.252
1.00	1.898	1.782	1.773	1.764	1.734	1.731	1.668

Table 3. Maximum delay upper bounds h comparisons with $\mu =0.9$.

		Methods				Methods
$K_P$	$K_I$	Corollary 3.2	Peng and Zhang (2016)	$K_P$	$K_I$	Corollary 3.2	Jiang et al. (2012)
0	0.05	38.735	33.621	0.05	0.05	32.241	20.031
0	0.1	24.475	16.470	0.05	0.1	20.179	10.173
0	0.2	13.395	7.796	0.05	0.2	13.144	4.769
0	0.4	6.654	3.318	0.05	0.4	6.493	1.880
0	0.6	4.518	2.008	0.05	0.6	4.484	1.048
0	1.0	3.169	1.020	0.05	1.0	3.049	0.468
0.1	0.05	33.785	27.487	0.2	0.05	20.180	10.489
0.1	0.1	20.414	14.663	0.2	0.1	11.686	5.803
0.1	0.2	9.979	7.628	0.2	0.2	8.807	3.236
0.1	0.4	6.500	3.240	0.2	0.4	5.350	1.436
0.1	0.6	4.421	2.004	0.2	0.6	4.048	0.963
0.1	1.0	3.119	1.059	0.2	1.0	3.035	0.507

Table 4. Maximum delay upper bounds h with $\gamma =1$.

h	$K_I$
$K_P$	0.05	0.1	0.15	0.2	0.4	0.6	1.0
0	0.705	0.713	0.714	0.708	0.684	0.667	0.641
0.05	0.712	0.705	0.698	0.692	0.672	0.657	0.633
0.10	0.688	0.683	0.679	0.674	0.659	0.646	0.625
0.20	0.650	0.648	0.646	0.643	0.634	0.625	0.609
0.40	0.601	0.601	0.600	0.599	0.594	0.590	0.580
0.60	0.571	0.570	0.570	0.569	0.566	0.563	0.555
1.00	0.528	0.528	0.527	0.527	0.525	0.522	0.516

Table 5. Maximum delay upper bounds h with $K_P=0.4$, $K_I=0.4$.

Methods	h
Corollary 3.1	0.594
Theorem 3.1 with ${\alpha }_0=0.4$,$h_0=1.0$	1.158
Theorem 3.1 with ${\alpha }_0=0.4$,$h_0=1.5$	1.649
Theorem 3.1 with ${\alpha }_0=0.6$,$h_0=1.5$	1.642
Theorem 3.1 with ${\alpha }_0=0.6$,$h_0=2.0$	2.138
Theorem 3.1 with ${\alpha }_0=0.8$,$h_0=2.0$	2.127

Table 6. Allowable minimum ${\gamma }_{min}$ with h = 2.

${\gamma }_{min}$	$K_I$
$K_P$	0.05	0.1	0.15	0.2	0.4	0.6	0.8
0	2.075	2.165	2.270	2.391	2.911	3.808	5.388
0.05	2.167	2.266	2.356	2.471	3.044	4.021	5.780
0.10	2.280	2.340	2.430	2.510	3.157	4.283	6.326
0.20	2.570	2.653	2.754	2.818	3.463	4.799	7.706
0.40	3.331	3.406	3.524	3.715	4.766	7.230	16.923
0.60	5.043	5.232	5.504	5.802	8.293	19.516	–
0.80	10.695	11.400	12.389	13.942	47.844	–	–

Table 7. Allowable minimum ${\gamma }_{min}$ with $K_P=0.2$, $K_I=0.6$.

Methods	${\gamma }_{min}$
Corollary 3.1	4.799
Theorem 3.1 with ${\alpha }_0=0.2$,$h_0=1.0$	1.572
Theorem 3.1 with ${\alpha }_0=0.2$,$h_0=1.5$	1.795
Theorem 3.1 with ${\alpha }_0=0.4$,$h_0=1.5$	1.896
Theorem 3.1 with ${\alpha }_0=0.4$,$h_0=1.8$	1.247
Theorem 3.1 with ${\alpha }_0=0.6$,$h_0=1.8$	1.378

Table 8. Allowable minimum ${\gamma }_{min}$ comparisons with h = 2.

		$h_0=1.0$	$h_0=1.5$	$h_0=1.5$
Methods	${\alpha }_0=0$	${\alpha }_0=0.6$	${\alpha }_0=0.6$	${\alpha }_0=0.8$
Jiang et al. (2012)	10.7503	–	–	–
Peng and Zhang (2016)	6.8638	3.6787	3.2720	3.8639
Corollary 3.1	2.4682	–	–	–
Theorem 3.1	–	1.3363	1.7325	1.7817

Figure 2. State response trajectory with $K_P=0.4$, $K_I=0.4$ by Corollary 3.1.

Figure 3. State response trajectories with $K_P=0.4$, $K_I=0.4$ by Theorem 3.1.

Figure 4. State response trajectory with $K_P=0.2$, $K_I=0.6$ by Corollary 3.1.

Figure 5. State response trajectories with $K_P=0.2$, $K_I=0.6$ by Theorem 3.1.

Seuret, A., & Gouaisbaut, F. (2018). Stability of linear systems with time-varying delays using Bessel-Legendre inequalities. IEEE Transactions on Automatic Control, 63(1), 225–232. https://doi.org/https://doi.org/10.1109/TAC.2017.2730485

 Web of Science ®Google Scholar

Lee, T. H., & Park, J. H. (2017). A novel Lyapunov functional for stability of time-varying delay systems via matrix-refined-function. Automatica, 80, 239–242. https://doi.org/https://doi.org/10.1016/j.automatica.2017.02.004

 Web of Science ®Google Scholar

Lee, W. I., Lee, S. Y., & Park, P. G. (2018). Affine Bessel–Legendre inequality: Application to stability analysis for systems with time-varying delays. Automatica, 93, 535–539. https://doi.org/https://doi.org/10.1016/j.automatica.2018.03.073

 Web of Science ®Google Scholar

Zeng, H. B., Liu, X. G., & Wang, W. (2019). A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems. Applied Mathematics and Computation, 354, 1–8. https://doi.org/https://doi.org/10.1016/j.amc.2019.02.009

 Web of Science ®Google Scholar

Li, Z. C., Yan, H. C., Zhang, H., Zhan, X. S., & Huang, C. Z. (2019). Improved inequality-based functions approach for stability analysis of time delay system. Automatica, 108. https://doi.org/https://doi.org/10.1016/j.automatica.2019.05.033

 Google Scholar

Chen, Y., & Chen, G. (2019). Stability analysis of systems with timevarying delay via a novel Lyapunov functional. IEEE/CAA Journal of Automatica Sinica, 6(4), 1068–1073. https://doi.org/https://doi.org/10.1109/JAS.6570654

 Google Scholar

Park, M. J., Kwon, O. M., & Ryu, J. H. (2018). Advanced stability criteria for linear systems with time-varying delays. Journal of the Franklin Institute, 355(1), 520–543. https://doi.org/https://doi.org/10.1016/j.jfranklin.2017.11.029

 Web of Science ®Google Scholar

Peng, C., & Zhang, J. (2016). Delay-distribution-dependent load frequency control of power systems with probabilistic interval delays. IEEE Transactions on Power Systems, 31(4), 3309–3317. https://doi.org/https://doi.org/10.1109/TPWRS.2015.2485272

 Web of Science ®Google Scholar

Jiang, L., Yao, W., Wu, Q. H., Wen, J. Y., & Cheng, S. J. (2012). Delay-dependent stability for load frequency control with constant and time-varying delays. IEEE Transactions on Power Systems, 27(2), 932–941. https://doi.org/https://doi.org/10.1109/TPWRS.2011.2172821

 Web of Science ®Google Scholar