# A New Method for Computing the Delay Margin for the Stability of Load Frequency Control Systems

^{*}

## Abstract

**:**

## 1. Introduction

_{P}and K

_{I}, on the delay margin was also investigated. In Reference [18], an improved and less conservative criterion for computing the delay margin was introduced. The Lyapunov–Krasovskii functional was used to reduce the conservativeness of the method, the Wirtinger inequality and Jenson integral inequality were applied to bound the derivative of the Lyapunov function. It was reported in References [10,19] that the number of decision variables were reduced compared to the number of decision variables in Reference [16], and this will lead to less conservative results for the delay margin.

## 2. Dynamic Model of One-Area LFC System with Time Delay

_{d}is the load deviation, ΔP

_{m}is the generator mechanical output deviation, ΔP

_{v}is the valve position deviation, Δf is the frequency deviation. M (=2H) is the inertia constant, D is the load damping coefficient [33,34], T

_{g}is the time constant of the governor, T

_{ch}is the time constant of the turbine, R is the speed drop, and β is the frequency bias factor. For a one-area LFC system, the area control error ACE is given as [16]:

_{P}is the proportional gain, K

_{I}is the integral gain, and ∫ ACE is the integration of the area control error. With the PI controller, the closed-loop system is expressed as follows:

_{d}to zero, the single-area LFC system becomes a linear time delay system, then Equation (4) becomes:

## 3. Delay Margin Computation Using the Sweeping Test

**Definition 1**

**([36]).**The spectral radius of two matrices pair is defined as:

_{d}.

**Theorem 1**

**([36]).**For the system (5) stable at τ

_{d}= 0, i.e., A + A

_{d}is stable and rank(A

_{d}) = q, we define

**Proof**

**([35,36,37,38]).**The system (5) is stable independent of the time delay if the following condition is satisfied:

**Corollary 1**

**([36]).**The system (5) is stable independent of delay if and only if:

- (i)
- A is stable,
- (ii)
- A + A
_{d}is stable, and - (iii)
- $\underset{\_}{\rho}(j\omega I-A,{A}_{d})>1,$$\forall \omega >0$

_{d}. Using the sweeping test, check if the system is stable independent of delay or not, that is $\underset{\_}{\rho}(j\omega I-A,{A}_{d})>1$ for $\omega \in (0,\infty )$. If for some values of $\omega $, $\underset{\_}{\rho}(j\omega I-A,{A}_{d})=1$, then proceed to step 2, otherwise the system is stable independent of the time delay.

## 4. Case Study: One-Area LFC System

_{ch}= 0.3, T

_{g}= 0.1, R = 0.05, D = 1.0, β = 21.0, and M = 10. Under an open communication network, the remote terminal unit (RTU) sends the signals to the central controller through the shared network, and then the controller sends the commands back. In most of the studies these two delays are aggregated into a single delay and this assumption is made in this paper. The ACE signals are updated every 2–4 s [10]. In power systems the data collection is in the order of 1−5 s [4]. The results of the delay margin with different values of the controller gains, K

_{P}and K

_{I}, are shown in Table 1 along with the results of the methods in References [16,18,32]. It should be noted that the method in Reference [32] gives the most accurate reported delay margins. Table 1 shows clearly that the proposed method gives almost exactly as the results of the method reported in Reference [32]; however, the proposed method is simpler with less computations.

_{P}= 0 and K

_{I}= 0.4): In Reference [32], it is reported that the delay margin with K

_{P}= 0 and K

_{I}= 0.4 is 3.382 s, with the proposed method it was 3.382 s. The proposed method gives accurate values of the delay margin as the method reported in Reference [32]. The proposed method gives less conservative results than the LMI methods reported in Reference [16,18]. To validate the results, simulations with Matlab/Simulink were carried out. The frequency response of the LFC system with K

_{P}= 0 and K

_{I}= 0.4 for different values of the time delay is shown in Figure 10. A 0.1 p.u change in the load occurs at 10 s. Figure 10 shows the frequency response with 3.3 s, 3.382 s, and 3.4 s, and it is clear that the system is stable with 3.3 s and unstable with 3.4 s. From the simulation the system was marginally stable with 3.384 s. The percentage error with the simulation-based delay margin was 0.071%. For this system, we have only one generalized eigenvalue. The spectral radius as a function of ω is shown in Figure 11. From Figure 11 the crossing frequency was 0.4025 rad/s, solving Equation (14), we have θ = 1.3678 rad which makes the delay margin equal 3.382 s. From Figure 10 the oscillating frequency was 0.4025 rad/s which proves the validity of the results.

_{P}= 0.6 and K

_{I}= 0.6): In Reference [32] it is reported with K

_{P}= 0.6 and K

_{I}= 0.6 that the delay margin is 2.281 s, while with our method it was 2.281 s. The frequency response with different delays is shown in Figure 12. The system is stable with 2.1 s; however, it becomes unstable with 2.4 s which is larger than the delay margin. The simulation-based delay margin was 2.282 s. The crossing frequency was 0.8015 rad/s, solving Equation (14) for θ, then; θ = 1.8283 rad. Figure 13 shows the spectral radius as a function of ω for K

_{P}= 0.6 and K

_{I}= 0.6.

_{P}= 0.05 and K

_{I}= 0.05): In Reference [32] it is reported with K

_{P}= 0.05, K

_{I}= 0.05 that the delay margin is 31.875 s, while with our method it was 31.8509 s. The frequency response with different delays is shown in Figure 14. The system is stable with 31.8509 s; however, it becomes unstable with 33 s which is larger than the delay margin. The simulation-based delay margin was 31.88 s. The relative percentage error was 0.0438%. The crossing frequency was 0.0502 rad/s, solving Equation (14) for θ then θ = 1.596 rad. Figure 15 shows the spectral radius as a function of ω for K

_{P}= 0.05 and K

_{I}= 0.05.

_{I}controller gains is shown in Figure 16. The delay margin dependence on K

_{I}, and K

_{P}shows similar behavior as observed in References [16,18,32]. The crossing frequencies and the crossing angles are shown in Table 2 and Table 3 respectively. The variations of the crossing angle and the crossing frequency give more details on the dependence of the delay margin on K

_{P}and K

_{I}. From Figure 16, the delay margin decreases with increasing K

_{I}if K

_{P}is kept constant. The delay margin increases as K

_{P}increase in the range K

_{P}< 0.4, then the delay margin decreases as K

_{P}becomes larger than 0.4. This is the same behavior observed in References [16,18,32]. The delay margin becomes large for small values of K

_{P}and K

_{I}.

_{P}and K

_{I}. The LFC system tends to oscillate with higher frequency with large values of K

_{P}and K

_{I}.

_{I}when K

_{P}= 0 are shown in Table 4. It is interesting to observe that the crossing frequency with K

_{P}= 0 numerically equals K

_{I}. As K

_{P}is made larger than 0.2, the crossing frequency increases with increasing K

_{P}.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 10.**The frequency response for different values of the time delay with K

_{P}= 0 and K

_{I}= 0.4.

**Figure 12.**The frequency response for different values of the time delay with K

_{P}= 0.6 and K

_{I}= 0.6.

**Figure 14.**The frequency response for different values of the time delay with K

_{P}= 0.05 and K

_{I}= 0.05.

K_{P} | τ_{d}, s | K_{I} | ||||||
---|---|---|---|---|---|---|---|---|

Method | 0.05 | 0.1 | 0.15 | 0.2 | 0.4 | 0.6 | 1 | |

0 | Theorem 1 | 30.928 | 15.207 | 9.961 | 7.338 | 3.382 | 2.042 | 0.923 |

[32] | 30.915 | 15.201 | 9.96 | 7.335 | 3.382 | 2.042 | 0.923 | |

[18] | 30.853 | 15.172 | 9.942 | 7.323 | 3.377 | 2.04 | 0.922 | |

[16] | 27.927 | 13.778 | 9.056 | 6.692 | 3.124 | 1.91 | 0.886 | |

0.05 | Theorem 1 | 31.851 | 15.687 | 10.277 | 7.573 | 3.502 | 2.122 | 0.97 |

[32] | 31.875 | 15.681 | 10.279 | 7.575 | 3.501 | 2.122 | 0.97 | |

[18] | 31.498 | 15.647 | 10.258 | 7.561 | 3.496 | 2.119 | 0.969 | |

[16] | 27.874 | 14.061 | 9.284 | 6.866 | 3.215 | 1.974 | 0.927 | |

0.1 | Theorem 1 | 32.769 | 16.127 | 10.575 | 7.793 | 3.61 | 2.194 | 1.012 |

[32] | 32.751 | 16.119 | 10.571 | 7.794 | 3.61 | 2.194 | 1.012 | |

[18] | 30.415 | 15.765 | 10.547 | 7.777 | 3.604 | 2.191 | 1.011 | |

[16] | 27.038 | 13.682 | 9.22 | 6.941 | 3.29 | 2.029 | 0.963 | |

0.2 | Theorem 1 | 34.198 | 16.86 | 11.06 | 8.16 | 3.792 | 2.313 | 1.079 |

[32] | 34.226 | 16.856 | 11.062 | 8.162 | 3.792 | 2.313 | 1.079 | |

[18] | 28.01 | 14.597 | 10.107 | 7.821 | 3.784 | 2.309 | 1.077 | |

[16] | 25.114 | 12.76 | 8.617 | 6.535 | 3.32 | 2.108 | 1.016 | |

0.4 | Theorem 1 | 35.802 | 17.661 | 11.596 | 8.559 | 3.981 | 2.426 | 1.118 |

[32] | 35.834 | 17.658 | 11.594 | 8.559 | 3.98 | 2.426 | 1.118 | |

[18] | 22.457 | 11.835 | 8.287 | 6.505 | 3.718 | 2.419 | 1.116 | |

[16] | 20.364 | 10.426 | 7.065 | 5.384 | 2.832 | 1.912 | 1.017 | |

0.6 | Theorem 1 | 34.906 | 17.198 | 11.28 | 8.311 | 3.826 | 2.281 | 0.947 |

[32] | 34.922 | 17.195 | 11.278 | 8.312 | 3.826 | 2.281 | 0.947 | |

[18] | 16.033 | 8.624 | 6.209 | 4.997 | 3.038 | 2.178 | 0.964 | |

[16] | 14.618 | 7.477 | 5.157 | 3.958 | 2.13 | 1.475 | 0.827 | |

1 | Theorem 1 | 0.595 | 0.586 | 0.575 | 0.564 | 0.516 | 0.463 | 0.361 |

[32] | 0.596 | 0.586 | 0.575 | 0.564 | 0.516 | 0.463 | 0.361 | |

[18] | 0.594 | 0.584 | 0.574 | 0.563 | 0.515 | 0.463 | 0.36 | |

[16] | 0.546 | 0.538 | 0.53 | 0.522 | 0.482 | 0.438 | 0.348 |

θ/rad | K_{I} | ||||||
---|---|---|---|---|---|---|---|

K_{P} | 0.05 | 0.1 | 0.15 | 0.2 | 0.4 | 0.6 | 1.0 |

0 | 1.546 | 1.521 | 1.496 | 1.471 | 1.368 | 1.257 | 0.989 |

0.05 | 1.596 | 1.571 | 1.546 | 1.521 | 1.418 | 1.307 | 1.041 |

0.1 | 1.646 | 1.621 | 1.596 | 1.571 | 1.468 | 1.358 | 1.092 |

0.2 | 1.747 | 1.722 | 1.696 | 1.671 | 1.567 | 1.456 | 1.187 |

0.4 | 1.956 | 1.929 | 1.902 | 1.875 | 1.765 | 1.647 | 1.349 |

0.6 | 2.184 | 2.153 | 2.123 | 2.092 | 1.968 | 1.828 | 1.419 |

1.0 | 1.435 | 1.413 | 1.390 | 1.365 | 1.262 | 1.152 | 0.934 |

ω/(rad/s) | K_{I} | ||||||
---|---|---|---|---|---|---|---|

K_{P} | 0.05 | 0.1 | 0.15 | 0.2 | 0.4 | 0.6 | 1.0 |

0 | 0.0500 | 0.1000 | 0.1502 | 0.2005 | 0.4045 | 0.6153 | 1.0714 |

0.05 | 0.0501 | 0.1002 | 0.1505 | 0.2009 | 0.4050 | 0.6161 | 1.0732 |

0.1 | 0.0502 | 0.1005 | 0.1509 | 0.2016 | 0.4067 | 0.6187 | 1.0784 |

0.2 | 0.0511 | 0.1021 | 0.1534 | 0.2048 | 0.4133 | 0.6295 | 1.1004 |

0.4 | 0.0546 | 0.1092 | 0.1640 | 0.2191 | 0.4434 | 0.6789 | 1.2065 |

0.6 | 0.0626 | 0.1252 | 0.1882 | 0.2518 | 0.5143 | 0.8015 | 1.4981 |

1.0 | 2.4102 | 2.4120 | 2.4151 | 2.4193 | 2.4462 | 2.4861 | 2.5867 |

K_{I} | θ | ω | τ_{d} |
---|---|---|---|

0.05 | 1.5460 | 0.0500 | 30.9283 |

0.1 | 1.5212 | 0.1000 | 15.2066 |

0.15 | 1.4963 | 0.1502 | 9.9614 |

0.2 | 1.4712 | 0.2005 | 7.3375 |

0.4 | 1.3678 | 0.4045 | 3.3816 |

0.6 | 1.2566 | 0.6153 | 2.0422 |

1.0 | 0.9889 | 1.0714 | 0.9230 |

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**MDPI and ACS Style**

Khalil, A.; Swee Peng, A.
A New Method for Computing the Delay Margin for the Stability of Load Frequency Control Systems. *Energies* **2018**, *11*, 3460.
https://doi.org/10.3390/en11123460

**AMA Style**

Khalil A, Swee Peng A.
A New Method for Computing the Delay Margin for the Stability of Load Frequency Control Systems. *Energies*. 2018; 11(12):3460.
https://doi.org/10.3390/en11123460

**Chicago/Turabian Style**

Khalil, Ashraf, and Ang Swee Peng.
2018. "A New Method for Computing the Delay Margin for the Stability of Load Frequency Control Systems" *Energies* 11, no. 12: 3460.
https://doi.org/10.3390/en11123460