# Model-Independent Derivative Control Delay Compensation Methods for Power Systems

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Literature Review

#### 1.2.1. Communication Delays in Power Systems

#### 1.2.2. Compensation of Delays in Control Loops

#### 1.2.3. Compensation of Delays in on-Line Monitoring

#### 1.2.4. D-Term-Based Delay Compensation

#### 1.3. Contributions

- A discussion on the maximum fidelity that the two delay-compensation methods for wide-area controllers can achieve is presented.
- A technique to solve small-signal stability analysis of power systems with inclusion of either constant or realistically-modeled communication delays in the derivatives of the state variables is proposed.
- A new theorem on the stability of the NTDS is derived.
- A thorough comparison of the performance of PD and predictor-based delay compensation methods in power systems, including both open-loop and closed-loop scenarios, is shown.

#### 1.4. Organization

## 2. D-Based Delay Compensation Methods and Small Signal Stability

#### 2.1. Power System Model with Inclusion of Delays

#### 2.2. PD Delay Compensation

#### 2.3. Predictor-Based Delay Compensation

#### 2.4. Power System Model with Inclusion of Delay Compensation

**Theorem**

**1.**

**Remark**

**1.**

## 3. Fidelity Comparison

#### Illustrative Example

## 4. Wide-Area Measurement Delay Compensation

#### 4.1. Wide-Area Measurement Delay Model

#### 4.2. Implementation of the Delay Compensation

## 5. Case Study

#### 5.1. IEEE 14-Bus System

#### 5.1.1. Constant Delay

#### 5.1.2. WAMS Delay

#### 5.2. New England System

#### 5.2.1. Constant Delay

#### 5.2.2. WAMS Delay

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof.**

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**Figure 2.**Illustrative Example. Performance index p for $t\in [\tau ,5\pi ]$ s of D-based compensation methods as the compensation gains vary: (

**top**) PD-based compensation; and (

**bottom**) predictor-based compensation.

**Figure 3.**Illustrative Example. Trajectories of sending (x), delayed (${x}_{d}$), PD-based compensated (${x}_{{\mathrm{com}}_{\mathrm{A}}}$), and predictor-based compensated (${x}_{{\mathrm{com}}_{\mathrm{B}}}$) signals.

**Figure 8.**Stability maps $\tau -{K}_{s}$ of delay compensation methods for the IEEE 14-bus system. Shaded regions are stable. Dark shaded regions indicate $\xi >5\%$. The dashed and dashed-dotted vertical lines indicate the stability delay margin and the delay margin for $\xi >5\%$, respectively, of the system without delay compensation.

**Figure 10.**Best compensation effects of the PLL at bus 39 with different constant delays for the New England system.

**Figure 11.**Trajectories of the measured signals of the PMU installed at bus 39 of the New England system.

**Table 1.**Rightmost eigenvalues of the IEEE 14-bus system with WAMS delay and delay compensation. ${K}_{m}=2$ for the PD and ${K}_{m}=15$ for the predictor-based compensation.

Compensation Gain | PD | Predictor-Based |
---|---|---|

$0.2\phantom{\rule{0.166667em}{0ex}}{K}_{m}$ | $0.1030\pm j11.2144$ | $0.1841\pm j11.1718$ |

$0.4\phantom{\rule{0.166667em}{0ex}}{K}_{m}$ | $0.0133\pm j11.2987$ | $0.1989\pm j11.1762$ |

$0.6\phantom{\rule{0.166667em}{0ex}}{K}_{m}$ | $-0.1138\pm j11.4314$ | $0.2285\pm j11.1854$ |

$0.8\phantom{\rule{0.166667em}{0ex}}{K}_{m}$ | $-0.1328\pm j0.0343$ | $0.2885\pm j11.2056$ |

$1.0\phantom{\rule{0.166667em}{0ex}}{K}_{m}$ | $-0.1228\pm j15.2758$ | $0.3494\pm j11.2279$ |

$1.2\phantom{\rule{0.166667em}{0ex}}{K}_{m}$ | $0.0626\pm j15.5234$ | $0.4736\pm j11.2796$ |

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**MDPI and ACS Style**

Liu, M.; Dassios, I.; Tzounas, G.; Milano, F.
Model-Independent Derivative Control Delay Compensation Methods for Power Systems. *Energies* **2020**, *13*, 342.
https://doi.org/10.3390/en13020342

**AMA Style**

Liu M, Dassios I, Tzounas G, Milano F.
Model-Independent Derivative Control Delay Compensation Methods for Power Systems. *Energies*. 2020; 13(2):342.
https://doi.org/10.3390/en13020342

**Chicago/Turabian Style**

Liu, Muyang, Ioannis Dassios, Georgios Tzounas, and Federico Milano.
2020. "Model-Independent Derivative Control Delay Compensation Methods for Power Systems" *Energies* 13, no. 2: 342.
https://doi.org/10.3390/en13020342