#
H_{∞} Robust Load Frequency Control for Multi-Area Interconnected Power System with Hybrid Energy Storage System

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## Abstract

**:**

_{∞}robust controller proposed in this paper is effective.

## 1. Introduction

_{∞}robust controller based on the LMI and state-feedback method to calculate how much power the HESS should provide to the power system. Through the analysis and comparison of simulation results from PID control, it is identified that the H

_{∞}controller can improve the stability of power frequency under external load disturbance and the HESS can convert the output power demand signal from the H

_{∞}controller into the corresponding power flow effectively.

_{∞}state feedback controller is designed for power system with the HESS in the ith control area in Section 4, based on the mathematical model established in Section 3. Further simulation verification and results analysis of the method proposed in this paper are presented in Section 5. Finally, the conclusion is given in Section 6.

## 2. Multi-Area Interconnected Power System with the HESS

- One objective is to make sure that the deviation of load frequency is zero or fluctuate in a certain really small range near zero.
- Another objective is to ensure that the switching power of the tie line returns to the set value.

#### 2.1. Load Frequency Control

_{∞}robust state feedback controller is designed. The controller calculates the power supply in real time according to the variation of user load in system, and then the power control signal is transmitted to the HESS. The regulation of power in the system is realized by the effective charge and discharge of the HESS, which can ensure the stability of the load frequency.

#### 2.2. Hybrid Energy Storage System

_{∞}state feedback controller designed in this paper. ${P}_{Bat}$ and ${P}_{Sc}$ are the power should be provided to power system by the battery and super-capacitor. Moreover, ${P}_{Sc}$ is positive when the SC is in discharging mode and negative in charging mode. There are two working modes, namely, the voltage-reduction or charging mode and the boosting or discharge mode [30]. The power instruction ${P}_{Sc}$ and ${P}_{Bat}$ converted the corresponding current instruction ${i}_{Sc}$ and ${i}_{Bat}$ by calculation [16]. The HESS absorbs or emits corresponding power according to the difference value between the power generated by the power grid and the power required on the user side, thus the stabilization of frequency fluctuation power in power grid can be realized.

## 3. Mathematical Model of Power System with the HESS in the ith Control Area

## 4. H_{∞} State Feedback Controller Design for Power System with the HESS in the ith Control Area

_{∞}controller to the HESS. ${P}_{ci}$ denotes the power flow provided to power system by the HESS. ${y}_{i}$ is the output of power system in the ith control area which denotes the area control error (ACE). ${x}_{i1}$, … , ${x}_{i6}$ (i = 1, 2, 3, 4) present six inner state quantities of the ith control area. The specific description and meaning of each state which have been given in Section 2 and Section 3.

_{∞}control theory lies in: solving appropriate state feedback controller promising the closed-loop system is stable and the H

_{∞}norm of the closed-loop transfer function matrix is minimum or less than a given value. Let ${T}_{zw}$ be the transfer function from input w to output z. The goal of H

_{∞}design is guaranteeing the H

_{∞}norm of ${T}_{zw}$ to be less than $\gamma $ and it can be expressed as follows:

_{∞}controller design problems can be differentiated according to the selection of $\gamma $. If the state feedback controller K(s) can guarantee the closed-loop transfer function of the control system to be internally stable and the value of ${\Vert {T}_{zw}\Vert}_{\infty}$ is minimum. That is, ${\Vert {T}_{zw}\Vert}_{\infty}={\gamma}_{0}$ and it is called H

_{∞}controller optimal design. If ${K}_{i}$ satisfies the equation ${\Vert {T}_{zw}\Vert}_{\infty}=\gamma >{\gamma}_{0}$ based on satisfying the sufficient stability condition of $({A}_{i},{B}_{i},{C}_{2i})$, then this kind of problem is H

_{∞}controller suboptimal design problem. If ${\Vert {T}_{zw}\Vert}_{\infty}=\gamma \le 1$ exists, then the corresponding H

_{∞}controller standard design problem [31,32,33].

_{∞}controller design:

_{∞}state feedback controller of the system and the controller coefficient matrix is ${K}_{i}={W}_{i}^{*}{({X}_{i}^{*})}^{-1}$. The control coefficient matrix ${K}_{i}$ can be obtained by using the MATLAB LMI toolbox, feasp() or mincx() function to solve the linear matrix inequality in Equation (12).

## 5. Simulation and Result Analysis

- Case 1: Traditional LFC, i.e., power system without the HESS.
- Case 3: Power system with the HESS and external H
_{∞}robust state feedback controller.

_{∞}robust state feedback controller design, the weighting coefficients and performance indices are given as Table 2. Combining Section 4, the state feedback control coefficient matrix of Area 1 can be obtained by Matlab as ${K}_{1}$ = [−10.8625, 25.5883 5.2364, −8.8031, −7.6875, −9.2341] and coefficient matrices for other three control areas can be easily obtained by using the method above.

_{∞}state feedback controller designed in this paper. These control signals denote the power instruction transmitted to the HESS from H

_{∞}controller. More attention should be paid to the magnitude order of the axis because they are different in each figure. It can be noticed that the orders of magnitude in Figure 8b–d are really small, respectively, ${10}^{-5}$, ${10}^{-5}$ and ${10}^{-6}$.

_{∞}robust state feedback controller, in Figure 8 we can see that the load frequency ${f}_{i}(i=1,2,3,4)$ restores stability rapidly in seconds and $\Delta {f}_{i}(i=1,2,3,4)$ was guaranteed to be within ±0.0006 Hz since the maximum interference is almost 10 Kw. The result comparison of PID controller and H

_{∞}state-feedback controller is shown in Figure 10 under the condition of LFC with the HESS.

_{∞}robust state feedback control, it can be clearly seen in Figure 10 that H

_{∞}robust state feedback controller can effectively enhance the robustness and stability of the system in the condition of disturbance, keep the system output $AC{E}_{i}$ fluctuating within the range of ±0.0006 Hz near zero and restore the system frequency stability within 10 s.

## 6. Conclusions

_{∞}robust state feedback controller based on LMI is designed to further enhance the robustness and stability of the system for the LFC. Finally, the feasibility and effectiveness of the proposed scheme are proven by the simulation and results analysis. Moreover, the coupling problem in large power system is not considered and the study of coupling quantity in large area power system will be the next research direction.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 8.**Control signals from controller of four areas in Case 3: (

**a**) control signal ${u}_{1}$; (

**b**) control signal ${u}_{2}$; (

**c**) control signal ${u}_{3}$; and (

**d**) control signal ${u}_{4}$.

**Figure 10.**$AC{E}_{i}(i=1,2,3,4)$ of each control area under load disturbance in three different cases: (

**a**) $AC{E}_{1}$ of Control Area 1; (

**b**) $AC{E}_{2}$ of Control Area 2; (

**c**) $AC{E}_{3}$ of Control Area 3; and (

**d**) $AC{E}_{4}$ of Control Area 4.

${\mathit{T}}_{\mathit{g}\mathit{i}}$(s) | ${\mathit{T}}_{\mathit{r}\mathit{i}}$(s) | ${\mathit{T}}_{\mathit{t}\mathit{i}}$(s) | ${\mathit{R}}_{\mathit{i}}$(Hz/p.u.Mw) | ${\mathit{T}}_{\mathbf{1}\mathit{j}\mathbf{(}\mathit{j}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{4})}$(s) |

0.08 | 4.2 | 0.3 | 2.4 | 0.0707 |

${\mathit{K}}_{\mathit{p}\mathbf{1}}$(s) | ${\mathit{T}}_{\mathit{p}\mathbf{1}}$(s) | ${\mathit{c}}_{\mathbf{1}}$ | ${\mathit{\beta}}_{\mathbf{1}}$(Hz/p.u.Mw) | ${\mathit{K}}_{\mathit{A}\mathit{F}\mathbf{1}}$ |

120 | 20 | 0.35 | 0.425 | 1.1 |

$\mathbf{(}\mathit{i}\mathbf{=}\mathbf{1}\mathbf{)}$ | ${\mathit{q}}_{\mathit{i}\mathbf{1}}$ | ${\mathit{q}}_{\mathit{i}\mathbf{2}}$ | ${\mathit{q}}_{\mathit{i}\mathbf{3}}$ | ${\mathit{q}}_{\mathit{i}\mathbf{4}}$ | ${\mathit{q}}_{\mathit{i}\mathbf{5}}$ | ${\mathit{q}}_{\mathit{i}\mathbf{6}}$ | ${\mathit{\rho}}_{\mathbf{1}}$ | ${\mathit{\gamma}}_{\mathbf{1}}^{\mathbf{2}}$ |
---|---|---|---|---|---|---|---|---|

Area 1 | 0.0125 | 0.01 | 0.001 | 0.005 | 0.001 | 0.013 | 0.09 | 0.86 |

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## Share and Cite

**MDPI and ACS Style**

Yan, W.; Sheng, L.; Xu, D.; Yang, W.; Liu, Q.
H_{∞} Robust Load Frequency Control for Multi-Area Interconnected Power System with Hybrid Energy Storage System. *Appl. Sci.* **2018**, *8*, 1748.
https://doi.org/10.3390/app8101748

**AMA Style**

Yan W, Sheng L, Xu D, Yang W, Liu Q.
H_{∞} Robust Load Frequency Control for Multi-Area Interconnected Power System with Hybrid Energy Storage System. *Applied Sciences*. 2018; 8(10):1748.
https://doi.org/10.3390/app8101748

**Chicago/Turabian Style**

Yan, Wenxu, Lina Sheng, Dezhi Xu, Weilin Yang, and Qian Liu.
2018. "H_{∞} Robust Load Frequency Control for Multi-Area Interconnected Power System with Hybrid Energy Storage System" *Applied Sciences* 8, no. 10: 1748.
https://doi.org/10.3390/app8101748