# On transient stability of multi-machine power systems through Takagi–Sugeno fuzzy-based sliding mode control approach

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## Abstract

The present research focuses on transient stability of multi-machine power systems in a full consideration regarding the performances of the Takagi–Sugeno fuzzy-based sliding mode control approach in association with the conventional sliding mode and also the optimal control approaches to improve the last finding outcomes in this area. Hereinafter, concerning the robustness of the sliding mode control approach toward parametric uncertainties and environment perturbations, in fact, a couple of different sliding mode control approaches are designed for mutual comparison, after a number of state-of-the-art technique considerations. To increase the control performance, the Takagi–Sugeno fuzzy-based approach is devised to provide the appropriate coefficients. Finally, the three control approaches are all carried out in the six-machine power system under the same condition and the investigated results are correspondingly provided to be analyzed. The results indicate that the proposed fuzzy-based control approach is well behaved with respect to other related ones.

## Keywords

Transient stability Six-machine power system Takagi–Sugeno fuzzy-based sliding mode control approach Optimal control approach## List of symbols

*H*Inertial constant (s)

*M*Inertial coefficient (s)

*D*Convergence coefficient (pu)

- \(T_s \)
Time constant of input control system (s)

- \(T_g \)
Servo-motor time constant (s)

- \(P_e \)
Electrical power (pu)

- \(P_m \)
Mechanical power (pu)

- \(\omega \)
Angular speed (rad/s)

- \(\delta \)
Rotor angle (rad)

- \(E^{\prime }_q \)
Internal transient voltage (pu)

- \(Y_{ij} \)
\(i{-}j\) line transmission admittance (pu)

## Introduction

Transient stability is a concept to be considered for the purpose of measuring the performance of synchronous machines, which has the highest importance for long-distance grids. From a physical point of view, transient stability can be defined as the ability of a system to remain with synchronous outcomes during the occurrence of large perturbations. On its own, in fact, the stability is addressed as the property of a power system that enables it to maintain a stable equilibrium and return to an acceptable state, when faced with large perturbation for normal performance situations. A large number of investigations are dedicated to the types of stability of the power systems, where some of them have directly focused on the transient stability, in-depth. With this goal, a set of potential related works in this area are now listed.

### Related works

State-of-the-art investigations in the area of transient stability with its specific application to synchronous machines in a wide range of structure variations have been recently proposed. In one such case, Ashraf et al. have explored a Takagi–Sugeno fuzzy-based control approach in dealing with transient stability augmentation of multi-machine power system, while Bakhshi et al. have considered fuzzy-based damping control approach though local measurements for the purpose of enhancing transient stability in power systems [1, 2].

Schaab et al. have considered robust control for voltage and transient stability of power grids focusing on wind power, and also Mazhari et al. have addressed a frequency-domain approach for distributed harmonic analysis in multi-area interconnected power systems [3, 4]. Darabian et al. have presented a power control strategy, to improve power system stability in the presence of wind farms by designing predictive control and Shah et al. have studied the performance improvement of intrusion detection with fusion of multiple sensors, while Wuthishuwong et al. have focused on consensus-based local information coordination for the networked control of the autonomous intersection management [5, 6, 7]. In Yipeng et al’s. works, an integrated high side voltage control approach is presented to improve short-term voltage stability regarding the receiving-end power systems, while in Yan et al.’s research work, trajectory sensitivity analysis on the equivalent one-machine-infinite-bus in case of multi-machine systems for preventive transient stability control is researched [8, 9].

In Godpromesse et al’s. research, online simplified nonlinear control approach for transient stabilization enhancement of multi-machine power systems is considered, whereas in Haotian et al.’s research, switching excitation control approach for enhancement of transient stability of such systems is investigated [10, 11]. Jiebei et al.’s research is to deal with generic inertia emulation controller for multi-terminal voltage-source-converter high voltage direct current systems. Shahgholian et al.’s research copes with power system stabilizer and flexible alternating current transmission systems control approach coordinated design via adaptive velocity update relaxation particle swarm optimization algorithm. Hui et al.’s research handles Lyapunov-based decentralized excitation control for global asymptotic stability and voltage regulation of the same multi-machine power systems, and subsequently Hongshan et al.’s investigation designs excitation prediction control in case of the aforementioned multi-machine power systems through balanced reduced model [10, 11, 12, 13, 14]. Shi et al. have proposed stabilizing control with transmission losses based on the pseudo-generalized Hamiltonian theory. Agrawal et al. have addressed support vector clustering-based direct coherency identification of generators and Ningqiang et al. have described damping Torques during transient behaviors as well [15, 16, 17, 18]. Du et al.’s work considers robustness of an energy storage system-based stabilizer to suppress inter-area oscillations. Shojaeian et al.’s work explores damping of low-frequency oscillations in case of multi-machine power systems, based on adaptive input–output feedback linearization control. Sheng-Kuan et al.’s work realizes the objective function and algorithm for optimal design. Son et al.’s study is on the direct stability analysis, and finally Muyeen et al.’s work explains the reduction of frequency fluctuation for wind farm-connected power systems by an adaptive artificial neural network controlled energy capacitor system [19, 20, 21, 22, 23]. Seung-Ju et al.’s research discusses the passivity-based output synchronization of port-controlled Hamiltonian and general linear interconnected systems, and Casagrande et al.’s work describes a solution to the multi-machine transient stability problem and simulated validation in realistic scenarios [24, 25].

Thereafter, Dragosavac et al. have proposed practical implementation of coordinated control, and Qiqi et al. have investigated the power angle control in case of grid-connected doubly fed induction generator wind turbines for fault ride-through, while Chaudhuri et al. have addressed system frequency support via multi-terminal direct current grids [26, 27, 28]. In Bijami et al.’s research, stabilizing signals for power system damping using generalized predictive control is designed through a new hybrid shuffled frog leaping algorithm, while in Chun-Feng et al.’s research, the coordinated control of flexible AC transmission system devices via an evolutionary fuzzy lead-lag control approach is realized under advanced continuous ant colony optimization [29, 30]. In Wang et al.’s investigations, a number of aspects of stability enhancement based on offshore wind farm fed are addressed to deal with a multi-machine system [31, 32]. Finally, in Yija et al.’s research, a nonlinear variable structure stabilizer in power system stability is discussed. In Qiang et al.’s research, nonlinear stabilizing control in the aforementioned multi-machine systems is described [32, 33, 34].

The rest of the paper is organized as follows: the proposed control approaches are given in Sect. 3, where the simulation results are all illustrated in Sect. 1. The research concludes the investigated outcomes in Sect. 2.

## The proposed control approaches

### The preliminary information

*q*of voltage of transmission reactance, \(M_i \) is the moment inertia, \(D_i \) is the mechanical damping, \(x_{di} \) is the synchronous reactance of axis

*d*and \(x^{\prime }_{di} \) is the transient reactance of axis

*d*(all) in

*i*th machine. Additionally, \(T^{\prime }_{doi} \), \(G_{ij} \) and \(B_{ij} \) are the transient time constant of axis

*d*, the transmission admittance between

*i*th and

*j*th machines and the transmission conductance between

*i*th and

*j*th machines, respectively. Finally, \(\Delta x_{di} =x_{di} -{x}'_{di} \) and \(\alpha _{ij} =\pi /2-\arcsin (B_{ij} /Y_{ij} )\) are defined and \(U_i \) is indicated to be the power control for

*i*th machine of

*U*. Subsequently, the state variables to be chosen are taken as

### The conventional sliding mode control approach

### The optimal control approach

### The Takagi–Sugeno fuzzy-based sliding-mode control approach

Regarding the Takagi–Sugeno fuzzy-based control approach, the input membership functions; IMF, of two inputs, including the sliding surface and also their derivatives, are composed of the five behavioral types of the triangular that are equally given in the normalized span of \([-1, 1]\) as follows:

\({\text {IMf}}\_1: {\text {Very}}\, {\text {small}}, {\text {IMf}}\_2: {\text {Small}}, {\text {IMf}}\_3: {\text {Zero}}, {\text {IMf}}\_4: {\text {Large}}, {\text {IMf}}\_5: {\text {Very}}\, {\text {large}}.\)

The rule based realized in the Takagi–Sugeno fuzzy-based control approach

\(S\backslash {S}'\) | IMf_1 | IMf_2 | IMf_3 | IMf_4 | IMf_5 |
---|---|---|---|---|---|

IMf_1 | OMF_5 | OMF_4 | OMF_4 | OMF_2 | OMF_1 |

IMf_2 | OMF_4 | OMF_4 | OMF_4 | OMF_2 | OMF_1 |

IMf_3 | OMF_2 | OMF_2 | OMF_3 | OMF_2 | OMF_1 |

IMf_4 | OMF_2 | OMF_2 | OMF_2 | OMF_2 | OMF_1 |

IMf_5 | OMF_1 | OMF_1 | OMF_1 | OMF_1 | OMF_1 |

## The simulation results

### The outcomes of the proposed control approaches

For this scheme, the power follows the desired value and is stabilized in nearly 10 s, while the frequency of variations seems to be unacceptable. In other words, the main liability of the sliding mode control approach focusing on the chattering barely affects the output power. Hereinafter, the second control approach shows the better performance in time and accuracy, as its investigated outcomes are all illustrated in Fig. 3.

Although the acquired convergence rate investigated here is better than the optimal control approach, its rate of variations may be problematic.

### The outcomes of the proposed control approaches by focusing on the instantaneous perturbation

The performance comparison regarding the proposed control approaches in accordance with the settling time and the variation frequency

The approaches | Settling time (s) | Variation frequency |
---|---|---|

The optimal control | 10 | Medium |

The sliding mode control | 5 | Severe |

The Takagi–Sugeno fuzzy-based control | 1 | Benign |

## Conclusion

The six-machine power system is considered to be controlled through the conventional and Takagi–Sugeno fuzzy-based sliding mode control approaches, while the optimal control approach is also designed to have the merit of comparison. It is clearly concluded that the conventional one is not as efficient as the proposed Takagi–Sugeno fuzzy-based integral control approaches in general. As long as the same conventional control approach involves the high levels of the chattering on the sliding surface, the Takagi–Sugeno fuzzy-based control approach tries to suppress the phenomenon. Besides, the results of the aforementioned Takagi–Sugeno fuzzy-based control approach are much quicker than other related considered ones. It is worth to noting that both the conventional and the Takagi–Sugeno fuzzy-based control approaches stabilize the system successfully after the occurrence of the grid faults to show acceptable efficiency. Stabilizing in a few seconds, the good accuracy and the ability to trace and follow the desired levels, prior to and after the fault, can be considered from the positive aspects of these control approaches. Although the higher control parameters in optimal control approach yields quicker stability, it fails to efficiently stabilize the system in the presence of perturbations. Regarding the supremacy of the robust control approach in overcoming the perturbations, it is shown that by realizing these types of control approaches in parallel, the effectiveness of the proposed Takagi–Sugeno fuzzy-based control approach is tangibly visible.

## Notes

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