## 1 Introduction
\(P_{ab}\) in line power flow is specified as  
\[P_{ab}\text{=}\frac{V_{a}V_{b}}{x_{t}-x_{c}}\sin\delta \tag{2}\]  
where \(x_{c}\) is the TCSC reactance, and \(x_{t}\) is the transmission line reactance. In recent years, oscillations in power networks have been reduced by using a variety of FACTS devices to build controllers that inject or absorb reactive power. Two FACTS shunt devices--SVC and STATCOM--have been used for dampening to increase the angle stability of FACTS [7; 8]. SVC is best used in the middle of transmission lines or at the sending end, as well as at buses and branches, and has applications in wind energy with a focus on high voltage drops. SVC is not permitted on PV buses [9]. STATCOM offers a stability buffer for transients and a single power factor for wind power systems. A damping performance analysis UPFC [10], which is a dynamic control technique to promote low-voltage ride-through capability in power systems [11], is also described in the literature. The UPFC is a multipurpose device that controls bus voltage; transmission line impedance, phase angle, line active, and reactive power flow; and bus impedance. To reduce local and intermode oscillations in power systems [9], PSS and FACTS devices with efficient controller coordination have been incorporated. The TCSC and SVC with PSS coordination control design are discussed in [12; 13]. The optimal reactive power injection is used in the TCSC model, which also has variable reactance. It is not compatible with transformer use at branches [9]. Furthermore, STATCOM, a UPFC, and an IPFC have coordinated PSS [14; 15; 16]. The design of damping controllers has been studied by a number of researchers. Some authors have taken into account lead lag controllers, and proportional integral (PI) and proportional integral derivative (PID) with various formulations of the objective function for stability analysis of the single machine infinite bus (SMIB) system [17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. The proportional integral derivative with filter (PIDF) is used to control the load frequency of a deregulated power system and to mitigate reactive losses in multi-phase flow regimes in pipelines and risers at offshore oilfields [27], buck converters for DC-DC converters, and industrial processes to obtain a true solution for high-level process oscillation [28] and to compensate for the reactive power for steady state and the transient stability of power systems [29]. To improve the efficiency of a hybrid microgrid system, a fuzzy PIDF load frequency controller was optimised using the marine predator algorithm [30]. The installation of a controller utilising the proper formulation may result in a damping scheme that contributes a sufficient amount. A suitable formulation of the objective function and a suitable damping scheme are essential for the robust design of the damping controller. To improve the stability of the SMIB system and the multimachine power system, an optimum optimisation technique needs to be found to create an objective function by comparing the performances of various formulations of objective functions. In recent years, the use of heuristic algorithms as optimisation techniques to build damping controllers for the stability of power systems has become widespread and well recognised [18; 31; 32]. Many conventional heuristic algorithms have been used in research, and their performance is application focused. These heuristic algorithms are used extensively in SMIB systems and small multimachine power systems to construct damping controllers. Optimisation parameters have been reduced in many studies to reduce design complexity. However, examining the variance in design performance caused by the addition of more problem dimensions and all-optimisation parameters is crucial. The path of convergence is altered each time using a heuristic algorithm, which finds solutions through trial and error using stochastic approaches.