On the optimal multi-objective design of fractional order PID controller with antlion optimization

Abstract

This study introduces an optimal multi-objective design approach for a robust multimachine Fractional Order PID Controller (FOPID) using the Antlion algorithm. The research focuses on the need for effective stabilizers in multimachine power systems by employing traditional speed-based lead-lag FOPID controllers. The study formulates a multi-objective problem, optimizing the damping factor and damping ratio of lightly damped electromechanical modes to maximize a composite set of objective functions, tackled through the Antlion algorithm. Stability analysis of Single-Machine Infinite-Bus (SMIB) and multimachine power systems is conducted based on rotor speed and power deviation minimization in the time domain response, along with damping ratio and eigenvalue analysis. The proposed approach is implemented and tested on three IEEE test cases, showcasing significant improvements in stability through the reduction of maximum overshoot (M_p) and settling time (t_s) of speed deviation. Comparative analysis with other optimization-based FOPID controllers underscores the superiority of the proposed approach in enhancing stability in multimachine power systems. The main impact of this research lies in its contribution to the advancement of stability enhancement techniques in multimachine power systems, offering a systematic framework for optimal FOPID controller design and empowering decision-making processes in power engineering.

Keywords

Antlion optimization ant colony algorithm fractional order proportional integral derivative controller genetic algorithm power system stability

1. Introduction

In the world of power systems engineering, the continual search of stability and efficiency presents an ongoing and dynamic challenge. With the increasing demand for a dependable electricity supply, modern power networks are encountering unprecedented challenges due to variables such as the integration of renewable energy, shifting demand patterns, and the expanding interconnectedness of regional grids. In this ever-changing environment, finding efficient control solutions to reduce power system oscillations and maintain grid stability becomes crucial.

The traditional framework for power system stability is centered on the concept of preserving synchronous operation among generators, guaranteeing that the system stays resistant to disturbances and maintains a stable equilibrium. Nevertheless, when power systems increase in size and intricacy, conventional control methods have restrictions in successfully reducing oscillations, especially those arising from weakly damped electromechanical modes. These oscillations, which have low frequencies, can spread across the grid, causing voltage instability, changes in frequency, and in severe instances, cascading failures. Although conventional Proportional-Integral-Derivative (PID) controllers have been widely used in power system control, their ability to reduce low-frequency oscillations has been doubted. Researchers have used more sophisticated control paradigms, such as Fractional Order PID (FOPID) controllers, to better capture the complex dynamics of power systems. Nevertheless, the design and calibration of FOPID controllers still pose a challenging issue, especially in multimachine power systems that exhibit various operating circumstances and system configurations.

The primary challenge is to design an ideal FOPID controller that can effectively reduce low-frequency oscillations in multimachine power systems. This must be done while considering the various operating conditions and system configurations that are commonly found in real-world scenarios. This research issue highlights the lack of current knowledge and emphasizes the necessity for a methodical approach to designing and adjusting FOPID controllers, specifically for the complexities of multimachine power systems.

This study suggests a novel strategy to formulate and solve this problem by using a multi-objective design method for FOPID controllers. The approach makes use of the Antlion algorithm, which is a metaheuristic optimization tool inspired by the foraging behavior of antlions. Our objective is to create FOPID controllers that can effectively stabilize multimachine power systems under various operating conditions by solving a multi-objective optimization problem. This problem aims to maximize both the damping factor and damping ratio of lightly damped electromechanical modes.

The scope of this research includes multiple primary goals. The primary objective is to get a thorough comprehension of the fundamental mechanisms driving power system oscillations and their impact on stability. This involves completing a comprehensive examination of current literature on power system stability, control tactics, and optimization methodologies, which will establish the foundation for the suggested methodology.

Expanding on this theoretical basis, the multi-objective optimization problem for FOPID controller design has been established. This will involve specifying the goals, limitations, and design parameters that are relevant to the challenge. A highly effective nature-inspired optimization technique, Antlion algorithm has been applied to develop an iterative optimization framework. This framework will be used to search for the best controller parameters that maximize stability metrics while also meeting system constraints. Simultaneously, a computational representation of a complex power system has been simulated consisting of several generators, transmission lines, and loads. This model will be used to assess the performance of the proposed FOPID controller in different operating scenarios, such as normal operation, contingencies, and transient disturbances.

Then, the effectiveness of the proposed FOPID controller have resulted in reducing power system oscillations and improving stability thorough simulation studies on benchmark IEEE test systems. We will compare our suggested strategy to existing optimization strategies for FOPID controller design, such as Genetic Algorithms and Ant Colony Optimization, in order to determine whether approach is best.

The ultimate outcome of this research project will be a strong and comprehensive foundation for designing FOPID controllers that are specifically suited to the complexities of multimachine power systems. Our work seeks to provide power system engineers and operators with a robust tool to improve grid stability and resilience. This will ensure the dependable and efficient operation of contemporary power grids by bridging the gap between theory and practice.

In line with the said objectives and in order to lay the foundation of this paper, an outline of this paper is structured as follows. After this section justifies the importance of this research and provides background knowledge on the optimal multi-objective design approach for a robust multimachine Fractional Order PID Controller (FOPID) for the novice readers. Section 2 contains descriptions of standard test systems used for testing various methodologies. The best selection of PID parameters can achieve the objectives. For better PID performance, multi-objective functions are considered, and in addition to the novel Antlion Optimization (ALO), the various aspects of the objective function and constraints are also discussed. Section 3 presents different techniques used for optimal tuning of the PID controller to improve performance under various operating conditions of the power system network. Finally, Section 4 provides concluding remarks.

2. Background work

Over the past two decades, there has been significant interest in using power system stabilizers to improve the dynamic stability of power systems [1]. When a power system network experiences moderate disturbance, low-frequency oscillations may occur. To ensure stable operation, voltage regulation, and internal and external disturbance rejection, electrical power systems must meet specific requirements. Low-frequency oscillations arise in synchronous generators due to rotor angle oscillations, with a frequency range of 0.1–2 Hz [2]. These oscillations occur due to an imbalance between power demand and available power over a specific time period [3]. During the early stages of power system operation, when generators and loads were closely connected, power oscillations were barely noticeable. However, in today’s complex power system networks, transmitting significant amounts of power over long transmission lines to meet high demand at the system’s far end can result in increased power system oscillations. To mitigate these oscillations, one technique is to use a high-speed excitation mechanism. In the past, excitation controllers were designed using trial and error methods with a transfer function model, but more recently, a systematic technique based on optimum control theory has been presented in various literature. The choice of stabilizing signal in a Power System Stabilizer (PSS) is unconstrained and can be any easily measurable signal, such as speed, frequency, angle, real power, reactive power, armature current, and so on.

In recent years, researchers worldwide have presented various intelligent controllers, such as fuzzy controllers, neural network-based controllers, hybrid fuzzy-neuro controllers, and optimally tuned controllers, to improve dynamic stability [4]. Furthermore, recent literature proposes various optimally tuned FOPID controllers for improved stability. The FOPID controller’s design is an evolutionary optimization problem that aims to identify most, if not all, of the multiple locally optimal solutions that lead to a single best solution. The FOPID controller is a hybrid controller that is increasingly being used to enhance system performance, particularly low-frequency oscillations in the power system, and improve system stability. The most common stability domain boundaries are the real, imaginary, and complex boundaries [5].

Majidabad et al. (2015) demonstrated that a fractional order controller provides better results in a multi-machine power system compared to a simple tuned PID controller [6]. Lal et al. (2019) presented a moth flame-based controller design to reduce low frequency oscillations, focusing on terminal voltage control and load frequency control using an optimization technique to tune the values of the FOPID controller [7]. Das et al. (2011) proposed a methodology for selecting the tuning of FOPID controllers for higher order control systems, emphasizing the importance of appropriate order of the FOPID controller to enhance the stability of the system [8]. Zamani et al. (2007) presented a particle swarm-based FOPID controller design to improve the robustness of the system and enhance system performance and stability [9]. Sambariya et al. (2016) introduced a novel Bat algorithm optimization technique for tuning PID controllers in various power systems [10]. Ray et al. (2019) proposed an IT2FOFPID-based PSS to dampen oscillations and hybrid FAPSO-based techniques to optimize parameters for improved stability performance [11]. Alejandro et al. (2020) discussed the determination of appropriate controller parameters for multi-objective optimization problems using various computational intelligence and soft computing techniques in their survey paper [12]. Devarapalli et al. (2020) proposed a novel hybrid technique called the Enhanced Grey Wolf Optimization-Sine Cosine Algorithm-Cuckoo Search (EGWO-SCA-CS) algorithm for optimal controller parameter tuning [13]. Sabo et al. (2020) proposed an eigenvalue-based objective function with GA, PSO, and FFA optimization methods applied to search for optimal control parameters of PSSs in a multi-machine system [14]. Paital et al. (2021) presented an Adaptive Fractional Fuzzy Sliding Mode Controller (AFFSMC) based Power System Stabilizer (PSS) for stability analysis using Eigenvalue, integral time absolute error, and shape of demerit [15]. Ekinci et al. (2021) proposed Henry Gas Solubility Optimization (HGSO), a recent novel metaheuristic algorithm, to achieve optimal parameters for a power system stabilizer (PSS) employed in a power system [16].

Samir et al. proposed an Antlion Optimization (ALO) algorithm to optimize Fractional Order PID (FOPID) controller performance across various systems, including first-order delay, second-order unstable delay, and higher-order systems. Simulation results demonstrate improved margin parameters and robustness, particularly challenging in non-minimum phase systems. ALO’s superior transient performance and robustness make it a promising alternative for FOPID controller design, outperforming other methods in parameter tuning. The study underscores ALO’s potential as a viable solution for enhancing FOPID controller efficacy [17].

The study carried out by Ibrahim et al. aims to optimize the settings of a power system stabilizer (PSS) in coordination with a static variable-reluctance compensator (SVC) using a coordinated fractional-order proportional integral-derivative (FOPID) approach. The Moth Flame Optimization technique is employed to optimize controller parameters, ensuring coordinated device operation. Performance analysis, based on Integral Time Absolute Error (ITAE), reveals improved stability with the proposed coordinated PSS&SVC-FOPID-POD controller compared to other configurations. Simulation results demonstrate significant enhancements in performance indices, including settling time, overshoot, and various error measures, indicating the superiority of the proposed scheme in power system stability and oscillation damping [18].

The PI controller was optimized using Bat optimization, resulting in over a 90% reduction in settling time compared to the open-loop system. Although it achieved the minimum settling time among all controllers, it exhibited a longer rise time and the highest overshoot. Conversely, the PID controller tuned by ALO significantly improved system response, reducing rise time by 71%, settling time by 77%, and peak time by 97% compared to the open-loop system, albeit with some overshoot. The FOPID controller, optimized using ALO, further reduced settling time and overshoot to zero, offering a viable alternative for DFIG controller design [19].

Maximum overshoot and Eigenvalues have been identified as the two key parameters affecting system stability in interconnected dynamic multi-machine power systems. Therefore, the tuned PID controller parameters with the proposed objective function have been applied in the SMIB and multi-machine power system network, and system stability has significantly improved.

The new multi-objective fitness function is formulated, taking into consideration the damping ratio and minimizing overshoot to increase system stability, as well as an Eigenvalue analysis with several constraints to comply. A new fractional order PID controller was tuned with a novel nature-inspired optimization paradigm on real-time IEEE test cases, and its efficacy was analyzed with that of other meta-heuristic techniques.

After formulating the objective function, a novel method known as Antlion Optimization (ALO) is implemented. ALO algorithm is adept at optimization in continuous spaces, drawing inspiration from antlions’ hunting behaviour. It demonstrates rapid convergence and efficiently manages large-scale optimization problems. ALO strikes a balance between exploration and exploitation, conducive to finding optimal solutions in multi-objective optimization scenarios. However, its efficacy may be hindered by the need for parameter fine-tuning, leading to potential time and computational resource constraints. Additionally, ALO may encounter challenges with complex multimodal functions or problems featuring high-dimensional search spaces. The ALO-based PID has become a significant tool for improving the performance of traditional controller tuning methods. The method used here overcomes the drawbacks of Genetic Algorithm (GA) and Ant Colony Optimization (ACO) methods. This optimization strategy stands out because it takes less time to execute, is simple to implement, and avoids trapping in local minima because the point is optimal. A multi-machine power system was used to test the proposed design method. To evaluate the effectiveness of the suggested PSSs under various disturbances, loading conditions, and system configurations, eigenvalue analysis and nonlinear simulation results were used.

3. System modelling

This section focuses on the mathematical modeling of the state space for all configurations present in the test systems, including both single-machine and multi-machine power systems (based on the selected state variables). State matrices have been developed for all three models considered, which are used for stability analysis.

3.1 Test system configuration

Models of power system components must be selected according to the purpose of the system study. Therefore, it’s necessary to be aware of which models, in terms of accuracy and complexity, are appropriate for a particular type of system study, while keeping computational burden as low as possible. Selecting inappropriate models for power system components can lead to erroneous conclusions [20].

3.1.1 SMIB power system

A classical model for synchronous machine is developed with the following assumptions.

The mechanical power input is taken as constant.

Natural damping (D) in the system is included in modelling.

Synchronous generator is assumed as a voltage source with constant magnitude with reactance.

Figure 1.

Synchronous generator.

In power system operation, synchronous machine is essential [21]. The equivalent circuit of synchronous generator having terminal voltage $E_{t}$ connected to the infinite bus having voltage $E_{b}$ through transmission line having impedance $(R_{T} + j X_{T})$ is shown in Fig. 1.

Figure 2.

Model of Heffron-Philip generator with connected with infinite bus PID Controller.

Figure 2 represents the Single Machine Infinite Bus power system (SMIB) with complete state space Heffron Phillips model. Here, the generator is equipped with governor-turbine model and power damping controller. The optimization scheme along with scheme of sensing signals is shown. The model in Fig. 2 shows generator (mechanical loop), PID controller, and excitation system (electrical loop) governed by a set of nonlinear differential equations. To cover all operating conditions, the power system with generators, stabilizers, and excitation systems can be modeled by a set of nonlinear differential equations. In Fig. 2, the blocks have terms given by $K_{1} - K_{6}$ are constants, $H$ is inertia constant of machine in p.u, $T_{0 d 0}, T_{0 q 0}$ are direct and quadrature axis transient flux decay constant, $E_{0 q}$ is quadrature axis induced voltage behind transient reactance, $E_{𝑓𝑑}$ is field voltage, $Δ T_{M}$ is mechanical power input of generator, $Δ V_{𝑟𝑒𝑓}$ is desired terminal voltage, $Δ ω$ is change in rotor speed, $Δ δ$ is change in load angle and change in internal voltage is $(Δ E_{0 q})$ and field voltage is $(Δ E_{𝑓𝑑})$ .

3.1.2 Two area kundur power system network

Figure 3.

A ten bus power system with two areas and four machines.

Figure 4.

Thirty-nine bus power system with ten machines.

The power system being tested [22] comprises two areas, specifically Area 1 and Area 2, with four machines: G1, G2, G3, and G4. Figure 3 depicts the system. Each of the four synchronous generators has a capacity of 900 MVA, with the connected loads treated as impedances of constant values. These loads are divided between the two areas in such a way that there is a power transfer of 500 MVA from Area 1 to Area 2. The system includes two areas known as Machine Area 1 and Machine Area 2. Each area is equipped with four identical non-salient pole generators, each with a capacity of 900 MVA and a voltage of 20 kV. All generating units have identical parameters considered as their default values. Both areas are connected by two 230 kV tie lines, each with a length of 220 km. Machine Area 1 is modeled with a transmission line length of 110 km, including two generators represented as M1 and M2 connected to a turbine and regulators, with a total of 5 buses. The synchronous generator is connected to the transmission line through a 900 MVA, 20 kV/230 kV transformer (T1). Additionally, two loads are connected in Area 1, specifically inductive and capacitive loads. Area 2 has a similar configuration as Area 1, as both areas exhibit mirror symmetry with respect to each other.

3.1.3 Ten machine thirty-nine bus power systems

Figure 4 illustrates a system comprising 10 machines with a 39-bus system [23]. All ten synchronous generators are connected to 36 transmission lines through 12 transformers, with 19 loads being supplied. The generating units from G1 to G9 have static excitation and local Power System Stabilizers (PSSs) incorporated. For analysis [24], we have selected generating unit G10 and have utilized the default parameters. A controller is installed in all generators to dampen oscillations, except for generator 1.

3.2 Damping controller

The power mismatch in a power system network creates a change in shaft torque leading to the rotor oscillations. We can suppress these oscillations by providing proper damping torque; otherwise, the oscillations would persist to extend and hence the system would be forced to unstable condition resulting in blackouts which is catastrophic. Hence, the oscillations of the range of 0.2 to 3 Hz have to be suppressed. All synchronous generators are equipped with damping controllers [25]. In this work, FOPID controller is used as power damping controller. It has five control parameters, $K_{p}, K_{i}, K_{d}, λ$ and $μ$ which are to be tuned. When we increase the value of $K_{p}$ , the percentage peak overshoot $(% M_{p})$ increases and steady state error $(e_{𝑠𝑠})$ decreases simultaneously. This implies that as we increase the value of $K_{p}$ more and more, the system becomes unstable, due to the increase in overshoot. On the other hand, when $K_{i}$ increases, the overshoot becomes smaller, and the speed of response becomes slower. Again, when we increase the value of $K_{d}$ , the overshoot is very small with a slow rise time but the settling time is similar. In practical applications, the pure derivative action is never used as it produces a derivative kick due to which an undesirable noise is also produced. The “pure derivative” term implies $[\frac{d error (t)}{d t}]$ , but the derivative of the output $[\frac{d output (t)}{d t}]$ is used in practice to avoid a phenomena termed “derivative kick”. Derivative kick occurs because the value of the error changes suddenly whenever the reference signal is adjusted. The derivative of a sudden jump in the error causes the derivative of the error to be instantaneously large and causes the controller output to saturate for one cycle at either an upper or lower bound.

3.2.1 Fractional order PID (FOPID) controller

The FOPID controller is recognized for its superior performance compared to integer order PID equivalents, thanks to its uniformity in step margin input and narrower bandwidth. This results in higher damping and lower overshoot, making it ideal for dampening low-frequency device oscillations in multi-machine power systems under configuration uncertainty and various operational conditions [26]. This translates to better dynamic performance and greater resistance to uncertainty compared to conventional PID controllers [27]. The basic parameters of FOPID controllers are $K_{p}, K_{i}, K_{d}, λ$ and $μ$ . Unlike the integer order controller, both the velocity and the shape of resetting can be controlled by the integration order, $λ (I^{λ})$ . Fractional derivative controllers do respond to constant error signals; the differentiation of $μ$ -th order $(D^{μ})$ of the error curve at the point $e (t)$ of a constant is different from zero. These parameters are tuned with the help of optimization techniques to achieve best values. Since the conventional values does not provide stability to the power system, FOPID controller is used to tune the values of the multi machine power system [28, 29]. The mathematical definition of the differintegral operator of fractional order has been the subject of different approaches. The most used are the Riemann-Liouville (RL) given by Eq. (1) and the Grünwald-Letnikov (GL) given by Eq. (2) which provides the most basic concept of a FOPID controller. The definition of Riemann-Liouville (RL) can be represented as Eq. (1).

$a D_{t}^{γ} f (t) = \frac{1}{Γ (n - 1)} \frac{d^{n}}{d t^{n}} \int_{a}^{t} \frac{f (τ)}{f {(t - τ)}^{γ - n + 1}} d τ$ (1)

Where $(n - 1 < γ < n)$ and $Γ (γ)$ is the familier gamma function defined by Eq. (2).

$Γ (γ) = \int_{0}^{\infty} e^{- u} u^{γ - 1} d u, ℜ (γ) > 0$ (2)

Here, in this paper the Grunwald-Letnikov definition is used and is expressed by Eq. (3).

$a D_{t}^{γ} f (t) = 𝑚𝑖𝑠𝑠𝑖𝑛𝑔 l i \underset{h \to 0}{m} h^{- γ} \sum_{j = 0}^{[(t - a) / h]} {(- 1)}^{j} (\begin{matrix} γ \\ j \end{matrix}) f (t - j h)$ (3)

where $(n - 1 < γ < n)$ , $[(t - a) / h]$ is integer part, and $a$ , $t$ are the limits of operator. The binomial coefficient is evaluated by the gamma function that generalizes the factorial operator as in Eq. (4).

$(\begin{matrix} γ \\ j \end{matrix}) = \frac{γ!}{j! (γ - j)!} = \frac{Γ (γ + j)}{Γ (j + 1) Γ (γ - j + 1)}$ (4)

An ITAE criterion is used as the design criteria in the optimization algorithm for designing of FOPID controller [30]. It stands for Integral Time Absolute Error Criteria as shown in Eq. (5).

$𝐼𝑇𝐴𝐸 = \int_{0}^{\infty} t | e (t) | d t$ (5)

where $t$ represents time and $e (t)$ is time difference to be controlled between variables.

Figure 5 depicts the graphical representation of FOPID [31], which expands the integer order PID from a point to a plane, providing more flexibility in controller design and enabling accurate control of real-time problems. PID controllers can operate in manual or automatic modes. In manual mode, the operator manipulates the controller output directly by pressing buttons to increase or decrease the output. Controllers can also operate in conjunction with other controllers, such as in a cascade or ratio connection, or with nonlinear elements, such as multipliers and selectors. In automatic mode, PID parameters can be adjusted during operation. To avoid switching transients during mode and parameter changes, careful consideration is needed. Figure 6 shows the block diagram of the exciter, terminal voltage transducer, and PID controller.

Each parameter has a gain value by which we control the contribution of the block and hence, entire control system expressed by Eq. (6).

$U (s) = K_{p} + K_{i} s^{- λ} + K_{d} s^{μ}$ (6)

Table 1 depicts the FOPID parameters and upper limit and lower limits for optimum design of FOPID controller. Limits are set so that at specified range, it gives best desired values so that the stability is maintained, and its performance is enhanced.

Table 1

Parameters of PID with their upper and lower limit

Parameters of FOPID	P	I	D	$λ$	$μ$
Upper limit				0.1	0.1
Lower limit	50	30	20	1	1

Figure 5.

FOPID representation.

Figure 6.

Excitation system equipped with FOPID.

The state space matrix for Single Machine Infinite Bus (SMIB) system [32] is represented by the Eq. (7).

$[\begin{matrix} 0 & ω_{0} & 0 & 0 & 0 \\ - \frac{K_{1}}{M} & 0 & - \frac{K_{2}}{M} & 0 & 0 \\ - \frac{K_{4}}{T_{d 0}^{'}} & 0 & - \frac{K_{3}}{T_{d 0}^{'}} & \frac{1}{T_{d 0}^{'}} & 0 \\ - \frac{K_{A} K_{5}}{T_{A}} & 0 & - \frac{K_{A} K_{6}}{T_{A}} & - \frac{K_{A}}{T_{A}} & \frac{K_{A}}{T_{A}} \\ - \frac{K_{1} K_{p}}{M} + \frac{K_{2} K_{4} K_{d}}{M T_{d 0}^{'}} & K_{i} - \frac{K_{1} K_{d} ω_{0}}{M} & - \frac{K_{2} K_{p}}{M} + \frac{K_{2} K_{3} K_{d}}{M T_{d 0}^{'}} & \frac{K_{2} K_{d}}{M T_{d 0}^{'}} & 0 \end{matrix}]$ (7)

where $K_{p}$ is proportional gain and has no effect on steady-state error but influences minimizing the rise time. $K_{i}$ is integral gain may result into poor transient response. $K_{i}$ further has the effect on eradication of steady-state error. $K_{d}$ is derivation control helps in reduction of overshoot and it helps in improvement of the transient response.

3.3 Fitness function

The formulation of the fitness function aims to minimize the error signal, specifically the deviation in rotor speed, to enhance system stability. The error signal is minimized by utilizing error performance criteria. Our work utilizes a multi-objective objective function that considers both damping ratio and overshoot to achieve this goal [33].

Figure 7.

Plot of Eigenvalues in S-plane ( $J_{1}$ ).

Firstly, we have considered the damping ratio which is an index of stability of system as given in Eq. (8). If the damping factor is specified, then the condition of stability is that the closed loop Eigen values should lie in the left half of a s-plane.

$J_{1} = \sum_{j = 1}^{N} \sum_{ξ_{i, j} ⩾ ξ_{0}} {(ξ_{0} - ξ_{i j})}^{2}$ (8)

where, $N$ represents number of operating points, $ξ_{i j}$ indicates the damping ratio of the $i^{t h}$ Eigen value at the $j^{t h}$ operating point. If we consider the placement of the Eigen values (closed loop) in the s plane, for $ξ_{i j} ⩾ ξ_{0}$ , our system will be stable and the plot for the considered values will be having one thick end and tapering to a thin edge towards origin, similar to a wedge-shape sector as shown in Fig. 7.

For $J_{1}$ , the desired least value of $ξ_{0}$ for damping has to be attained. It is to be noted here that for calculations, only those Eigenvalues are considered for the objective function which need to be relocated.

Secondly, we consider the minimization of the system maximum overshoot expression as $J_{2}$ shown in Eq. (9).

$J_{2} = \sum_{j = 1}^{N} \sum_{σ_{i, j} ⩾ σ_{0}} {(σ_{0} - σ_{i_{j}})}^{2}$ (9)

where, $N$ represent number of operating points, $σ_{i j}$ represents the represents the real part/damping factor of the $i^{t h}$ Eigen value at the $j^{t h}$ operating point; starting values of damping factor $ξ_{0}$ and overshoot $σ_{0}$ are chosen to be 0.2 & $-$ 1.0, respectively. In Eq. (10), the $i^{t h}$ Eigenvalue has been considered to be composed of two components; $σ_{i}$ represents the real part and $σ_{0}$ is a chosen threshold which is a measure of damping at the desirable level. If the dominant Eigenvalues are now relocated, beyond s $=$ $σ_{0}$ line, where $σ_{i_{j}} ⩽ σ_{0}$ as shown in Fig. 8, then the desired level can be attained. This fortifies the relative stability to a great extent.

The fitness function obtained by the combination of objective functions and [34] given by Eqs (8) and (9) lead to a multi-objective function, minimizing overshoot and damping factor, and is given by Eq. (10).

$J = J_{1} + a J_{2}$ (10)

where, $a$ is weighting factor and we have assumed the value of $a =$ 10. The constraints on the FOPID controller parameters are given by Eq. (8).

Figure 8.

Placement of $σ_{1}$ in S-plane ( $J_{2}$ ).

Subjected to,

$\begin{matrix} K_{p}^{\min} ⩽ K_{p} ⩽ K_{p}^{\max} \\ K_{i}^{\min} ⩽ K_{i} ⩽ K_{i}^{\max} \\ \begin{matrix} K_{d}^{\min} ⩽ K_{d} ⩽ K_{d}^{\max} \\ λ^{\min} ⩽ λ ⩽ λ^{\max} \\ μ^{\min} ⩽ μ ⩽ μ^{\max} \end{matrix} \end{matrix}} (for FOPID)$ (11)

When both the constraints are fulfilled for our multi objective function, then Eigenvalues $σ_{i_{j}} ⩽ σ_{0}$ and the damping ratios $ξ_{i j} ⩾ ξ_{0}$ potray the region as shown in Fig. 9 as the stable region.

The Eigenvalues should have larger negative real parts and higher damping ratios to increase stability. Soft computing approaches such as GA, ACO and ALO are used to tune the above parameters for enhanced transient performance.

3.4 Antlion optimization

Antlions, also known as doodlebugs, are part of the Myrmeleontid family, which belongs to the Neuropteran order of net-winged insects. An Antlion’s life cycle has two stages: larvae and adults. The adult stage lasts only 35 weeks and is solely for reproductive purposes [35]. In contrast, the lengthy larval stage is used for hunting their preferred prey, mostly other insects. Antlion larvae exhibit a peculiar and fascinating hunting behavior, which is briefly explained here. The Antlion Optimization (ALO) is a novel stochastic search strategy that mimics the foraging activity of ants in nature. The ALO uses ants and Antlions as search agents to tackle prey hunting stages such as random ant walks, trap construction, ant trapping, prey capture, and trap rebuilding [36, 37].

Table 2
Pseudo code for antlion optimization algorithm

S no.	Steps
1.	Initialize randomly the initial population on antlions and ants
2.	Compute the fitness of antlions and ants
3.	Locate the best antlions and its position (elite antlion)
4.	while (iteration $<$ maximum iteration)
5.	for (each antlion)
6.	Select an antlion using roulette wheel method
7.	Update c and d using (12) and (13)
8.	Create and normalize random walk using (14) and (15)
9.	Update the location of ant using (16)
10.	end for
11.	Calculate the fitness value of all ants
12.	Sort according to their fitness and substitute an antlion with its equivalent fittest ant using (17)
13.	Update elite antlion
14.	end while
15.	Return elite

Figure 9.

Plot for minimum $J$ in $S$ plane.

The pseudo code for ALO is given in Table 2. Ants walks probabilistically in search of foods, a local search for an Ant is described as below at each point of optimization:

Step I: Random walk of ants

The optimal solution for such a dynamically modifiable problem is a difficult challenge. Ants change positions at each point of optimization with random steps. Because each search field is limited, thus, by using a min – max normalize, the arbitrary path is constrained within the search space given by Eq. (12).

$X_{i} = [0; r (1); r (1) + r (2) \dots; \sum_{J = 1}^{T - 1} r (j); \sum_{J = 1}^{T} r (j);]$ (12)

where $i = 1 \dots n_{𝑑𝑖𝑚}$ , $d_{𝑖𝑚} =$ number of dimensions in search space for ALO, $T =$ represents maximum number of iterations to achieve best value.

Here, $X = [X_{1}; \dots; X_{𝑑𝑖𝑚}],$ and $r (j)$ can be represented as in Eq. (13).

$r (j) = {\begin{matrix} 1 𝑖𝑓 𝑟𝑎𝑛𝑑 (\max_{i t r}, 1) > 0.5 \\ 0 𝑖𝑓 𝑟𝑎𝑛𝑑 (\max_{i t r}, 1) ⩽ 0.5 \end{matrix}$ (13)

where $j$ shows the step of random walk (iteration in this study), $r (j)$ is a stochastic function and rand is a random number generated with uniform distribution in the interval of [0, 1].

In order to normalize the random variables in search space Eq. (14) is used.

$Y_{i} = [\frac{(X_{i} - a_{i}) (d_{i} - c_{i})}{b_{i} - c_{i}} + c_{i}]$ (14)

where $a_{1}$ and $b_{1}$ denotes the lowest and highest limits of and denotes the lowest and highest values of Antlion in $i^{t h}$ dimension, respectively.

Step II: Construction of traps

Antlion constructs a barrier (i.e. pit) to catch the target. The outside edge of the pit is so blunt that it allows the tiny ant to tumble down instantly into the pit. The size of the pit varies from 1 cm to 3 cm, based on their appetite behaviour. Step II is mathematically represented by Eqs (15) and (16).

$c_{i}^{t} = {𝐴𝑛𝑡𝑙𝑖𝑜𝑛}_{j}^{t} + c^{t}$ (15) $d_{i}^{t} = {𝐴𝑛𝑡𝑙𝑖𝑜𝑛}_{j}^{t} + d^{t}$ (16)

where,

$c^{t} =$ Lowest of all parameters at t^th iteration

$d^{t} =$ Highest of all parameters at t^th iteration

$c_{i}^{t} =$ Lowest of all parameters at i^th iteration

$d_{i}^{t} =$ Highest of all parameters at i^th iteration

And c^t is the minimum of i^th variables at t^th iteration and d^t indicates the vector including the maximum of i^th variables at t^th iteration.

Step III: Trapping of ants in tarps

Ant lion is in the pit looking back and watching for the food. ALO uses roulette wheel operation. Roulette wheel is an operation which is used to identify the Antlion based on its best fitness. Antlion dig their holes to trap the food, in relation to their best fitness value.

Step IV: Tumbling ants leading to ant lion

When an ant is in a net, the Antlion begins scattering sands in the middle of the pit to the outer edge to roll down the fleeing ant. Random walks pseudo-sphere is diminished dynamically range of ants to simulate this stage mathematically given by Eqs (17) and (18).

$C_{t} = \frac{c^{t}}{I}$ (17) $D_{t} = \frac{d^{t}}{I}$ (18)

where $I$ is a ratio, $c^{t}$ is the minimum of all variables at t^th iteration, and $d^{t}$ indicates the vector including the maximum of all variables at t^th iteration.

Step V: Capturing predators and reconstructing the trap

Inevitably as it’s fallen to the lower part of the trap, an ant gets trapped in the ant lion’s jaw. This cycle of capturing prey is calculated by ensuring that the ant is healthier than the Antlion it belongs to. The Antlion then changes its location to the captured ant’s most probable location to maximize the chance of capturing fresh prey as shown in Eq. (19).

${𝐴𝑛𝑡𝑙𝑖𝑜𝑛}_{j}^{t} = {𝐴𝑛𝑡}_{i}^{t}, 𝑖𝑓 f ({𝐴𝑛𝑡}_{i}^{t}) > f ({𝐴𝑛𝑡𝑙𝑖𝑜𝑛}_{j}^{t})$ (19)

where $t$ shows the current iteration, ${𝐴𝑛𝑡𝑙𝑖𝑜𝑛}_{j}^{t}$ shows the position of selected $j_{t h}$ antlion at $t^{t h}$ iteration, and ${𝐴𝑛𝑡}_{i}^{t}$ indicates the position of $i_{t h}$ ant at $t^{t h}$ iteration.

Step VI: Elitism

Elitism retains the strongest approach achieved in every cycle, and here the most suitable Antlion obtained thus far throughout each iteration is preserved and called an elite Antlion. Now this elite Antlion controls all the ants’ movements across iterations. Ants’ motion is often guided at the same period by the roulette wheels randomly selected Antlion. Elitism is a critical method to find the right answer at every point of the optimization cycle for certain type of evolutionary computation.

Figure 10.

Block diagram of proposed ALO-FOPID controller.

3.5 Proposed ALO-FOPID controller design

The proposed ALO-FOPID control structure is shown in Fig. 10. It consists of an Antlion optimizer with PID controller for a particular test system under consideration.

The suggested controller’s detailed steps are outlined below.

Input: System with adjustable FOPID controller parameter with randomly generated ant and Antlion size $n$ in 5 dimensions.

Output: The optimum value of FOPID controller parameter with the help of best solutions.

Consider the boundary values of the controller parameters while initializing the Ant population, Antlions, and number of iterations. $(K_{p}, K_{i}, K_{d}, λ, μ)$ .

The number of ants and the number of variables or dimensions are indicated in the position matrix $(M_{𝑎𝑛𝑡})$ , where $n$ and $d$ are the number of ants and the number of variables or dimensions, respectively. The number of variables is five in this example because the five control parameters must be modified.

The fitness (objective) function of each ant is evaluated and stored in a fitness matrix termed as $M_{𝐹𝐴}$ , where $n$ indicates the number of ants in the population, $d$ represents the number of variables, and $f$ represents the fitness or objective function.

$M_{𝐹𝐴} = [\begin{matrix} f (A_{1, 1} A_{1, 2} \dots . . A_{1, d}) \\ f (A_{2, 1} A_{2, 2} \dots . . A_{2, d}) \\ \begin{matrix} ⋮ ⋮ ⋮ ⋮ \\ f (A_{n, 1} A_{n, 2} \dots . . A_{n, d}) \end{matrix} \end{matrix}]$

Because the Antlions are hidden within the search space, their position and fitness value are also kept in the matrices $M_{𝐴𝑛𝑡𝑙𝑖𝑜𝑛}$ and $M_{𝐹𝐴𝐿}$ respectively. The numbers n and d denote the number of Antlions and variables, respectively.

$M_{𝐴𝑛𝑡𝑙𝑖𝑜𝑛} = [\begin{matrix} A L_{1, 1} A L_{1, 2} \dots . . A L_{1, d} \\ A L_{2, 1} A L_{2, 2} \dots . . A L_{2, d} \\ \begin{matrix} ⋮ ⋮ ⋮ ⋮ \\ A L_{n, 1} A L_{n, 2} \dots . . A L_{n, d} \end{matrix} \end{matrix}]$

$M_{𝐹𝐴𝑙} = [\begin{matrix} f (A L_{1, 1} A L_{1, 2} \dots . . A L_{1, d}) \\ f (A L_{2, 1} A L_{2, 2} \dots . . A L_{2, d}) \\ \begin{matrix} ⋮ ⋮ ⋮ ⋮ \\ f (A L_{n, 1} A L_{n, 2} \dots . . A L_{n, d}) \end{matrix} \end{matrix}]$

The best fitness function is picked from $M_{𝐹𝐴𝐿}$ using a roulette wheel, and then the appropriate Antlion is chosen as the best.

An Antlion is chosen using a roulette wheel in each round. The boundary positions are updated using Eqs (12) and (13) which are proportionate to the current iteration, according to the Antlion.

For each ant, a random walk is produced, and its step is normalized over the search space of a selected Antlion using Eq. (9) for each ant. Following that, the ants’ positions are modified by taking Eq. (17) into account.

The fitness values of all the ants are then determined, and the Antlion is replaced with its associated ant if necessary, as indicated in Eq. (16).

If an Antlion has a higher fitness function than the ideal, the position of the optimum Antlion is updated.

The method is repeated until the maximum number of iterations is reached. After all of the iterations have been completed, the best or elite answer is determined.

Figure 11 represents flow chart for ALO algorithm. It represents step by step process for running the optimization algorithm. Algorithm is run till the termination criteria is satisfied and as the desired solution as given by Eq. (20) is obtained & it will terminate.

${𝐴𝑛𝑡}_{i}^{t} = \frac{R_{A}^{t} + R_{E}^{t}}{2}$ (20)

where; $R_{A}$ represents roulette wheel selection of Antlion walk randomly in search space for $t^{t h}$ iteration and $R_{E}$ represents roulette wheel selection of elite walk randomly in search space for $t^{t h}$ iteration.

The simulation parameters in Table 3 are used to determine the optimal gain of the proposed FOPID controller. The lower bound (lb) and the upper bound (ub) varies depending on the gain of the FOPID controller to be optimized. When optimizing the fractional part of the controller i.e., $λ$ and $μ$ , the lower bound was set to 0.1 and the upper bound was set to be 1.0. However, when optimizing the gain of the integer part of the controller i.e., Proportional, Integral and Derivative (PID), the lower bound was set to 0 and the upper bound was set to be 50, 30 and 20, respectively. The parameters for different test systems considered for design of damping controllers are given in Table 3. It is to be noted that all the parameters are similar for all algorithms.

Table 3

ALO algorithm parameters

S. no.	Antlion algorithm parameters	Values
1.	Number of variables	5
2.	Maximum number of iterations	100
3.	Number of search agents	50
4.	Upper bound of variables	$[P = 50, I = 30, D = 20, λ = 1, μ = 1]$
5.	Lower bound of variables	$[P = 0, I = 0, D = 0, λ = 0.1, μ = 0.1]$

Figure 11.

Flow chart for ALO algorithm.

4. Results and discussion

Application of different optimization techniques for estimation of controller parameters is analyzed. An efficient programmer with a support of effective programming language can solve any complex optimization task with his programming skills. The implementation of the proposed PID controller is measured with the various optimization techniques like GA, ACO, and Antlion Optimization (ALO) in different networks.

The analysis of stability in SMIB and multi machine power system network is based on following criteria [38].

Rotor speed and power deviation minimization (time domain response) analysis.

Damping ratio and Eigenvalues analysis

Based on above criterion, different optimization techniques based PID controllers are compared such that they minimize the overshoots and increase the damping ratio. The proposed work is implemented and tested on three IEEE test cases as discussed in the sub-sections below. In test case-1, three operating regimes, light load, normal load and heavy load have been taken. The time domain specifications of different intelligent optimization techniques for all loading condition are listed in Table 4 for test case 1. The parameters of ALO based FOPID controller, viz., $K_{p}$ , $K_{i}$ , $K_{d}$ , $λ$ , $μ$ with other FOPID control designs based on GA and ACO optimization techniques are shown in Table 4. In test case-2 and test case-3, two operating regimes, three-phase fault at mid-point of the line outage of line 8–9 have been taken. The time domain specifications of different intelligent optimization techniques for all loading condition are listed in Table 6 for test case 2 and Tables 8 and 9 for test case-3. Performance comparison of different optimization-based controllers is analyzed in terms of maximum overshoot $(M_{p})$ and settling time $(t_{s})$ of speed deviation which is a function of change in rotor angle. We have compared the ALO-FOPID controller results with other FOPID control designs based on GA and ACO optimization techniques.

Table 4
Loading condition of SMIB system

Operating points for test case 1
Types of power	Light load	Normal load	Heavy load
P	0.6	1.0	1.4
Q	0.3	0.2	0.3

Table 5

Comparison of overshoot and settling time in parameters for test case-1

Operating	Optimization	PID parameters					Peak	Settling time
point	technique	$K_{p}$	$K_{i}$	$K_{d}$	$λ$	$μ$	overshoot (p.u.)	(sec)
Light load	GA-PID	28.5090	0.4860	0.3541	0.3586	0.3281	0.59 $\times$ 10^{- 3}	4.2
	ACO-PID	21.5221	0.2103	0.1689	0.6851	0.4586	0.78 $\times$ 10^{- 3}	5.5
	ALO-PID	38.3261	0.1153	0.2321	0.3224	0.7531	0.32 $\times$ 10^{- 3}	2.9
Normal load	GA-PID	18.284	0.124	0.196	0.9689	0.7054	1.01 $\times$ 10^{- 3}	5.7
	ACO-PID	15.7044	0.1023	0.187	0.5541	0.3682	1.21 $\times$ 10^{- 3}	5.5
	ALO-PID	24.335	0.4321	0.2162	0.2987	0.3981	0.58 $\times$ 10^{- 3}	2.2
Heavy load	GA-PID	14.826	0.983	0.253	0.4512	0.5264	1.43 $\times$ 10^{- 3}	5.8
	ACO-PID	20.002	0.256	0.532	0.6232	0.6258	0.98 $\times$ 10^{- 3}	3.9
	ALO-PID	22.315	0.835	0.332	0.5187	0.5471	0.33 $\times$ 10^{- 3}	2.6

4.1 Test case-1

In this test system, linearized model of a single machine connected to an infinite bus is examined for optimization of PID parameters. To assess the efficacy and robustness of the proposed intelligent control methods for parameters of PID controller, three loading regimes (Light, Medium, and Heavy) are considered. Eigenvalues PID controller designed using intelligent optimization algorithms are obtained for analysis.

4.1.1 Operating Conditions for test case-1

For a SMIB system, dynamic response with PID controller is simulated for 0.1 p.u. step change in reference voltage $(Δ V_{𝑟𝑒𝑓})$ and mechanical torque $(Δ T_{m})$ respectively. The time response curve for change in speed deviation $(Δ ω)$ and change in rotor angle deviation $(Δ δ)$ are plotted for three loading regimes as shown in Table 4.

4.1.2 Time domain analysis

Table 6
Comparison of weakly damped damping ratio and Eigenvalue for test system-1

Loading conditions	Optimization technique	Weakly damped values
		Eigenvalues	Damping ratio
Light load	GA-PID	$- 0.9021 \pm j 7.821$	0.1145
	ACO-PID	$- 0.8317 \pm 𝒋 6.923$	0.1192
	ALO-PID	$- 1.1080 \pm 𝒋 7.932$	0.1382
Normal load	GA-PID	$- 0.9281 \pm j 6.962$	0.1321
	ACO-PID	$- 0.7723 \pm 𝒋 5.234$	0.1459
	ALO-PID	$- 1.1321 \pm 𝒋 7.256$	0.1541
Heavy load	GA-PID	$- 0.8321 \pm j 7.452$	0.1110
	ACO-PID	$- 0.8006 \pm 𝒋 6.5932$	0.1205
	ALO-PID	$- 1.0321 \pm j 7.192$	0.1420

Figure 12.

Deviation in speed and power angle under light load.

For establishing the efficacy of the proposed PID controller, the SMIB system for the light loading condition has been analyzed with respect to time response and Eigenvalues. For different loading conditions, with values of P & Q specified, the response curves showing the deviation in speed and power angle are shown in Figs 12, 13 and 14, respectively.

Performance comparison of different optimization-based controllers is analyzed in terms of maximum overshoot $(M_{p})$ and settling time $(t_{s})$ of speed deviation which is a function of change in rotor angle. The time domain specifications of different intelligent optimization techniques for all loading condition are listed in Table 5.

From the results obtained, it is evident that ALO-PID controller designed provides better response than the GA-PID and ACO-PID. Further, it is observed that Antlion Optimization (ALO) based PID provides better dynamic response with minimum peak overshoot and reduced settling time.

4.1.3 Eigenvalue analysis

The Eigen values of the system are found by using its state-space matrix. Eigenvalue and damping ratio is computed at the different operating conditions with optimal damping controller parameters and are shown in Table 6.

Figure 13.

Deviation in speed and power angle under normal load.

Figure 14.

Deviation in speed and power angle under heavy load.

Figure 15.

Deviation in speed and power of G1 under fault.

All the Eigenvalues indicate the stable locations in s-plane. In light load condition, real part of the Eigenvalue in GA-PID is $-$ 0.9021 is minimized to $-$ 0.8317 and $-$ 1.1080 for intelligent PID controllers implemented as shown in Table 8. This clearly satisfies the objective function formulated to achieve stability. The Eigenvalues for ALO-PID for light, normal and heavy load are $-$ 1.1080, $-$ 1.1321 and $-$ 1.0321, respectively, which are on the left hand side of s-plane indicating stability and are better in comparison with GA-PID and ACO-PID; thus, enhancing the SMIB power system stability to a greater extent. Same result is obtained for other operating conditions. For stable operation of power systems, threshold limit of the damping ratios recommended is 0.05. The minimum value of damping ratio in GA-PID analysis is 0.1145 and is maximized to 0.1192 and 0.1382 for optimized PID controllers with ACO and ALO, respectively. It is seen from that the ALO-PID controllers provide better damping as compared to other controllers for SMIB power system.

4.2 Test case-2

We have taken a test system comprising of four machines, two areas and developed its Simulink model to be simulated by conventional methods. Thus, the parameters of the power damping controller are obtained.

4.2.1 Operating conditions for test system-2

Under transient conditions, the following operating conditions are applied, and the effectiveness of the controller is determined. The conditions are as follows [39]:

Three-phase fault at mid-point of the line

Outage of line 8–9

Figure 16.

Deviation in speed and power of G4 under fault.

Figure 17.

Deviation in speed and power of G2 under temporary line outage at bus 8–9.

4.2.2 Time response stability analysis

In first operating condition, a three-phase fault is simulated in mid-point of the lines of the nine-bus system, i.e., three phase short circuit fault. Controller parameters obtained for first case by all the intelligent optimization methods are used for the simulation of the four generator ten bus power system network. The power and speed deviation response curves obtained for generators G1 and G4 with the intelligent optimization techniques based PID controllers are shown in Figs 15 and 16.

The responses curve of Generators 1 and 4 depicting the deviation in speed and power are shown in Figs 15 and 16, by executing the controllers in the four machine test case for first disturbance. From the dynamic response curves, it is observed that the steady state error is zero for all the controllers.

Table 7
Comparison of overshoot and settling time for test case-2

Operating	Optimization	Gen	PID parameters					Peak over	Settling time
point	technique		$K_{p}$	$K_{i}$	$K_{d}$	$λ$	$μ$	shoot (pu)	(sec)
Three-phase	GA-PID	G1	14.23	2.36	0.22	0.3561	0.3231	$- 8.35 \times 10^{- 3}$	9.54
short circuit		G2	12.56	2.32	1.33	0.5251	0.5421	$- 7.80 \times 10^{- 3}$	8.23
		G3	21.34	3.33	0.39	0.3584	0.8541	$- 8.10 \times 10^{- 3}$	10.54
		G4	8.35	1.17	0.37	0.9614	0.3215	$- 7.53 \times 10^{- 3}$	10.25
	ACO-PID	G1	36.54	2.32	0.43	0.1154	0.6521	$- 8.01 \times 10^{- 3}$	4.37
		G2	28.33	3.68	0.97	0.3258	0.4532	$- 7.80 \times 10^{- 3}$	5.30
		G3	29.65	2.85	0.41	0.6134	0.3786	$- 6.26 \times 10^{- 3}$	4.95
		G4	15.57	0.98	0.86	0.7853	0.7862	$- 7.53 \times 10^{- 3}$	5.89
	ALO-PID	G1	6.332	0.293	0.336	0.1148	0.2541	$- 5.15 \times 10^{- 3}$	7.75
		G2	12.523	0.121	0.236	0.3232	0.6559	$- 5.95 \times 10^{- 3}$	6.90
		G3	3.224	0.883	0.623	0.6547	0.3214	$- 6.35 \times 10^{- 3}$	8.20
		G4	2.229	0.523	0.113	0.8921	0.9961	$- 5.35 \times 10^{- 3}$	9.05
Outage of line	GA-PID	G1	12.564	13.523	0.986	0.6154	0.8994	$4.55 \times 10^{- 4}$	5.25
between 8–9		G2	21.452	9.567	0.238	0.2514	0.3254	$4.23 \times 10^{- 4}$	5.25
		G3	17.563	19.256	0.249	0.9478	0.5241	$3.92 \times {𝟏𝟎}^{- 4}$	6.35
		G4	22.862	13.553	0.329	0.5412	0.9953	$- 4.05 \times {𝟏𝟎}^{- 4}$	5.89
	ACO-PID	G1	33.742	17.883	0.837	0.6521	0.9873	$- 3.95 \times {𝟏𝟎}^{- 4}$	4.37
		G2	29.238	12.382	0.772	0.3237	0.5641	$- 3.80 \times {𝟏𝟎}^{- 4}$	4.35
		G3	24.350	9.059	0.561	0.5223	0.6532	$- 2.46 \times {𝟏𝟎}^{- 4}$	4.95
		G4	29.2185	11.032	0.908	0.9664	0.3324	$- 3.77 \times {𝟏𝟎}^{- 4}$	5.10
	ALO-PID	G1	23.523	13.652	0.982	0.8445	0.5847	$- 3.35 \times {𝟏𝟎}^{- 4}$	3.20
		G2	36.228	9.237	0.886	0.7142	0.6325	$- 3.22 \times {𝟏𝟎}^{- 4}$	3.78
		G3	18.621	12.862	0.872	0.6552	0.9947	$- 2.15 \times {𝟏𝟎}^{- 4}$	2.90
		G4	21.723	11.024	1.021	0.9636	0.3693	$- 2.25 \times {𝟏𝟎}^{- 4}$	4.30

Figure 18.

Deviation in speed and power of G3 under temporary line outage at bus 8–9.

Figures 17 and 18 indicate the speed and power angle deviations of Generator 3 under conditions of outage of line at bus 8–9. From these speed and power deviation curves, it is clear that the implementation of optimization based PID controllers damp the oscillations in an effective manner, as the deviation overshoots are reduced and the deviations settle at the earliest possible time. The Antlion Optimization (ALO) based power damping controller provides considerably better performance compared to other methods. It is further analyzed that the undershoots, overshoots and settling time are more or less same for the generators at first operating condition. However, best dynamic response curves are obtained using Antlion Optimization (ALO) technique followed by GA and ACO techniques. Comparison of overshoot and settling time in different controller parameter are measured and listed in Table 7.

From Table 7, the maximum under shoot obtained for Generator 1 using GA-PID is $8.35 \times 10^{- 3}$ , ACO-PID is $7.35 \times 10^{- 4}$ and $5.15 \times 10^{- 4}$ p.u for ALO based PID. It is observed that Antlion Optimization (ALO) based PID provides better damping with reduced settling time for first operating condition. For overshoot, ALO-PID and ACO-PID provide minimum overshoot as compared to GA-PID.

4.2.3 Eigenvalue analysis

The damping ratios and Eigenvalue corresponding to the optimized PID controller are given in Table 8.

Table 8
Comparison of weakly damped Eigen value and damping ratio for test case-2

Operating point	Optimization	Weakly damped values
	technique	Eigenvalues	Damping ratios
Three-phase short circuit	GA-PID	$- 1.627 \pm j 5.091, - 2.053 \pm j 9.721$ $- 2.176 \pm j 8.827, - 1.738 \pm j 11.251$	0.3045, 0.2067, 0.2393, 0.1526
	ACO-PID	$- 1.054 \pm j 4.328, - 1.908 \pm j 8.819$ $- 2.412 \pm j 10.382, - 1.321 \pm j 9.211$	0.2366, 0.2114 0.2263, 0.1419
	ALO-PID	$- 2.018 \pm j 7.221, - 3.192 \pm j 9.023$ $- 2.018 \pm j 6.282, - 3.229 \pm j 4.392$	0.2691, 0.3335, 0.3058, 0.5932
Outage of line between 8–9	GA-PID	$- 0.952 \pm j 6.621, - 2.295 \pm j 9.521$ $- 1.983 \pm j 7.331, - 1.854 \pm j 8.835$	0.1423, 0.2343, 0.2611, 0.2050
	ACO-PID	$- 1.721 \pm j 6.318, - 2.412 \pm j 8.441$ $- 1.821 \pm j 6.231, - 1.912 \pm j 8.212$	0.2628, 0.2747, 0.2805, 0.2267
	ALO-PID	$- 1.909 \pm j 6.821, - 3.631 \pm j 7.023$ $- 2.109 \pm j 7.128, - 2.993 \pm j 8.753$	0.2695, 0.4592, 0.2837, 0.3235

In Table 8, the computed Eigenvalues consist of both real and complex values. The Eigenvalues that contain only real part represent non-oscillatory mode in the system. The oscillatory frequency will be zero and this will not contribute to any oscillations in the system. The complex Eigenvalues represent the oscillatory modes. The Eigenvalues for all operating condition confirm that the system stability is improved, as all the Eigenvalues are well placed at stable locations in s-plane. These results provide the better stability performance. In Table 6, Antlion Optimization (ALO) based PID controller provides the best possible locations for the Eigenvalues compared to the other bio inspired optimization-based controller designs. Using all PID parameters, damping ratios are computed for all operating condition of multi-machine test system as shown in Table 8. Though the damping ratios for GA and ACO based controllers are more and sufficient for stability.

4.3 Test system-3 (New England Ten Machine Power System Network)

The performance of PID controller has been assessed on single machine and multi machine system as test cases. It is observed that the evaluation of the performance of these algorithms becomes cumbersome and tedious, when the test system becomes complex consisting of many generators and buses constituting multi areas. Hence, in order to analyze the reliability of operation of proposed PID controller, the system taken as test case is known as New England system consisting of 10 generating units, 19 loads, 12 transformers, 39 buses and 36 transmission lines with PID parameters are considered.

4.3.1 Operating conditions for test system-3

The effectiveness and robustness of the proposed PID controller is determined by applying the following operating conditions:

Three phase fault between bus 1 and bus 2.

Outage of line between 21–22.

Figure 19.

Deviation in speed and power of G4 under fault condition.

Figure 20.

Deviation in speed and power of G8 under fault condition.

4.3.2 Time response stability analysis

To analyse the effectiveness of the PID controller in mitigating observed oscillations for different operating conditions, time response analysis is carried out to prove the robustness of the designed PID. In first case, a temporary three-phase fault occurs between bus1 and bus 2 at 1 second and is cleared after 1.2 second and the system’s behaviour is evaluated for 20 seconds. In this condition, the dynamic response curves obtained for Generators 4 and 8 are depicted in Figs 19 and 20.

Figures 19 and 20 show the speed and power deviation response of Generator 4 and Generator 8 when three phase fault occur between bus 1 and bus 2. It is evident that oscillations are suppressed, and the generators responses reach the steady state when optimization based PIDs are employed in the system. In second case, a temporary line outage is applied between lines 21–22 at 1 second and is cleared after 1.2 second and the system’s behaviour is evaluated for 20 second. In this condition the time response curves obtained for Generators 1 and 6 are shown in Figs 20 to 21.

Figure 21.

Deviation in speed and power of G1 under temporary line outage at bus 21–22.

Figure 22.

Deviation in speed and power of G6 under temporary line outage at bus 21–22.

Figures 21 and 22 shows the speed and power deviation curve of Generator 1 and Generator 6 in outage condition. The oscillations of the power system are suppressed when Antlion optimization (ALO) based PID is incorporated in the system. Further, while GA and ACO based PID controller provide better responses. Comparison of overshoot and settling time is listed in Tables 9 and 10.

Table 9

Comparison of overshoot and settling time for test case-3

Operating point	Optimization	Gen	PID parameters					Peak	Settling
	technique		$K_{p}$	$K_{i}$	$K_{d}$	$λ$	$μ$	overshot (p.u.)	time (sec)
Three phase fault	GA-PID	G1	8.563	3.242	1.523	0.6541	0.5487	$4.23 \times 10^{- 3}$	9.25
between bus 1 and		G2	9.235	4.523	8.268	0.7524	0.6287	$5.58 \times 10^{- 3}$	7.38
bus 2		G3	7.359	3.338	8.237	0.3231	0.4961	$5.15 \times 10^{- 3}$	7.95
		G4	10.023	4.602	4.284	0.3569	0.6987	$3.25 \times 10^{- 3}$	7.90
		G5	9.554	6.857	4.892	0.5587	0.4125	$3.92 \times 10^{- 3}$	6.65
		G6	12.296	7.234	14.820	0.9687	0.5532	$5.56 \times 10^{- 3}$	9.60
		G7	8.734	4.027	3.552	0.5648	0.9968	$4.49 \times 10^{- 3}$	9.10
		G8	12.003	3.250	5.562	0.3265	0.5689	$15.58 \times 10^{- 3}$	6.25
		G9	14.032	2.353	9.234	0.7845	0.8956	$6.35 \times 10^{- 3}$	8.86
		G10	12.362	4.563	7.065	0.4512	0.5623	$9.25 \times 10^{- 3}$	8.25
	ACO-PID	G1	21.356	12.882	9.238	0.8754	0.9865	$4.20 \times 10^{- 3}$	8.45
		G2	29.596	13.002	11.352	0.5421	0.6532	$5.30 \times 10^{- 3}$	6.20
		G3	24.053	9.238	14.983	0.7946	0.4613	$5.10 \times 10^{- 3}$	6.10
		G4	31.205	16.008	22.069	0.9764	0.6431	$8.25 \times 10^{- 3}$	5.85
		G5	23.056	7.886	17.025	0.8523	0.9632	$3.90 \times 10^{- 3}$	5.05
		G6	30.523	16.028	12.382	0.7412	0.8521	$5.47 \times 10^{- 3}$	7.15
		G7	27.887	13.782	11.006	0.8432	0.2975	$4.40 \times 10^{- 3}$	5.30
		G8	19.925	18.235	23.098	0.7530	0.9512	$15.50 \times 10^{- 3}$	6.38
		G9	23.582	9.938	13.397	0.6491	0.1530	$6.30 \times 10^{- 3}$	7.25
		G10	32.960	18.237	22.027	0.1452	0.5529	$9.20 \times 10^{- 3}$	7.13
	ALO-PID	G1	11.238	13.253	9.883	0.1996	0.2014	$4.19 \times 10^{- 3}$	7.11
		G2	32.652	9.326	16.652	0.3229	0.5423	$5.10 \times 10^{- 3}$	6.00
		G3	23.563	12.983	16.032	0.1207	0.2496	$5.10 \times 10^{- 3}$	6.00
		G4	32.553	20.021	22.235	0.1395	0.5864	$8.10 \times 10^{- 3}$	5.80
		G5	25.632	18.923	9.536	0.3269	0.2075	$3.80 \times 10^{- 3}$	5.10
		G6	22.352	11.212	21.553	0.3164	0.6108	$5.30 \times 10^{- 3}$	6.90
		G7	19.053	17.251	16.128	0.1401	0.9654	$4.30 \times 10^{- 3}$	6.94
		G8	42.536	16.553	22.362	0.1905	0.2708	$15.60 \times 10^{- 3}$	6.50
		G9	23.562	18.634	18.689	0.2456	0.3244	$6.35 \times 10^{- 3}$	7.65
		G10	42.321	19.025	29.029	0.3658	0.9963	$9.15 \times 10^{- 3}$	6.92

At the outset, Antlion optimization (ALO) based PID performs better than ACO as dynamic response and performance indices are concerned. Thus, it is concluded that by providing PID controller in all the generators with real power given as the input to the stabilizer, it is possible to damp out unwanted oscillations in the system when it is perturbed.

4.3.3 Eigenvalue analysis

Table 11 shows the comparison of Eigenvalues corresponding to the electromechanical modes with damping ratio for the all generators, considered GA-PID, ACO-PID and Antlion optimization (ALO) based PID.

Figure 23.

Eigenvalues plot comparison for New England System for GA-FOPID, ACO-PID and ALO-PID for line outages between lines 8–9 and lines 21–22.

Table 10

Comparison of overshoot and settling time for test case-3

Outage of line between	GA-PID	G1	16.234	6.882	9.563	0.2080	0.1998	$10.10 \times 10^{- 5}$	7.10
21–22		G2	10.235	6.237	7.228	0.8991	0.2547	$8.25 \times 10^{- 3}$	4.25
		G3	21.983	9.687	3.339	0.7865	0.8870	$6.79 \times 10^{- 3}$	3.21
		G4	14.237	5.060	4.203	0.1996	0.9591	$6.26 \times 10^{- 3}$	8.87
		G5	10.130	2.032	8.592	0.9987	0.5521	$4.88 \times 10^{- 3}$	7.10
		G6	17.230	3.521	7.021	0.3328	0.5213	$1.28 \times 10^{- 3}$	3.98
		G7	17.052	8.892	4.583	0.6083	0.9564	$11.50 \times 10^{- 3}$	5.56
		G8	11.228	11.392	6.083	0.2214	0.2007	$7.90 \times 10^{- 3}$	5.45
		G9	12.552	6.005	9.982	0.5228	0.5691	$6.85 \times 10^{- 3}$	7.81
		G10	21.021	9.928	5.238	0.9923	0.7708	$2.78 \times 10^{- 3}$	5.63
	ACO-PID	G1	14.332	5.238	8.836	0.6625	0.2937	$10.09 \times 10^{- 5}$	6.98
		G2	12.832	8.332	9.023	0.5087	0.6532	$8.14 \times 10^{- 3}$	4.10
		G3	24.556	7.802	4.251	0.8597	0.2087	$5.51 \times 10^{- 3}$	2.97
		G4	21.098	7.889	5.603	0.6535	0.2221	$6.10 \times 10^{- 3}$	8.71
		G5	13.523	8.523	7.036	0.9997	0.6338	$4.50 \times 10^{- 3}$	6.42
		G6	21.001	8.251	5.731	0.7523	0.4452	$1.35 \times 10^{- 3}$	4.11
		G7	18.986	7.059	3.523	0.3369	0.0852	$9.45 \times 10^{- 3}$	3.15
		G8	16.099	9.231	8.553	0.3256	0.8521	$7.80 \times 10^{- 3}$	4.90
		G9	21.531	12.237	11.519	0.9685	0.4571	$5.20 \times 10^{- 3}$	6.23
		G10	32.520	13.577	8.239	0.3358	0.5241	$2.30 \times 10^{- 3}$	3.89
	ALO-PID	G1	22.256	9.213	9.653	0.9689	0.5548	$6.20 \times 10^{- 5}$	5.09
		G2	19.365	11.025	12.053	0.5478	0.6523	$8.12 \times 10^{- 3}$	4.10
		G3	12.369	8.632	19.281	0.5547	0.5583	$4.20 \times 10^{- 3}$	2.75
		G4	32.692	19.294	6.532	0.9923	0.5001	$5.60 \times 10^{- 3}$	7.09
		G5	42.653	16.821	11.024	0.2501	0.3029	$4.11 \times 10^{- 3}$	6.21
		G6	21.321	21.521	14.853	0.2121	0.9987	$1.09 \times 10^{- 3}$	2.50
		G7	31.025	22.213	13.531	0.3325	0.5460	$8.03 \times 10^{- 3}$	3.21
		G8	24.542	9.282	19.867	0.5642	0.2056	$7.65 \times 10^{- 3}$	4.66
		G9	33.243	19.918	9.538	0.2035	0.5016	$5.08 \times 10^{- 3}$	5.84
		G10	21.328	23.562	11.532	0.5861	0.2096	$2.50 \times 10^{- 3}$	4.45

Figure 24.

Nonparametric statistical tests results for GA, ACO and ALO algorithms.

Table 11

Comparison of weakly damped Eigenvalues and damping ratio for Test case-3

Operating point	Optimization	Weakly damped values
	technique	Eigenvalues	Damping ratios
Three phase faults between	GA-PID	$- 1.145 \pm j 2.101$ , $- 1.953 \pm j 4.227$ , $- 2.287 \pm j 5.923$ , $- 1.094 \pm j 4.983$ , $- 2.438 \pm j 5.592$ , $- 2.337 \pm j 6.256$ , $- 2.091 \pm j 7.784$ , $- 1.874 \pm j 8.231$ , $- 2.725 \pm j 4.321$ , $- 8.883 \pm j 9.927$	0.4785, 0.4194, 0.377 0.2144, 0.399, 0.0348 0.2594, 0.2219, 0.3958 0.6693
bus 1 and bus 2	ACO-PID	$- 1.168 \pm 𝒋 1.982$ , $- 2.328 \pm 𝒋 5.675$ , $- 2.882 \pm 𝒋 6.025$ , $- 1.782 \pm 𝒋 5.821$ , $- 3.023 \pm 𝒋 5.921$ , $- 2.822 \pm 𝒋 5.212$ , $- 1.532 \pm 𝒋 5.772$ , $- 2.812 \pm 𝒋 8.221$ , $- 2.291 \pm 𝒋 6.521$ , $- 9.212 \pm 𝒋 7.519$	0.5077, 0.3795, 0.4315 0.2927, 0.4547, 0.4761 0.2565, 0.3236, 0.3314 0.7747
	ALO-PID	$- 1.109 \pm 𝒋 2.118$ , $- 1.923 \pm 𝒋 4.312$ , $- 1.823 \pm 𝒋 4.021$ , $- 2.913 \pm 𝒋 6.023$ , $- 4.131 \pm 𝒋 6.937$ , $- 2.983 \pm 𝒋 5.012$ , $- 1.997 \pm 𝒋 6.901$ , $- 2.792 \pm 𝒋 8.131$ , $- 4.183 \pm 𝒋 9.391$ , $- 9.532 \pm 𝒋 8.838$	0.4638, 0.4072, 0.4129 0.4353, 0.5116, 0.5114 0.2779, 0.3247, 0.4068 0.7332
Outage of line between 21–22	GA-PID	$- 2.002 \pm j 4.521$ , $- 1.232 \pm j 2.995$ , $- 1.541 \pm j 3.337$ , $- 2.332 \pm j 5.521$ , $- 2.985 \pm j 7.749$ , $- 2.245 \pm j 4.338$ , $- 3.002 \pm j 6.014$ , $- 1.943 \pm j 9.562$ , $- 3.662 \pm j 5.012$ , $- 8.825 \pm j 9.838$	0.4048, 0.3804, 0.4192 0.3892, 0.3594, 0.4596 0.4467, 0.2213, 0.5899 0.6677
	ACO-PID	$- 2.119 \pm 𝒋 3.212$ , $- 1.289 \pm 𝒋 2.810$ , $- 2.512 \pm 𝒋 4.991$ , $- 1.782 \pm 𝒋 6.310$ , $- 1.779 \pm 𝒋 7.012$ , $- 5.728 \pm 𝒋 3.121$ , $- 2.206 \pm 𝒋 8.782$ , $- 2.212 \pm 𝒋 8.721$ , $- 4.012 \pm 𝒋 4.889$ , $- 8.913 \pm 𝒋 9.771$	0.5506, 0.4169, 0.4495 0.2717, 0.2459, 0.8781 0.2436, 0.2458, 0.6343 0.6739
	ALO-PID	$- 1.332 \pm 𝒋 2.438$ , $- 1.393 \pm 𝒋 2.382$ , $- 1.538 \pm 𝒋 3.998$ , $- 3.083 \pm 𝒋 5.112$ , $- 2.238 \pm 𝒋 5.183$ , $- 2.039 \pm 𝒋 3.832$ , $- 3.012 \pm 𝒋 5.082$ , $- 1.983 \pm 𝒋 9.131$ , $- 2.183 \pm 𝒋 6.321$ , $- 8.939 \pm 𝒋 8.101$	0.4794, 0.5048, 0.3590 0.5164, 0.3964, 0.4697 0.5098, 0.2122, 0.3264 0.7409

It is evident that when PID designed by using Bio-inspired optimization are incorporated in the system, the Eigenvalues with least stable state are as shown in Table 10. It is seen that the electro mode of Eigenvalues lies in LHP of the s-plane and damping ratio is improved from negative to positive value which in turn increases the system dynamic stability. The damping ratios for all the generators for both cases are shown in Table 10. All these results confirm that the proposed Bio-inspired controllers can make the multi-machine system more stable for all operating conditions.

Figure 25.

Comparative convergence of GA, ACO and ALO algorithms.

Based on the numerical simulation findings of Tables 9, 10 and 11 and Fig. 23 shows the Eigenvalues plot comparison for the New England test system with the proposed ALO-FOPID controller in addition to GA-FOPID and ACO-FOPID. The ALO-FOPID will reliably manage the power system low frequency oscillations as discussed.

Nonlinear simulations were run by 15 iterations for each algorithm since the consistency of the solution found by the GA, ACO, and ALO algorithms is crucial for evaluating their performance. The results of 15 iterations of the Integral of the time weighted absolute error (ITAE) objective function for the Single Machine Infinite Bus (SMIB) power system are shown as boxplots in Fig. 24. According to an examination of the boxplots, the value of ITAE obtained by the ALO technique has a more consistent distribution in a tighter region than that acquired by the GA and ACO.

Figure 25 represents the comparative convergence of three optimization techniques which are implemented in our work. From the overall view of implementation of ACO in FOPID, it is observed that it requires a number of variables to be handled which increases the memory capacity, dimension of the problem and time for execution. In GA, the objective function is much sensitive to adaptive mutation parameter; a slight change in the parameters makes a vast change in objective function. Among the above said evolutionary computation techniques, Antlion optimization (ALO) consistently obtains better solution for all the test systems and in all the operating conditions, Antlion optimization (ALO) based FOPID controller obtains better settling time reduction and improving the stability power system network. From the simulation results and performance analysis, it may be concluded that Antlion optimization (ALO) leads to better solution than other methods like GA and ACO in terms of solution quality and computational time. For the test case-3, the performance of ALO-PID for objective functions J1, J2 and J are shown in Table 12 along with damping ratios. As evident from Table 12 that all the eigen values obtained for J by combination of J1 $+$ 10 * J2 are towards the more negative side of LHS of s plane as compared to the eigen values for the objective functions J1 and J2, separately. Also, the damping ratios validate the above claim.

Table 12

Comparison of weakly damped Eigenvalues and damping ratio for test case-3 for J1, J2 and J

	Three phase fault between bus 1 and bus 2	Outage of line between 21–22
Without PID	$- 0.2701 \pm j 5.141, 0.052$	$- 0.028 \pm j 12.945, 0.0022$
	$- 0.1951 \pm j 6.089, 0.0320$	$- 0.270 \pm j 9.581, 0.0281$
	$- 0.2887 \pm j 12.798, 0.022$	$0.195 \pm j 5.556, - 0.035$
	$0.205 \pm j 5.378, - 0.0380$	$- 0.101 \pm j 6.692, 0.0150$
	$- 0.172 \pm j 10.992, 0.0156$	$- 0.167 \pm j 9.727, 0.017$
	$- 0.172 \pm j 9.492, 0.0181$	$0.126 \pm j 4.485, - 0.028$
	$+ 0.0.88 \pm j 3.496, - 0.0251$	$- 0.172 \pm j 9.693, 0.018$
	$- 0.178 \pm j 9.992, 0.0173$	$- 0.269 \pm j 9.342, 0.029$
	$- 0.220 \pm j 8.113, 0.0271$	$- 0.195 \pm j 6.901, 0.0322$
ALO-PID with only J1	$- 1.014 \pm j 6.817, 0.160$	$- 1.135 \pm j 6.109, 0.1826$
	$- 1.229 \pm j 11.021, 0.110$	$- 1.435 \pm j 4.309, 0.3158$
	$- 1.382 \pm j 9.939, 0.1377$	$- 1.134 \pm j 5.802, 0.1921$
	$- 2.153 \pm j 9.989, 0.2106$	$- 1.312 \pm j 3.711, 0.3381$
	$- 2.952 \pm j 9.908, 0.2831$	$- 1.136 \pm j 5.611, 0.1986$
	$- 1.892 \pm j 5.389, 0.3312$	$- 2.414 \pm j 9.818, 0.2388$
	$- 1.005 \pm j 8.209, 0.1215$	$- 2.404 \pm j 7.8112, 0.2941$
	$- 1.985 \pm j 5.999, 0.3141$	$- 1.135 \pm j 11.009, 0.1021$
	$- 3.125 \pm j 9.125, 0.3239$	$- 2.275 \pm j 11.123, 0.2193$
ALO-PID with only J2	$- 1.158 \pm j 3.001, 0.3599$	$- 1.435 \pm j 3.396, 0.3892$
	$- 1.421 \pm j 3.991, 0.3348$	$- 2.005 \pm j 3.549, 0.4911$
	$- 1.815 \pm j 3.901, 0.4218$	$- 2.673 \pm j 9.869, 0.2615$
	$- 2.118 \pm j 5.451, 0.3621$	$- 1.005 \pm j 2.001, 0.4826$
	$- 3.921 \pm j 8.001, 0.4400$	$- 3.395 \pm j 8.651, 0.3658$
	$- 2.552 \pm j 5.228, 0.4386$	$- 1.029 \pm j 2.281, 0.4128$
	$- 1.186 \pm j 5.271, 0.2195$	$- 2.005 \pm j 3.601, 0.4864$
	$- 2.015 \pm j 6.719, 0.2872$	$- 1.134 \pm j 5.805, 0.1921$
	$- 2.938 \pm j 6.439, 0.4151$	$- 1.435 \pm j 4.317, 0.315$
ALO-PID with J	$- 1.109 \pm j 2.118, 0.4638$	$- 1.332 \pm j 2.438, 0.4794$
	$- 1.923 \pm j 4.312, 0.4072$	$- 1.393 \pm j 2.382, 0.5048$
	$- 1.823 \pm j 4.021, 0.4129$	$- 1.538 \pm j 3.998, 0.3590$
	$- 2.913 \pm j 6.023, 0.4353$	$- 3.083 \pm j 5.112, 0.5164$
	$- 4.131 \pm j 6.937, 0.5116$	$- 2.238 \pm j 5.183, 0.3964$
	$- 2.983 \pm j 5.012, 0.5114$	$- 2.039 \pm j 3.832, 0.4697$
	$- 1.997 \pm j 6.901, 0.2779$	$- 3.012 \pm j 5.082, 0.5098$
	$- 2.792 \pm j 8.131, 0.3247$	$- 1.983 \pm j 9.131, 0.2122$
	$- 4.183 \pm j 9.391, 0.4068$	$- 2.183 \pm j 6.321, 0.3264$

The optimized parameters values for test case 3 for all the ten generators along with the overshoot and settling time when a three-phase fault occurs between bus 1 and bus2 are shown in Table 13 with both single objective functions J1 and J2 and the multiobjective function J for ALO-PID.

Table 13

Comparison of overshoot and settling time for test case-3 for J1, J2 and J

Operating point	Optimization	Gen	PID parameters
	technique		$K_{p}$	$K_{i}$	$K_{d}$	$λ$	$μ$
Three phase fault between	ALO-PID with only J1	G1	28.5090	5.311	3.242	0.4860	0.3541
bus 1 and bus 2		G2	21.5221	8.426	4.523	0.2103	0.1689
		G3	41.3600	6.5285	3.338	0.1052	0.1123
		G4	61.1800	8.335	4.602	0.1003	0.122
		G5	18.284	3.562	6.857	0.124	0.196
		G6	15.704	6.235	7.234	0.1023	0.187
		G7	36.257	4.238	4.027	0.1003	0.107
		G8	56.52	5.568	3.250	0.121	0.1086
		G9	14.826	17.883	2.353	0.983	0.253
		G10	20.002	12.382	4.563	0.256	0.532
	ALO-PID with only J2	G1	31.523	9.059	12.882	0.325	0.159
		G2	36.925	11.032	13.002	0.118	0.973
		G3	14.23	21.521	9.238	0.22	0.312
		G4	12.56	18.732	16.008	1.33	0.423
		G5	21.34	15.862	7.886	0.39	0.553
		G6	8.35	17.872	16.028	0.37	0.324
		G7	9.311	19.831	13.782	0.324	0.825
		G8	7.426	20.552	18.235	0.765	0.886
		G9	8.5285	22.676	9.938	0.995	0.531
		G10	9.335	19.002	18.237	0.235	0.736
	ALO-PID with J	G1	11.238	13.253	9.883	0.1996	0.2014
		G2	32.652	9.326	16.652	0.3229	0.5423
		G3	23.563	12.983	16.032	0.1207	0.2496
		G4	32.553	20.021	22.235	0.1395	0.5864
		G5	25.632	18.923	9.536	0.3269	0.2075
		G6	22.352	11.212	21.553	0.3164	0.6108
		G7	19.053	17.251	16.128	0.1401	0.9654
		G8	42.536	16.553	22.362	0.1905	0.2708
		G9	23.562	18.634	18.689	0.2456	0.3244
		G10	42.321	19.025	29.029	0.3658	0.9963

Figure 26.

Comparative convergence of ALO algorithm for J1, J2 and J.

The convergence rate of the single objective functions J1 and J2 and the multiobjective function J for the ALO algorithm is shown in Fig. 26.

To detect significant differences between ALO, ACO and GA optimization techniques, we make use of the Friedman’s test [40] is to rank the algorithms and ALO has the best rank. We have applied an additional Bonferroni-Dunn’s method as a post hoc procedure in order to evaluate the significance level of all the algorithms by calculating the Critical Difference (CD) for comparing their differences with $α =$ 0.05 and $α =$ 0.1 as shown in Eq. (21).

$𝐶𝐷 = q_{α} \sqrt{\frac{k (k + 1)}{6 N}}$ (21)

where parameters $k$ and $N$ represent the number of algorithms and the number of instances, respectively. In our work, $k =$ 6 and $N =$ 120. When $α = 0.1$ , $q_{α}$ is 2.327, and when $α =$ 0.05, $q_{α}$ is 2.576 [41]. Table 8 presents the ranking calculated by Friedman’s test and the critical difference of Bonferroni-Dunn’s procedure.

Table 14

Friedman’s test ranking and critical difference of Bonferroni-Dunn’s procedure

Algorithm	Ranking
Genetic Algorithm	2.4265
Antcolony Optimization	1.9825
Antlion Optimization	0.4238
Critical difference (CD) when $α$ $=$ 0.05, 0.3643
Critical difference (CD) when $α$ $=$ 0.1, 0.3290

Figure 27.

Bonferroni-Dunn’s test.

The graphical results of Bonferroni-Dunn’s test are given in Fig. 27. ALO and comparison algorithms have significant differences at $α =$ 0.05 and $α =$ 0.1.

5. Conclusion

This research has presented a novel approach for the optimal design of Fractional Order PID (FOPID) controllers tailored to the complexities of multimachine power systems. Through a systematic investigation encompassing theoretical analysis, algorithm development, and computational simulations, our study has made significant contributions to the field of power system stability enhancement. This study has addressed a critical gap in current knowledge by proposing an innovative multi-objective design approach for FOPID controllers using the Antlion algorithm. By formulating a comprehensive optimization problem that simultaneously maximizes the damping factor and damping ratio of lightly damped electromechanical modes, we have provided a systematic framework for synthesizing robust controllers capable of effectively dampening power system oscillations across diverse operating conditions. This novel approach represents a paradigm shift in FOPID controller design, offering greater flexibility and adaptability to the intricacies of multimachine power systems. Through extensive computational simulations conducted on benchmark IEEE test systems, the efficacy of the proposed FOPID controller have been demonstrated with enhancing power system stability. The proposed controller design outperforms existing optimization techniques in terms of damping oscillations and minimizing transient deviations, thereby mitigating the risk of voltage instability, frequency deviations, and cascading failures. These findings underscore the practical significance of the research in addressing the pressing need for reliable control strategies in modern power grids. Furthermore, this study has provided valuable insights into the dynamics underlying power system oscillations and the role of excitation control in maintaining grid stability. By conducting a thorough analysis of system response under various operating scenarios, we have deepened our understanding of the factors influencing stability and identified opportunities for further refinement in controller design. This nuanced understanding lays the groundwork for future research endeavours aimed at advancing the state-of-the-art in power system stability enhancement. The adoption of our proposed approach stands to enhance the resilience and reliability of modern power grids, ultimately benefiting society as a whole by ensuring uninterrupted access to electricity.

In conclusion, this research represents a significant contribution to the field of power system stability enhancement, offering a systematic framework for the optimal design of FOPID controllers in multimachine power systems. By bridging the gap between theory and practice, our research paves the way for advancements in control strategies that are critical to ensuring the reliable and efficient operation of modern power grids. We hope that our findings will inspire further research in this area and contribute to the ongoing efforts to address the challenges facing the global energy landscape.

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On the optimal multi-objective design of fractional order PID controller with antlion optimization

Abstract

Keywords

1. Introduction

2. Background work

3. System modelling

3.1 Test system configuration

3.1.1 SMIB power system

3.2 Damping controller

3.2.1 Fractional order PID (FOPID) controller

Table 2 Pseudo code for antlion optimization algorithm

Step I: Random walk of ants

Step II: Construction of traps

Step III: Trapping of ants in tarps

Step IV: Tumbling ants leading to ant lion

Step V: Capturing predators and reconstructing the trap

Step VI: Elitism

Table 4 Loading condition of SMIB system

4.1.1 Operating Conditions for test case-1

4.1.2 Time domain analysis

Table 6 Comparison of weakly damped damping ratio and Eigenvalue for test system-1

4.2.1 Operating conditions for test system-2

Table 7 Comparison of overshoot and settling time for test case-2

Table 8 Comparison of weakly damped Eigen value and damping ratio for test case-2

4.3.1 Operating conditions for test system-3

References

Table 2
Pseudo code for antlion optimization algorithm

Table 4
Loading condition of SMIB system

Table 6
Comparison of weakly damped damping ratio and Eigenvalue for test system-1

Table 7
Comparison of overshoot and settling time for test case-2

Table 8
Comparison of weakly damped Eigen value and damping ratio for test case-2