Improving Interarea Mode Oscillation Damping in Multi-Machine Energy Systems through a Coordinated PSS and FACTS Controller Framework

Abstract

Power system stability is of paramount importance in the context of energy sustainability. The reliable and efficient operation of power systems is crucial for supporting modern societies, economies, and the growing demand for electricity while minimizing environmental impact and increasing sustainability. Due to the insufficient effect of power system stabilizers (PSSs) on damping the inter-area mode oscillations, Flexible AC Transmission System (FACTS) devices are utilized for damping this mode and stabilizing power systems. In the present study, a novel optimization framework considering different and variable weight coefficients based on eigenvalue locations is presented, and the parameters of PSS and variable impedance devices, including static Volt-Ampere Reactive (VAR) compensator (SVC) and Thyristor-Controlled Series Compensator (TCSC) (comprising amplifying gain factor and time constants of phase-compensating blocks), are optimized in a coordinated manner using the proposed optimization framework built based on genetic algorithm (GA). Moreover, in the suggested optimization framework, the locations of FACTS devices and control signals are considered optimization parameters. Numerical results for the IEEE 69-bus power system demonstrated an effective improvement in the damping of inter-area modes utilizing the offered approach.

1. Introduction

Power system stability is a pivotal pillar in the realm of energy sustainability, twisting the reliable delivery of electricity with the imperative to decrease environmental consequences. In an era where distributed generation and renewable energy sources are progressively supplanting conventional fossil fuel-based energy systems, ensuring power system stability takes on heightened significance. The equilibrium between electricity generation, transmission, and consumption becomes vital to preventing grid failures and blackouts, which not only disrupt daily life but also lead to substantial societal disarray and economic losses [1,2]. By upholding power system stability, sustainable energy transitions are facilitated, as a resilient grid can integrate intermittent renewable sources, thereby curbing greenhouse gas emissions and fostering a more ecologically harmonious energy landscape for future generations [3,4]. The United Nations Sustainable Development Goals (UN SDGs) provide a comprehensive framework for promoting worldwide sustainability. Several objectives align significantly with the crucial aspect of maintaining power system stability within this particular framework. The aforementioned items encompass the following goals: Goal (7) Ensuring widespread availability of cost-effective and environmentally sustainable energy sources for all individuals; Goal (9) Facilitating progress in the realms of industry, innovation, and infrastructural development; Goal (11) Implementing proactive measures to effectively mitigate the impacts of climate change; Goal (13) Fostering the Development of Sustainable Urban Areas and Communities. The mentioned interconnected objectives jointly emphasize the importance of reliable electricity infrastructure in the wider endeavor of achieving sustainable development. Under normal conditions, the voltage and frequency of the busbar and electrical power passing transmission lines in a unified power system should be constant [5,6,7]. Power system oscillations are one of the inherent characteristics of energy systems, and their occurrence in the system is unavoidable [8,9]. As long as they are damped on time, these oscillations are acceptable for the power system [10,11,12]. Due to the interaction of their various parts, power systems have numerous oscillation modes. The damping control of oscillations in the time area used for a special transmission line is extremely important in terms of controllability and observability [13,14]. As a result, Flexible AC Transmission System (FACTS) devices are a better choice for damping inter-area oscillation modes than power system stabilizers (PSS) because they are located on transmission lines, as well as control and regular transmission system parameters [15,16]. In recent years, supplementary modulation controllers (SMC) have been added to FACTS devices so that they could dampen inter-area oscillations [17,18]. As sequential regulation methods do not ensure the best performance of power system controllers, a new solution for the simultaneous regulation of PSS parameters is required. So far, several methods have been offered for the coordinated regulation of PSS and FACTS devices’ parameters, most of which are based on parametric optimization models [19,20]. These models include the transient state model [21] and the closed-loop residuals model [22], which employ different gradient-based non-linear optimization methods to solve the optimization problem. These methods are computationally fast but have problems finding the general system optima point or starting points.

Soft computing methods have been used to overcome the disadvantages of regular optimization methods in different sources, including innovative methods such as simulated annealing [23], evolutionary methods such as the genetic algorithm (GA) [24,25,26], evolutionary programming [27,28], particle swarm optimization (PSO) [29], and other intelligent methods such as neural network [30] and fuzzy logic [31]. These methods are highly powerful in solving non-linear and non-differentiable optimization problems [32]. Intelligent GA methods are considered more often than other methods due to their easy implementation and availability of computational software. To improve controllability, stability, and efficiency in electrical power systems, FACTS devices are cutting-edge power electronics-based technologies. To enhance power transfer efficiency and system performance, these devices are able to quickly and dynamically alter several electrical grid characteristics like voltage, current, and phase angle [33]. FACTS devices such as Static VAR Compensator (SVC) [34], Thyristor-Controlled Series Compensator (TCSC) [35], Static Synchronous Compensator (STATCOM) [36], and Static Synchronous Series Compensator (SSSC) [37] have been used to increase local and inter-area damping of power system [38,39]. Although these devices are used in studies to enhance the damping of these modes, in practice, SVC and TCSC are often incorporated [40]. In [41], inter-area mode damping was attempted to be increased using the optimal design of High Voltage Direct Current (HVDC) transmission line parameters. In this study, optimization was performed using a linear model for the system and DC transmission line. Moreover, to send and receive signals from different points of the system, synchrophasors were utilized, and the effect of delay in sending and receiving on control system performance was considered. In [42], inter-area damping was increased using SVC and synchrophasor, employing the system linearized method for modeling and optimization. Authors in [43] utilized three controllers of PSS, TCSC, and SVC simultaneously to boost the stability and damping of non-linear oscillations of a multi-machine system. Controller parameters were optimized using the adaptive velocity update relaxation particle swarm optimization (AVURPSO) algorithm in order to enhance the performance of the designed control system and increase the stability and damping of power system oscillations. The introduced design was unstable in the face of three-phase-to-earth short-circuit faults, but the system stability was improved, and oscillations were damped by the simultaneous use of three controllers. In [44], a coordinated design between PSS and the Unified Power Flow Controller (UPFC) was proposed using the GA. The issue was formulated as a multi-objective optimization problem to maximize the damping coefficients of electromechanical modes such that a large number of UPFCs and PSSs would correspond to one another. This method was experimented on the New York/New England 16-machine 68-bus power system, and its effectiveness for suppressing local and inter-area oscillations was confirmed. In [45], the coordinated design of SVC, PSS, and TCSC controllers was performed. Linearized control coefficients of the energy system were simultaneously optimized using the GA. Optimization was in the form of a maximum function, which optimized the general system damping coefficients. Finally, the performance of the controllers was experimented with utilizing eigenvalue investigation and time-domain optimization, and the improved damping of the tested system was shown. In [46], a coordinated PSS–Power Oscillation Damping (POD) controller was designed employing the PSO algorithm in a two-area four-machine system. Results revealed that this method increased the damping ratio and coefficients. Moreover, the time-domain simulation of a multi-machine system under various loading conditions and turbulences showed that the coordinated PSS–POD controller quickly damped electromechanical oscillations and had minimum overshoot and undershoot. In [47], inter-area oscillations were damped by designing a closed-loop controller. This method was based on the formation of the open-loop transfer function in the Nyquist diagram. The difference between the expected and actual open-loop transfer functions was minimized. The designed controller was adapted to a two-area, four-machine power system containing a Doubly Fed Induction Generator (DFIG) and SVC with promising results, including the acceptable damping of power transmission line oscillations under different wind conditions and operating points.

In this paper, PSS, SVC, and TCSC parameters are optimized in a coordinated manner by introducing a novel optimization framework to suppress inter-area modes and improve power system stability. The optimization procedure based on GA is employed to find the optimized characteristic of a multi-machine large-scale power system equipped with PSS and two types of FACT devices. Different scenarios are carried out, and the numerical results are analogized with those obtained from the conventional method. In this framework, the optimization zone is divided into three parts in order to enhance the effectiveness of the optimization algorithm. Moreover, weight coefficients allocated to each pole are determined based on its position. In the optimization procedure, poles, which are in a more critical condition, are targeted by the optimization algorithm based on their higher allocated weight coefficients, and modes with better damping receive less attention. Since the location of FACTs devices has a significant effect on optimization results and considering that the location of these devices has been assumed constant or determined utilizing the conventional method in the literature, the present study considered the location of these devices as a variable integer in the optimization procedure. The number of buses is limited to install FACTS devices in the proposed framework. Furthermore, appropriate bus determination has been provided by system eigenvalues, and solving the transient state is not required. As a result, the interaction effect of these devices and PSS is considered in this optimization approach. The numerical outcomes of the proposed method are analogized to those of a 68-bus system. Therefore, this paper’s main contribution can be summarized as follows:

✓

It introduces a unique optimization framework that coordinates the optimization of parameters for PSSs and two types of FACTS devices—SVC and TCSC;

✓

It addresses the challenge of insufficient energy system stabilizers’ effectiveness in damping inter-area mode oscillations by utilizing FACTS devices to enhance damping and stabilize power systems, contributing to improved power system stability and voltage regulation performance;

✓

It innovatively employs variable weight coefficients based on eigenvalue locations within the optimization framework, effectively prioritizing critical modes and optimizing PSS and FACTS device parameters for enhanced performance;

✓

It deviates from conventional methods by treating the location of FACTS devices and control signals as optimization parameters, recognizing their significant influence on energy system performance and enabling a more comprehensive optimization procedure;

✓

It validates the proposed approach through a large-scale multi-machine power system, showcasing tangible improvements in damping inter-area mode oscillations and highlighting the efficiency of the optimization strategy.

This paper is structured as follows: Section 2 presents the model of the FACTS device. In Section 3, the design of the proposed formulation is described. Section 4 investigates the optimization framework, considering different and variable weight coefficients based on eigenvalue locations. The system description and simulation results are provided in Section 5. Eventually, this paper will be concluded in Section 6.

2. FACTS Devices Model

In order to investigate the effective performance of the presented framework, two types of FACTS devices, i.e., TCSC and SVC, have been utilized. Similar to PSS modeling, the model of FACTS devices has two main parts: small-signal dynamics of FACTS devices and their effects on the power system load flow. The common TCSC model includes a Thyristor-Controlled Reactor (TCR) parallel to a fixed series compensator, which can be found in [48]. In addition, there are several types of SVC models. In this work, an SVC model containing a TCR parallel to a TSC is used; more information about this FACTS device can be discovered in [34].

The thyristor-controlled series capacitor is one of the FACTS devices used in power systems to control the transfer of power and improve the stability of this system. The TCSC consists of a series capacitor and a TCR. The series capacitor is connected in series with the transmission line, and the TCR is connected in parallel with the series capacitor.

The TCR can vary its impedance by controlling the firing angle of the thyristors. By varying the impedance of the TCR, the TCSC can control the power flow in the transmission line and improve the system stability by damping oscillations. The circuit schematic diagram of a TCSC is shown in Figure 1.

The Static Var Compensator is also one of the FACTS devices utilized in power systems to regulate voltage and control reactive power. The SVC consists of a shunt reactor (or a thyristor-controlled series capacitor (TSC)), a TCR, and a capacitor bank. The shunt reactor is connected in parallel with the transmission line, and the TCR and capacitor bank are connected in series with the shunt reactor. The TCR can vary its impedance by controlling the firing angle of the thyristors. The capacitor bank is used to compensate for the reactive power generated by the TCR. By varying the impedance of the TCR, the SVC can control the power factor of the system and regulate the voltage at the point of connection. The schematic diagram of an SVC is shown in Figure 2.

3. Interconnected Power System

In this section, to define the problem formulation, the model of PSS and FACTS devices used in the implemented multi-machine energy system is illustrated. The PSS transfer function is in the form of the following equation:

$$H_{PSS}(s)=K_{PSS}\left(\frac{s{T}_W}{1+s{T}_W}\right)\left(\frac{1+s{T}_1}{1+s{T}_2}\right)\left(\frac{1+s{T}_3}{1+s{T}_4}\right)$$

(1)

where K_PSS is a positive constant gain; T_w is the washout time constant; T₁ and T₃ lead time constant, and T₂ and T₄ lag time constant. This transfer function includes a washout filter to remove the steady state error, a gain block amplifier, and two pre-phase post-phase blocks. The latter block provides a pre-phase characteristic that compensates for the post-phase between the generator’s stimulation system and electric torque, which may be useful for damping local modes. Furthermore, the transfer function of FACTS devices is represented by the following equation:

$$H_{FACTS}(s)=K_{FACTS}\left(\frac{s{T}_W}{1+s{T}_W}\right){\left(\frac{1+s{T}_1}{1+s{T}_2}\right)}^2$$

(2)

where K_FACTS is a positive constant gain. T₁ and T₂ are the lead and lag time constants, and T_w is the washout time constant. This transfer function includes a washout filter, a gain block amplifier, and a phase-compensating block [20,25]. In this function and the PSS transfer function, the T_w parameter is usually pre-defined and is not involved in optimization processes. In the proposed optimization-based coordinated regulation framework, the optimal performance of controllers is obtained by minimizing an objective function. The objective function must be defined so that its minimization would correspond to gaining an optimal performance in the set of controllers. The optimization problem in this paper is solved using an optimization algorithm built based on the well-known GA.

4. Objective Function

The major goal of the objective function considered for the suggested optimization algorithm is a minimum of 5% damping for all modes. If ξ_i ≥ 5%, the optimization procedure will be finished. Parameters considered in this optimization include T_1,i, T_2,i, T_3,i, T_4,i, and K_PSS,i for the PSS, and T_1,j, T_2,j, and K_FACTS,j for FACTS devices. In these parameters, the values of j and i are the numbers of FACTS and PSS devices, respectively.

The aim of the coordinated regulation of parameters is to generally optimize the damping performance. This can be performed by minimizing the defined objective function or obtaining the minimum location for the critical state. The damping ratio of an oscillation mode is offered by

$$\xi =-\sigma /\sqrt{{\sigma }^2+{\omega }^2}$$

(3)

where σ and ω are real and imaginary parts of eigenvalues. Thus, if the real part of the oscillation frequency is kept constant, its damping rate will increase. The objective function used here is as follows:

$$\min f(z)={\displaystyle \sum _{i=1}^n{w}_i}{({\sigma }_{des}-{\sigma }_i)}^2$$

(4)

where σ_i is the real part of critical eigenvalues; σ_des is the optimal value of the real part of critical eigenvalues, and z is the optimization variable, which includes an integer for bus number or line and a series of continuous numbers for PSS and FACTS devices’ parameters. When the actual part of σ_i oscillation mode is in the expected actual part σ_des, f(z) can reach its minimum value. Constraints are applied for practical considerations. Time constants are limited, so it is easier to find the optimal solution. Therefore, the set of objectives f(z), according to [19,20,21], is subjected to a set of boundary constraints as follows:

$$T_{1,i}^{\min }\le T_{1,i}\le T_{1,i}^{\max },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,N_{PSS}^{\max }$$

(5)

$$T_{2,i}^{\min }\le T_{2,i}\le T_{2,i}^{\max },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,N_{PSS}^{\max }$$

(6)

$$T_{3,i}^{\min }\le T_{3,i}\le T_{3,i}^{\max },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,N_{PSS}^{\max }$$

(7)

$$T_{4,i}^{\min }\le T_{4,i}\le T_{4,i}^{\max },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,N_{PSS}^{\max }$$

(8)

$$K_{PSS,i}^{\min }\le K_{PSS,i}\le K_{PSS,i}^{\max },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,N_{PSS}^{\max }$$

(9)

$$T_{1,j}^{\min }\le T_{1,j}\le T_{1,j}^{\max },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\dots ,N_{FACTs}^{\max }$$

(10)

$$T_{2,j}^{\min }\le T_{2,j}\le T_{2,j}^{\max },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\dots ,N_{FACTs}^{\max }$$

(11)

$$K_{FACTs,j}^{\min }\le K_{FACTs,j}\le K_{FACTs,j}^{\max },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\dots ,N_{FACTs}^{\max }$$

(12)

where $N_{FACTs}^{\max }$ and $N_{PSS}^{\max }$ are the maximum numbers of FACTS and PSS devices, respectively. Moreover, the controller gain is limited to 50 in order to prevent the enhancement of noise and controller saturation. Weight coefficients play a pivotal role in the control strategy, strategically guiding the optimizer operator’s focus toward the modes in the power system that exhibit low damping. In the realm of power system stability analysis, these modes are typically visualized on the complex plane, where the positioning of eigenvalues serves as a powerful indicator of how various components within the power system react to external disturbances. This complex plane is partitioned into three distinct zones, as illustrated in Figure 3. The first of these regions encompasses the critical modes, the eigenvalues of which demand prompt relocation to the left-hand side of the complex plane; given the highest priority, these modes are crucial for overall system stability. In the second region, relatively scattered oscillating modes are found, while the third region houses the damped modes, characterized by damping rates exceeding 20%. For modes within this third region, the need for further control is obviated as their placement already resides on the stable left-hand side of the complex plane. The weight coefficients assigned to these regions not only underscore the paramount importance of addressing more critical modes within the function operator but also serve as an intelligent constraint. This constraint prevents the optimizer operator from expending unnecessary control efforts on modes that inherently require no further intervention. As a result, the system’s stability and efficiency are harmoniously optimized.

In the proposed framework, the optimal actual value is determined based on the location in each area. These values are determined and placed by the designer. In this study, the optimal actual value or σ_des is −1 for the first area and 1 for the second and third areas. Also, to reduce computations, modes with an actual part larger than −1.5 are not placed inside the objective function. Only at the end of each generation of the GA-based algorithm the damping coefficient of all poles of each gene is examined. If this minimal damping coefficient is lower than the optimal value for at least one of the genes in each generation, the proposed method will be stopped. Otherwise, the next generation will be created. The flowchart in Figure 4 demonstrates this process.

What is applied in the above objective function as an input to the optimization framework is the determination of an appropriate bus for the installation of SVC and the determination of a suitable line for the installation of TCSC, which will be involved in the proposed framework as an integer. In this algorithm, the proper bus affects objective function indirectly. First, a bus is determined using the proposed optimization algorithm; then, system eigenvalues are obtained based on a desired bus position and are placed in the objective function, and the objective function will be calculated for this gene.

5. Studied System

In order to compare the results, all the analyses will be examined on an IEEE 68-bus system. Figure 5, adapted from [21,44,46], shows the structure of this multi-machine large-scale power system, and

Table 1. Inter-area modes and their damping.

Inter-Area Mode	$\mathit{\sigma }\pm \mathit{j}\mathit{\omega }$	$\frac{-\mathit{\xi }}{\sqrt{{\mathit{\omega }}^2+{\mathit{\xi }}^2}}$	$\frac{\mathit{\omega }}{2\mathit{\pi }}$
(1)	−0.035 ± j2.421	1.43%	38.53%
(2)	−0.066 ± j3.721	1.78%	59.22%
(3)	−0.102 ± j4.313	2.36%	68.65%
(4)	−0.139 ± j4.981	2.79%	79.27%

presents the inter-area modes of this system. Different states of the coordinated design are compared in this work. One of these methods is the residual method. In this scheme, based on Lyapunov stability, all the eigenvalues of the system’s state matrix must have negative actual values so that the system would be stable around the equilibrium point. This concept is the central idea of the controller design method. The second method examined in this paper with the results compared to those of the residual method is the coordinate regulation method.

5.1. Scenario I: Coordinated Design for the IEEE 68-Bus Power System

In this section, the coordinated design of PSS and FACTS devices is introduced for the standard IEEE 68-bus power system, and all the outcomes are analogized to those of the classic design using the residual method. In this network, all the generators, except the G₁₃, have AVR and PSS, and active power feedback and bus voltage are used for TCSC and SVC. Figure 6 demonstrates the system’s zero location and poles before the optimization and installation of FACTS devices. The zero locations correspond to the roots of the characteristic equation of the power system, which is obtained from the mathematical model of the system. If any of the zeros are located in the right-half plane of the complex plane, it may indicate instability or oscillatory activity in the system. Conversely, the poles in the characteristic equation of the power system refer to the solutions that elicit either growth or decay in the system’s reaction as time progresses. The poles of the system are intricately linked to its eigenvalues, and their locations on the complex plane dictate the system’s transient and oscillatory behaviors. Poles that are in the left-half plane are considered stable, as they imply that the response of the system will ultimately converge to a steady state. The presence of unstable poles located in the right-half plane indicates that the system’s reaction will exhibit unbounded growth, ultimately resulting in instability. In addition, inter-area modes are shown in this figure. FACTS devices are installed to enhance the damping of these modes. For this network, a more difficult condition for optimization is created. Therefore, the damping ratio is considered equal to 5%.

From among the resulting modes for all lines, the above lines are selected as the optimization variables. Lines 51–50 are determined as the proper lines for the installation of the TCSC after the optimization process.

5.2. Scenario II: Coordinated Design for TCSC

Within the optimization framework, the determination of the optimal TCSC installation site emerges as a crucial factor. In this section, a strategic approach is adopted to expedite the optimization procedure by selectively focusing on a subset of transmission lines. This selection is predicated upon the discernment of system residuals. To provide a comprehensive overview,

Table 2. System modes after installing TCSC controller.

Normalized Residues
Buses	Mode (1)	Mode (2)	Mode (3)	Mode (4)
27	0.016	0.019	0.153	0.004
41	0.009	0.009	0.007	0.002
42	0.001	0.001	0.002	0.003
50	0.406	1.000	0.494	1.000
51	0.059	0.867	0.137	0.964
52	0.030	0.092	0.008	0.279
53	0.038	0.370	0.430	0.081
60	0.007	0.026	0.207	0.013
61	1.000	0.052	1.000	0.608

enumerates various potential locations for the TCSC deployment, accompanied by their corresponding normalized residual values. The parameterization of the TCSC, however, is a nuanced endeavor reliant upon the residual-based methodology. Figure 7 visually encapsulates the evolution of system poles both prior to and subsequent to the incorporation of the TCSC, a transformation brought about by the application of both classical and genetic algorithm (GA) optimization techniques. Notably, this depiction emphasizes the transition of all system poles beyond the 5% damping threshold. Further insight is gleaned from

Table 3. PSS parameters after optimization using the coordinated method (Scenario II).

PSS(i)	Twi	KPSSi	T1i	T2i	T3i	T4i
i = 1	10	22.209	0.091	0.595	0.958	0.351
i = 2	10	11.781	0.635	0.222	0.801	0.589
i = 3	10	37.785	0.887	0.157	0.669	0.645
i = 4	10	38.326	0.438	0.919	0.855	0.514
i = 5	10	13.872	0.279	0.879	0.299	0.463
i = 6	10	0.628	0.032	0.980	0.022	0.280
i = 7	10	18.948	0.439	0.709	0.675	0.014
i = 8	10	4.265	0.464	0.966	0.452	0.877
i = 9	10	23.547	0.307	0.162	0.987	0.602
i = 10	10	16.559	0.878	0.766	0.624	0.872
i = 11	10	33.122	0.725	0.139	0.594	0.833
i = 12	10	39.690	0.565	0.071	0.089	0.883
i = 13	-	-	-	-	-	-
i = 14	10	39.216	0.912	0.049	0.938	0.900
i = 15	10	16.012	0.217	0.683	0.390	0.472
i = 16	10	29.823	0.730	0.191	0.328	0.666

and

Table 4. TCSC parameters after optimization using the coordinated method.

FACTS Controller	Tw	KFACTS	T1,lead	T2,lag
TCSC	10	397.40%	15.21%	81.18%

, which meticulously present the outcomes of the optimization process utilizing the GA methodology. These tabular representations furnish an intricate insight into the achieved optimization results.

The worst situation for every power system is the three-phase short circuit, which causes the highest probability of instability. In terms of the system behavior against voltage range and load fluctuation changes, the three-phase short circuit itself makes a type of extreme changes in the network voltage. In this work, the considered line is removed from the network after the short circuit, which is a type of change in the network load, and then, the system behavior against these changes is investigated. To examine the resulting parameters, a three-phase short-circuit fault in the system occurs for 80 ms. This fault is created at t = 1 s.

In this fault, a short-circuit three-phase is made in bus 42, and lines 42–41 exit after 80 ms. It is evident that the coordinate method has managed to stabilize the system far better. Note that the fitness function in this scenario demonstrated convergence to a steady state after 43 iterations.

5.3. Scenario III: Coordinated Design for SVC

In the pursuit of selecting an optimal bus for SVC installation, a strategy based on residuals is employed due to the extensive number of buses. This involves the initial identification of several potential buses. These identified buses are subsequently treated as integer variables within the algorithm.

Detailed information regarding the buses and their corresponding residual values for inter-area modes is presented in

Table 5. System modes after installing SVC controller.

Normalized Residues
Grid Lines	Mode (1)	Mode (2)	Mode (3)	Mode (4)
52–42	0.048	0.012	0.031	0.015
42–41	0.002	1.000	0.957	0.955
53–47	0.018	0.001	0.001	0.001
53–54	0.129	0.036	0.773	0.006
53–27	0.009	0.002	0.052	0.001
46–49	0.043	0.025	0.008	0.037
60–61	0.121	0.056	1.000	0.011
50–51	1.000	0.370	0.553	1.000

. Ultimately, through meticulous analysis, bus 50 emerges as the most suitable candidate for the installation of the SVC system. A visual representation of the pre-and post-optimization distribution of system poles is showcased in Figure 8. To further refine the system’s performance,

Table 6. PSS parameters after optimization using the coordinated method (Scenario III).

PSS(i)	Twi	KPSSi	T1i	T2i	T3i	T4i
i = 1	10	10.265	0.442	0.819	0.751	0.235
i = 2	10	7.675	0.270	0.207	0.183	0.531
i = 3	10	36.995	0.436	0.648	0.637	0.481
i = 4	10	31.031	0.292	0.005	0.953	0.943
i = 5	10	32.859	0.135	0.309	0.831	0.233
i = 6	10	34.000	0.227	0.286	0.275	0.994
i = 7	10	19.545	0.222	0.183	0.978	0.311
i = 8	10	37.984	0.694	0.256	0.555	0.644
i = 9	10	26.417	0.615	0.053	0.997	0.921
i = 10	10	14.960	0.825	0.146	0.264	0.714
i = 11	10	12.776	0.874	0.409	0.070	0.335
i = 12	10	39.635	0.884	0.603	0.942	0.550
i = 13	-	-	-	-	-	-
i = 14	10	37.225	0.676	0.215	0.732	0.530
i = 15	10	21.429	0.832	0.425	0.944	0.720
i = 16	10	22.086	0.120	0.175	0.943	0.605

and

Table 7. SVC parameters after optimization using the coordinated method.

FACTS Controller	Tw	KFACTS	T1,lead	T2,lag
SVC	10	709%	93.01%	83.19%

outline the optimized PSS and SVC parameters achieved through the coordinated approach.

This scenario involves the introduction of a fault into the power system, enabling an assessment of the control system’s effectiveness under such circumstances. This fault event commences at t = 1 s and concludes after 80 milliseconds. Specifically, a three-phase short circuit is initiated at bus 53, causing lines 47–53 to disconnect from the circuit after 80 ms. Figure 9 visually contrasts the angular discrepancies of the G1–G2 generators between two states during the fault condition. Notably, the coordinated design exhibits superior performance compared to the conventional method, as evidenced by these results. Note that the fitness function in this scenario demonstrated convergence to a steady state after 56 iterations.

5.4. Scenario IV: Coordinated Design for SVC and TCSC Simultaneously

In this scenario, the optimization procedure involves the simultaneous adjustment of parameters for SVC, PSS, and TCSC. However, it is important to note that the bus and line of interest are excluded from the optimization variables. This exclusion is a deliberate choice, as in the preceding section, the optimal placement of each FACTS device was determined using the residual values and the coordinated regulation approach.

Figure 10 demonstrates the location of system poles before and after optimization, and

Table 8. PSS parameters after optimization using the coordinated method (Scenario IV).

PSS(i)	Twi	KPSSi	T1i	T2i	T3i	T4i
i = 1	10	31.359	0.761	0.333	0.085	0.481
i = 2	10	38.293	0.947	0.170	0.600	0.917
i = 3	10	34.548	0.524	0.213	0.241	0.205
i = 4	10	36.449	0.907	0.404	0.406	0.379
i = 5	10	31.827	0.370	0.287	0.490	0.819
i = 6	10	27.427	0.417	0.722	0.696	0.159
i = 7	10	32.247	0.995	0.857	0.370	0.424
i = 8	10	38.905	0.432	0.772	0.500	0.322
i = 9	10	32.780	0.885	0.353	0.650	0.745
i = 10	10	36.959	0.878	0.245	0.651	0.394
i = 11	10	35.567	0.288	0.431	0.061	0.046
i = 12	10	35.072	0.929	0.163	0.242	0.501
i = 13	-	-	-	-	-	-
i = 14	10	38.025	0.920	0.488	0.925	0.361
i = 15	10	36.769	0.949	0.090	0.025	0.382
i = 16	10	35.306	0.834	0.421	0.952	0.456

and

Table 9. SVC and TCSC parameters after optimization using the coordinated method.

FACTS Controller	Tw	KFACTS	T1,lead	T2,lag
SVC	10	709%	93.01%	83.19%
TCSC	10	32.72%	5.19%	96.87%

present the optimized parameters of PSS, TCSC, and SVC.

As shown in Figure 10, the proposed method has caused the inter-area modes to move to the left side more. In order to examine the system’s dynamic efficiency, a fault was created, like in the previous two states. This fault occurs at t = 1 s and is resolved after 80 ms by removing the line.

In this fault, a short circuit is created in bus 46, and lines 49–46 and 57–53 exit after 80 ms. Figure 11 shows the angle difference between G1 and G16 for the created fault. Note that the fitness function in this scenario demonstrated convergence to a steady state after 76 iterations. Based on simulation results, the proposed coordinated approach for the PSS and FACTS controllers has demonstrated the potential to enhance the transient and dynamic stability of multimachine energy systems. This advancement holds the promise of bolstering energy sustainability while concurrently providing critical support to contemporary societies and economies facing escalating electricity demands. Additionally, it plays a pivotal role in reducing environmental repercussions, thereby contributing to an overall increase in sustainability.

6. Conclusions

The significance of the power system stability is echoed in the wide field of sustainability, which includes important areas like the use of renewable energy, the guarantee of a steady supply of energy, the reduction in negative effects on the environment, the promotion of efficiency and energy conservation, the protection against events that cause problems, and the journey of grid modernization. In this study, we presented a novel optimization framework based on the coordinated regulation of power systems featuring PSS and FACTS device controllers. To this end, we defined a weighted objective function and determined weight functions that prioritize the movement of modes with low damping. Employing an optimization strategy based on the genetic algorithm, we optimized the objective function. A distinctive paradigm was introduced in this study by considering the location of FACTS devices and control signals as optimization parameters, deviating from conventional approaches. This framework yielded optimal placements that significantly enhanced system performance. The effectiveness of the proposed framework in damping inter-area oscillations and improving power system stability was demonstrated through the simulation results. Comparisons with the residual method underscored the efficacy of the introduced framework.

Consequently, this paper introduces a unique optimization framework that enhances power system stability by coordinating PSS and FACTS device parameters. This approach addresses the challenge of inadequate energy system stabilizers and employs innovative techniques for improved performance. It deviates from traditional methods by considering FACTS device locations and control signals in optimization. Our approach is validated on a multi-machine power system, demonstrating tangible improvements in damping inter-area mode oscillations and overall efficiency. Notably, the results indicate that TCSC outperformed SVC in damping inter-area modes, with the possibility of achieving superior outcomes for large-scale power system stability when simultaneously employing SVC and TCSC. The coordinated regulation of control parameters and their location with PSS further solidified the framework’s effectiveness. For future work, considering emerging technologies and advanced machine learning techniques for real-time power system optimization and control presents a promising direction.