# Mayfly Optimization Algorithm Applied to the Design of PSS and SSSC-POD Controllers for Damping Low-Frequency Oscillations in Power Systems

Authors: Elenilson V. Fortes; Luís Fabiano Barone Martins; Marcus V. S. Costa; Luis Carvalho; Leonardo H. Macedo; Rubén Romero
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

In this paper, it is proposed to apply the mayfly optimization algorithm (MOA) to perform the coordinated and simultaneous tuning of the parameters of supplementary damping controllers, i.e., power system stabilizer (PSS) and power oscillation damping (POD), that actuate together with the automatic voltage regulators of the synchronous generators and the static synchronous series compensator (SSSC), respectively, for damping low-frequency oscillations in power systems. The performance of the MOA is compared with the performances of the genetic algorithm (GA) and particle swarm optimization (PSO) algorithm for solving this problem. The dynamics of the power system is represented using the current sensitivity model, and, because of that, a current injections model is proposed for the SSSC, which uses proportional-integral (PI) controllers and the residues of the current injections at the buses, obtained from the Newton–Raphson method. Tests were carried out using the New England system and the two-area symmetrical system. Both static and dynamic analyses of the operation of the SSSC were performed. To validate the proposed optimization techniques, two sets of tests were conducted: first, with the purpose of verifying the performance of the most effective algorithm for tuning the parameters of PSSs, PI, and POD controllers, and second, with the purpose of performing studies focused on small-signal stability. The results have validated the current injections model for the SSSC, as well as have indicated the superior performance of the MOA for solving the problem, accrediting it as a powerful tool for small-signal stability studies in power systems.


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## Body

## 1. Introduction

The increase in the demand for electricity, the industrial expansion, and the interconnection of large systems with high numbers of generating units contributed to potentiate the appearance of problems related to the operation of electric power systems, including stability issues.According to [1], the stability concept is related to the capacity that power systems have to remain in equilibrium in normal operating conditions. Besides, the system operation should evolve to a new acceptable equilibrium point after being submitted to small- or large-magnitude disturbances.Typically, low-frequency oscillations occur in the range of0.2−2.0 Hz and are associated with electromechanical torque imbalances in synchronous generators, resulting in power transfer with oscillatory characteristics in transmission lines that interconnect two systems. Oscillations in the frequency range of 0.8−2.0 Hz are classified as local, while those occurring in the range of 0.2−0.7 Hz are classified as interarea [1].One of the problems addressed in this work is to develop techniques capable of designing supplementary damping controllers for damping these types of oscillations present in power systems. [2] already used, in 1969, the Heffron and Phillips’s model [3] to represent a single machine infinite bus system and explored the important concepts of synchronizing torque and damping, through which they laid the foundation for understanding the phenomenon of angular stability under small-signal disturbances. In the work presented by [2], it was observed that automatic voltage regulators (AVRs) with high gains negatively affected the stability of the system. Since then, power system stabilizers (PSSs) controllers have been widely used to insert supplementary stabilizing signals into the generators’ excitation control systems in order to provide additional damping torques to rotors’ oscillations [4–6]. It should be noted that, in general, PSSs have good performance for damping local oscillatory modes [7, 8], while they cannot adequately damp interarea modes. On the other hand, depending on the settings provided to the control parameters of the PSSs (time constants and gains) they can turn previously stable modes into unstable modes, in particular, the interarea ones [9].The recent advances in power electronics propitiated the emergence of flexible AC transmission systems (FACTS) that have been widely used in power systems with the most diverse purposes. In general, FACTS can increase power transfer and consequently improve the stability of a system [10, 11]. Recent studies show that when a power oscillation damping (POD) controller is coupled to the control loop of these devices, the FACTS-POD set can act inserting additional damping to the interarea modes as can be analyzed in the works of [7, 12, 13]. In these works, the POD was coupled to the unified power flow controller (UPFC), generalized unified power flow controller (GUPFC), and the interline power flow controller (IPFC) with the goal of inserting additional damping to the interarea oscillatory modes present in power systems.In this work, the static synchronous series compensator (SSSC) FACTS is modeled and employed to perform studies that address both static and dynamic analyses of the power system. In general terms, the SSSC performs a series compensation of the electric power transmission system. Its operation consists of inserting (by means of a coupling transformer) a synchronous quasi-sinusoidal voltage source of controllable amplitude and phase angle in series with the transmission line. This provides an effect capable of reproducing an inductive or capacitive reactance in series with the transmission line where the device is connected, thus allowing effective control of the active and reactive power flows in the line where the SSSC is installed [10, 11].Existing works in the literature that perform the control of active and reactive power flows using the SSSC, usually model it by power injections [14–16]. In this paper, a new model based on current injections is presented for the SSSC in order to perform both static and dynamic analyses of the device in the New England system [17]. The model will allow the SSSC operation to be represented in the expanded power flow algorithm expressed as a function of the residues of the current injections at the buses using the Newton-Raphson method [18]. The control structure used to model the SSSC’s dynamics is based on proportional-integral (PI) controllers, as proposed by [17, 19]. The proposed model can be used both for the control of active and reactive power flows by the SSSC and for obtaining the dynamic equations of the device that are included in the model used to represent the power system dynamics, i.e., the current sensitivity model (CSM) [7, 12, 17]. Furthermore, the proposed model for the SSSC, when compared to other representations available in the literature [20, 21], has the significant advantage of being independent of the phase angle of the voltage source of the SSSC (see Section 2) eliminating, in this way, the necessity for its previous calculations. More details about the SSSC modeling will be presented in Section 2.Regarding the CSM, it is a linear model whose fundamental principle is based on the nodal balance of currents. This balance must be satisfied in all dynamic processes of the power system and, therefore, can be used in small-signal stability studies [7, 12, 17]. Thus, all devices present in the system should be modeled by current injections, which justifies the new modeling proposed for the SSSC (see Subsection 2.1).The AVR-PSS and SSSC-POD sets are only able to effectively damp the poles of interest of the system if the parameters of the corresponding controllers (PSS and POD) are adequately adjusted. In the literature, different optimization techniques have been used for this purpose. In 2010, [22] developed a design procedure using the particle swarm optimization (PSO) algorithm for the simultaneous and coordinated design of the thyristor-controlled series capacitor (TCSC) damping controller and PSSs in multi-machine power systems. Tests were conducted using the two-area symmetrical system, with four generators. In 2014, [23] presented a coordinated approach for the allocation and design of PSSs and UPFC using genetic algorithms (GA). The GA algorithm determines the optimal location for the UPFC installation while simultaneously tuning its control parameters, resulting in the optimization of the number, locations, and parameters of the PSSs. The problem is formulated with the objective of maximizing the damping ratios of the electromechanical modes. In 2016, [24] presented a hybrid method is to damp oscillations in the power systems equipped with UPFC and PSS controllers. The hybrid method consists of an offline and an online stage. In the offline stage, the parameters of the PSS and UPFC controllers are determined through a PSO algorithm; then in the second stage, an online fuzzy controller is proposed to determine parameters of the PSS and UPFC controllers coefficients according to the operating point of the system. Tests were conducted using the single machine infinite bus system, with only one generator. In 2018, [25] proposed to explore the process of tuning controllers for conditions using control theory. Tests were conducted using the two-area symmetrical system.In general, more recent works found in the literature do not use the CSM to represent the power system dynamics, which is, therefore, a differential of the proposed work. In this context, it is possible to cite the works [26–29]. In [28], three optimization algorithms, i.e., GA, PSO, and the farmland fertility algorithm (FFA) are used to tune the control parameters of the PSSs in the New England system using a linearized power system model. In [27], the differential evolution (DE) algorithm, PSO algorithm, gray wolf optimizer (GWO), whale optimization algorithm (WOA), and chaotic whale optimization algorithm (CWOA) were used in a single machine infinite bus (SMIB) system in order to adjust the parameters of the proportional-integral (PI) controller. It is worth noting that in this article, the simulated FACTS was the generalized unified power flow controller (GUPFC). In [26], the modified sine cosine algorithm (MSCA) was used to tune the control parameters of PSSs and SSSC in a SMIB system and a multi-machine system. Finally, in [29], the parameters of the PSSs were adjusted in a coordinated and simultaneous way using the backtracking search algorithm (BSA). The results are obtained from simulations performed on two multi-machine systems and are compared to the results of the PSO algorithm.In this work, a recently proposed optimization technique, known as the mayfly optimization algorithm (MOA) [30], is implemented for designing the supplementary damping controllers. This technique is based on the PSO algorithm [31] and has the advantages of combining the main features of the PSO algorithm, GA [32, 33], and the firefly algorithm (FA) [34]. It provides a powerful hybrid algorithmic framework, based on the behavior of mayflies, the performance of PSO algorithm, with crossover techniques [35], and local search [36], since it has been proven that the PSO algorithm needs some modifications to achieve an optimal solution for high-dimensional search spaces [37].From the above, the main contributions of this work are as follows:(i)
To present a new current injections model for the SSSC(ii)
To use an expanded power flow tool based on the residues of the currents to determine the current injections of the SSSC(iii)
To include PI controllers in the control loop of the shunt converter of the SSSC(iv)
To represent the power system and its supplementary damping controllers (PSSs and POD) using the CSM(v)
To implement the PSO algorithm, GA, and MOA to perform the design of the supplementary damping controllersThe remainder of this work is organized as follows: in Section2, the SSSC is modeled through current injections; in Section 3, the CSM, used to represent the power system dynamics, is presented together with details about the representation of the SSSC; Section 4 presents details about the structure of the supplementary damping controllers (PSS and POD) and their respective differential equations representing their dynamics; Section 5 presents some details about the implementation of the GA, PSO algorithm, and MOA; Section 6 presents details about the problem formulation, discussing the representation of a candidate solution and the objective function used in the optimization methods; Section 7 presents the simulations and results for the implementations using the New England system and the two-area symmetrical system; finally, and Section 8 presents the main conclusions of the work.


## 2. Static Synchronous Series Compensator

The SSSC FACTS can be represented by a voltage source converter (VSC) connected in series with the transmission line. This device can vary the effective impedance of the line by injecting a voltage in phase with respect to the line current, thus allowing the exchange of active and reactive powers with the transmission system [38]. In Figures 1(a) and 1(b), the schematic diagram and equivalent circuit of the SSSC are shown, respectively.Figure 1
(a) Representation of the SSSC FACTS and (b) equivalent circuit of the SSSC.
(a)(b)In Figure1(a), the SSSC is represented using a VSC in series with the transmission line. The VSC is connected to the system through a coupling transformer with a reactance xkn, and by means of gate turn-off thyristors, it modulates a DC voltage coming from an external source. In Figure 1(b), the phase angle γ of the voltage source V˜s is represented by (1). Furthermore, bkn=−1/xkn is the susceptance of the coupling transformer, and V˜k and V˜m are, respectively, the voltage phasors at buses k and m. Finally, V˜n is the voltage phasor at the fictitious bus (FB), n, included in the system to perform simulations with the SSSC, and Z¯nm is the impedance of the transmission line between buses n and m:(1)γ=arctanVpVq,where 0≤γ≤2π, Vq and Vp are, respectively, the in-phase and quadrature components of the voltage source V˜s [39, 40].


### 2.1. Current Injections Model for the SSSC

From (1) and the equivalent circuit of the SSSC, shown in Figure 1(b), (2)–(5) are obtained. These are decomposed into real, r, and imaginary, i, components:(2)Ikr=bkm−Vksinθk+Vmsinθm+bkmVpcosθk+Vqsinθk︸Iinjkrsssc,(3)Iki=bkmVkcosθk−Vmcosθm+bkmVpsinθk−Vqcosθk︸Iinjkisssc,(4)Ikmr=bkmVksinθk−Vmsinθm−bkmVpcosθk+Vqsinθk︸Iinjmrsssc,(5)Ikmi=bkm−Vkcosθk+Vmcosθm−bkmVpsinθk−Vqcosθk︸Iinjmisssc,where Iinjkrsssc, Iinjkisssc and Iinjmrsssc, Iinjmisssc are, respectively, the real, in (2) and (4), and imaginary, in (3) and (5), components of the currents injected into buses k and m by the SSSC at the common buses of its installation. In Figure 2, the single-line diagram of the proposed current injections model for the SSSC is illustrated.Figure 2
Current injection model of the SSSC.The great advantage of using the current injections model for the SSSC, as shown (2)–(5) and depicted in Figure 2, is the fact that this model keeps the current flow equations unchanged and considers the device contributions at the common buses to its installation as if they were current injections. This facilitates the inclusion of the SSSC current injections model in conventional power flow models, such as in Newton–Raphson method.


### 2.2. Control System for the SSSC

To perform the control of the active and reactive power flows by the SSSC (Figure3) on the common buses to its installation, the PI controllers are used [7, 12, 17].Figure 3
Structure of the control system of the SSSC.
(a)(b)The PI controllers shown in Figures3(a) and 3(b), are used, respectively, to modulate the synchronous voltage source control variables, Vp and Vq. The time constant Tmpi is the inherent delay of the control device and is determined in the range between 1 and 10 ms [10]. In addition, X1, X2, and X3 are input signals and, in this case, are specified. The parameters of the PI controllers are the gains K1pi and K2pi, and the time constants, T1pi and T2pi. The supplementary signal Vsuppod comes from a POD controller (see Section 4) and, will be used to modulate the quadrature axis component, Vp.The dynamic behavior of the control structure shown in Figure3 is given by (6)–(9), which are obtained by inspection:(6)V˙p=K1piTmpiPkmref−Pkmctrl+1TmpiX1+Vsuppod−Vp,(7)X˙1=1T1piPkmref−Pkmctrl,(8)V˙q=K2piTmpiQmref−Qmctrl+1TmpX2−Vq,(9)X˙2=1T2piQmref−Qmctrl,where the quantities Pmctrl, Qmctrl, Pmref, and Qmref are, respectively, the control and specified values of the active and reactive power flows on the line between buses k−m.


## 2.1. Current Injections Model for the SSSC

From (1) and the equivalent circuit of the SSSC, shown in Figure 1(b), (2)–(5) are obtained. These are decomposed into real, r, and imaginary, i, components:(2)Ikr=bkm−Vksinθk+Vmsinθm+bkmVpcosθk+Vqsinθk︸Iinjkrsssc,(3)Iki=bkmVkcosθk−Vmcosθm+bkmVpsinθk−Vqcosθk︸Iinjkisssc,(4)Ikmr=bkmVksinθk−Vmsinθm−bkmVpcosθk+Vqsinθk︸Iinjmrsssc,(5)Ikmi=bkm−Vkcosθk+Vmcosθm−bkmVpsinθk−Vqcosθk︸Iinjmisssc,where Iinjkrsssc, Iinjkisssc and Iinjmrsssc, Iinjmisssc are, respectively, the real, in (2) and (4), and imaginary, in (3) and (5), components of the currents injected into buses k and m by the SSSC at the common buses of its installation. In Figure 2, the single-line diagram of the proposed current injections model for the SSSC is illustrated.Figure 2
Current injection model of the SSSC.The great advantage of using the current injections model for the SSSC, as shown (2)–(5) and depicted in Figure 2, is the fact that this model keeps the current flow equations unchanged and considers the device contributions at the common buses to its installation as if they were current injections. This facilitates the inclusion of the SSSC current injections model in conventional power flow models, such as in Newton–Raphson method.


## 2.2. Control System for the SSSC

To perform the control of the active and reactive power flows by the SSSC (Figure3) on the common buses to its installation, the PI controllers are used [7, 12, 17].Figure 3
Structure of the control system of the SSSC.
(a)(b)The PI controllers shown in Figures3(a) and 3(b), are used, respectively, to modulate the synchronous voltage source control variables, Vp and Vq. The time constant Tmpi is the inherent delay of the control device and is determined in the range between 1 and 10 ms [10]. In addition, X1, X2, and X3 are input signals and, in this case, are specified. The parameters of the PI controllers are the gains K1pi and K2pi, and the time constants, T1pi and T2pi. The supplementary signal Vsuppod comes from a POD controller (see Section 4) and, will be used to modulate the quadrature axis component, Vp.The dynamic behavior of the control structure shown in Figure3 is given by (6)–(9), which are obtained by inspection:(6)V˙p=K1piTmpiPkmref−Pkmctrl+1TmpiX1+Vsuppod−Vp,(7)X˙1=1T1piPkmref−Pkmctrl,(8)V˙q=K2piTmpiQmref−Qmctrl+1TmpX2−Vq,(9)X˙2=1T2piQmref−Qmctrl,where the quantities Pmctrl, Qmctrl, Pmref, and Qmref are, respectively, the control and specified values of the active and reactive power flows on the line between buses k−m.


## 3. Power System Operation Model

Suppose an SSSC is installed in a power system, beingk the common bus to its installation. The residue of the current injected at bus k, ΔI˜k, is determined from the current balance, as illustrated in Figure 4.Figure 4
Current balance at busk.From Figure4, by inspection, (10) is obtained, representing the nodal current balance at bus k:(10)I˜kG+I˜injksssc−∑j∈ΩkI˜kj−I˜kL=ΔI˜k,where I˜kG is the current phasor injected at node k, as shown in (11), ∑j∈ΩkI˜kj are the current phasors on the lines connected to bus k, as shown in (12), in which Ωk is set of all neighboring buses of bus k, I˜kL is the current phasor drained by the load at bus k, as calculated in (13), and I˜injksssc is the current phasor injected at bus k by the SSSC, as shown in (14):(11)I˜kG=1VkPkGcosθk+QkGsinθk+j1VkPkGsinθk−QkGcosθk,(12)∑j∈ΩkI˜kj=∑j∈ΩlVjGkj+jBkjcosθj+jsinθj,(13)I˜kL=1VkPkLcosθk+QkLsinθk+j1VkPkLsinθk−QkLcosθl,(14)I˜injksssc=bkmVpcosθk+Vqsinθk+jbkmVpsinθk−Vqcosθk,where PkG and QkG are the active and reactive powers injected by the generator at bus k, Gkj and Bkj are the conductance and susceptance between buses k and j, and PkL and QkL are the active and reactive power drained by the load connected at bus k.For the SSSC to control the power flows, it is necessary thatPkmctrl and Qkmctrl are included as functions of the power system’s voltages and reactances, and the device’s control variables. For this purpose, the active and reactive power balances must be analyzed, according to Figure 5.Figure 5
Active and reactive power balances for the SSSC.In Figure5, the active and reactive power flows that are transmitted from bus m towards bus k, Pmk and Qmk, the active and reactive power flows controlled by the SSSC, Pmctrl and Qmctrl, and the active and reactive powers injected by the device at bus m are obtained by inspection, as shown, respectively, in (15) and (16):(15)Pmctrl=−bkmVkVmsinθkm︸Pmk−bkmVmVpcosθkm+Vqsinθkm︸Pinjmsssc,(16)Qmctrl=−bkmVm2−VkVmcosθkm︸Qmk−bkmVmVqcosθkm−Vpsinθkm︸Qinjmsssc.For the SSSC to actuate in the power system, its equations must be inserted in the power flow formulation using the expanded power flow technique, proposed by Kopcak et al. [18]. To do so, it is necessary to assume that its state variables are considered constant with respect to time, which makes its temporal derivatives equal to zero. This observation allows, at an operation point of the power system, its differential equations to be considered to be algebraic. Thus, the problem is limited to determining the zeros of a set of nonlinear algebraic expressions that can be found using Newton-Raphson algorithm, just as in the conventional power flow. Since in this work the power flow modeling was done by current injections, the essence of the problem is to satisfy all current residues between the buses in which the SSSC is installed, as shown in Figure 4.The power flow equations formulated by current injections are shown in the following equation:(17)The Jacobian matrix shown in (17) is divided into four blocks, J1sssc, J2sssc, J3sssc, and J4sssc. It is worth noting that J4sssc has its construction similar to the Jacobian of the conventional power flow formulation. However, the difference is in the elements marked with the “∗”, in which the derivatives that relate the injections of the real and imaginary components of the currents injected by the SSSC are included.


### 3.1. Current Sensitivity Model

The CSM is a linear analysis tool for power systems based on Kirchhoff’s current law, which must be met when the system is disturbed [7, 12, 17]. Its modeling for multi-machine systems composed of ng generators and nb buses is shown in (18)–(21). In this formulation, Δx represents the state variables, Δu represents the input variables, and Δz represents the algebraic variables:(18)Δx=[Δω1…ΔωngΔδ1…ΔδngΔEq′1…ΔEq′ngΔEfd1…ΔEfdng]t,(19)Δu=[ΔPm1…ΔPmngΔVref1…ΔVrefngΔPL1…ΔPLnbΔQL1…ΔQLnb]t,(20)Δz=[Δθ1…ΔθnbΔV1…ΔVnb]t,(21)Δx.0=J1J2J3J4ΔxΔz+B1B2Δu,where Δω represents the variations of the angular speed, Δδ represents the variations of the internal angle of the rotor, ΔEq′ represents the variations of the internal voltage in quadrature, and ΔEfd represents the variations of the generator’s field voltage, ΔPm represents the variations of the input mechanical power, ΔVref represents the variations of the reference voltage of the AVR, ΔPL and ΔQL are, respectively, the variations of the active and reactive power demands of the loads, and ΔV and Δθ are, respectively, the voltage magnitude and voltage phase variations at each bus of the power system.The representation in the state space, in the following equation, is obtained by eliminatingΔz from (21):(22)Δx.=AΔx+BΔu,where A and B are the state and input matrix, respectively, in which A=J1−J2J4−1J3 and B=B1−J2J4−1B2.


### 3.2. Inclusion of the SSSC in the CSM

The dynamic characteristics of the SSSC have already been incorporated into the power flow modeled by current injections and are shown in the submatricesJ1sssc, J2sssc, J3sssc, and J4sssc in (17). These can be directly included in the CSM, as shown in the following equations:(23)(24)Δx.c=J1c−J2cJ4sssc−1J3cΔxc+B1c−J2cJ4sssc−1B2cΔuc,where k=1,…,nb, Ac=J1c−J2cJ4sssc−1J3c and Bc=B1c−J2cJ4sssc−1B2c are, respectively, the new state and input matrices of the CSM.


## 3.1. Current Sensitivity Model

The CSM is a linear analysis tool for power systems based on Kirchhoff’s current law, which must be met when the system is disturbed [7, 12, 17]. Its modeling for multi-machine systems composed of ng generators and nb buses is shown in (18)–(21). In this formulation, Δx represents the state variables, Δu represents the input variables, and Δz represents the algebraic variables:(18)Δx=[Δω1…ΔωngΔδ1…ΔδngΔEq′1…ΔEq′ngΔEfd1…ΔEfdng]t,(19)Δu=[ΔPm1…ΔPmngΔVref1…ΔVrefngΔPL1…ΔPLnbΔQL1…ΔQLnb]t,(20)Δz=[Δθ1…ΔθnbΔV1…ΔVnb]t,(21)Δx.0=J1J2J3J4ΔxΔz+B1B2Δu,where Δω represents the variations of the angular speed, Δδ represents the variations of the internal angle of the rotor, ΔEq′ represents the variations of the internal voltage in quadrature, and ΔEfd represents the variations of the generator’s field voltage, ΔPm represents the variations of the input mechanical power, ΔVref represents the variations of the reference voltage of the AVR, ΔPL and ΔQL are, respectively, the variations of the active and reactive power demands of the loads, and ΔV and Δθ are, respectively, the voltage magnitude and voltage phase variations at each bus of the power system.The representation in the state space, in the following equation, is obtained by eliminatingΔz from (21):(22)Δx.=AΔx+BΔu,where A and B are the state and input matrix, respectively, in which A=J1−J2J4−1J3 and B=B1−J2J4−1B2.


## 3.2. Inclusion of the SSSC in the CSM

The dynamic characteristics of the SSSC have already been incorporated into the power flow modeled by current injections and are shown in the submatricesJ1sssc, J2sssc, J3sssc, and J4sssc in (17). These can be directly included in the CSM, as shown in the following equations:(23)(24)Δx.c=J1c−J2cJ4sssc−1J3cΔxc+B1c−J2cJ4sssc−1B2cΔuc,where k=1,…,nb, Ac=J1c−J2cJ4sssc−1J3c and Bc=B1c−J2cJ4sssc−1B2c are, respectively, the new state and input matrices of the CSM.


## 4. Dynamic Model of the PSS and POD Controllers

The purposes of the PSS and POD controllers are to insert, respectively, additional damping torque to the local and interarea oscillatory modes present in the power system. In this work, both the PSSs, Figure6(a), and the POD, Figure 6(b), have similar structures, differing only in the input and output signals adopted for each controller. Both are represented by two lead-lag compensation blocks, characterized by the time constants T1pssT1pod, T2pssT2pod, T3pssT3pod, and T4pssT4pod, a gain KpssKpod and a washout block, represented by the time constants TωpssTωpod.Figure 6
(a) Dynamic model of the PSS controller and (b) dynamic model of the POD controller.
(a)(b)The input signal chosen for a PSS is the variation of the angular speed of thekth generator, Δωk, while the output of a PSS, ΔVsuppss, actuates on the control loop of the AVR of the kth generator (see Figure 6(a)). For the POD controller, the chosen input signal is the active power deviation of the transmission line adjacent to the installation of the device, ΔPkm, and the output signal, ΔVsuppod, actuates on the control loop of the SSSC, used to modulate the quadrature component, Vp, of the series converter of the SSSC (see Figure 6(b)).In Figure6(a), the AVR installed at the kth generator is represented by a gain Krk and a time constant Trk. The excitation voltage of the synchronous machine is Efdk, the variations of the terminal voltage are represented by ΔVk, and the variations of the reference voltage are represented by Vrefk. In Figure 6(b), ΔVp is the quadrature axis component of the series converter, ΔXref is an input signal and Tmpod is a time constant, being that both must be specified.It is adopted thatT1pssT1pod=T3pssT3pod and T2pssT2pod=T4pssT4pod [1]. The time constant Tmpod is the inherent delay of the control device with values between 1−10 ms [10].From the block diagram shown in Figure6(a), the expressions of the new state variables of the PSS that will be included in the CSM are written as shown in (25)–(27), and of the output signal, Vsuppss, according to (28):(25)ΔV˙1k=−1TωkKpssΔωk+ΔV1k,(26)ΔV˙2k=−1T21−T1pssT2pssKpssΔωk+ΔV1k−ΔV2k,(27)ΔV˙3k=−1T41−T3pssT4pssΔV2k+T1pssT2pssKpssΔωk+ΔV1k−ΔV3k,(28)ΔVsuppss=ΔV3k+T3pssT4pssΔV2k+T1pssT2pssKpssΔωk+ΔV1k.The procedure for including the PSS in the CSM at thekth generator is terminated by adding the supplementary signal, according to (28), to the control loop of the AVR shown from Figure 6(a). In this way, it is possible to obtain the following equation:(29)ΔE˙fdk=−1TrkΔEfdk+KrkTrkΔVrefk−KrkTrkΔVk+ΔVsuppss.Analogously to the procedure performed for the PSS, from Figure6(b), the expressions of the state variables of the POD that will be included in the CSM are obtained as shown in (30)–(32), and the supplementary output signal of the POD controller, Vsuppss, is obtained according to (33):(30)ΔY˙1=−1TωpodKpodΔPkm+ΔY1,(31)ΔY˙2=−1T2pod1−T1podT2podKpodΔPkm+ΔY1−ΔY2,(32)ΔY˙3=−1T4pod1−T3podT4podΔY2+T1podT2podKpodΔPkm+ΔY1−ΔY3,(33)ΔVsuppod=ΔY3+T3podT4podΔY2+T1podT2podKpodΔPkm−ΔY1.Finally, (33), which represents the supplementary output signal of the POD controller, is replaced into (6), which describes the dynamics of the quadrature control voltage of the series converter of the SSSC, giving rise to the following equation:(34)ΔV˙p=K1piTmpiΔPmref−ΔPmctrl+1TmpodΔX1−Vsuppod−ΔVp.


## 5. Optimization Techniques

This section will present the details the three optimization methods that were used for the coordinated parameter tuning design of the PI, PSS, and SSSC-POD controllers. The GA with elitism, Subsection5.1, the PSO algorithm, Subsection 5.2, and the MOA, Subsection 5.3, will be discussed.


### 5.1. Genetic Algorithm

The first GA was proposed by [33] in the 1970s. He analyzed the phenomenon of the natural evolution of species and applied operators in order to reproduce this phenomenon when solving a complex problem. In the particular case of this paper, the problem consists of performing the coordinated tuning of the parameters of PI controllers and supplementary damping controllers (PSS and POD). The goal is to insert the desired damping to the poles of interest present in the power system.Figure 7
Two-point crossover.Basically, in this work, the GA performs the seven steps outlined below:(1)
Generation of the initial population: properly represent the set of candidate solutions. The GA starts by randomly generating a predefined number of initial solutions, forming the initial population.(2)
Evaluation function and constraints: evaluate the objective function or its equivalent (fitness) for each individual. In this step, the evaluation of individuals, or chromosomes, takes place, using the objective function for the problem (see (53)–(55)).(3)
Selection: the probabilistic operatorproportional selection used in the GA, draws individuals from a population for crossover, in which each has drawn chances proportional to their respective evaluation function values. This selection method is also known as the roulette rule [32].(4)
Crossover: two individuals are chosen (P1 and P2), called parents, and the crossover operator is applied. Two new individuals are obtained, known as offspring (P1′ and P2′). The crossover occurs until a new population of offspring is formed. In this work, two recombination points are used, and these points are chosen randomly at each crossover step (see Figure 7).(5)
Mutation: the mutation operator randomly changes one or more variables of a chromosome with a probability and within a range. This operator is used to allow diversification in the search process and introduce diversity.(6)
Elitism: in the basic GA, the individuals obtained in the previous generation are discarded, and only the new descendants are considered in the next generation. Elitism consists in reintroducing the best-evaluated individual from one generation to the next, avoiding the loss of important information present in individuals and that may be lost during the crossover and mutation procedures.(7)
Verification of the convergence criterion: the goal is to solve a constraint satisfaction problem [41]. Thus, the desired dampings are considered to be constraints of the problem and therefore any settings determined for the parameters of the controllers within the pre-established limits, (50)–(52), that provide the desired damping, is considered a solution to the problem, see Table 1.Table 1
Parameters for the evaluation of the objective function.
[1.5pt]ξdes−tol (p.u.)ωniloc (Hz)ωninter (Hz)0.145Min. 0.80Min. 0.50Max. 1.40Max. 0.70


### 5.2. Particle Swarm Optimization

Initially consider a search space of dimensionnv. Suppose that each particle i, in the search space, can be represented according to its current position xi,psot=x1i,psot,…,xnvi,psot and a velocity vi,psot=v1i,psot,…,vnvi,psot. During the iterative process, at each step t, the current position of each particle is measured using an objective function according to (53)–(55). From the analysis of the evaluation function, it is verified, at each iteration t, if the position of the particle at iteration t is better than the other positions previously verified. If so, the position of this particle is stored in a vector pbesti,psot=pbest1i,psot,…,pbestnvi,psot. In the course of the iterative process, at each best evaluation, this vector is updated. Finally, the best-found position measured by the evaluation function stored in pbesti,psot is stored in another vector called gbestpsot.In the PSO algorithm, what determines the motion of a given particlei at step t+1 is its velocity vi,psot calculated in the following equation:(35)vi,psot+1=wvi,psot+C1r1coit+C2r2soit,where r1 and r2 are random values over the interval 0,1,C1 and C2 are called acceleration constants and are responsible, respectively, for weighting the cognitive factor, coit, and the social factor, soit. Furthermore, w=wmax−wmin/tmaxt is known as the inertia factor and is meant to control the impact of velocities on the particles of the PSO algorithm and tmax is the maximum number of iterations [42].The cognitive and social factors, shown in (35), are represented by the following equations:(36)coit=pbesti,psot−xi,psot,(37)soit=gbestpsot−xi,psot,where the ith cognitive factor, coit, is related to each of the individuals in the ith best position pbestit, the social factor, soit, is related to the best position found among all individuals in the swarm gbestt and xi,psot is the position vector of the ith particle at iteration t. At iteration t+1, the position of the ith particle, xi,psot+1, is updated according to the following equation:(38)xi,psot+1=xi,psot+vi,psot+1.


### 5.3. Mayfly Optimization Algorithm

As stated before, the MOA is an optimization algorithm proposed by [30] in 2020, based on concepts from the PSO algorithm, the GA, and the FA. Initially, two sets of mayflies are randomly generated, one containing a male and the other a female population. Suppose that each mayfly i, in the search space, can be randomly represented by a nv-dimensional vector xi,moat=x1i,moat,…,xnvi,moat and by a velocity vi,moat=v1i,moat,…,vnvi,moat. During the iterative process, at each iteration t, the current position of each mayfly is measured using an evaluation function, according to (53)–(55). From the analysis of the evaluation function, at each iteration t, each mayfly adjusts its trajectory toward its personal best position, pbesti,moat=pbest1i,moat,…,pbestnvi,moat. In the course of the iterative process, when a better candidate solution is found, this vector is updated. Finally, at each iteration, the best-found position measured by the evaluation function stored in pbesti,moat is stored in another vector, gbestmoat.


#### 5.3.1. Movement of Male Mayflies

Assuming thatxi,moat is the current position of the male mayfly i in the search space at iteration t, its position is changed by adding a velocity vmlit+1 to the current position, according to following equation:(39)xi,moat+1=xi,moat+vmli,moat+1,where vi,moat+1 is given according to the following equation:(40)vmli,moat+1=gvmli,moat+C1e−βrp2coit+C2e−βrg2soit,where C1 and C2 can be defined as in (35), g is called the gravity coefficient and has values determined in the interval 0,1, while coit and soit were addressed, respectively, in (36) and (37). Finally, β is called the visibility coefficient and has its value determined empirically, while rp and rg are, respectively, the calculated cartesian distance from xi,moat to pbestit and between xi,moat and gbestt according to the following equation:(41)xi,moat−x¯i,moat=∑j=1nvxij,moat−x¯ij,moat2,where xi,moat=pbesti,moat or xi,moat=gbestmoat.The velocity control, shown in (40), of each mayfly in the MOA is performed according to the following equation:(42)vmli,moat+1=vmlmax,ifvmli,moat+1>vmlmax,ρ2vmlmax,ifvmlit+1<ρ2vmlmax,where ρ2 is an empirically fixed random value, vmax=ρ1xmax−xmin, in which xmax and xmin are the upper and lower bounds of the control parameters of the devices installed in the power system shown in (50)–(52) and ρ1 is calibrated from values selected from the range 0,1.To the step that reproduces the nuptial dance, in which the best mayflies have to keep changing their velocities, the following equation is presented:(43)vmli,moat+1=vmli,moat+d⋅r,where d is the coefficient of the nuptial dance and r is a vector with random values within −1,1.


#### 5.3.2. Movement of Female Mayflies

Assuming thatyi,moat is the current position of the female mayfly i in the search space, at iteration t, its position is changed by adding a velocity vi,moat+1 according to the following equation:(44)yit+1=yit+vfmli,moat+1,where vfmlit+1 is determined using the following equation:(45)vfmli,moat+1=gvfmli,moat+C3e−βrmf2xi,moat−yit,if male dominates femalegvfmli,moat+flt⋅r,otherwise,where flt is a random walk coefficient, rmf is the Cartesian distance between the male and female mayflies, formulated according to (41), and C3 is a positive attraction constant.


#### 5.3.3. Mating of Mayflies

The crossover operator is used in this algorithm to represent the mating process between two mayflies and is used as follows: the best female crosses with the best male, the second-best female with the second-best male, and so on. The results obtained after these crosses are two offspring generated according to the following equations:(46)offspring1=L⋅male+1−L⋅female,(47)offspring2=L⋅female+1−L⋅male,where male is the male parent, female is the female parent, and L is a random value within a certain interval. The initial velocities of the offspring are set to zero.


#### 5.3.4. Reduction of the Nuptial Dance and Random Walk Coefficients

The nuptial dance performed by male mayflies represented by the coefficientdt, as well as the random walk performed by females, are based on local searches shown in (43) and (45). However, random flights can produce poor results during a random exploration. This problem occurs from the fact that the nuptial dance dt or random walk flt coefficients usually assume large initial values. This can be mitigated by gradually reducing the dance coefficient dt and the random walk coefficient flt in each iteration as shown in the following equations:(48)dt=d0δdt,(49)flt=fl0δflt,where at each iteration t of the optimization problem, δdt and δdt take fixed values in the interval 0,1.


## 5.1. Genetic Algorithm

The first GA was proposed by [33] in the 1970s. He analyzed the phenomenon of the natural evolution of species and applied operators in order to reproduce this phenomenon when solving a complex problem. In the particular case of this paper, the problem consists of performing the coordinated tuning of the parameters of PI controllers and supplementary damping controllers (PSS and POD). The goal is to insert the desired damping to the poles of interest present in the power system.Figure 7
Two-point crossover.Basically, in this work, the GA performs the seven steps outlined below:(1)
Generation of the initial population: properly represent the set of candidate solutions. The GA starts by randomly generating a predefined number of initial solutions, forming the initial population.(2)
Evaluation function and constraints: evaluate the objective function or its equivalent (fitness) for each individual. In this step, the evaluation of individuals, or chromosomes, takes place, using the objective function for the problem (see (53)–(55)).(3)
Selection: the probabilistic operatorproportional selection used in the GA, draws individuals from a population for crossover, in which each has drawn chances proportional to their respective evaluation function values. This selection method is also known as the roulette rule [32].(4)
Crossover: two individuals are chosen (P1 and P2), called parents, and the crossover operator is applied. Two new individuals are obtained, known as offspring (P1′ and P2′). The crossover occurs until a new population of offspring is formed. In this work, two recombination points are used, and these points are chosen randomly at each crossover step (see Figure 7).(5)
Mutation: the mutation operator randomly changes one or more variables of a chromosome with a probability and within a range. This operator is used to allow diversification in the search process and introduce diversity.(6)
Elitism: in the basic GA, the individuals obtained in the previous generation are discarded, and only the new descendants are considered in the next generation. Elitism consists in reintroducing the best-evaluated individual from one generation to the next, avoiding the loss of important information present in individuals and that may be lost during the crossover and mutation procedures.(7)
Verification of the convergence criterion: the goal is to solve a constraint satisfaction problem [41]. Thus, the desired dampings are considered to be constraints of the problem and therefore any settings determined for the parameters of the controllers within the pre-established limits, (50)–(52), that provide the desired damping, is considered a solution to the problem, see Table 1.Table 1
Parameters for the evaluation of the objective function.
[1.5pt]ξdes−tol (p.u.)ωniloc (Hz)ωninter (Hz)0.145Min. 0.80Min. 0.50Max. 1.40Max. 0.70


## 5.2. Particle Swarm Optimization

Initially consider a search space of dimensionnv. Suppose that each particle i, in the search space, can be represented according to its current position xi,psot=x1i,psot,…,xnvi,psot and a velocity vi,psot=v1i,psot,…,vnvi,psot. During the iterative process, at each step t, the current position of each particle is measured using an objective function according to (53)–(55). From the analysis of the evaluation function, it is verified, at each iteration t, if the position of the particle at iteration t is better than the other positions previously verified. If so, the position of this particle is stored in a vector pbesti,psot=pbest1i,psot,…,pbestnvi,psot. In the course of the iterative process, at each best evaluation, this vector is updated. Finally, the best-found position measured by the evaluation function stored in pbesti,psot is stored in another vector called gbestpsot.In the PSO algorithm, what determines the motion of a given particlei at step t+1 is its velocity vi,psot calculated in the following equation:(35)vi,psot+1=wvi,psot+C1r1coit+C2r2soit,where r1 and r2 are random values over the interval 0,1,C1 and C2 are called acceleration constants and are responsible, respectively, for weighting the cognitive factor, coit, and the social factor, soit. Furthermore, w=wmax−wmin/tmaxt is known as the inertia factor and is meant to control the impact of velocities on the particles of the PSO algorithm and tmax is the maximum number of iterations [42].The cognitive and social factors, shown in (35), are represented by the following equations:(36)coit=pbesti,psot−xi,psot,(37)soit=gbestpsot−xi,psot,where the ith cognitive factor, coit, is related to each of the individuals in the ith best position pbestit, the social factor, soit, is related to the best position found among all individuals in the swarm gbestt and xi,psot is the position vector of the ith particle at iteration t. At iteration t+1, the position of the ith particle, xi,psot+1, is updated according to the following equation:(38)xi,psot+1=xi,psot+vi,psot+1.


## 5.3. Mayfly Optimization Algorithm

As stated before, the MOA is an optimization algorithm proposed by [30] in 2020, based on concepts from the PSO algorithm, the GA, and the FA. Initially, two sets of mayflies are randomly generated, one containing a male and the other a female population. Suppose that each mayfly i, in the search space, can be randomly represented by a nv-dimensional vector xi,moat=x1i,moat,…,xnvi,moat and by a velocity vi,moat=v1i,moat,…,vnvi,moat. During the iterative process, at each iteration t, the current position of each mayfly is measured using an evaluation function, according to (53)–(55). From the analysis of the evaluation function, at each iteration t, each mayfly adjusts its trajectory toward its personal best position, pbesti,moat=pbest1i,moat,…,pbestnvi,moat. In the course of the iterative process, when a better candidate solution is found, this vector is updated. Finally, at each iteration, the best-found position measured by the evaluation function stored in pbesti,moat is stored in another vector, gbestmoat.


### 5.3.1. Movement of Male Mayflies

Assuming thatxi,moat is the current position of the male mayfly i in the search space at iteration t, its position is changed by adding a velocity vmlit+1 to the current position, according to following equation:(39)xi,moat+1=xi,moat+vmli,moat+1,where vi,moat+1 is given according to the following equation:(40)vmli,moat+1=gvmli,moat+C1e−βrp2coit+C2e−βrg2soit,where C1 and C2 can be defined as in (35), g is called the gravity coefficient and has values determined in the interval 0,1, while coit and soit were addressed, respectively, in (36) and (37). Finally, β is called the visibility coefficient and has its value determined empirically, while rp and rg are, respectively, the calculated cartesian distance from xi,moat to pbestit and between xi,moat and gbestt according to the following equation:(41)xi,moat−x¯i,moat=∑j=1nvxij,moat−x¯ij,moat2,where xi,moat=pbesti,moat or xi,moat=gbestmoat.The velocity control, shown in (40), of each mayfly in the MOA is performed according to the following equation:(42)vmli,moat+1=vmlmax,ifvmli,moat+1>vmlmax,ρ2vmlmax,ifvmlit+1<ρ2vmlmax,where ρ2 is an empirically fixed random value, vmax=ρ1xmax−xmin, in which xmax and xmin are the upper and lower bounds of the control parameters of the devices installed in the power system shown in (50)–(52) and ρ1 is calibrated from values selected from the range 0,1.To the step that reproduces the nuptial dance, in which the best mayflies have to keep changing their velocities, the following equation is presented:(43)vmli,moat+1=vmli,moat+d⋅r,where d is the coefficient of the nuptial dance and r is a vector with random values within −1,1.


### 5.3.2. Movement of Female Mayflies

Assuming thatyi,moat is the current position of the female mayfly i in the search space, at iteration t, its position is changed by adding a velocity vi,moat+1 according to the following equation:(44)yit+1=yit+vfmli,moat+1,where vfmlit+1 is determined using the following equation:(45)vfmli,moat+1=gvfmli,moat+C3e−βrmf2xi,moat−yit,if male dominates femalegvfmli,moat+flt⋅r,otherwise,where flt is a random walk coefficient, rmf is the Cartesian distance between the male and female mayflies, formulated according to (41), and C3 is a positive attraction constant.


### 5.3.3. Mating of Mayflies

The crossover operator is used in this algorithm to represent the mating process between two mayflies and is used as follows: the best female crosses with the best male, the second-best female with the second-best male, and so on. The results obtained after these crosses are two offspring generated according to the following equations:(46)offspring1=L⋅male+1−L⋅female,(47)offspring2=L⋅female+1−L⋅male,where male is the male parent, female is the female parent, and L is a random value within a certain interval. The initial velocities of the offspring are set to zero.


### 5.3.4. Reduction of the Nuptial Dance and Random Walk Coefficients

The nuptial dance performed by male mayflies represented by the coefficientdt, as well as the random walk performed by females, are based on local searches shown in (43) and (45). However, random flights can produce poor results during a random exploration. This problem occurs from the fact that the nuptial dance dt or random walk flt coefficients usually assume large initial values. This can be mitigated by gradually reducing the dance coefficient dt and the random walk coefficient flt in each iteration as shown in the following equations:(48)dt=d0δdt,(49)flt=fl0δflt,where at each iteration t of the optimization problem, δdt and δdt take fixed values in the interval 0,1.


## 5.3.1. Movement of Male Mayflies

Assuming thatxi,moat is the current position of the male mayfly i in the search space at iteration t, its position is changed by adding a velocity vmlit+1 to the current position, according to following equation:(39)xi,moat+1=xi,moat+vmli,moat+1,where vi,moat+1 is given according to the following equation:(40)vmli,moat+1=gvmli,moat+C1e−βrp2coit+C2e−βrg2soit,where C1 and C2 can be defined as in (35), g is called the gravity coefficient and has values determined in the interval 0,1, while coit and soit were addressed, respectively, in (36) and (37). Finally, β is called the visibility coefficient and has its value determined empirically, while rp and rg are, respectively, the calculated cartesian distance from xi,moat to pbestit and between xi,moat and gbestt according to the following equation:(41)xi,moat−x¯i,moat=∑j=1nvxij,moat−x¯ij,moat2,where xi,moat=pbesti,moat or xi,moat=gbestmoat.The velocity control, shown in (40), of each mayfly in the MOA is performed according to the following equation:(42)vmli,moat+1=vmlmax,ifvmli,moat+1>vmlmax,ρ2vmlmax,ifvmlit+1<ρ2vmlmax,where ρ2 is an empirically fixed random value, vmax=ρ1xmax−xmin, in which xmax and xmin are the upper and lower bounds of the control parameters of the devices installed in the power system shown in (50)–(52) and ρ1 is calibrated from values selected from the range 0,1.To the step that reproduces the nuptial dance, in which the best mayflies have to keep changing their velocities, the following equation is presented:(43)vmli,moat+1=vmli,moat+d⋅r,where d is the coefficient of the nuptial dance and r is a vector with random values within −1,1.


## 5.3.2. Movement of Female Mayflies

Assuming thatyi,moat is the current position of the female mayfly i in the search space, at iteration t, its position is changed by adding a velocity vi,moat+1 according to the following equation:(44)yit+1=yit+vfmli,moat+1,where vfmlit+1 is determined using the following equation:(45)vfmli,moat+1=gvfmli,moat+C3e−βrmf2xi,moat−yit,if male dominates femalegvfmli,moat+flt⋅r,otherwise,where flt is a random walk coefficient, rmf is the Cartesian distance between the male and female mayflies, formulated according to (41), and C3 is a positive attraction constant.


## 5.3.3. Mating of Mayflies

The crossover operator is used in this algorithm to represent the mating process between two mayflies and is used as follows: the best female crosses with the best male, the second-best female with the second-best male, and so on. The results obtained after these crosses are two offspring generated according to the following equations:(46)offspring1=L⋅male+1−L⋅female,(47)offspring2=L⋅female+1−L⋅male,where male is the male parent, female is the female parent, and L is a random value within a certain interval. The initial velocities of the offspring are set to zero.


## 5.3.4. Reduction of the Nuptial Dance and Random Walk Coefficients

The nuptial dance performed by male mayflies represented by the coefficientdt, as well as the random walk performed by females, are based on local searches shown in (43) and (45). However, random flights can produce poor results during a random exploration. This problem occurs from the fact that the nuptial dance dt or random walk flt coefficients usually assume large initial values. This can be mitigated by gradually reducing the dance coefficient dt and the random walk coefficient flt in each iteration as shown in the following equations:(48)dt=d0δdt,(49)flt=fl0δflt,where at each iteration t of the optimization problem, δdt and δdt take fixed values in the interval 0,1.


## 6. Problem Formulation

All algorithms are implemented considering an initial population consisting of 20 individuals. Each individual is anv-dimensional vector of variables, this quantity being determined by the number of parameters of PI, PSSs, and the SSSC-POD controllers installed in the power system. Each variable in an individual of the optimization problem in all the algorithms must satisfy the bounds shown in the following equations:(50)0.10≤T1rpss≤1.25;0.01≤T2rpss≤0.20;1.00≤Krpss≤10.00,(51)0.10≤T1pod≤0.40;0.10≤T2pod≤0.40;0.10≤Kpod≤0.50,(52)0.001≤T1pi=T2pi≤0.90;0.001≤K1pi=K2pi≤1.00,where the units of the time constants and gains of the controllers are, respectively, seconds and p.u., while r=1,…,9.Theith individual of a given population zi, is shown in Figure 8. As it will be justified in Section 6, the simulations will be performed using the New England system (see [17]) and the system will be equipped with one PI controller, an SSSC-POD, and nine PSSs, totaling 32 variables, i.e., nv=32. Moreover, tests will also be conducted using the two-area symmetrical system equipped with one PI controller, an SSSC-POD, and two PSSs, totaling 11 variables, i.e., nv=11.Figure 8
Representation of a solution proposal.The objective function of the problem is defined in the following equations for all algorithms:(53)F1zi=∑q=1nΔξq,(54)F2zi=∑q=1nΔωq,(55)minimizeFzi=η1F1zi+η2F2zi.where n is the number of eigenvalues present in the power system, η1=η2=1, Δξq=ξdes−ξqcalc and Δωq=ωdes−ωqcalc, in which ξdes and ξqcalc are, respectively, the desired and calculated damping of the qth eigenvalue of interest of the power system, while ωdes and ωqcalc are, respectively, the desired and calculated frequency of the qth eigenvalue of interest of the power system.The settings provided to the control parameters by the algorithms to the PSSs, the SSSC-POD, and PI controllers must ensure the minimum desired damping for all poles of interest of the power system as well as that the oscillation frequencies of all existing poles remain within predefined design ranges, as shown in Figure9.Figure 9
Desired location for the eigenvalues of interest.In Figure9, σi and ωi are, respectively, the real and imaginary parts of the ith eigenvalue λi=σi±jωi. With the bounds given in (50)–(52), the objective function of a solution, shown in (55), should satisfy the constraints defined in the project, as shown in Table 1.In Table1, ωniloc is related to the oscillation frequency of local modes, while ωninter is associated with the interarea mode. Finally, Tωpod=Tωpss=1 s, while Tmpi=Tmpod=0.001 s are prespecified.


## 7. Simulations and Results

This Section presents the results of simulations performed using the New England system [17] and the two-area symmetrical system [1].


### 7.1. New England System

To validate the models and optimization algorithms presented in this work, small-signal stability simulations will be carried out using the New England system whose complete data is available in [17]. This is a multimachine system composed of 10 generators, 40 buses, and 47 transmission lines, already accounting for one new line (38− FB) and one new bus (FB, between buses 37−38), included to perform simulations with the SSSC, as can be analyzed in Figure 10.Figure 10
Single-line diagram of the New England system with one SSSC.For the simulations, the algorithms were calibrated according to data shown in Table2, considering the set of bounds shown in (50)–(52).Table 2
Parameters of the algorithms.
Search Engine StructureParameters of the AlgorithmParameters of the SimulationsFitness FunctionRoulette selection:—GATwo-point crossoverMutation (23% of the value):Elitism:10% probability rate10% probability rate best individualPSOFormulation(35)–(38)C1=2.55C2=1.55wmax=0.89wmin=0.0025tmax=2000Population size: 20 Stopping criterion: Table1Max. number of iterations:2000(55)MOAFormulation(39)–(48)g=0.851C1=1.99, C2=2.00C3=2.50, β=2.00ρ1=0.10, ρ2=−1.10d0=950, δdt=0.9999fl0=1.50, δflt=0.99The presentation of the results is organized in two cases: first, a static analysis is performed, and second, a dynamic analysis is conducted. In the first case, the objective is to validate the proposed formulation for the power flow modeled by current injections for the SSSC. The second case is subdivided into two parts: in the first part, a statistical analysis is performed considering the results of the three simulated optimization techniques (GA, PSO algorithm, and MOA) in order to determine which of the three algorithms best applies to the discussed problem; second, an analysis focused on small-signal stability using the New England system is performed. The proposed models and optimization algorithms were implemented in MATLAB R2019a using a computer with a 3.19 GHz Intel®Core™i7-8700 processor and 16 GB of RAM.


#### 7.1.1. Static Analysis

Initially, it can be seen that the SSSC was installed between buses37−38. The reasons for installing the device at this location are: first, the installation of the SSSC should occur at locations close to the buses that presented voltage magnitudes below 0.95 p.u. According to Figure 11, the buses with voltage magnitudes below 0.95 p.u. are 12, 36, and 37 (0.9491 p.u., 0.9486 p.u., and 0.9475 p.u.) and are close to the site chosen for SSSC installation; second, the location should allow for the controllability and observability over the interarea mode present in the power system. Indeed, according to Figure 10, buses 38 and 37 are characterized by being located near to Area 1 (New York system that is represented by the equivalent bus 10). It is known in the literature that interconnection areas favor the emergence of interarea oscillatory modes.Figure 11
Voltage profiles for the New England system.Case I will represent the situation in which the New England system is equipped with an SSSC, but the device does not control the active and reactive power flows in the line in which it is installed, and Case II will represent the case in which the SSSC controls the active and reactive power flows. For the first case,Pmctrl=−15.96 MW and Qmctrl=−111.27 MVAr and, for Case II, Pmctrl=−25.00 MW and Qmctrl=−125.00 MVAr. In both cases, for simulation purposes, the coupling reactance of the transformer, xkm, connecting the VSC to the transmission line (positioned between buses 38 and FB) is equal to 0.01 p.u.In Figure11, the voltage profiles before (Case I) and after (Case II) the SSSC actuation for controlling the active and reactive power flows are shown.By analyzing Figure11, it can be seen that before the control of the active and reactive power flows performed by the SSSC (Case I), the system had voltage problems at buses 12 (0.9491 p.u.), 36 (0.9486 p.u.), and 37 (0.9475 p.u.). However, the same cannot be said for Case II. After controlling the flows, the voltage profiles were adjusted, being equal to 0.9505 p.u. for bus 12, 0.9513 p.u. for bus 36, and 0.9506 p.u. for bus 37.In Table3, the values of the SSSC control variables, Vp and Vq, are presented.Table 3
Variables of the SSSC’s control structure of the new England system.
SSSCVpVqCase I−7.11283×10−072.78607×10−08Case II−0.0155367−0.0150003When analyzing Table3, it is possible to verify that the values of Vp and Vq are close to zero for Case I, demonstrating that the SSSC is included in the system, but it is not actuating. In relation to Case II, it can be seen that after the SSSC has actuated to control the active and reactive power flows, the values of the control variables are different from zero, which corroborates the SSSC’s actuation in the New England system.Finally, in Figure12, the active and reactive power flows, generations, loads, and the power injections performed by the SSSC, for the buses and lines near to the installation site of the SSSC, are presented. The results for Case I are represented in Figure 12(a) while the results for Case II are shown in Figure 12(b).Figure 12
Control of the active and reactive power flows performed by the SSSC in the New England system.
(a)(b)By analyzing Figure12, it is possible to verify that, in both cases, the control of the power flows is performed according to what was established for Cases I and II. It is important to highlight that, in Case II, in order to control the flows specified in the project (Pmctrl=−25.00 MW and Qmctrl=−125.00 MVAr) for improving the voltage magnitudes at buses 12, 36, and 37, the SSSC needed to inject at buses 37 and FB, respectively, 147.70 MW and 142.60 MVAr, and −149.38 MW and −140.30 MVAr. Finally, it is possible to perform the nodal balance of active and reactive powers at the buses indicated in Figure 12 and thus validate the control of the power flows performed by the SSSC.


#### 7.1.2. Dynamic Analysis

After validating the current injections model for the SSSC and confirming its ability to manage the active and reactive power flows, as shown in Subsection7.1.1, a dynamic analysis involving the SSSC, PI controllers, and supplementary damping controllers (PSS and POD), will be performed. In Table 4, the eigenvalues of interest of the New England system are presented before (Case I) and after (Case II) the actuation of the SSSC in the control of the active and reactive power flows.Table 4
Dominant eigenvalues, damping coefficients, and natural undamped frequencies of the New England system.
ModeCase ICase IIEigenvaluesξi (p.u.)ωni (Hz)Eigenvaluesξi (p.u.)ωni (Hz)λ10.0729±j6.8461−0.01061.08970.0767±j6.8448−0.01121.0894λ2−0.2064±j7.23480.02851.1519−0.2052±j7.23350.02841.1517λ3−0.1903±j8.27310.02301.3171−0.1880±j8.27250.02271.3169λ40.1799±j5.9142−0.03040.94170.1745±j5.9090−0.02950.9409λ5−0.1291±j6.49680.01991.0342−0.1245±j6.48290.01921.0320λ6−0.2690±j8.10710.03321.2910−0.2687±j8.10590.03311.2908λ7−0.2413±j8.32040.02901.3248−0.2412±j8.31870.02901.3245λ80.1622±j6.3862−0.02541.01670.1587±j6.3764−0.02491.0152λ90.0037±j3.5625−0.00100.5670−0.0812±j3.46380.02340.5514By analyzing the frequencies,ωni=λi/2π, of the modes of interest, λi=σi±jωi, shown in Table 4, it can be concluded that the New England system is characterized by the presence of eight local modes, λ1 and λ8, and one interarea mode, λ9. In both Case I and Case II, the system is unstable. In Case I, the instability of the system is caused by four oscillatory modes with positive real parts, λ1, λ4, λ8, and λ9, while in Case II, the instability of the system is caused by three oscillatory modes with positive real parts, λ1, λ4, and λ8. Importantly, the interarea mode in Case I has negative damping, of −0.0010 p.u., and, in Case II, after the control of the active and reactive power flows by the SSSC, this mode changes from unstable to stable with damping of 0.0234 p.u.The inclusion of the SSSC dynamics in the CSM and the correct control of the active and reactive power flows performed by this device in the New England system provided improvements in the voltage profile of the system (see Figure11) as well as small damping increments in some poles of interest of the system. In fact, it is possible to verify that in Case II, for the considered operation point, the New England system started to operate with three poles responsible for the system instability, one less when compared to Case I (see Table 4). Due to these observations, Case II, from this point on, will be used as the base case in the following simulations.(1) Statistical Analyses of the Performances of the Algorithms. Initially, an SSSC-POD installed in the system according with the arguments presented in Subsection 7.1.1 and nine PSSs coupled to generators G1 to G9 are considered.The parameter tuning of the PI and supplementary damping controllers, PSSs and POD, for Case II, is performed using the PSO algorithm, GA, and MOA, and these are calibrated based on the data provided in Table2. In this first scenario, one hundred tests limited to two thousand iterations in each one are performed. Under these conditions, the values given in Table 1 should be checked. The results of these simulations can be verified in Table 5.Table 5
Comparison of the performances of the algorithms for the New England system.
AlgorithmDamping ratioTests with convergence%Number of iterationsTime (s)AverageMin.Max.AverageMin.Max.PSOTable187≈5981172000≈41≈8≈145GATable191≈7651632000≈107≈23≈284MOATable1100≈5816549≈14≈4≈127When analyzing the results presented in Table5, it is possible to make the following comments: (i) the MOA achieved success in 100% of the tests performed, against 91% of GA and 87% of PSO algorithm; (ii) both in average or when the maximum and minimum numbers of iterations are compared, it is possible to observe that the MOA was more efficient than the other two optimization algorithms; (iii) finally, when the convergence times of the algorithms were compared, again the MOA had better performance than the others, being the best in the average, minimum, and maximum times observed.The results analyzed so far would already allow saying that the MOA is the most efficient algorithm for tuning the parameters of the controllers for small-signal stability improvement, and therefore, would allow accrediting it as a powerful tool in this type of study. However, to corroborate the results presented in Table5, the radar plot of the three simulated algorithms is shown in Figure 13. In it, the observed results for four quantitative variables obtained in the one hundred simulated tests are presented: time, numbers of iterations, minimum damping, and the smallest values found for the objective function for each simulated test.Figure 13
Radar plot for the GA, PSO algorithm, and MOA for the New England system.It is clear from Figure13 that the radar plot for the MOA is the one that presents the best results. It is possible to observe that the respective curves presented for the MOA radar plot are concentrated closer to the center, corroborating the superiority of the MOA in all analyzed aspects compared to the other algorithms.(2) Small-Signal Stability Studies. In order to perform studies focused on the small-signal stability, simulations were performed with the three algorithms proposed in this paper (MOA, PSO algorithm, and GA). Again, the New England system equipped with nine PSSs installed at generators G1 to G9 and an SSSC-POD between buses 37 and 38 is considered. All algorithms are simulated and calibrated with the settings shown in Table 2. It is worth noting that, in this case, the same randomly generated initial population was used for the three proposed algorithms in each trial. In this case, 50000 iterations were considered, without specifying the desired minimum damping. Table 6 shows the results obtained for the parameters of the PSSs, PI, and POD controllers.Table 6
Gains and Time Constants for the PSS, PI, and POD Controllers Tuned by the MOA, PSO algorithm, and GA for the New England System.
DeviceMOAPSOGAT1pss (s)T2pss (s)Kpss (p.u.)T1pss (s)T2pss (s)Kpss (p.u.)T1pss (s)T2pss (s)Kpss (p.u.)PSS G11.25000.067910.0001.25000.073810.0001.24810.07089.7607PSS G20.93030.09387.16391.25000.13165.95940.99710.14548.1228PSS G31.02960.09166.37800.82610.085810.0001.14790.08705.6172PSS G40.68870.06859.49271.25000.07465.81201.05100.05327.0592PSS G50.42250.10477.74500.52630.11442.00000.71550.20009.6881PSS G61.18760.11818.96211.25000.13977.91741.10670.07958.2647PSS G70.47580.10584.54300.05000.01009.64110.28440.07133.9547PSS 80.86470.10376.72720.85960.09707.25400.85240.08349.2116PSS G90.56180.20004.53540.59490.20006.05370.43520.20007.3625PODT1podT2podkpodT1podT2podkpodT1podT2podkpod0.23950.27650.36220.32470.40000.50000.28450.40000.3979PIT1pi=T2piK1pi=K2piT1pi=T2piK1pi=K2piT1pi=T2piK1pi=K2pi0.61190.07300.76040.10780.31850.0781With the parameters presented in Table6, the PI and supplementary damping controllers (PSSs and POD) are calibrated and the CSM is evaluated. In this way, new damping levels ξi=−σi/λi associated with the eigenvalues of interest of the system can be obtained, as well as their respective undamped natural frequencies, as presented in Table 7.Table 7
Eigenvalues of interest, damping coefficients, and natural undamped frequencies determined by the MOA, PSO, and GA for the New England system.
ModeMOAPSOGAEigenvaluesξi (p.u.)ωni (Hz)Eigenvaluesξi (p.u.)ωni (Hz)Eigenvaluesξi (p.u.)ωni (Hz)λ1−2.6438±j6.61690.37101.1341−1.9419±j5.79000.31800.9720−2.1065±j7.42440.27301.2283λ2−2.2244±j5.55630.37170.9525−1.6865±j5.02840.31800.8441−1.8755±j6.48840.27771.0749λ3−3.1582±j7.87260.37231.3500−2.7814±j8.27690.31851.3897−2.3732±j8.09210.28141.3421λ4−2.5701±j6.39060.37311.0963−2.0599±j6.11440.31931.0269−1.5898±j5.08680.29830.8482λ5−2.3988±j5.92380.37531.0172−2.1980±j6.43610.32321.0824−1.7815±j5.62380.30200.9389λ6−1.9515±j4.79990.37660.8247−2.8851±j8.29770.32841.3982−1.8132±j5.20910.32870.8778λ7−2.4977±j5.38620.42070.9449−2.4205±j5.12570.42700.9022−2.5926±j7.27890.33551.2298λ8−2.6836±j4.55230.50780.8410−2.5751±j4.47370.49890.8215−3.0614±j4.61780.55260.8818λ9−1.5229±j3.74590.37660.6436−1.3002±j3.87650.31800.6507−1.1888±j3.37150.33250.5690By observing Table7 it is possible to verify that, for the operating point considered, from the adjustments obtained in Table 6, the frequencies of all poles of interest of the New England system are within the maximum and minimum limits stipulated in design (see Table 1). Regarding the values found for the dampings, it can be stated that the system operates with high damping margins, with 0.3710 p.u., 0.3180 p.u., and 0.2730 p.u. being the minimum damping levels, achieved from the MOA, PSO algorithm, and GA, respectively. Finally, on average, the MOA also has shown better results, as it achieved average damping of 0.3939 p.u., compared to 0.3521 p.u. for the PSO algorithm and 0.3313 p.u. for the GA.Power systems are often subject to small variations in loads and, therefore, consequent adjustments in generations. Figure14 shows the variations of the angular speeds of all generating units (except G2, the reference of the system) of the New England system for the simulated case presented in Tables 6 and 7.Figure 14
Variation of the angular speeds of the generators G1 and G3–G10 of the New England system.The disturbance in the mechanical power at G2 (+0.05 p.u.) is similar to a small adjustment in the generation, which can be caused by a small increase in the system loading. It can be seen in Figure 14 that even after the disturbance, the system is stabilized in approximately two seconds in all cases, graphically evidencing its high stability margin for small-signal disturbances.(3) Sensitivity Analysis on the Number of PSSs in the System. This section investigates the influence of the number of PSSs in the system for damping the local oscillation modes. In these tests, the number of PSSs installed in the New England system will be systematically reduced, and the parameters of the controllers will be optimized to maximize the minimum damping in the system. In all simulated cases, the FACTS-POD set is installed in the defined location of Subsection 7.1.1 and the parameters that were used to tune the algorithm are available in Table 2. Table 8 shows the mode with minimum damping obtained in each simulation.Table 8
Modes with minimum damping for each simulation obtained when removing PSSs from the New England system.
SimulationMode with minimum dampingminξi (p.u.)ωni (Hz)I−1.2807±j8.65390.14641.3923II−0.8888±j5.95550.14760.9583III−0.5260±j6.07140.08630.9699IV−0.4654±j7.01810.06621.1194V−0.2340±j7.20370.03251.1471VI−0.2180±j7.22600.03021.1506VII−0.2135±j7.23120.02951.1514VIII−0.2162±j8.28600.02611.3192In the simulations shown in Table8, the PSSs of the following generators were removed for each simulation:(i)
Simulation I: G7(ii)
Simulation II: G7, and G5(iii)
Simulation III: G7, G5, and G2(iv)
Simulation IV: G7, G5, G2, and G1(v)
Simulation V: G7, G5, G2, G1, and G3(vi)
Simulation VI: G7, G5, G2, G1, G3, and G8(vii)
Simulation VII: G7, G5, G2, G1, G3, G8, and G4(viii)
Simulation VIII: G7, G5, G2, G1, G3, G8, G4, and G6The order for removing the PSSs from the system was obtained by systematically removing one PSS at a time and verifying the removal that led to the smallest impact on the lowest damping after re-optimizing the parameters of the controllers. For example, it was verified that the PSS of G7 presented the lowest impact on the minimum damping of the system, therefore, it was the first one that was removed.From the results presented in Table8, it can be verified that the reduction of the number of PSSs make the system more susceptible to low-frequency oscillations, since the minimum damping of the system reduces from 0.1464 p.u. to 0.0261 p.u. from Simulation I, with 8 PSSs in the system, to Simulation VIII, with only one PSS.Based on these results, it can be concluded that the proposed algorithm is robust, since it was capable of obtaining stable solutions even when a small number of PSSs was considered in the system.


### 7.2. Two-Area Symmetrical System

The two-area symmetrical system, characterized by 2 areas, 4 generators, 16 branches, and 11 buses (see Figure15), already includes the new bus FB and a new transmission line, specially included to carry out simulations with the SSSC. Furthermore, this system is characterized by having long transmission lines between buses 7 and 8 with a highly inductive nature. This fact contributes to the appearance of modes with oscillatory characteristics (inter-area mode). Complete data for this system can be found in [1].Figure 15
Single-line diagram of the two-area symmetrical system with one SSSC.To improve the voltage profile of the two-areaQmctrl=−70.00 symmetrical system, the following was considered: (Pmctrl=70 MW and MVAr). In this situation, the CSM is simulated and the results obtained can be seen in Table 9. It can be verified that the system is unstable due to λ1.Table 9
Dominant eigenvalues, damping coefficients, and natural undamped frequencies of the two-area symmetrical system.
ModeEigenvaluesξi (p.u.)ωni (Hz)λ10.0487±4.3299−0.01120.6892λ2−0.2444±5.91880.04130.9428λ3−0.3650±6.29380.05791.0034The results of the simulations for evaluating the performance of the algorithms can be verified in Table10.Table 10
Comparison of the performances of the algorithms for the two-area symmetrical system.
AlgorithmDamping ratioTests with convergence%Number of iterationsTime (s)AverageMin.Max.AverageMin.Max.PSOTable192≈17962000≈13≈0.67≈144GATable199≈109102000≈16≈2≈301MOATable1100≈113167≈3≈1≈41When analyzing the results presented in Table10, it is possible to make the following comments: (i) the MOA achieved success in 100% of the tests performed, against 99% of GA and 92% of PSO algorithm; (ii) both in average or when the maximum and minimum numbers of iterations are compared, it is possible to observe that the MOA was more efficient than the other two optimization algorithms; (iii) finally, when the convergence times of the algorithms were compared, again the MOA had better performance than the others, being the best in the average, minimum, and maximum times observed.The radar plot of the three simulated algorithms is shown in Figure16 for the two-area symmetrical system.Figure 16
Radar plot for the GA, PSO algorithm, and MOA for the two-area symmetrical system.Again, as for the New England system, it is clear from Figure16 that the radar plot for the MOA is the one that presents the best results for the two-area symmetrical system. These results, therefore, show the effectiveness of the proposed approach for the designing of the controllers.


## 7.1. New England System

To validate the models and optimization algorithms presented in this work, small-signal stability simulations will be carried out using the New England system whose complete data is available in [17]. This is a multimachine system composed of 10 generators, 40 buses, and 47 transmission lines, already accounting for one new line (38− FB) and one new bus (FB, between buses 37−38), included to perform simulations with the SSSC, as can be analyzed in Figure 10.Figure 10
Single-line diagram of the New England system with one SSSC.For the simulations, the algorithms were calibrated according to data shown in Table2, considering the set of bounds shown in (50)–(52).Table 2
Parameters of the algorithms.
Search Engine StructureParameters of the AlgorithmParameters of the SimulationsFitness FunctionRoulette selection:—GATwo-point crossoverMutation (23% of the value):Elitism:10% probability rate10% probability rate best individualPSOFormulation(35)–(38)C1=2.55C2=1.55wmax=0.89wmin=0.0025tmax=2000Population size: 20 Stopping criterion: Table1Max. number of iterations:2000(55)MOAFormulation(39)–(48)g=0.851C1=1.99, C2=2.00C3=2.50, β=2.00ρ1=0.10, ρ2=−1.10d0=950, δdt=0.9999fl0=1.50, δflt=0.99The presentation of the results is organized in two cases: first, a static analysis is performed, and second, a dynamic analysis is conducted. In the first case, the objective is to validate the proposed formulation for the power flow modeled by current injections for the SSSC. The second case is subdivided into two parts: in the first part, a statistical analysis is performed considering the results of the three simulated optimization techniques (GA, PSO algorithm, and MOA) in order to determine which of the three algorithms best applies to the discussed problem; second, an analysis focused on small-signal stability using the New England system is performed. The proposed models and optimization algorithms were implemented in MATLAB R2019a using a computer with a 3.19 GHz Intel®Core™i7-8700 processor and 16 GB of RAM.


### 7.1.1. Static Analysis

Initially, it can be seen that the SSSC was installed between buses37−38. The reasons for installing the device at this location are: first, the installation of the SSSC should occur at locations close to the buses that presented voltage magnitudes below 0.95 p.u. According to Figure 11, the buses with voltage magnitudes below 0.95 p.u. are 12, 36, and 37 (0.9491 p.u., 0.9486 p.u., and 0.9475 p.u.) and are close to the site chosen for SSSC installation; second, the location should allow for the controllability and observability over the interarea mode present in the power system. Indeed, according to Figure 10, buses 38 and 37 are characterized by being located near to Area 1 (New York system that is represented by the equivalent bus 10). It is known in the literature that interconnection areas favor the emergence of interarea oscillatory modes.Figure 11
Voltage profiles for the New England system.Case I will represent the situation in which the New England system is equipped with an SSSC, but the device does not control the active and reactive power flows in the line in which it is installed, and Case II will represent the case in which the SSSC controls the active and reactive power flows. For the first case,Pmctrl=−15.96 MW and Qmctrl=−111.27 MVAr and, for Case II, Pmctrl=−25.00 MW and Qmctrl=−125.00 MVAr. In both cases, for simulation purposes, the coupling reactance of the transformer, xkm, connecting the VSC to the transmission line (positioned between buses 38 and FB) is equal to 0.01 p.u.In Figure11, the voltage profiles before (Case I) and after (Case II) the SSSC actuation for controlling the active and reactive power flows are shown.By analyzing Figure11, it can be seen that before the control of the active and reactive power flows performed by the SSSC (Case I), the system had voltage problems at buses 12 (0.9491 p.u.), 36 (0.9486 p.u.), and 37 (0.9475 p.u.). However, the same cannot be said for Case II. After controlling the flows, the voltage profiles were adjusted, being equal to 0.9505 p.u. for bus 12, 0.9513 p.u. for bus 36, and 0.9506 p.u. for bus 37.In Table3, the values of the SSSC control variables, Vp and Vq, are presented.Table 3
Variables of the SSSC’s control structure of the new England system.
SSSCVpVqCase I−7.11283×10−072.78607×10−08Case II−0.0155367−0.0150003When analyzing Table3, it is possible to verify that the values of Vp and Vq are close to zero for Case I, demonstrating that the SSSC is included in the system, but it is not actuating. In relation to Case II, it can be seen that after the SSSC has actuated to control the active and reactive power flows, the values of the control variables are different from zero, which corroborates the SSSC’s actuation in the New England system.Finally, in Figure12, the active and reactive power flows, generations, loads, and the power injections performed by the SSSC, for the buses and lines near to the installation site of the SSSC, are presented. The results for Case I are represented in Figure 12(a) while the results for Case II are shown in Figure 12(b).Figure 12
Control of the active and reactive power flows performed by the SSSC in the New England system.
(a)(b)By analyzing Figure12, it is possible to verify that, in both cases, the control of the power flows is performed according to what was established for Cases I and II. It is important to highlight that, in Case II, in order to control the flows specified in the project (Pmctrl=−25.00 MW and Qmctrl=−125.00 MVAr) for improving the voltage magnitudes at buses 12, 36, and 37, the SSSC needed to inject at buses 37 and FB, respectively, 147.70 MW and 142.60 MVAr, and −149.38 MW and −140.30 MVAr. Finally, it is possible to perform the nodal balance of active and reactive powers at the buses indicated in Figure 12 and thus validate the control of the power flows performed by the SSSC.


### 7.1.2. Dynamic Analysis

After validating the current injections model for the SSSC and confirming its ability to manage the active and reactive power flows, as shown in Subsection7.1.1, a dynamic analysis involving the SSSC, PI controllers, and supplementary damping controllers (PSS and POD), will be performed. In Table 4, the eigenvalues of interest of the New England system are presented before (Case I) and after (Case II) the actuation of the SSSC in the control of the active and reactive power flows.Table 4
Dominant eigenvalues, damping coefficients, and natural undamped frequencies of the New England system.
ModeCase ICase IIEigenvaluesξi (p.u.)ωni (Hz)Eigenvaluesξi (p.u.)ωni (Hz)λ10.0729±j6.8461−0.01061.08970.0767±j6.8448−0.01121.0894λ2−0.2064±j7.23480.02851.1519−0.2052±j7.23350.02841.1517λ3−0.1903±j8.27310.02301.3171−0.1880±j8.27250.02271.3169λ40.1799±j5.9142−0.03040.94170.1745±j5.9090−0.02950.9409λ5−0.1291±j6.49680.01991.0342−0.1245±j6.48290.01921.0320λ6−0.2690±j8.10710.03321.2910−0.2687±j8.10590.03311.2908λ7−0.2413±j8.32040.02901.3248−0.2412±j8.31870.02901.3245λ80.1622±j6.3862−0.02541.01670.1587±j6.3764−0.02491.0152λ90.0037±j3.5625−0.00100.5670−0.0812±j3.46380.02340.5514By analyzing the frequencies,ωni=λi/2π, of the modes of interest, λi=σi±jωi, shown in Table 4, it can be concluded that the New England system is characterized by the presence of eight local modes, λ1 and λ8, and one interarea mode, λ9. In both Case I and Case II, the system is unstable. In Case I, the instability of the system is caused by four oscillatory modes with positive real parts, λ1, λ4, λ8, and λ9, while in Case II, the instability of the system is caused by three oscillatory modes with positive real parts, λ1, λ4, and λ8. Importantly, the interarea mode in Case I has negative damping, of −0.0010 p.u., and, in Case II, after the control of the active and reactive power flows by the SSSC, this mode changes from unstable to stable with damping of 0.0234 p.u.The inclusion of the SSSC dynamics in the CSM and the correct control of the active and reactive power flows performed by this device in the New England system provided improvements in the voltage profile of the system (see Figure11) as well as small damping increments in some poles of interest of the system. In fact, it is possible to verify that in Case II, for the considered operation point, the New England system started to operate with three poles responsible for the system instability, one less when compared to Case I (see Table 4). Due to these observations, Case II, from this point on, will be used as the base case in the following simulations.(1) Statistical Analyses of the Performances of the Algorithms. Initially, an SSSC-POD installed in the system according with the arguments presented in Subsection 7.1.1 and nine PSSs coupled to generators G1 to G9 are considered.The parameter tuning of the PI and supplementary damping controllers, PSSs and POD, for Case II, is performed using the PSO algorithm, GA, and MOA, and these are calibrated based on the data provided in Table2. In this first scenario, one hundred tests limited to two thousand iterations in each one are performed. Under these conditions, the values given in Table 1 should be checked. The results of these simulations can be verified in Table 5.Table 5
Comparison of the performances of the algorithms for the New England system.
AlgorithmDamping ratioTests with convergence%Number of iterationsTime (s)AverageMin.Max.AverageMin.Max.PSOTable187≈5981172000≈41≈8≈145GATable191≈7651632000≈107≈23≈284MOATable1100≈5816549≈14≈4≈127When analyzing the results presented in Table5, it is possible to make the following comments: (i) the MOA achieved success in 100% of the tests performed, against 91% of GA and 87% of PSO algorithm; (ii) both in average or when the maximum and minimum numbers of iterations are compared, it is possible to observe that the MOA was more efficient than the other two optimization algorithms; (iii) finally, when the convergence times of the algorithms were compared, again the MOA had better performance than the others, being the best in the average, minimum, and maximum times observed.The results analyzed so far would already allow saying that the MOA is the most efficient algorithm for tuning the parameters of the controllers for small-signal stability improvement, and therefore, would allow accrediting it as a powerful tool in this type of study. However, to corroborate the results presented in Table5, the radar plot of the three simulated algorithms is shown in Figure 13. In it, the observed results for four quantitative variables obtained in the one hundred simulated tests are presented: time, numbers of iterations, minimum damping, and the smallest values found for the objective function for each simulated test.Figure 13
Radar plot for the GA, PSO algorithm, and MOA for the New England system.It is clear from Figure13 that the radar plot for the MOA is the one that presents the best results. It is possible to observe that the respective curves presented for the MOA radar plot are concentrated closer to the center, corroborating the superiority of the MOA in all analyzed aspects compared to the other algorithms.(2) Small-Signal Stability Studies. In order to perform studies focused on the small-signal stability, simulations were performed with the three algorithms proposed in this paper (MOA, PSO algorithm, and GA). Again, the New England system equipped with nine PSSs installed at generators G1 to G9 and an SSSC-POD between buses 37 and 38 is considered. All algorithms are simulated and calibrated with the settings shown in Table 2. It is worth noting that, in this case, the same randomly generated initial population was used for the three proposed algorithms in each trial. In this case, 50000 iterations were considered, without specifying the desired minimum damping. Table 6 shows the results obtained for the parameters of the PSSs, PI, and POD controllers.Table 6
Gains and Time Constants for the PSS, PI, and POD Controllers Tuned by the MOA, PSO algorithm, and GA for the New England System.
DeviceMOAPSOGAT1pss (s)T2pss (s)Kpss (p.u.)T1pss (s)T2pss (s)Kpss (p.u.)T1pss (s)T2pss (s)Kpss (p.u.)PSS G11.25000.067910.0001.25000.073810.0001.24810.07089.7607PSS G20.93030.09387.16391.25000.13165.95940.99710.14548.1228PSS G31.02960.09166.37800.82610.085810.0001.14790.08705.6172PSS G40.68870.06859.49271.25000.07465.81201.05100.05327.0592PSS G50.42250.10477.74500.52630.11442.00000.71550.20009.6881PSS G61.18760.11818.96211.25000.13977.91741.10670.07958.2647PSS G70.47580.10584.54300.05000.01009.64110.28440.07133.9547PSS 80.86470.10376.72720.85960.09707.25400.85240.08349.2116PSS G90.56180.20004.53540.59490.20006.05370.43520.20007.3625PODT1podT2podkpodT1podT2podkpodT1podT2podkpod0.23950.27650.36220.32470.40000.50000.28450.40000.3979PIT1pi=T2piK1pi=K2piT1pi=T2piK1pi=K2piT1pi=T2piK1pi=K2pi0.61190.07300.76040.10780.31850.0781With the parameters presented in Table6, the PI and supplementary damping controllers (PSSs and POD) are calibrated and the CSM is evaluated. In this way, new damping levels ξi=−σi/λi associated with the eigenvalues of interest of the system can be obtained, as well as their respective undamped natural frequencies, as presented in Table 7.Table 7
Eigenvalues of interest, damping coefficients, and natural undamped frequencies determined by the MOA, PSO, and GA for the New England system.
ModeMOAPSOGAEigenvaluesξi (p.u.)ωni (Hz)Eigenvaluesξi (p.u.)ωni (Hz)Eigenvaluesξi (p.u.)ωni (Hz)λ1−2.6438±j6.61690.37101.1341−1.9419±j5.79000.31800.9720−2.1065±j7.42440.27301.2283λ2−2.2244±j5.55630.37170.9525−1.6865±j5.02840.31800.8441−1.8755±j6.48840.27771.0749λ3−3.1582±j7.87260.37231.3500−2.7814±j8.27690.31851.3897−2.3732±j8.09210.28141.3421λ4−2.5701±j6.39060.37311.0963−2.0599±j6.11440.31931.0269−1.5898±j5.08680.29830.8482λ5−2.3988±j5.92380.37531.0172−2.1980±j6.43610.32321.0824−1.7815±j5.62380.30200.9389λ6−1.9515±j4.79990.37660.8247−2.8851±j8.29770.32841.3982−1.8132±j5.20910.32870.8778λ7−2.4977±j5.38620.42070.9449−2.4205±j5.12570.42700.9022−2.5926±j7.27890.33551.2298λ8−2.6836±j4.55230.50780.8410−2.5751±j4.47370.49890.8215−3.0614±j4.61780.55260.8818λ9−1.5229±j3.74590.37660.6436−1.3002±j3.87650.31800.6507−1.1888±j3.37150.33250.5690By observing Table7 it is possible to verify that, for the operating point considered, from the adjustments obtained in Table 6, the frequencies of all poles of interest of the New England system are within the maximum and minimum limits stipulated in design (see Table 1). Regarding the values found for the dampings, it can be stated that the system operates with high damping margins, with 0.3710 p.u., 0.3180 p.u., and 0.2730 p.u. being the minimum damping levels, achieved from the MOA, PSO algorithm, and GA, respectively. Finally, on average, the MOA also has shown better results, as it achieved average damping of 0.3939 p.u., compared to 0.3521 p.u. for the PSO algorithm and 0.3313 p.u. for the GA.Power systems are often subject to small variations in loads and, therefore, consequent adjustments in generations. Figure14 shows the variations of the angular speeds of all generating units (except G2, the reference of the system) of the New England system for the simulated case presented in Tables 6 and 7.Figure 14
Variation of the angular speeds of the generators G1 and G3–G10 of the New England system.The disturbance in the mechanical power at G2 (+0.05 p.u.) is similar to a small adjustment in the generation, which can be caused by a small increase in the system loading. It can be seen in Figure 14 that even after the disturbance, the system is stabilized in approximately two seconds in all cases, graphically evidencing its high stability margin for small-signal disturbances.(3) Sensitivity Analysis on the Number of PSSs in the System. This section investigates the influence of the number of PSSs in the system for damping the local oscillation modes. In these tests, the number of PSSs installed in the New England system will be systematically reduced, and the parameters of the controllers will be optimized to maximize the minimum damping in the system. In all simulated cases, the FACTS-POD set is installed in the defined location of Subsection 7.1.1 and the parameters that were used to tune the algorithm are available in Table 2. Table 8 shows the mode with minimum damping obtained in each simulation.Table 8
Modes with minimum damping for each simulation obtained when removing PSSs from the New England system.
SimulationMode with minimum dampingminξi (p.u.)ωni (Hz)I−1.2807±j8.65390.14641.3923II−0.8888±j5.95550.14760.9583III−0.5260±j6.07140.08630.9699IV−0.4654±j7.01810.06621.1194V−0.2340±j7.20370.03251.1471VI−0.2180±j7.22600.03021.1506VII−0.2135±j7.23120.02951.1514VIII−0.2162±j8.28600.02611.3192In the simulations shown in Table8, the PSSs of the following generators were removed for each simulation:(i)
Simulation I: G7(ii)
Simulation II: G7, and G5(iii)
Simulation III: G7, G5, and G2(iv)
Simulation IV: G7, G5, G2, and G1(v)
Simulation V: G7, G5, G2, G1, and G3(vi)
Simulation VI: G7, G5, G2, G1, G3, and G8(vii)
Simulation VII: G7, G5, G2, G1, G3, G8, and G4(viii)
Simulation VIII: G7, G5, G2, G1, G3, G8, G4, and G6The order for removing the PSSs from the system was obtained by systematically removing one PSS at a time and verifying the removal that led to the smallest impact on the lowest damping after re-optimizing the parameters of the controllers. For example, it was verified that the PSS of G7 presented the lowest impact on the minimum damping of the system, therefore, it was the first one that was removed.From the results presented in Table8, it can be verified that the reduction of the number of PSSs make the system more susceptible to low-frequency oscillations, since the minimum damping of the system reduces from 0.1464 p.u. to 0.0261 p.u. from Simulation I, with 8 PSSs in the system, to Simulation VIII, with only one PSS.Based on these results, it can be concluded that the proposed algorithm is robust, since it was capable of obtaining stable solutions even when a small number of PSSs was considered in the system.


## 7.1.1. Static Analysis

Initially, it can be seen that the SSSC was installed between buses37−38. The reasons for installing the device at this location are: first, the installation of the SSSC should occur at locations close to the buses that presented voltage magnitudes below 0.95 p.u. According to Figure 11, the buses with voltage magnitudes below 0.95 p.u. are 12, 36, and 37 (0.9491 p.u., 0.9486 p.u., and 0.9475 p.u.) and are close to the site chosen for SSSC installation; second, the location should allow for the controllability and observability over the interarea mode present in the power system. Indeed, according to Figure 10, buses 38 and 37 are characterized by being located near to Area 1 (New York system that is represented by the equivalent bus 10). It is known in the literature that interconnection areas favor the emergence of interarea oscillatory modes.Figure 11
Voltage profiles for the New England system.Case I will represent the situation in which the New England system is equipped with an SSSC, but the device does not control the active and reactive power flows in the line in which it is installed, and Case II will represent the case in which the SSSC controls the active and reactive power flows. For the first case,Pmctrl=−15.96 MW and Qmctrl=−111.27 MVAr and, for Case II, Pmctrl=−25.00 MW and Qmctrl=−125.00 MVAr. In both cases, for simulation purposes, the coupling reactance of the transformer, xkm, connecting the VSC to the transmission line (positioned between buses 38 and FB) is equal to 0.01 p.u.In Figure11, the voltage profiles before (Case I) and after (Case II) the SSSC actuation for controlling the active and reactive power flows are shown.By analyzing Figure11, it can be seen that before the control of the active and reactive power flows performed by the SSSC (Case I), the system had voltage problems at buses 12 (0.9491 p.u.), 36 (0.9486 p.u.), and 37 (0.9475 p.u.). However, the same cannot be said for Case II. After controlling the flows, the voltage profiles were adjusted, being equal to 0.9505 p.u. for bus 12, 0.9513 p.u. for bus 36, and 0.9506 p.u. for bus 37.In Table3, the values of the SSSC control variables, Vp and Vq, are presented.Table 3
Variables of the SSSC’s control structure of the new England system.
SSSCVpVqCase I−7.11283×10−072.78607×10−08Case II−0.0155367−0.0150003When analyzing Table3, it is possible to verify that the values of Vp and Vq are close to zero for Case I, demonstrating that the SSSC is included in the system, but it is not actuating. In relation to Case II, it can be seen that after the SSSC has actuated to control the active and reactive power flows, the values of the control variables are different from zero, which corroborates the SSSC’s actuation in the New England system.Finally, in Figure12, the active and reactive power flows, generations, loads, and the power injections performed by the SSSC, for the buses and lines near to the installation site of the SSSC, are presented. The results for Case I are represented in Figure 12(a) while the results for Case II are shown in Figure 12(b).Figure 12
Control of the active and reactive power flows performed by the SSSC in the New England system.
(a)(b)By analyzing Figure12, it is possible to verify that, in both cases, the control of the power flows is performed according to what was established for Cases I and II. It is important to highlight that, in Case II, in order to control the flows specified in the project (Pmctrl=−25.00 MW and Qmctrl=−125.00 MVAr) for improving the voltage magnitudes at buses 12, 36, and 37, the SSSC needed to inject at buses 37 and FB, respectively, 147.70 MW and 142.60 MVAr, and −149.38 MW and −140.30 MVAr. Finally, it is possible to perform the nodal balance of active and reactive powers at the buses indicated in Figure 12 and thus validate the control of the power flows performed by the SSSC.


## 7.1.2. Dynamic Analysis

After validating the current injections model for the SSSC and confirming its ability to manage the active and reactive power flows, as shown in Subsection7.1.1, a dynamic analysis involving the SSSC, PI controllers, and supplementary damping controllers (PSS and POD), will be performed. In Table 4, the eigenvalues of interest of the New England system are presented before (Case I) and after (Case II) the actuation of the SSSC in the control of the active and reactive power flows.Table 4
Dominant eigenvalues, damping coefficients, and natural undamped frequencies of the New England system.
ModeCase ICase IIEigenvaluesξi (p.u.)ωni (Hz)Eigenvaluesξi (p.u.)ωni (Hz)λ10.0729±j6.8461−0.01061.08970.0767±j6.8448−0.01121.0894λ2−0.2064±j7.23480.02851.1519−0.2052±j7.23350.02841.1517λ3−0.1903±j8.27310.02301.3171−0.1880±j8.27250.02271.3169λ40.1799±j5.9142−0.03040.94170.1745±j5.9090−0.02950.9409λ5−0.1291±j6.49680.01991.0342−0.1245±j6.48290.01921.0320λ6−0.2690±j8.10710.03321.2910−0.2687±j8.10590.03311.2908λ7−0.2413±j8.32040.02901.3248−0.2412±j8.31870.02901.3245λ80.1622±j6.3862−0.02541.01670.1587±j6.3764−0.02491.0152λ90.0037±j3.5625−0.00100.5670−0.0812±j3.46380.02340.5514By analyzing the frequencies,ωni=λi/2π, of the modes of interest, λi=σi±jωi, shown in Table 4, it can be concluded that the New England system is characterized by the presence of eight local modes, λ1 and λ8, and one interarea mode, λ9. In both Case I and Case II, the system is unstable. In Case I, the instability of the system is caused by four oscillatory modes with positive real parts, λ1, λ4, λ8, and λ9, while in Case II, the instability of the system is caused by three oscillatory modes with positive real parts, λ1, λ4, and λ8. Importantly, the interarea mode in Case I has negative damping, of −0.0010 p.u., and, in Case II, after the control of the active and reactive power flows by the SSSC, this mode changes from unstable to stable with damping of 0.0234 p.u.The inclusion of the SSSC dynamics in the CSM and the correct control of the active and reactive power flows performed by this device in the New England system provided improvements in the voltage profile of the system (see Figure11) as well as small damping increments in some poles of interest of the system. In fact, it is possible to verify that in Case II, for the considered operation point, the New England system started to operate with three poles responsible for the system instability, one less when compared to Case I (see Table 4). Due to these observations, Case II, from this point on, will be used as the base case in the following simulations.(1) Statistical Analyses of the Performances of the Algorithms. Initially, an SSSC-POD installed in the system according with the arguments presented in Subsection 7.1.1 and nine PSSs coupled to generators G1 to G9 are considered.The parameter tuning of the PI and supplementary damping controllers, PSSs and POD, for Case II, is performed using the PSO algorithm, GA, and MOA, and these are calibrated based on the data provided in Table2. In this first scenario, one hundred tests limited to two thousand iterations in each one are performed. Under these conditions, the values given in Table 1 should be checked. The results of these simulations can be verified in Table 5.Table 5
Comparison of the performances of the algorithms for the New England system.
AlgorithmDamping ratioTests with convergence%Number of iterationsTime (s)AverageMin.Max.AverageMin.Max.PSOTable187≈5981172000≈41≈8≈145GATable191≈7651632000≈107≈23≈284MOATable1100≈5816549≈14≈4≈127When analyzing the results presented in Table5, it is possible to make the following comments: (i) the MOA achieved success in 100% of the tests performed, against 91% of GA and 87% of PSO algorithm; (ii) both in average or when the maximum and minimum numbers of iterations are compared, it is possible to observe that the MOA was more efficient than the other two optimization algorithms; (iii) finally, when the convergence times of the algorithms were compared, again the MOA had better performance than the others, being the best in the average, minimum, and maximum times observed.The results analyzed so far would already allow saying that the MOA is the most efficient algorithm for tuning the parameters of the controllers for small-signal stability improvement, and therefore, would allow accrediting it as a powerful tool in this type of study. However, to corroborate the results presented in Table5, the radar plot of the three simulated algorithms is shown in Figure 13. In it, the observed results for four quantitative variables obtained in the one hundred simulated tests are presented: time, numbers of iterations, minimum damping, and the smallest values found for the objective function for each simulated test.Figure 13
Radar plot for the GA, PSO algorithm, and MOA for the New England system.It is clear from Figure13 that the radar plot for the MOA is the one that presents the best results. It is possible to observe that the respective curves presented for the MOA radar plot are concentrated closer to the center, corroborating the superiority of the MOA in all analyzed aspects compared to the other algorithms.(2) Small-Signal Stability Studies. In order to perform studies focused on the small-signal stability, simulations were performed with the three algorithms proposed in this paper (MOA, PSO algorithm, and GA). Again, the New England system equipped with nine PSSs installed at generators G1 to G9 and an SSSC-POD between buses 37 and 38 is considered. All algorithms are simulated and calibrated with the settings shown in Table 2. It is worth noting that, in this case, the same randomly generated initial population was used for the three proposed algorithms in each trial. In this case, 50000 iterations were considered, without specifying the desired minimum damping. Table 6 shows the results obtained for the parameters of the PSSs, PI, and POD controllers.Table 6
Gains and Time Constants for the PSS, PI, and POD Controllers Tuned by the MOA, PSO algorithm, and GA for the New England System.
DeviceMOAPSOGAT1pss (s)T2pss (s)Kpss (p.u.)T1pss (s)T2pss (s)Kpss (p.u.)T1pss (s)T2pss (s)Kpss (p.u.)PSS G11.25000.067910.0001.25000.073810.0001.24810.07089.7607PSS G20.93030.09387.16391.25000.13165.95940.99710.14548.1228PSS G31.02960.09166.37800.82610.085810.0001.14790.08705.6172PSS G40.68870.06859.49271.25000.07465.81201.05100.05327.0592PSS G50.42250.10477.74500.52630.11442.00000.71550.20009.6881PSS G61.18760.11818.96211.25000.13977.91741.10670.07958.2647PSS G70.47580.10584.54300.05000.01009.64110.28440.07133.9547PSS 80.86470.10376.72720.85960.09707.25400.85240.08349.2116PSS G90.56180.20004.53540.59490.20006.05370.43520.20007.3625PODT1podT2podkpodT1podT2podkpodT1podT2podkpod0.23950.27650.36220.32470.40000.50000.28450.40000.3979PIT1pi=T2piK1pi=K2piT1pi=T2piK1pi=K2piT1pi=T2piK1pi=K2pi0.61190.07300.76040.10780.31850.0781With the parameters presented in Table6, the PI and supplementary damping controllers (PSSs and POD) are calibrated and the CSM is evaluated. In this way, new damping levels ξi=−σi/λi associated with the eigenvalues of interest of the system can be obtained, as well as their respective undamped natural frequencies, as presented in Table 7.Table 7
Eigenvalues of interest, damping coefficients, and natural undamped frequencies determined by the MOA, PSO, and GA for the New England system.
ModeMOAPSOGAEigenvaluesξi (p.u.)ωni (Hz)Eigenvaluesξi (p.u.)ωni (Hz)Eigenvaluesξi (p.u.)ωni (Hz)λ1−2.6438±j6.61690.37101.1341−1.9419±j5.79000.31800.9720−2.1065±j7.42440.27301.2283λ2−2.2244±j5.55630.37170.9525−1.6865±j5.02840.31800.8441−1.8755±j6.48840.27771.0749λ3−3.1582±j7.87260.37231.3500−2.7814±j8.27690.31851.3897−2.3732±j8.09210.28141.3421λ4−2.5701±j6.39060.37311.0963−2.0599±j6.11440.31931.0269−1.5898±j5.08680.29830.8482λ5−2.3988±j5.92380.37531.0172−2.1980±j6.43610.32321.0824−1.7815±j5.62380.30200.9389λ6−1.9515±j4.79990.37660.8247−2.8851±j8.29770.32841.3982−1.8132±j5.20910.32870.8778λ7−2.4977±j5.38620.42070.9449−2.4205±j5.12570.42700.9022−2.5926±j7.27890.33551.2298λ8−2.6836±j4.55230.50780.8410−2.5751±j4.47370.49890.8215−3.0614±j4.61780.55260.8818λ9−1.5229±j3.74590.37660.6436−1.3002±j3.87650.31800.6507−1.1888±j3.37150.33250.5690By observing Table7 it is possible to verify that, for the operating point considered, from the adjustments obtained in Table 6, the frequencies of all poles of interest of the New England system are within the maximum and minimum limits stipulated in design (see Table 1). Regarding the values found for the dampings, it can be stated that the system operates with high damping margins, with 0.3710 p.u., 0.3180 p.u., and 0.2730 p.u. being the minimum damping levels, achieved from the MOA, PSO algorithm, and GA, respectively. Finally, on average, the MOA also has shown better results, as it achieved average damping of 0.3939 p.u., compared to 0.3521 p.u. for the PSO algorithm and 0.3313 p.u. for the GA.Power systems are often subject to small variations in loads and, therefore, consequent adjustments in generations. Figure14 shows the variations of the angular speeds of all generating units (except G2, the reference of the system) of the New England system for the simulated case presented in Tables 6 and 7.Figure 14
Variation of the angular speeds of the generators G1 and G3–G10 of the New England system.The disturbance in the mechanical power at G2 (+0.05 p.u.) is similar to a small adjustment in the generation, which can be caused by a small increase in the system loading. It can be seen in Figure 14 that even after the disturbance, the system is stabilized in approximately two seconds in all cases, graphically evidencing its high stability margin for small-signal disturbances.(3) Sensitivity Analysis on the Number of PSSs in the System. This section investigates the influence of the number of PSSs in the system for damping the local oscillation modes. In these tests, the number of PSSs installed in the New England system will be systematically reduced, and the parameters of the controllers will be optimized to maximize the minimum damping in the system. In all simulated cases, the FACTS-POD set is installed in the defined location of Subsection 7.1.1 and the parameters that were used to tune the algorithm are available in Table 2. Table 8 shows the mode with minimum damping obtained in each simulation.Table 8
Modes with minimum damping for each simulation obtained when removing PSSs from the New England system.
SimulationMode with minimum dampingminξi (p.u.)ωni (Hz)I−1.2807±j8.65390.14641.3923II−0.8888±j5.95550.14760.9583III−0.5260±j6.07140.08630.9699IV−0.4654±j7.01810.06621.1194V−0.2340±j7.20370.03251.1471VI−0.2180±j7.22600.03021.1506VII−0.2135±j7.23120.02951.1514VIII−0.2162±j8.28600.02611.3192In the simulations shown in Table8, the PSSs of the following generators were removed for each simulation:(i)
Simulation I: G7(ii)
Simulation II: G7, and G5(iii)
Simulation III: G7, G5, and G2(iv)
Simulation IV: G7, G5, G2, and G1(v)
Simulation V: G7, G5, G2, G1, and G3(vi)
Simulation VI: G7, G5, G2, G1, G3, and G8(vii)
Simulation VII: G7, G5, G2, G1, G3, G8, and G4(viii)
Simulation VIII: G7, G5, G2, G1, G3, G8, G4, and G6The order for removing the PSSs from the system was obtained by systematically removing one PSS at a time and verifying the removal that led to the smallest impact on the lowest damping after re-optimizing the parameters of the controllers. For example, it was verified that the PSS of G7 presented the lowest impact on the minimum damping of the system, therefore, it was the first one that was removed.From the results presented in Table8, it can be verified that the reduction of the number of PSSs make the system more susceptible to low-frequency oscillations, since the minimum damping of the system reduces from 0.1464 p.u. to 0.0261 p.u. from Simulation I, with 8 PSSs in the system, to Simulation VIII, with only one PSS.Based on these results, it can be concluded that the proposed algorithm is robust, since it was capable of obtaining stable solutions even when a small number of PSSs was considered in the system.


## 7.2. Two-Area Symmetrical System

The two-area symmetrical system, characterized by 2 areas, 4 generators, 16 branches, and 11 buses (see Figure15), already includes the new bus FB and a new transmission line, specially included to carry out simulations with the SSSC. Furthermore, this system is characterized by having long transmission lines between buses 7 and 8 with a highly inductive nature. This fact contributes to the appearance of modes with oscillatory characteristics (inter-area mode). Complete data for this system can be found in [1].Figure 15
Single-line diagram of the two-area symmetrical system with one SSSC.To improve the voltage profile of the two-areaQmctrl=−70.00 symmetrical system, the following was considered: (Pmctrl=70 MW and MVAr). In this situation, the CSM is simulated and the results obtained can be seen in Table 9. It can be verified that the system is unstable due to λ1.Table 9
Dominant eigenvalues, damping coefficients, and natural undamped frequencies of the two-area symmetrical system.
ModeEigenvaluesξi (p.u.)ωni (Hz)λ10.0487±4.3299−0.01120.6892λ2−0.2444±5.91880.04130.9428λ3−0.3650±6.29380.05791.0034The results of the simulations for evaluating the performance of the algorithms can be verified in Table10.Table 10
Comparison of the performances of the algorithms for the two-area symmetrical system.
AlgorithmDamping ratioTests with convergence%Number of iterationsTime (s)AverageMin.Max.AverageMin.Max.PSOTable192≈17962000≈13≈0.67≈144GATable199≈109102000≈16≈2≈301MOATable1100≈113167≈3≈1≈41When analyzing the results presented in Table10, it is possible to make the following comments: (i) the MOA achieved success in 100% of the tests performed, against 99% of GA and 92% of PSO algorithm; (ii) both in average or when the maximum and minimum numbers of iterations are compared, it is possible to observe that the MOA was more efficient than the other two optimization algorithms; (iii) finally, when the convergence times of the algorithms were compared, again the MOA had better performance than the others, being the best in the average, minimum, and maximum times observed.The radar plot of the three simulated algorithms is shown in Figure16 for the two-area symmetrical system.Figure 16
Radar plot for the GA, PSO algorithm, and MOA for the two-area symmetrical system.Again, as for the New England system, it is clear from Figure16 that the radar plot for the MOA is the one that presents the best results for the two-area symmetrical system. These results, therefore, show the effectiveness of the proposed approach for the designing of the controllers.


## 8. Conclusions

This work presented comparisons involving three algorithms, i.e., genetic algorithm (GA), particle swarm optimization (PSO) algorithm, and the mayfly optimization algorithm (MOA), to perform the coordinated tuning of the parameters of the proportional and integral (PI) and supplementary damping controllers, power system stabilizers (PSSs) and power oscillation damping (POD). The current sensitivity model was used to represent the electric power system, the reason why all devices present in the system were modeled by current injections, including the static synchronous series compensator (SSSC) flexible AC transmission systems. The performances of the proposed algorithms were evaluated using the New England system and the two-area symmetrical system.To validate the proposed current injections model for the SSSC, a static analysis was performed. Finally, simulations were performed in order to compare the three proposed algorithms to analyze which one has the best performance for damping the oscillatory modes of the system in the small-signal stability problem.Based on the results, it was possible to validate the current injections model proposed for the SSSC. Finally, the MOA had its performance compared with the GA and PSO algorithm for two different scenarios, being unbeatable in the two simulated cases. On this occasion, the coordinated adjustments of PI and supplementary damping controllers (PSSs and POD) parameters were performed, and therefore it is possible to accredit it as a powerful tool in the small-signal stability in power systems.The authors would like to thank the Goiás Federal Institute of Education, Science, and Technology (IFG), the Paraná Federal Institute of Education, Science and Technology (IFPR), the Federal Rural University of the Semi-Arid Region (UFERSA).



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