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Hindawi International Transactions on Electrical Energy Systems Volume 2022, Article ID 5612334, 23 pages https://doi.org/10.1155/2022/5612334 Research Article Mayfly Optimization Algorithm Applied to the Design of PSS and SSSC-POD Controllers for Damping Low-Frequency Oscillations in Power Systems 1 2 3 Elenilson V. Fortes , Luı´s Fabiano Barone Martins , Marcus V. S. Costa , 3 4 4 Luis Carvalho , Leonardo H. Macedo , and Rube´n Romero Goia´s Federal Institute of Education, Science, and Technology, Rua Maria Vieira Cunha, 775, Residencial Flamboyant, 75804-714 Jataı, GO, Brazil Parana Federal Institute of Education, Science, and Technology, Av. Doutor Tito, s/n, Jardim Panorama, 86400-000 Jacarezinho, PR, Brazil Federal Rural University of the Semi-Arid Region, Rua Francisco Mota, 572, Pres. Costa e Silva, 59625-900 Mossoro´, RN, Brazil São Paulo State University, Avenida Brasil, 56, Centro, 15385-000 Ilha Solteira, SP, Brazil Correspondence should be addressed to Elenilson V. Fortes; elenilson.fortes@ifg.edu.br Received 28 October 2021; Revised 24 December 2021; Accepted 14 February 2022; Published 26 April 2022 Academic Editor: Mouloud Azzedine Denaı Copyright © 2022 . +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. +e performance of the MOA is compared with the performances of the genetic algorithm (GA) and particle swarm optimization (PSO) algorithm for solving this problem. +e 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. +e 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. should evolve to a new acceptable equilibrium point after 1. Introduction being submitted to small- or large-magnitude disturbances. +e increase in the demand for electricity, the industrial Typically, low-frequency oscillations occur in the range expansion, and the interconnection of large systems with of 0.2 − 2.0 Hz and are associated with electromechanical high numbers of generating units contributed to potentiate torque imbalances in synchronous generators, resulting in the appearance of problems related to the operation of power transfer with oscillatory characteristics in transmis- electric power systems, including stability issues. sion lines that interconnect two systems. Oscillations in the According to [1], the stability concept is related to the frequency range of 0.8 − 2.0 Hz are classiﬁed as local, while capacity that power systems have to remain in equilibrium in those occurring in the range of 0.2 − 0.7 Hz are classiﬁed as normal operating conditions. Besides, the system operation interarea [1]. 2 International Transactions on Electrical Energy Systems method [18]. +e control structure used to model the SSSC’s One of the problems addressed in this work is to develop techniques capable of designing supplementary damping dynamics is based on proportional-integral (PI) controllers, as proposed by [17, 19]. +e proposed model can be used controllers for damping these types of oscillations present in power systems. [2] already used, in 1969, the Heﬀron and both for the control of active and reactive power ﬂows by the Phillips’s model [3] to represent a single machine inﬁnite bus SSSC and for obtaining the dynamic equations of the device system and explored the important concepts of synchro- that are included in the model used to represent the power nizing torque and damping, through which they laid the system dynamics, i.e., the current sensitivity model (CSM) foundation for understanding the phenomenon of angular [7, 12, 17]. Furthermore, the proposed model for the SSSC, stability under small-signal disturbances. In the work pre- when compared to other representations available in the literature [20, 21], has the signiﬁcant advantage of being sented by [2], it was observed that automatic voltage reg- ulators (AVRs) with high gains negatively aﬀected the independent of the phase angle of the voltage source of the SSSC (see Section 2) eliminating, in this way, the necessity stability of the system. Since then, power system stabilizers (PSSs) controllers have been widely used to insert supple- for its previous calculations. More details about the SSSC modeling will be presented in Section 2. mentary stabilizing signals into the generators’ excitation control systems in order to provide additional damping Regarding the CSM, it is a linear model whose funda- torques to rotors’ oscillations [4–6]. It should be noted that, mental principle is based on the nodal balance of currents. in general, PSSs have good performance for damping local +is balance must be satisﬁed in all dynamic processes of the oscillatory modes [7, 8], while they cannot adequately damp power system and, therefore, can be used in small-signal interarea modes. On the other hand, depending on the stability studies [7, 12, 17]. +us, all devices present in the settings provided to the control parameters of the PSSs (time system should be modeled by current injections, which justiﬁes the new modeling proposed for the SSSC (see constants and gains) they can turn previously stable modes into unstable modes, in particular, the interarea ones [9]. Subsection 2.1). +e AVR-PSS and SSSC-POD sets are only able to ef- +e recent advances in power electronics propitiated the emergence of ﬂexible AC transmission systems (FACTS) fectively damp the poles of interest of the system if the parameters of the corresponding controllers (PSS and POD) that have been widely used in power systems with the most diverse purposes. In general, FACTS can increase power are adequately adjusted. In the literature, diﬀerent optimi- transfer and consequently improve the stability of a system zation techniques have been used for this purpose. In 2010, [10, 11]. Recent studies show that when a power oscillation [22] developed a design procedure using the particle swarm damping (POD) controller is coupled to the control loop of optimization (PSO) algorithm for the simultaneous and these devices, the FACTS-POD set can act inserting addi- coordinated design of the thyristor-controlled series ca- tional damping to the interarea modes as can be analyzed in pacitor (TCSC) damping controller and PSSs in multi- machine power systems. Tests were conducted using the the works of [7, 12, 13]. In these works, the POD was coupled to the uniﬁed power ﬂow controller (UPFC), generalized two-area symmetrical system, with four generators. In 2014, [23] presented a coordinated approach for the allocation and uniﬁed power ﬂow controller (GUPFC), and the interline power ﬂow controller (IPFC) with the goal of inserting design of PSSs and UPFC using genetic algorithms (GA). additional damping to the interarea oscillatory modes +e GA algorithm determines the optimal location for the present in power systems. UPFC installation while simultaneously tuning its control In this work, the static synchronous series compensator parameters, resulting in the optimization of the number, (SSSC) FACTS is modeled and employed to perform studies locations, and parameters of the PSSs. +e problem is for- that address both static and dynamic analyses of the power mulated with the objective of maximizing the damping system. In general terms, the SSSC performs a series ratios of the electromechanical modes. In 2016, [24] pre- compensation of the electric power transmission system. Its sented a hybrid method is to damp oscillations in the power systems equipped with UPFC and PSS controllers. +e operation consists of inserting (by means of a coupling transformer) a synchronous quasi-sinusoidal voltage source hybrid method consists of an oﬄine and an online stage. In the oﬄine stage, the parameters of the PSS and UPFC of controllable amplitude and phase angle in series with the transmission line. +is provides an eﬀect capable of controllers are determined through a PSO algorithm; then in reproducing an inductive or capacitive reactance in series the second stage, an online fuzzy controller is proposed to with the transmission line where the device is connected, determine parameters of the PSS and UPFC controllers thus allowing eﬀective control of the active and reactive coeﬃcients according to the operating point of the system. power ﬂows in the line where the SSSC is installed [10, 11]. Tests were conducted using the single machine inﬁnite bus Existing works in the literature that perform the control system, with only one generator. In 2018, [25] proposed to explore the process of tuning controllers for conditions of active and reactive power ﬂows using the SSSC, usually model it by power injections [14–16]. In this paper, a new using control theory. Tests were conducted using the two- area symmetrical system. model based on current injections is presented for the SSSC in order to perform both static and dynamic analyses of the In general, more recent works found in the literature do not use the CSM to represent the power system dynamics, device in the New England system [17]. +e model will allow the SSSC operation to be represented in the expanded power which is, therefore, a diﬀerential of the proposed work. In ﬂow algorithm expressed as a function of the residues of the this context, it is possible to cite the works [26–29]. In [28], current injections at the buses using the Newton-Raphson three optimization algorithms, i.e., GA, PSO, and the International Transactions on Electrical Energy Systems 3 Section 3, the CSM, used to represent the power system farmland fertility algorithm (FFA) are used to tune the control parameters of the PSSs in the New England system dynamics, is presented together with details about the representation of the SSSC; Section 4 presents details about using a linearized power system model. In [27], the dif- ferential evolution (DE) algorithm, PSO algorithm, gray wolf the structure of the supplementary damping controllers (PSS optimizer (GWO), whale optimization algorithm (WOA), and POD) and their respective diﬀerential equations rep- and chaotic whale optimization algorithm (CWOA) were resenting their dynamics; Section 5 presents some details used in a single machine inﬁnite bus (SMIB) system in order about the implementation of the GA, PSO algorithm, and to adjust the parameters of the proportional-integral (PI) MOA; Section 6 presents details about the problem for- controller. It is worth noting that in this article, the simu- mulation, discussing the representation of a candidate so- lated FACTS was the generalized uniﬁed power ﬂow con- lution and the objective function used in the optimization troller (GUPFC). In [26], the modiﬁed sine cosine algorithm methods; Section 7 presents the simulations and results for the implementations using the New England system and the (MSCA) was used to tune the control parameters of PSSs and SSSC in a SMIB system and a multi-machine system. Finally, two-area symmetrical system; ﬁnally, and Section 8 presents the main conclusions of the work. in [29], the parameters of the PSSs were adjusted in a co- ordinated and simultaneous way using the backtracking search algorithm (BSA). +e results are obtained from 2. Static Synchronous Series Compensator simulations performed on two multi-machine systems and +e SSSC FACTS can be represented by a voltage source are compared to the results of the PSO algorithm. converter (VSC) connected in series with the transmission In this work, a recently proposed optimization tech- line. +is device can vary the eﬀective impedance of the line nique, known as the mayﬂy optimization algorithm (MOA) by injecting a voltage in phase with respect to the line [30], is implemented for designing the supplementary current, thus allowing the exchange of active and reactive damping controllers. +is technique is based on the PSO powers with the transmission system [38]. In Figures 1(a) algorithm [31] and has the advantages of combining the and 1(b), the schematic diagram and equivalent circuit of the main features of the PSO algorithm, GA [32, 33], and the SSSC are shown, respectively. ﬁreﬂy algorithm (FA) [34]. It provides a powerful hybrid In Figure 1(a), the SSSC is represented using a VSC in algorithmic framework, based on the behavior of mayﬂies, series with the transmission line. +e VSC is connected to the performance of PSO algorithm, with crossover tech- the system through a coupling transformer with a reactance niques [35], and local search [36], since it has been proven x , and by means of gate turn-oﬀ thyristors, it modulates a that the PSO algorithm needs some modiﬁcations to achieve kn DC voltage coming from an external source. In Figure 1(b), an optimal solution for high-dimensional search spaces [37]. the phase angle c of the voltage source V is represented by From the above, the main contributions of this work are (1). Furthermore, b � −1/x is the susceptance of the as follows: kn kn coupling transformer, and V and V are, respectively, the k m (i) To present a new current injections model for the voltage phasors at buses k and m. Finally, V is the voltage SSSC phasor at the ﬁctitious bus (FB), n, included in the system to (ii) To use an expanded power ﬂow tool based on the perform simulations with the SSSC, and Z is the im- nm residues of the currents to determine the current pedance of the transmission line between buses n and m: injections of the SSSC (iii) To include PI controllers in the control loop of the c � arctan , (1) shunt converter of the SSSC (iv) To represent the power system and its supple- where 0≤ c≤ 2π, V and V are, respectively, the in-phase q p mentary damping controllers (PSSs and POD) using and quadrature components of the voltage source V the CSM [39, 40]. (v) To implement the PSO algorithm, GA, and MOA to perform the design of the supplementary damping 2.1. Current Injections Model for the SSSC. From (1) and the controllers equivalent circuit of the SSSC, shown in Figure 1(b), (2)–(5) +e remainder of this work is organized as follows: in are obtained. +ese are decomposed into real, r, and Section 2, the SSSC is modeled through current injections; in imaginary, i, components: I � b −V sin θ + V sin θ + b V cos θ + V sin θ , k km k k m m km p k q k √√√√√√√√√√√√√√√√√√√√√√ (2) sssc inj I � b V cos θ − V cos θ + b V sin θ − V cos θ , k km k k m m km p k q k √√√√√√√√√√√√√√√√√√√√√√ (3) sssc inj i 4 International Transactions on Electrical Energy Systems ˜ ˜ ˜ ˜ ˜ V ˜ V V V V k m k SSSC n m n Transmission Line Transmission Line V ∠γ ˜ ˜ Z s b Z nm kn nm SSSC kn n n m m (a) (b) Figure 1: (a) Representation of the SSSC FACTS and (b) equivalent circuit of the SSSC. I � b V sin θ − V sin θ − b V cos θ + V sin θ , km km k k m m km p k q k √√√√√√√√√√√√√√√√√√√√√√ (4) sssc inj I � b −V cos θ + V cos θ − b V sin θ − V cos θ , km km k k m m km p k q k √√√√√√√√√√√√√√√√√√√√√√ (5) sssc inj pi K 1 ref ctrl pod sssc sssc sssc sssc _ V � P − P + X1 + V − V , (6) where I , I and I , I are, respectively, the real, in p km km sup p inj inj inj inj pi pi k k m m r i r i T T m m (2) and (4), and imaginary, in (3) and (5), components of the currents injected into buses k and m by the SSSC at the ref ctrl X � P − P , (7) common buses of its installation. In Figure 2, the single-line 1 km km pi diagram of the proposed current injections model for the SSSC is illustrated. pi K 1 ref ctrl +e great advantage of using the current injections 2 V � Q − Q + X2 − V , (8) q q pi m m model for the SSSC, as shown (2)–(5) and depicted in T m p Figure 2, is the fact that this model keeps the current ﬂow equations unchanged and considers the device ref ctrl X � Q − Q , 2 (9) pi m m contributions at the common buses to its installation as if they were current injections. +is facilitates the in- ctrl ctrl ref ref clusion of the SSSC current injections model in con- where the quantities P , Q , P , and Q are, respec- m m m m ventional power ﬂow models, such as in tively, the control and speciﬁed values of the active and Newton–Raphson method. reactive power ﬂows on the line between buses k − m. 3. Power System Operation Model 2.2. Control System for the SSSC. To perform the control of the active and reactive power ﬂows by the SSSC (Figure 3) on Suppose an SSSC is installed in a power system, being k the the common buses to its installation, the PI controllers are common bus to its installation. +e residue of the current used [7, 12, 17]. injected at bus k, ΔI , is determined from the current +e PI controllers shown in Figures 3(a) and 3(b), are balance, as illustrated in Figure 4. used, respectively, to modulate the synchronous voltage From Figure 4, by inspection, (10) is obtained, repre- pi source control variables, V and V . +e time constant T is p q senting the nodal current balance at bus k: the inherent delay of the control device and is determined in the range between 1 and 10 ms [10]. In addition, X1, X2, and G sssc L I + I − I − I � ΔI , k inj kj k k (10) X3 are input signals and, in this case, are speciﬁed. +e k pi pi j∈Ω parameters of the PI controllers are the gains K and K , 1 2 pi pi and the time constants, T and T . +e supplementary 1 2 where I is the current phasor injected at node k, as shown pod signal V comes from a POD controller (see Section 4) sup in (11), I are the current phasors on the lines con- j∈Ω kj and, will be used to modulate the quadrature axis compo- nected to bus k, as shown in (12), in which Ω is set of all nent, V . neighboring buses of bus k, I is the current phasor drained sssc +e dynamic behavior of the control structure shown in by the load at bus k, as calculated in (13), and I is the inj Figure 3 is given by (6)–(9), which are obtained by current phasor injected at bus k by the SSSC, as shown in inspection: (14): VSC + International Transactions on Electrical Energy Systems 5 ˜ ˜ V V k m V ∠γ kn sssc sssc sssc sssc sssc sssc =+ I I ˜ j I ˜ =+ j I I I inj inj inj inj inj inj kr kj mr mj k m Figure 2: Current injection model of the SSSC. pi ref 1 V km X1 + 1 ∑ ERROR pi ∑ ∑ pi OUTPUT sT 1 + sT + m pod ct rl V sup km (a) pi ref km 1 1 q X2 ∑ ERROR pi ∑ pi OUTPUT sT 1 + sT + m ct rl km (b) Figure 3: Structure of the control system of the SSSC. ∑ I kj j∊Ω k Ω I˜ sssc inj Figure 4: Current balance at bus k. 1 1 G G G G I � P cos θ + Q sin θ + j P sin θ − Q cos θ , (11) k k k k k k k k k V V k k I � V G + jB cos θ + j sin θ , kj j kj kj j j (12) j∈Ω j∈Ω k l 1 1 L L L L I � P cos θ + Q sin θ + j P sin θ − Q cos θ , (13) k k k k k k k k l V V k k sssc I � b V cos θ + V sin θ + jb V sin θ − V cos θ , (14) km p k q k km p k q k inj + 6 International Transactions on Electrical Energy Systems G G where P and Q are the active and reactive powers injected k k k by the generator at bus k, G and B are the conductance kj kj L L and susceptance between buses k and j, and P and Q are ctrl ctrl k k km P + jQ P + jQ mk mk m m the active and reactive power drained by the load connected at bus k. For the SSSC to control the power ﬂows, it is necessary ctrl ctrl sssc sssc that P and Q are included as functions of the power km km inj + jQ inj mm system’s voltages and reactances, and the device’s control Figure 5: Active and reactive power balances for the SSSC. variables. For this purpose, the active and reactive power balances must be analyzed, according to Figure 5. In Figure 5, the active and reactive power ﬂows that are device at bus m are obtained by inspection, as shown, re- transmitted from bus m towards bus k, P and Q , the mk mk spectively, in (15) and (16): ctrl active and reactive power ﬂows controlled by the SSSC, P ctrl and Q , and the active and reactive powers injected by the ctrl P � − b V V sin θ − b V V cos θ + V sin θ , km k m km km m p km q km √√√√√√√√√√√√√ √ √√√√√√√√√√√√√ √√√√√√√√√√√√√√ √ (15) P sssc mk P inj ctrl 2 Q � − b V − V V cos θ − b V V cos θ − V sin θ . m km m k m km km m q km p km √√√√√√√√√ √√√√√√√√√√ √ √√√√√√√√√√√√√ √√√√√√√√√√√√√√ √ (16) sssc mk Q inj For the SSSC to actuate in the power system, its equa- nonlinear algebraic expressions that can be found using tions must be inserted in the power ﬂow formulation using Newton-Raphson algorithm, just as in the conventional the expanded power ﬂow technique, proposed by Kopcak power ﬂow. Since in this work the power ﬂow modeling was et al. [18]. To do so, it is necessary to assume that its state done by current injections, the essence of the problem is to satisfy all current residues between the buses in which the variables are considered constant with respect to time, which makes its temporal derivatives equal to zero. +is obser- SSSC is installed, as shown in Figure 4. vation allows, at an operation point of the power system, its +e power ﬂow equations formulated by current in- diﬀerential equations to be considered to be algebraic. +us, jections are shown in the following equation: the problem is limited to determining the zeros of a set of (17) +e Jacobian matrix shown in (17) is divided into four diﬀerence is in the elements marked with the “∗ ”, in sssc sssc sssc sssc blocks, J1 , J2 , J3 , and J4 . It is worth noting that which the derivatives that relate the injections of the real sssc J4 has its construction similar to the Jacobian of the and imaginary components of the currents injected by the conventional power ﬂow formulation. However, the SSSC are included. International Transactions on Electrical Energy Systems 7 3.1. Current Sensitivity Model. +e CSM is a linear analysis generators and nb buses is shown in (18)–(21). In this tool for power systems based on Kirchhoﬀ’s current law, formulation, Δx represents the state variables, Δu repre- which must be met when the system is disturbed [7, 12, 17]. sents the input variables, and Δz represents the algebraic Its modeling for multi-machine systems composed of ng variables: ng ′1 ′ng 1 t [Δx] � [ Δω . . .Δω Δδ . . .Δδ ΔE . . .ΔE ΔE . . .ΔE ] , (18) 1 ng 1 ng q q fd fd 1 ng 1 ng 1 nb 1 nb t [Δu] � [ΔP . . .ΔP ΔV . . .ΔV ΔP . . .ΔP ΔQ . . .ΔQ ] , (19) m m ref ref L L L L [Δz] � [Δ θ . . .Δ θ ΔV . . .ΔV ] , (20) 1 nb 1 nb J1 J2 Δx B1 Δx � + Δu , (21) J3 J4 Δz B2 Δx � AΔx + BΔu, (22) where Δω represents the variations of the angular speed, Δδ represents the variations of the internal angle of the where A and B are the state and input matrix, respectively, in − 1 − 1 rotor, ΔE represents the variations of the internal voltage which A � J1 − J2J4 J3 and B � B1 − J2J4 B2. in quadrature, and ΔE represents the variations of the fd generator’s ﬁeld voltage, ΔP represents the variations of 3.2. Inclusion of the SSSC in the CSM. +e dynamic char- the input mechanical power, ΔV represents the varia- ref tions of the reference voltage of the AVR, ΔP and ΔQ acteristics of the SSSC have already been incorporated into L L the power ﬂow modeled by current injections and are shown are, respectively, the variations of the active and reactive sssc sssc sssc sssc power demands of the loads, and ΔV and Δθ are, re- in the submatrices J1 , J2 , J3 , and J4 in (17). +ese can be directly included in the CSM, as shown in the fol- spectively, the voltage magnitude and voltage phase lowing equations: variations at each bus of the power system. +e representation in the state space, in the following equation, is obtained by eliminating Δz from (21): (23) c − 1 − 1 c c sssc c c c c sssc c c (24) Δx � J1 − J2 J4 J3 Δx + B1 − J2 J4 B2 Δu , c c c sssc − 1 c where k � 1, . . . , nb, A � J1 − J2 (J4 ) J3 and Figure 6(b), have similar structures, diﬀering only in the − 1 c c c sssc c input and output signals adopted for each controller. Both B � B1 − J2 (J4 ) B2 are, respectively, the new state and input matrices of the CSM. are represented by two lead-lag compensation blocks, pss pod pss pod characterized by the time constants T (T ), T (T ), 1 1 2 2 pss pod pss pod pss pod T (T ), and T (T ), a gain K (K ) and a 4. Dynamic Model of the PSS and 3 3 4 4 washout block, represented by the time constants POD Controllers pss pod T (T ). ω ω +e input signal chosen for a PSS is the variation of the +e purposes of the PSS and POD controllers are to insert, th angular speed of the k generator, Δω , while the output of a respectively, additional damping torque to the local and k pss PSS, ΔV , actuates on the control loop of the AVR of the interarea oscillatory modes present in the power system. sup th k generator (see Figure 6(a)). For the POD controller, the In this work, both the PSSs, Figure 6(a), and the POD, 8 International Transactions on Electrical Energy Systems ΔV K ΔE ΔV + ref r fd k k k 1 + sT pss ΔV sup pss pss pss 1 + sT 1 + sT sT ΔV ΔV ΔV Δω 3 3 1 2 ω 1 k pss pss pss pss 1 + sT 1 + sT 1 + sT 4 2 Lead-Lag Washout Gain (a) ΔV ΔX 1 p ref pod 1 + sT pod ΔV sup pod pod pod 1 + sT ΔY 1 + sT ΔY sT ΔY ΔP 3 2 1 km 3 1 ω pod pod pod pod 1 + sT 1 + sT 1 + sT 2 ω Lead-Lag Washout Gain (b) Figure 6: (a) Dynamic model of the PSS controller and (b) dynamic model of the POD controller. chosen input signal is the active power deviation of the V . In Figure 6(b), ΔV is the quadrature axis component ref p pod transmission line adjacent to the installation of the device, of the series converter, ΔX is an input signal and T is a ref pod ΔP , and the output signal, ΔV , actuates on the control time constant, being that both must be speciﬁed. km sup pss pod pss pod loop of the SSSC, used to modulate the quadrature com- It is adopted that T (T ) � T (T ) and 1 1 3 3 pss pod pss pod pod ponent, V , of the series converter of the SSSC (see T (T ) � T (T ) [1]. +e time constant T is the p 2 2 4 4 Figure 6(b)). inherent delay of the control device with values between 1 − th In Figure 6(a), the AVR installed at the k generator is 10 ms [10]. represented by a gain K and a time constant T . +e From the block diagram shown in Figure 6(a), the ex- r r k k excitation voltage of the synchronous machine is E , the pressions of the new state variables of the PSS that will be fd variations of the terminal voltage are represented by ΔV , included in the CSM are written as shown in (25)–(27), and pss and the variations of the reference voltage are represented by of the output signal, V , according to (28): sup pss ΔV � − K Δω + ΔV , (25) 1 k 1 k k pss 1 T 1 pss ΔV � − 1 − K Δω + ΔV − ΔV , (26) 2 pss k 1 2 k k k 2 2 pss pss 1 T T 3 1 pss ΔV � − 1 − ΔV + K Δω + ΔV − ΔV , (27) 3 pss 2 pss k 1 3 k k k k T T 4 4 2 pss pss T T pss 3 1 pss ΔV � ΔV + ΔV + K Δω + ΔV . (28) sup 3 pss 2 pss k 1 k k k T T 4 2 K K r r pss k k ΔE � − ΔE + ΔV − ΔV + ΔV . (29) fd fd ref k +e procedure for including the PSS in the CSM at the sup k k k T T T r r r th k k k k generator is terminated by adding the supplementary signal, according to (28), to the control loop of the AVR Analogously to the procedure performed for the PSS, shown from Figure 6(a). In this way, it is possible to obtain from Figure 6(b), the expressions of the state variables of the the following equation: POD that will be included in the CSM are obtained as shown International Transactions on Electrical Energy Systems 9 in (30)–(32), and the supplementary output signal of the pss POD controller, V , is obtained according to (33): sup pod ΔY � − K ΔP + ΔY , (30) 1 km 1 pod pod 1 T pod ⎢ 1 ⎥ ⎡ ⎢⎝ ⎠ ⎤ ⎥ _ ⎣⎛ ⎞ ⎦ ΔY � − 1 − K ΔP + ΔY − ΔY , (31) 2 km 1 2 pod pod T T 2 2 pod pod ⎧ ⎨ ⎫ ⎬ 1 T T pod 3 1 ⎝ ⎠ _ ⎛ ⎞⎡ ⎣ ⎤ ⎦ ΔY � − 1 − ΔY + K ΔP + ΔY − ΔY , (32) 3 2 km 1 3 pod ⎩ pod pod ⎭ T T T 4 4 2 pod pod T T pod 3 1 pod ⎣ ⎦ ⎡ ⎤ (33) ΔV � ΔY + ΔY + K ΔP − ΔY . sup 3 2 km 1 pod pod T T 4 2 or chromosomes, takes place, using the objective Finally, (33), which represents the supplementary output signal of the POD controller, is replaced into (6), which function for the problem (see (53)–(55)). describes the dynamics of the quadrature control voltage of (3) Selection: the probabilistic operator proportional the series converter of the SSSC, giving rise to the following selection used in the GA, draws individuals from a equation: population for crossover, in which each has drawn pi chances proportional to their respective evaluation K 1 ref ctrl pod ΔV � ΔP − ΔP + ΔX − V − ΔV . function values. +is selection method is also known p 1 p pi m m sup pod T T m m as the roulette rule [32]. (34) (4) Crossover: two individuals are chosen (P1 and P2), called parents, and the crossover operator is applied. Two new individuals are obtained, known as oﬀ- 5. Optimization Techniques ′ ′ spring (P1 and P2 ). +e crossover occurs until a new population of oﬀspring is formed. In this work, +is section will present the details the three optimization two recombination points are used, and these points methods that were used for the coordinated parameter are chosen randomly at each crossover step (see tuning design of the PI, PSS, and SSSC-POD controllers. Figure 7). +e GA with elitism, Subsection 5.1, the PSO algorithm, Subsection 5.2, and the MOA, Subsection 5.3, will be (5) Mutation: the mutation operator randomly changes discussed. one or more variables of a chromosome with a probability and within a range. +is operator is used to allow diversiﬁcation in the search process and 5.1. Genetic Algorithm. +e ﬁrst GA was proposed by [33] in introduce diversity. the 1970s. He analyzed the phenomenon of the natural evolution of species and applied operators in order to re- (6) Elitism: in the basic GA, the individuals obtained in produce this phenomenon when solving a complex problem. the previous generation are discarded, and only the In the particular case of this paper, the problem consists of new descendants are considered in the next gener- performing the coordinated tuning of the parameters of PI ation. Elitism consists in reintroducing the best- controllers and supplementary damping controllers (PSS evaluated individual from one generation to the next, and POD). +e goal is to insert the desired damping to the avoiding the loss of important information present poles of interest present in the power system. in individuals and that may be lost during the Basically, in this work, the GA performs the seven steps crossover and mutation procedures. outlined below: (7) Veriﬁcation of the convergence criterion: the goal is (1) Generation of the initial population: properly rep- to solve a constraint satisfaction problem [41]. +us, resent the set of candidate solutions. +e GA starts by the desired dampings are considered to be con- randomly generating a predeﬁned number of initial straints of the problem and therefore any settings solutions, forming the initial population. determined for the parameters of the controllers within the pre-established limits, (50)–(52), that (2) Evaluation function and constraints: evaluate the provide the desired damping, is considered a solu- objective function or its equivalent (ﬁtness) for each tion to the problem, see Table 1. individual. In this step, the evaluation of individuals, 10 International Transactions on Electrical Energy Systems x x x 3 1 3 P1 P1′ 2 b b x 2 2 P2 P2′ b b b b 1 3 1 3 Figure 7: Two-point crossover. 5.2. Particle Swarm Optimization. Initially consider a search Table 1: Parameters for the evaluation of the objective function. space of dimension nv. Suppose that each particle i, in the des loc inter [1.5pt] ξ − tol (p.u.) ω (Hz) ω (Hz) search space, can be represented according to its current ni n i,pso i,pso i,pso position x (t)[x (t), ... ,x (t)] and a velocity Min. 0.80 Min. 0.50 nv 0.145 i,pso i,pso i,pso Max. 1.40 Max. 0.70 v (t)[v (t), ... ,v (t)]. During the iterative pro- nv cess, at each step t, the current position of each particle is measured using an objective function according to 5.3. May y Optimization Algorithm. As stated before, the (53)–(55). From the analysis of the evaluation function, it is MOA is an optimization algorithm proposed by [30] in 2020, veri�ed, at each iterationt, if the position of the particle at based on concepts from the PSO algorithm, the GA, and the iteration t is better than the other positions previously FA. Initially, two sets of may’ies are randomly generated, veri�ed. If so, the position of this particle is stored in a vector i,pso i,pso i,pso one containing a male and the other a female population. p (t) [p (t), ... ,p (t)]. In the course of the it- best best best 1 nv Suppose that each may’y i, in the search space, can be erative process, at each best evaluation, this vector is randomly represented by a nv-dimensional vector updated. Finally, the best-found position measured by the i,moa i,moa i,moa i,pso x (t) [x (t), ... ,x (t)] and by a velocity 1 nv evaluation function stored in p (t) is stored in another best i,moa i,moa i,moa pso v (t) [v (t), ... ,v (t)]. During the iterative vector called g (t). nv best process, at each iteration t, the current position of each In the PSO algorithm, what determines the motion of a i,pso may’y is measured using an evaluation function, according given particle i at step (t + 1) is its velocity v (t) calcu- to (53)–(55). From the analysis of the evaluation function, at lated in the following equation: each iteration t, each may’y adjusts its trajectory toward its i,pso i,pso i i i,moa i,moa i,moa (35) v (t + 1) wv (t)+ C r co (t)+ C r so (t), personal best position, p (t) [p (t), ... ,p (t)]. 1 1 2 2 best best best 1 nv In the course of the iterative process, when a better candidate where r and r are random values over the interval 1 2 solution is found, this vector is updated. Finally, at each [0, 1], C and C are called acceleration constants and are 1 2 iteration, the best-found position measured by the evalua- responsible, respectively, for weighting the cognitive i,moa tion function stored in p (t) is stored in another vector, best i i factor, co (t), and the social factor, so (t). Furthermore, moa g (t). best w w − w /t t is known as the inertia factor and is max min max meant to control the impact of velocities on the particles i,moa of the PSO algorithm and t is the maximum number of 5.3.1. Movement of Male May ies. Assuming that x (t) is max iterations [42]. the current position of the male may’y i in the search space Še cognitive and social factors, shown in (35), are at iteration t, its position is changed by adding a velocity represented by the following equations: v (t + 1) to the current position, according to following ml equation: i i,pso i,pso (36) co (t) p (t)− x (t), best i,moa i,moa i,moa (39) x (t + 1) x (t)+ v (t + 1), ml pso i i,pso (37) so (t) g (t)− x (t), i,moa best where v (t+)1 is given according to the following th i equation: where the i cognitive factor, co (t), is related to each of the th i 2 2 individuals in the i best position p (t), the social factor, i,moa i,moa −βr i − βr i p g best v (t + 1) gv (t)+ C e co (t)+ C e so (t), i ml ml 1 2 so (t), is related to the best position found among all in- i,pso (40) dividuals in the swarm g (t) and x (t) is the position best th vector of the i particle at iteration t. At iteration (t + 1), the where C and C can be de�ned as in (35), g is called the 1 2 th i,pso position of the i particle, x (t + 1), is updated according gravity coe“cient and has values determined in the interval to the following equation: i i (0, 1], while co (t) and so (t) were addressed, respectively, i,pso i,pso i,pso in (36) and (37). Finally, β is called the visibility coe“cient x (t + 1) x (t)+ v (t + 1). (38) and has its value determined empirically, while r and r are, p g International Transactions on Electrical Energy Systems 11 i,moa respectively, the calculated cartesian distance from x (t) installed in the power system shown in (50)–(52) and ρ is i i,moa to p (t) and between x (t) and g (t) according to the calibrated from values selected from the range (0, 1]. best best following equation: To the step that reproduces the nuptial dance, in which ���������������������� the best mayﬂies have to keep changing their velocities, the nv � � following equation is presented: � i,moa i,moa � ij,moa ij,moa � � x (t) − x (t) � x (t) − x (t) , � � i,moa i,moa (43) j�1 v (t + 1) � v (t) + d · r, ml ml (41) where d is the coeﬃcient of the nuptial dance and r is a vector with random values within [−1, 1]. i,moa i,moa i,moa moa where x (t) � p (t) or x (t) � g (t). best best +e velocity control, shown in (40), of each mayﬂy in the i,moa 5.3.2. Movement of Female Mayﬂies. Assuming that y (t) MOA is performed according to the following equation: is the current position of the female mayﬂy i in the search max i,moa max ⎧ ⎨ v , if v (t + 1)> v , i,moa ml ml ml space, at iteration t, its position is changed by adding a v (t + 1) � (42) ml i,moa max i max velocity v (t + 1) according to the following equation: ρ v , if v (t + 1)< ρ v , 2 ml ml 2 ml i i i,moa y (t + 1) � y (t) + v (t + 1), (44) fml where ρ is an empirically ﬁxed random value, v � ρ (x − x ), in which x and x are the upper max 1 max min max min where v (t + 1) is determined using the following fml and lower bounds of the control parameters of the devices equation: i,moa −βr i,moa i ⎧ ⎪ mf gv (t) + C e x (t) − y (t), fml i,moa v (t + 1) � , (45) if male dominates female fml ⎪ i,moa gv (t) + fl(t) · r, otherwise fml where fl(t) is a random walk coeﬃcient, r is the Car- reducing the dance coeﬃcient d(t) and the random walk mf tesian distance between the male and female mayﬂies, coeﬃcient fl(t) in each iteration as shown in the fol- formulated according to (41), and C is a positive attraction lowing equations: constant. d(t) � d δ (t), (48) 0 d 5.3.3. Mating of Mayﬂies. +e crossover operator is used in fl(t) � fl δ (t), (49) 0 fl this algorithm to represent the mating process between two mayﬂies and is used as follows: the best female crosses with where at each iteration t of the optimization problem, δ (t) the best male, the second-best female with the second-best and δ (t) take ﬁxed values in the interval (0, 1). male, and so on. +e results obtained after these crosses are two oﬀspring generated according to the following equations: 6. Problem Formulation All algorithms are implemented considering an initial offspring � L · male + (1 − L) · female, (46) population consisting of 20 individuals. Each individual is a nv-dimensional vector of variables, this quantity being de- offspring � L · female + (1 − L) · male, (47) termined by the number of parameters of PI, PSSs, and the SSSC-POD controllers installed in the power system. Each where male is the male parent, female is the female parent, variable in an individual of the optimization problem in all and L is a random value within a certain interval. +e initial the algorithms must satisfy the bounds shown in the fol- velocities of the oﬀspring are set to zero. lowing equations: pss pss pss 5.3.4. Reduction of the Nuptial Dance and Random Walk 0.10≤ T ≤ 1.25; 0.01≤ T ≤ 0.20; 1.00≤ K ≤ 10.00, 1 2 r r r Coeﬃcients. +e nuptial dance performed by male may- (50) ﬂies represented by the coeﬃcient d(t), as well as the random walk performed by females, are based on local pod pod pod 0.10≤ T ≤ 0.40; 0.10≤ T ≤ 0.40; 0.10≤ K ≤ 0.50, 1 2 searches shown in (43) and (45). However, random ﬂights can produce poor results during a random exploration. (51) +is problem occurs from the fact that the nuptial dance pi pi pi pi d(t) or random walk fl(t) coeﬃcients usually assume (52) 0.001≤ T � T ≤ 0.90; 0.001≤ K � K ≤ 1.00, 1 2 1 2 large initial values. +is can be mitigated by gradually 12 International Transactions on Electrical Energy Systems PI POD PSSs pss pss pss pss pss pss pi pi pi pi pod pod z = pod T T K T T K T = T K = K T T K 2 1 2 1 2 1 1 1 9 9 9 1 2 1 2 1 Figure 8: Representation of a solution proposal. where the units of the time constants and gains of the jω controllers are, respectively, seconds and p.u., while max r � 1, . . . , 9. ni calc des th ξ ≥− tol +e i individual of a given population (z ), is shown in i i min max ω ω ω ≤≤ ni ni ni Figure 8. As it will be justiﬁed in Section 6, the simulations min λ = σ + jω i i i ni 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 λ = σ - jω i i i variables, i.e., nv � 11. +e objective function of the problem is deﬁned in the following equations for all algorithms: Figure 9: Desired location for the eigenvalues of interest. F z � Δξ , (53) 1 i 7.1. New England System. To validate the models and op- q�1 timization algorithms presented in this work, small-signal n stability simulations will be carried out using the New F z � England system whose complete data is available in [17]. Δω , (54) 2 i q�1 +is is a multimachine system composed of 10 generators, 40 buses, and 47 transmission lines, already accounting for one minimize F z � η F z + η F z . (55) new line (38− FB) and one new bus (FB, between buses i 1 1 i 2 2 i 37 − 38), included to perform simulations with the SSSC, as where n is the number of eigenvalues present in the power can be analyzed in Figure 10. des calc system, η � η � 1, Δξ � |ξ − ξ | and 1 2 q q For the simulations, the algorithms were calibrated des calc des calc Δ ω � |ω − ω |, in which ξ and ξ are, respec- q q q according to data shown in Table 2, considering the set of th tively, the desired and calculated damping of the q ei- bounds shown in (50)–(52). des genvalue of interest of the power system, while ω and +e presentation of the results is organized in two cases: calc ω are, respectively, the desired and calculated fre- q ﬁrst, a static analysis is performed, and second, a dynamic th quency of the q eigenvalue of interest of the power analysis is conducted. In the ﬁrst case, the objective is to system. validate the proposed formulation for the power ﬂow +e settings provided to the control parameters by the modeled by current injections for the SSSC. +e second case algorithms to the PSSs, the SSSC-POD, and PI controllers is subdivided into two parts: in the ﬁrst part, a statistical must ensure the minimum desired damping for all poles of analysis is performed considering the results of the three interest of the power system as well as that the oscillation simulated optimization techniques (GA, PSO algorithm, and frequencies of all existing poles remain within predeﬁned MOA) in order to determine which of the three algorithms design ranges, as shown in Figure 9. best applies to the discussed problem; second, an analysis In Figure 9, σ and ω are, respectively, the real and i i focused on small-signal stability using the New England th imaginary parts of the i eigenvalue λ � σ ± jω . With the i i i system is performed. +e proposed models and optimization bounds given in (50)–(52), the objective function of a so- algorithms were implemented in MATLAB R2019a using a lution, shown in (55), should satisfy the constraints deﬁned computer with a 3.19 GHz Intel Core i7-8700 processor in the project, as shown in Table 1. and 16 GB of RAM. loc In Table 1, ω is related to the oscillation frequency of ni inter local modes, while ω is associated with the interarea pod pss pi pod mode. Finally, T � T � 1 s, while T � T � 0.001 s ω ω m m 7.1.1. Static Analysis. Initially, it can be seen that the SSSC are prespeciﬁed. was installed between buses 37 − 38. +e reasons for in- stalling the device at this location are: ﬁrst, the installation of the SSSC should occur at locations close to the buses 7. Simulations and Results that presented voltage magnitudes below 0.95 p.u. According to Figure 11, the buses with voltage magnitudes +is Section presents the results of simulations performed using the New England system [17] and the two-area 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 symmetrical system [1]. International Transactions on Electrical Energy Systems 13 G8 G1 Area 2 Area 1 28 29 G9 G6 33 21 ∼ 34 G10 37 35 20 5 7 ∼ ∼ G5 G7 G2 VSC FB G4 G3 Figure 10: Single-line diagram of the New England system with one SSSC. Table 2: Parameters of the algorithms. Search Engine Structure Parameters of the Algorithm Parameters of the Simulations Fitness Function Roulette selection: — Two-point crossover 10% probability rate GA Mutation (23% of the value): 10% probability rate best individual Elitism: C � 2.55 Population size: 20 C � 1.55 Formulation Stopping criterion: Table 1 PSO w � 0.89 (55) max (35)–(38) Max. number of iterations: w � 0.0025 min t � 2000 max g � 0.851 C � 1.99, C � 2.00 1 2 Formulation C � 2.50, β � 2.00 MOA (39)–(48) ρ � 0.10, ρ � −1.10 1 2 d � 950, δ � 0.9999 fl � 1.50, δ � 0.99 0 fl 1.04 1.03 1.02 1.01 1.00 0.99 0.98 0.97 0.96 0.95 0.94 10 20 40 5 15 25 30 35 Bus Case I Case II Figure 11: Voltage proﬁles for the New England system. Voltage Magnitude (p.u.) 14 International Transactions on Electrical Energy Systems Table 3: Variables of the SSSC’s control structure of the new installation; second, the location should allow for the control- England system. lability and observability over the interarea mode present in the power system. Indeed, according to Figure 10, buses 38 and 37 SSSC V V p q are characterized by being located near to Area 1 (New York − 07 − 08 Case I −7.11283 × 10 2.78607 × 10 system that is represented by the equivalent bus 10). It is known Case II −0.0155367 −0.0150003 in the literature that interconnection areas favor the emergence of interarea oscillatory modes. Case I will represent the situation in which the New Subsection 7.1.1, a dynamic analysis involving the SSSC, PI England system is equipped with an SSSC, but the device controllers, and supplementary damping controllers (PSS does not control the active and reactive power ﬂows in the and POD), will be performed. In Table 4, the eigenvalues of line in which it is installed, and Case II will represent the case interest of the New England system are presented before in which the SSSC controls the active and reactive power ctrl (Case I) and after (Case II) the actuation of the SSSC in the ﬂows. For the ﬁrst case, P � −15.96 MW and ctrl ctrl control of the active and reactive power ﬂows. Q � −111.27 MVAr and, for Case II, P � −25.00 MW m m ctrl By analyzing the frequencies, ω � |λ |/2π, of the modes and Q � −125.00 MVAr. In both cases, for simulation ni i of interest, λ � σ ± jω , shown in Table 4, it can be con- purposes, the coupling reactance of the transformer, x , i i i km cluded that the New England system is characterized by the connecting the VSC to the transmission line (positioned presence of eight local modes, λ and λ , and one interarea between buses 38 and FB) is equal to 0.01 p.u. 1 8 mode, λ . In both Case I and Case II, the system is unstable. In Figure 11, the voltage proﬁles before (Case I) and after In Case I, the instability of the system is caused by four (Case II) the SSSC actuation for controlling the active and oscillatory modes with positive real parts, λ , λ , λ , and λ , reactive power ﬂows are shown. 1 4 8 9 while in Case II, the instability of the system is caused by By analyzing Figure 11, it can be seen that before the three oscillatory modes with positive real parts, λ , λ , and control of the active and reactive power ﬂows performed by the 1 4 λ . Importantly, the interarea mode in Case I has negative SSSC (Case I), the system had voltage problems at buses 12 damping, of −0.0010 p.u., and, in Case II, after the control of (0.9491 p.u.), 36 (0.9486 p.u.), and 37 (0.9475 p.u.). However, the active and reactive power ﬂows by the SSSC, this mode the same cannot be said for Case II. After controlling the ﬂows, changes from unstable to stable with damping of 0.0234 p.u. the voltage proﬁles were adjusted, being equal to 0.9505 p.u. for +e inclusion of the SSSC dynamics in the CSM and the bus 12, 0.9513 p.u. for bus 36, and 0.9506 p.u. for bus 37. correct control of the active and reactive power ﬂows per- In Table 3, the values of the SSSC control variables, V formed by this device in the New England system provided and V , are presented. improvements in the voltage proﬁle of the system (see When analyzing Table 3, it is possible to verify that the Figure 11) as well as small damping increments in some values of V and V are close to zero for Case I, demonstrating p q poles of interest of the system. In fact, it is possible to verify that the SSSC is included in the system, but it is not actuating. that in Case II, for the considered operation point, the New In relation to Case II, it can be seen that after the SSSC has England system started to operate with three poles re- actuated to control the active and reactive power ﬂows, the sponsible for the system instability, one less when compared values of the control variables are diﬀerent from zero, which to Case I (see Table 4). Due to these observations, Case II, corroborates the SSSC’s actuation in the New England system. from this point on, will be used as the base case in the Finally, in Figure 12, the active and reactive power ﬂows, following simulations. generations, loads, and the power injections performed by the SSSC, for the buses and lines near to the installation site (1) Statistical Analyses of the Performances of the Algorithms. of the SSSC, are presented. +e results for Case I are rep- Initially, an SSSC-POD installed in the system according resented in Figure 12(a) while the results for Case II are with the arguments presented in Subsection 7.1.1 and nine shown in Figure 12(b). PSSs coupled to generators G1 to G9 are considered. By analyzing Figure 12, it is possible to verify that, in both +e parameter tuning of the PI and supplementary cases, the control of the power ﬂows is performed according to damping controllers, PSSs and POD, for Case II, is per- what was established for Cases I and II. It is important to highlight formed using the PSO algorithm, GA, and MOA, and these that, in Case II, in order to control the ﬂows speciﬁed in the project ctrl ctrl are calibrated based on the data provided in Table 2. In this (P � −25.00 MW and Q � −125.00 MVAr) for improving m m ﬁrst scenario, one hundred tests limited to two thousand the voltage magnitudes at buses 12, 36, and 37, the SSSC needed to iterations in each one are performed. Under these condi- inject at buses 37 and FB, respectively, 147.70 MW and 142.60 tions, the values given in Table 1 should be checked. +e MVAr, and −149.38 MW and −140.30 MVAr. Finally, it is results of these simulations can be veriﬁed in Table 5. possible to perform the nodal balance of active and reactive When analyzing the results presented in Table 5, it is powers at the buses indicated in Figure 12 and thus validate the possible to make the following comments: (i) the MOA control of the power ﬂows performed by the SSSC. 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 7.1.2. Dynamic Analysis. After validating the current in- compared, it is possible to observe that the MOA was more jections model for the SSSC and conﬁrming its ability to eﬃcient than the other two optimization algorithms; (iii) manage the active and reactive power ﬂows, as shown in International Transactions on Electrical Energy Systems 15 0.9569∠ − 8.89° (p.u.) 0.9589∠ − 8.90° (p.u.) 147.70 MW 142.60 MVAr 0.9603∠ − 8.13° (p.u.) 0.9475∠ − 11.10° (p.u.) 0.9622∠ − 8.14° (p.u.) 0.9506∠ − 11.07° (p.u.) 37 35 522 + j176 MVA 522 + j176 MVA 1.00∠ − 11.12° (p.u.) 1.00∠ − 10.56° (p.u.) 1000 + j177.12 MVA 1000 + j159.62 MVA 10 G10 0.9487∠ − 10.55° (p.u.) 0.9514∠ − 10.53° (p.u.) G10 VSC 233 + j84 MVA FB 233 + j84 MVA 0.9591∠ − 11.00° (p.u.) 1104.40 + j250 MVA 1104.40+ j250 MVA FB 0.9500∠ − 11.86° (p.u.) 149.38 MW 140.30 MVAr 0.9950∠ − 1.78° (p.u.) 0.9908∠ − 11.46° (p.u.) Case I Case II Case II SSSC SSSC Load Load (a) (b) Figure 12: Control of the active and reactive power ﬂows performed by the SSSC in the New England system. Table 4: Dominant eigenvalues, damping coeﬃcients, and natural undamped frequencies of the New England system. Case I Case II Mode Eigenvalues ξ (p.u.) ω (Hz) Eigenvalues ξ (p.u.) ω (Hz) i ni i ni λ 0.0729 ± j6.8461 −0.0106 1.0897 0.0767 ± j6.8448 −0.0112 1.0894 λ −0.2064 ± j7.2348 0.0285 1.1519 −0.2052 ± j7.2335 0.0284 1.1517 λ −0.1903 ± j8.2731 0.0230 1.3171 −0.1880 ± j8.2725 0.0227 1.3169 0.1799 ± j5.9142 −0.0304 0.9417 0.1745 ± j5.9090 −0.0295 0.9409 λ −0.1291 ± j6.4968 0.0199 1.0342 −0.1245 ± j6.4829 0.0192 1.0320 λ −0.2690 ± j8.1071 0.0332 1.2910 −0.2687 ± j8.1059 0.0331 1.2908 λ −0.2413 ± j8.3204 0.0290 1.3248 −0.2412 ± j8.3187 0.0290 1.3245 λ 0.1622 ± j6.3862 −0.0254 1.0167 0.1587 ± j6.3764 −0.0249 1.0152 λ 0.0037 ± j3.5625 −0.0010 0.5670 −0.0812 ± j3.4638 0.0234 0.5514 ﬁnally, when the convergence times of the algorithms were the results presented in Table 5, the radar plot of the three compared, again the MOA had better performance than the simulated algorithms is shown in Figure 13. In it, the ob- others, being the best in the average, minimum, and max- served results for four quantitative variables obtained in the imum times observed. one hundred simulated tests are presented: time, numbers of +e results analyzed so far would already allow saying iterations, minimum damping, and the smallest values that the MOA is the most eﬃcient algorithm for tuning the found for the objective function for each simulated test. parameters of the controllers for small-signal stability im- It is clear from Figure 13 that the radar plot for the MOA provement, and therefore, would allow accrediting it as a is the one that presents the best results. It is possible to powerful tool in this type of study. However, to corroborate observe that the respective curves presented for the MOA 16.20 MW 16.18 MW 120.60 MW 316.56 MW 317.46 MW 15.96 MW 15.96 MW 16.18 MW 15.96 MW 189.48 MW 189.64 MW 423.88 MW 422.64 MW 129.73 MW 25.33 MW 25.31 MW 124.38 MW 312.29 MW 313.17 MW 124.38 MW 25.31 MW 186.39 MW 186.54 MW 420.74 MW 419.54 MW 78.51 MVAr 40.73 MVAr 49.65 MVAr 56.09 MVAr 78.51 MVAr 111.27 MVAr 111.27 MVAr 109.90 MVAr 56.66 MVAr 9.44 MVAr 4.26 MVAr 96.93 MVAr 88.26 MVAr 48.60 MVAr 94.14 MVAr 24.28 MVAr 17.04 MVAr 94.14 MVAr 15.29 MVAr 48.43 MVAr 47.28 MVAr 2.00 MVAr 3.29 MVAr 80.71 MVAr 88.84 MVAr 16 International Transactions on Electrical Energy Systems Table 5: Comparison of the performances of the algorithms for the New England system. Number of iterations Time (s) Algorithm Damping ratio Tests with convergence (%) Average Min. Max. Average Min. Max. PSO Table 1 87 ≈598 117 2000 ≈41 ≈8 ≈145 GA Table 1 91 ≈765 163 2000 ≈107 ≈23 ≈284 MOA Table 1 100 ≈58 16 549 ≈14 ≈4 ≈127 Minimum Damping [0.073, 19.90] (p.u.) Number of Iterations [16, 2000] GA MOA PSO Figure 13: Radar plot for the GA, PSO algorithm, and MOA for the New England system. radar plot are concentrated closer to the center, corrobo- limits stipulated in design (see Table 1). Regarding the values rating the superiority of the MOA in all analyzed aspects found for the dampings, it can be stated that the system compared to the other algorithms. operates with high damping margins, with 0.3710 p.u., 0.3180 p.u., and 0.2730 p.u. being the minimum damping (2) Small-Signal Stability Studies. In order to perform studies levels, achieved from the MOA, PSO algorithm, and GA, focused on the small-signal stability, simulations were respectively. Finally, on average, the MOA also has shown performed with the three algorithms proposed in this paper better results, as it achieved average damping of 0.3939 p.u., (MOA, PSO algorithm, and GA). Again, the New England compared to 0.3521 p.u. for the PSO algorithm and 0.3313 system equipped with nine PSSs installed at generators G1 to p.u. for the GA. G9 and an SSSC-POD between buses 37 and 38 is con- Power systems are often subject to small variations in sidered. All algorithms are simulated and calibrated with the loads and, therefore, consequent adjustments in generations. settings shown in Table 2. It is worth noting that, in this case, Figure 14 shows the variations of the angular speeds of all the same randomly generated initial population was used for generating units (except G2, the reference of the system) of the three proposed algorithms in each trial. In this case, the New England system for the simulated case presented in 50000 iterations were considered, without specifying the Tables 6 and 7. desired minimum damping. Table 6 shows the results ob- +e disturbance in the mechanical power at G2 (+0.05 tained for the parameters of the PSSs, PI, and POD p.u.) is similar to a small adjustment in the generation, which controllers. can be caused by a small increase in the system loading. It With the parameters presented in Table 6, the PI and can be seen in Figure 14 that even after the disturbance, the supplementary damping controllers (PSSs and POD) are system is stabilized in approximately two seconds in all calibrated and the CSM is evaluated. In this way, new cases, graphically evidencing its high stability margin for damping levels (ξ � −σ /|λ |) associated with the eigen- small-signal disturbances. i i i values of interest of the system can be obtained, as well as their respective undamped natural frequencies, as presented (3) Sensitivity Analysis on the Number of PSSs in the System. in Table 7. +is section investigates the inﬂuence of the number of PSSs By observing Table 7 it is possible to verify that, for the in the system for damping the local oscillation modes. In operating point considered, from the adjustments obtained these tests, the number of PSSs installed in the New England in Table 6, the frequencies of all poles of interest of the New system will be systematically reduced, and the parameters of England system are within the maximum and minimum the controllers will be optimized to maximize the minimum Time [4, 284] (s) –11 Objective Function [5.09 × 10 ,0.0060] International Transactions on Electrical Energy Systems 17 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. MOA PSO GA Device pss pss pss pss pss pss pss pss pss T (s) T (s) K (p.u.) T (s) T (s) K (p.u.) T (s) T (s) K (p.u.) 1 2 1 2 1 2 PSS G1 1.2500 0.0679 10.000 1.2500 0.0738 10.000 1.2481 0.0708 9.7607 PSS G2 0.9303 0.0938 7.1639 1.2500 0.1316 5.9594 0.9971 0.1454 8.1228 PSS G3 1.0296 0.0916 6.3780 0.8261 0.0858 10.000 1.1479 0.0870 5.6172 PSS G4 0.6887 0.0685 9.4927 1.2500 0.0746 5.8120 1.0510 0.0532 7.0592 PSS G5 0.4225 0.1047 7.7450 0.5263 0.1144 2.0000 0.7155 0.2000 9.6881 PSS G6 1.1876 0.1181 8.9621 1.2500 0.1397 7.9174 1.1067 0.0795 8.2647 PSS G7 0.4758 0.1058 4.5430 0.0500 0.0100 9.6411 0.2844 0.0713 3.9547 PSS 8 0.8647 0.1037 6.7272 0.8596 0.0970 7.2540 0.8524 0.0834 9.2116 PSS G9 0.5618 0.2000 4.5354 0.5949 0.2000 6.0537 0.4352 0.2000 7.3625 pod pod pod pod pod pod pod pod pod T T k T T k T T k 1 2 1 2 1 2 POD 0.2395 0.2765 0.3622 0.3247 0.4000 0.5000 0.2845 0.4000 0.3979 pi pi pi pi pi pi pi pi pi pi pi pi T � T K � K T � T K � K T � T K � K 1 2 1 2 1 2 1 2 1 2 1 2 PI 0.6119 0.0730 0.7604 0.1078 0.3185 0.0781 Table 7: Eigenvalues of interest, damping coeﬃcients, and natural undamped frequencies determined by the MOA, PSO, and GA for the New England system. MOA PSO GA Mode Eigenvalues ξ (p.u.) ω (Hz) Eigenvalues ξ (p.u.) ω (Hz) Eigenvalues ξ (p.u.) ω (Hz) i ni i ni i ni λ −2.6438 ± j6.6169 0.3710 1.1341 −1.9419 ± j5.7900 0.3180 0.9720 −2.1065 ± j7.4244 0.2730 1.2283 λ −2.2244 ± j5.5563 0.3717 0.9525 −1.6865 ± j5.0284 0.3180 0.8441 −1.8755 ± j6.4884 0.2777 1.0749 λ −3.1582 ± j7.8726 0.3723 1.3500 −2.7814 ± j8.2769 0.3185 1.3897 −2.3732 ± j8.0921 0.2814 1.3421 λ −2.5701 ± j6.3906 0.3731 1.0963 −2.0599 ± j6.1144 0.3193 1.0269 −1.5898 ± j5.0868 0.2983 0.8482 λ −2.3988 ± j5.9238 0.3753 1.0172 −2.1980 ± j6.4361 0.3232 1.0824 −1.7815 ± j5.6238 0.3020 0.9389 λ −1.9515 ± j4.7999 0.3766 0.8247 −2.8851 ± j8.2977 0.3284 1.3982 −1.8132 ± j5.2091 0.3287 0.8778 λ −2.4977 ± j5.3862 0.4207 0.9449 −2.4205 ± j5.1257 0.4270 0.9022 −2.5926 ± j7.2789 0.3355 1.2298 λ −2.6836 ± j4.5523 0.5078 0.8410 −2.5751 ± j4.4737 0.4989 0.8215 −3.0614 ± j4.6178 0.5526 0.8818 λ −1.5229 ± j3.7459 0.3766 0.6436 −1.3002 ± j3.8765 0.3180 0.6507 −1.1888 ± j3.3715 0.3325 0.5690 damping in the system. In all simulated cases, the FACTS- presented the lowest impact on the minimum damping of POD set is installed in the deﬁned location of Subsection the system, therefore, it was the ﬁrst one that was removed. 7.1.1 and the parameters that were used to tune the algorithm From the results presented in Table 8, it can be veriﬁed are available in Table 2. Table 8 shows the mode with that the reduction of the number of PSSs make the system minimum damping obtained in each simulation. more susceptible to low-frequency oscillations, since the In the simulations shown in Table 8, the PSSs of the minimum damping of the system reduces from 0.1464 p.u. to following generators were removed for each simulation: 0.0261 p.u. from Simulation I, with 8 PSSs in the system, to Simulation VIII, with only one PSS. (i) Simulation I: G7 Based on these results, it can be concluded that the (ii) Simulation II: G7, and G5 proposed algorithm is robust, since it was capable of (iii) Simulation III: G7, G5, and G2 obtaining stable solutions even when a small number of PSSs was considered in the system. (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 7.2. Two-Area Symmetrical System. +e two-area symmet- rical system, characterized by 2 areas, 4 generators, 16 (vii) Simulation VII: G7, G5, G2, G1, G3, G8, and G4 branches, and 11 buses (see Figure 15), already includes the (viii) Simulation VIII: G7, G5, G2, G1, G3, G8, G4, and new bus FB and a new transmission line, specially included G6 to carry out simulations with the SSSC. Furthermore, this +e order for removing the PSSs from the system was system is characterized by having long transmission lines obtained by systematically removing one PSS at a time and between buses 7 and 8 with a highly inductive nature. +is verifying the removal that led to the smallest impact on the fact contributes to the appearance of modes with oscillatory lowest damping after re-optimizing the parameters of the characteristics (inter-area mode). Complete data for this controllers. For example, it was veriﬁed that the PSS of G7 system can be found in [1]. 18 International Transactions on Electrical Energy Systems −4 −4 −4 ×10 ×10 ×10 1 4 G1 G3 G4 -4 -2 -2 4 6 8 10 4 6 8 4 6 8 10 0 2 0 2 0 2 Time (s) Time (s) Time (s) −4 −4 −4 ×10 ×10 ×10 -2 -2 0 0 0 G5 G6 G7 -4 2 2 10 10 4 6 8 10 4 6 8 4 6 8 0 2 0 2 0 2 Time (s) Time (s) Time (s) −4 −4 −4 ×10 ×10 ×10 4 -5 -2 0 0 G8 G9 G10 -5 -4 2 10 8 10 6 8 10 4 6 8 0 2 4 6 0 2 4 0 2 Time (s) Time (s) Time (s) Open Loop PSO GA MOA Figure 14: Variation of the angular speeds of the generators G1 and G3–G10 of the New England system. Table 8: Modes with minimum damping for each simulation obtained when removing PSSs from the New England system. Simulation Mode with minimum damping min(ξ ) (p.u.) ω (Hz) i ni I −1.2807 ± j8.6539 0.1464 1.3923 II −0.8888 ± j5.9555 0.1476 0.9583 III −0.5260 ± j6.0714 0.0863 0.9699 IV −0.4654 ± j7.0181 0.0662 1.1194 V −0.2340 ± j7.2037 0.0325 1.1471 VI −0.2180 ± j7.2260 0.0302 1.1506 VII −0.2135 ± j7.2312 0.0295 1.1514 VIII −0.2162 ± j8.2860 0.0261 1.3192 FB 1 5 6 9 10 ~ ~ G4 G1 ~ ~ G2 G3 Area 1 Area 2 Figure 15: Single-line diagram of the two-area symmetrical system with one SSSC. Angular Speed (rad/s) Angular Speed (rad/s) Angular Speed (rad/s) Angular Speed (rad/s) Angular Speed (rad/s) Angular Speed (rad/s) Angular Speed (rad/s) Angular Speed (rad/s) Angular Speed (rad/s) VSC International Transactions on Electrical Energy Systems 19 Table 9: Dominant eigenvalues, damping coeﬃcients, and natural undamped frequencies of the two-area symmetrical system. Mode Eigenvalues ξ (p.u.) ω (Hz) i ni λ 0.0487 ± 4.3299 −0.0112 0.6892 λ −0.2444 ± 5.9188 0.0413 0.9428 λ −0.3650 ± 6.2938 0.0579 1.0034 Table 10: Comparison of the performances of the algorithms for the two-area symmetrical system. Number of iterations Time (s) Algorithm Damping ratio Tests with convergence (%) Average Min. Max. Average Min. Max. PSO Table 1 92 ≈179 6 2000 ≈13 ≈0.67 ≈144 GA Table 1 99 ≈109 10 2000 ≈16 ≈2 ≈301 MOA Table 1 100 ≈11 3 167 ≈3 ≈1 ≈41 the proposed approach for the designing of the Minimum Damping [0.039, 0.17] (p.u.) controllers. 8. Conclusions +is work presented comparisons involving three algo- rithms, i.e., genetic algorithm (GA), particle swarm opti- mization (PSO) algorithm, and the mayﬂy optimization algorithm (MOA), to perform the coordinated tuning of the parameters of the proportional and integral (PI) and sup- plementary damping controllers, power system stabilizers (PSSs) and power oscillation damping (POD). +e current sensitivity model was used to represent the electric power system, the reason why all devices present in the system were Number of Iterations [3, 2000] modeled by current injections, including the static syn- GA chronous series compensator (SSSC) ﬂexible AC trans- MOA PSO mission systems. +e performances of the proposed algorithms were evaluated using the New England system Figure 16: Radar plot for the GA, PSO algorithm, and MOA for the and the two-area symmetrical system. two-area symmetrical system. To validate the proposed current injections model for the SSSC, a static analysis was performed. Finally, simulations To improve the voltage proﬁle of the two- were performed in order to compare the three proposed ctrl areaQ � −70.00 symmetrical system, the following was algorithms to analyze which one has the best performance ctrl considered: (P � 70 MW and MVAr). In this situation, the for damping the oscillatory modes of the system in the small- CSM is simulated and the results obtained can be seen in signal stability problem. Table 9. It can be veriﬁed that the system is unstable due to λ . 1 Based on the results, it was possible to validate the current +e results of the simulations for evaluating the per- injections model proposed for the SSSC. Finally, the MOA had formance of the algorithms can be veriﬁed in Table 10. its performance compared with the GA and PSO algorithm for When analyzing the results presented in Table 10, it is two diﬀerent scenarios, being unbeatable in the two simulated possible to make the following comments: (i) the MOA cases. On this occasion, the coordinated adjustments of PI and achieved success in 100% of the tests performed, against 99% of supplementary damping controllers (PSSs and POD) param- GA and 92% of PSO algorithm; (ii) both in average or when the eters were performed, and therefore it is possible to accredit it maximum and minimum numbers of iterations are compared, as a powerful tool in the small-signal stability in power systems. it is possible to observe that the MOA was more eﬃcient than +e authors would like to thank the Goias ´ Federal In- the other two optimization algorithms; (iii) ﬁnally, when the stitute of Education, Science, and Technology (IFG), the convergence times of the algorithms were compared, again the Parana´ Federal Institute of Education, Science and Tech- MOA had better performance than the others, being the best in nology (IFPR), the Federal Rural University of the Semi- the average, minimum, and maximum times observed. Arid Region (UFERSA). +e radar plot of the three simulated algorithms is shown in Figure 16 for the two-area symmetrical system. List of Abbreviations Again, as for the New England system, it is clear from Figure 16 that the radar plot for the MOA is the one that AVR: Automatic voltage regulator presents the best results for the two-area symmetrical BSA: Backtracking search algorithm system. +ese results, therefore, show the eﬀectiveness of CSM: Current sensitivity model Time [0.67, 302] (s) –11 Objective Function [8.60 × 10 , 0.012] 20 International Transactions on Electrical Energy Systems sssc Real and imaginary components of the currents CWOA: Chaotic whale optimization algorithm I , inj injected at bus k by the SSSC DE: Diﬀerential evolution sssc I : inj FA: Fireﬂy algorithm sssc I : Current phasor injected at bus k by the SSSC FACTS: Flexible AC transmission systems inj FB: Fictitious bus ctrl P , Active and reactive controlled power ﬂows on the ctrl FFA: Farmland fertility algorithm Q : line between buses k − m GA: Genetic algorithm ref P , Speciﬁed values of the active and reactive power ref GUPFC: Generalized uniﬁed power ﬂow controller Q : ﬂows on the line between buses k − m. GWO: Gray wolf optimizer Nomenclature of the POD Controllers IPFC: Interline power ﬂow controller ΔP : Deviation in the active power of km MOA: Mayﬂy optimization algorithm branch km MSCA: Modiﬁed sine cosine algorithm pod ΔV : Output signal from the POD sup PI: Proportional-Integral ΔY , ΔY , ΔY : Intermediary signals of the POD’s 1 2 3 POD: Power oscillation damping dynamic model PSO: Particle swarm optimization pod K : Gain of the POD PSS: Power system stabilizer pod pod pod T , T , T , Time constants of the POD 1 2 3 SCBGA: Specialized Chu-Beasley’s genetic algorithm pod pod T , T : SSSC: Static synchronous series compensator pod T : Time constant of the washout block of TCSC: +yristor-controlled series capacitor the POD UPFC: Uniﬁed power ﬂow controller pod V : Supplementary signal of the POD sup VSC: Voltage source converter controller. WOA: Whale optimization algorithm. Nomenclature of the CSM Δδ: Variations of the internal angle of the rotor Nomenclature of the Power System Operation Δ θ: Variations of the voltage phase at each bus of the c: Voltage phase angle power system ΔI : Residue of the current injected at bus k Δω: Variations of the angular speed of the rotor of Ω : Set of all neighboring buses of bus k generator k b : Susceptance of the coupling transformer kn A: State matrix connected between buses k and n B: Input matrix G , B : Conductance and susceptance between buses k kj kj ΔE : Variations of the internal voltage in quadrature and j q ΔE : Variations of the generator’s ﬁeld voltage I : Current phasor injected at node k fd ΔP , Variations of the active and reactive power I : Current phasor of the load at bus k ΔQ : demands of the loads P , Active and reactive power ﬂows that are km ΔP : Variations of the input mechanical power Q : transmitted from bus k towards bus m km ΔV: Variations of the voltage magnitude at each bus of G G P , Q : Active and reactive power injected by the k k the power system generator at bus k ΔV : Variations of the terminal voltage of generator k L L P , Q : Active and reactive power of the load connected at k k ΔV : Variations of the reference voltage of the AVR ref bus k E : Excitation voltage of generator k fd V : Voltage phasor at bus k K : Gain of the AVR of generator k r V : Phasor of the voltage source T : Time constant of the AVR of generator k. V , V : In-phase and quadrature components of the q p Nomenclature of the PSS Controller voltage source V ΔV , ΔV , ΔV : Intermediary signals of the PSS’s dynamic 1 2 3 x : Reactance of coupling transformer kn model Z : Impedance of the transmission line between buses nm pss ΔV : Output signal from the PSS sup n and m. pss K : Gain of the PSS pss pss pss T , T , T , Time constants of the PSS 1 2 3 Nomenclature of the PI Controllers pss T : ΔV : Variations of the quadrature axis component of p pss : Time constants of the washout block of the series converter the PSS. ΔX : Input signal of a PI controller ref Nomenclature of the PSO pi pi K , K : Gains of a PI controller 1 2 C , C : Acceleration constants of the PSO 1 2 pi pi Time constants of the PI controllers co (t): Cognitive factor of the particle i of the PSO at T , T , 1 2 pi iteration t T : pso g (t): Global best position of the PSO at iteration t X1, X2, Input signals of the PI controllers. best i,pso X3: p (t): Best position of the particle i of the PSO at best iteration t Nomenclature of the SSSC International Transactions on Electrical Energy Systems 21 Disclosure r , r : Random values in the interval [0, 1] 1 2 v: Inertia factor of the PSO +e authors declare the following: (i) this material is the so (t): Social factor of the particle i of the PSO at iteration t authors’ own original work, which has not been previously i,pso v (t): Velocity of the particle i of the PSO at iteration t published elsewhere; (ii) the paper is not currently being i,pso x (t): Current position of particle i of the PSO at considered for publication elsewhere; (ii) the paper reﬂects iteration t. the authors’ own research and analysis in a truthful and Nomenclature of the MOA complete manner; (iv) the results are appropriately placed in β: Visibility coeﬃcient the context of prior and existing research; (v) all sources used δ (t), δ (t): Fixed values in the interval (0, 1) at d d are properly disclosed; (vi) all authors have been personally iteration t and actively involved in substantial work leading to the ρ : Random value in the range (0, 1] paper, and will take public responsibility for its content. ρ : Fixed random value C : Positive attraction constant Conflicts of Interest d: Coeﬃcient of the nuptial dance Female: Female Parent +e authors declare that they have no known conﬂicts of fl(t): Random walk coeﬃcient at iteration t interest or personal relationships that could have appeared g: Gravity coeﬃcient of the MOA to inﬂuence the work reported in this paper. moa g (t): Global best position of the MOA at best iteration t Acknowledgments L: Random value within a certain interval +is work was supported by the Coordination for the Im- Male: Male Parent provement of Higher Education Personnel (CAPES), Fi- i,moa p (t): Best position of the particle i of the nance Code 001, the Brazilian National Council for Scientiﬁc best MOA at iteration t and Technological Development (CNPq), grant 305852/ r: Vector with random values within 2017-5, and the São Paulo Research Foundation (FAPESP), [−1, 1] under grants 2015/21972-6 and 2018/20355-1. r : Cartesian distance between the male mf and female mayﬂies References i,moa r : Cartesian distance between x (t) and g (t) [1] P. 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International Transactions on Electrical Energy Systems – Hindawi Publishing Corporation
Published: Apr 26, 2022
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