Journal of Control, Automation and Electrical Systems (2023) 34:121–136 https://doi.org/10.1007/s40313-022-00942-x Bio-Inspired Metaheuristics Applied to the Parametrization of PI, PSS, and UPFC–POD Controllers for Small-Signal Stability Improvement in Power Systems Elenilson V. Fortes1 · Luís Fabiano Barone Martins2 · Ednei Luiz Miotto3 · Percival Bueno Araujo4 · Leonardo H. Macedo4 · Rubén Romero4 Received: 20 July 2021 / Revised: 12 April 2022 / Accepted: 5 August 2022 / Published online: 16 September 2022 © Brazilian Society for Automatics–SBA 2022 Abstract The firefly algorithm (FA) and the artificial bee colony (ABC) algorithm are used in this study to perform a coordinated parametrization of the proportional-integral and supplementary damping controllers, i.e., power system stabilizers (PSSs) and the unified power flow controller (UPFC)–power oscillation damping set. The parametrization obtained for the controllers should allow them to damp the low-frequency oscillatorymodes in the power system for different loading scenarios. The power system dynamics is represented using a model based on current injections, known as the current sensitivity model, which implies that a formulation by current injections for the UPFC should be formulated. To validate the proposed optimization techniques and the current injection model for the UPFC for small-signal stability, simulations are carried out under two distinct perspectives, namely, static and dynamic analysis, using the New England system. The results demonstrated the effectiveness of the UPFC’s current injection model. Moreover, it was possible to verify that the FA performed better than the ABC algorithm to solve the discussed problem, accrediting both the UPFC current injection model and the FA algorithm as new tools for small-signal stability analysis in electrical power systems. Keywords Artificial bee colony · Current sensitivity model · Firefly algorithm · POD · PSS · UPFC B Elenilson V. Fortes elenilson.fortes@ifg.edu.br Luís Fabiano Barone Martins luis.martins@ifpr.edu.br Ednei Luiz Miotto edneimiotto@utfpr.edu.br Percival Bueno Araujo percival@dee.feis.unesp.br Leonardo H. Macedo leohfmp@ieee.org Rubén Romero ruben.romero@unesp.br 1 Goiás Federal Institute of Education, Science, and Technology, Av. Presidente Juscelino Kubitschek 775, Residencial Flamboyant, 75804-714 Jataí, GO, Brazil 2 Paraná Federal Institute of Education, Science, and Technology, Av. Doutor Tito s/n, Jardim Panorama, Jacarezinho, PR 86400-000, Brazil 3 Federal Technological University of Paraná, R. Cristo Rei, 19, Vila Becker, Toledo, PR 85902-490, Brazil 1 Introduction Electric power networks are increasingly subjected to situa- tions of excessive loading as the demand for electrical energy grows, resulting in unacceptable voltage levels and frequency deviations. These facts, combined with the interconnection of different systems via long transmission lines and the oper- ation of automatic voltage regulators (AVRs) with high gains and low time constants (DeMello & Concordia, 1969), result in the emergence of low-frequency electromechanical oscil- lations capable of compromising power system stability and operation (Anderson & Fouad, 2003). The local (0.7–2.0 Hz) and interarea (0.1–0.8 Hz) electromechanical oscillatory modes are of higher importance Kundur (1994), and must be adequately damped for power systems to operate safely. DeMello and Concordia (1969) were pioneers in studies involving synchronizing and damping torques in weakly- damped power systems. In their article, it is possible to verify 4 São Paulo State University, Avenida Brasil, 56, Centro, Ilha Solteira, SP 15385-000, Brazil 123 http://crossmark.crossref.org/dialog/?doi=10.1007/s40313-022-00942-x&domain=pdf http://orcid.org/0000-0002-0683-3914 http://orcid.org/0000-0002-0862-0576 http://orcid.org/0000-0001-9179-6208 http://orcid.org/0000-0002-2491-6254 http://orcid.org/0000-0001-9178-0601 http://orcid.org/0000-0002-7744-254X 122 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 that the AVRs of fast actuation (with low time constants and high gains) provide significant improvements to the limits of stability of permanent regime for the systems when they are subjected to disturbances. However, under high loading conditions associated with weak transmission systems, the actuation of AVRs becomes inefficient. To mitigate the negative impacts of AVR actuation on power system stability, another controller, called power sys- tem stabilizer (PSS), is added to the synchronous machines’ excitation systems. Its goal is to add electric torque in phase with changes in the rotor’s angular speed (damping torque) (DeMello & Concordia, 1969; Larsen & Swann, 1981). When the PSSs’ control settings are properly parametrized, they perform well, especially for local modes (Fortes et al., 2018). In some situations, however, the PSSs may not be capable of damping the interarea oscillatory modes (Fortes et al., 2018). Flexible ac transmission systems (FACTS) appear to be an appealing alternative because, in addition to improving power system operation (Hingorani, 2000), they can damp interarea oscillations when used in conjunction with a power oscillation damping (POD) controller (Fortes et al., 2018; Miotto et al., 2018). Among the many existing FACTS, the unified power flow controller (UPFC) stands out as one of the most comprehen- sive, allowing the control (simultaneously or not) of active and reactive power flows in the transmission line in which its series converter is installed, as well as controlling the voltage on the bus to which the shunt converter is connected (Eslami et al., 2014). For the PSSs and the UPFC–POD controllers to perform satisfactorily, i.e., for them to be able to damp the local and interarea modes, their correct parameterization is essential, guaranteeing the stability of the power system. To achieve this goal, different approaches have been used over decades of research, including the residue method (Yang et al., 1998), the Nyquist stability criterion (Zhenenko & Farah, 1984), and the decentralizedmodal control (DMC)method (Valle & Araujo, 2015). Theparametrizationof these devices in a coor- dinated and simultaneous fashion is a difficult multimodal optimization problem. As a result, in addition to traditional control theory techniques, metaheuristics have been utilized to address optimization problems in recent studies in this field. In general, in the literature, studies related to small-signal stability analysis in power systems are restricted to the use of PSSs. These are intended to ensure additional damping to the poles of interests present in power systems, as can be seen in Devarapalli and Bhattacharyya (2021); Movahedi et al. (2019); Singh et al. (2019), and Guesmi et al. (2021). It is essential to note, however, that PSSs alone are not always successful in adding extra damping to the poles of interest in power systems, particularly for interarea modes, as observed in Fortes et al. (2016). The use of FACTS together with the POD controller has been an alternative to provide extra damping to the interarea modes, as can be seen in the works of Fortes et al. (2018); Gamino and Araujo (2017); Menezes et al. (2016); Valle and Araujo (2015), and Fortes et al. (2016). The correct parametrization of PSS and/or POD con- trollers in order to add additional damping to the poles of interest present in power systems is a current challenge and has been the focus of study in several works available in the literature. Among the techniques used for this purpose, it is possible to highlight bio-inspired metaheuristics such as the genetic algorithm (GA) (Movahedi et al., 2019; Singh et al., 2019), particle swarm optimization (PSO) (Verdejo et al., 2020), bacterial foraging optimization (BFO) (Miotto et al., 2018), artificial bee colony (ABC) (Fazeli-Nejad et al., 2019), harmony search (Naderipour et al., 2020), and the firefly algorithm (FA) (Naderipour et al., 2020; Singh et al., 2019). In Sabo et al. (2020), three optimization algorithms, i.e., GA, PSO, and a new metaheuristic called farmland fertil- ity algorithm (FFA) were implemented and compared. The objective was to adjust the parameters of the PSSs and verify which of the implemented algorithms has the best perfor- mance. In Rahman et al. (2021), differential evolution (DE), PSO, gray wolf optimizer (GWO), whale optimization algo- rithm (WOA), and chaotic whale optimization algorithm (CWOA) were used in a single machine infinite bus system (SMIB) to adjust the parameters of the proportional-integral (PI) controller. Finally, in Kar et al. (2021), the modified sine cosine algorithm (MSCA) was used to adjust the parameters of PSSs in a SMIB system and a multi-machine system. It should be mentioned that the power system is repre- sented in this study using the current sensitivitymodel (CSM) (Takahashi et al., 2018). The CSM has a significant advan- tage over other linear models available in the literature, for example when compared to the (Heffron & Phillips, 1952) model. Indeed, in the CSM there is no need to maintain the infinite bus nor to reduce the power system to the terminal buses of the generators, which is, therefore, a differential of this work compared to the others mentioned above. Differently from the works mentioned above, this work investigates the performance of the UPFC FACTS for con- trolling the active and reactive power flows, aswell as voltage magnitudes, at its installation terminal buses, in order to improve the voltage profile of the power system for a speci- fied loading range. Additionally, its influence onmaintaining the stability of the system for small-signal disturbances when acting togetherwith supplementary controllers, i.e., PSSs and POD, is evaluated. As a result, a linear model, based on cur- rent injections for the UPFC, which can be used for both static and dynamic analysis is used (Martins et al., 2017). 123 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 123 The currents are calculated by an expanded formulation of the power flow and differently from the model proposed in Kopcak et al. (2007), the presented proposal uses the residues of the current injection equations expressed in polar coordi- nates. The UPFC’s control system is based on PI controllers, similar to the one shown in Valle and Araujo (2015), with the difference that, in this work, it is connected to the shunt voltage source converter (VSC). The parametrization of the supplementary damping con- trollers and the PI controllers of theUPFC is formulated as an optimization problem composed of two objective functions. Differently from what was proposed in Martins et al. (2017), in which a modified ABC algorithm was used to parametrize the supplementary damping controllers, in this work, the parametrization is performed using the ABC algorithm and the FA, and their performances to solve the problem are eval- uated and compared. Two objective functions are considered for evaluating a solution proposal. The first one calculates the minimum desired damping for the oscillatory modes ana- lyzed, while the second one evaluates the distance between the desired and the calculated eigenvalues of interest. The optimization of these functions determines the parametriza- tion of the controllers, which will be able to simultaneously allocate all the electromechanical oscillation modes of the power system in a predetermined region of the left half-plane of the complex plane, guaranteeing the stability of the power system. The New England multimachine system (Fortes et al., 2016) considering multiple loading scenarios is used to evaluate the performances of the ABC algorithm and the FA. The main contributions of this work are: (i) Todeduce andpresent a current injectionmodel forUPFC to be used in the CSM for small-signal stability studies; (ii) To implement the deduced model for the UPFC and per- form a static analysis of its operation in order to validate the proposed model; (iii) To deduce and implement the dynamic models for PI, PSSs, and the UPFC–POD controllers in the CSM for multi-machine systems; (iv) To present and implement theABCalgorithm and the FA, so that these algorithms can perform, in a coordinated and simultaneous way, the adjustments of the parameters of the PI, PSSs, and the UPFC–POD controllers; (v) To validate the ABC algorithm and FA as tools in the study of small-signal stability in power systems. The remainder of this work is organized as follows: Sect. 2 describes the modeling of the electric power system, Sect. 3 presents the parametrization techniques for supplementary damping controllers, in Sect. 4 the problem formulation is described, with emphasis on the objective function used to evaluate a solution proposal. Section 5 presents the simula- Fig. 1 Balance of the current at bus k tions and results and Sect. 6 presents the conclusions of the work. 2 The Electric Power SystemModeling Subsections 2.1–2.5 present the main equations that repre- sent the expanded power flow modeled by current injections in the polar form. Besides that, the current injection model for the UPFC FACTS and its control structure, based on PI controllers, are also presented. 2.1 Expanded Current InjectionModel The residue of the currents injected at bus k, � Îk , is deter- mined from the balance between the specified, Î spek , and calculated, Î calck , current phasors, according to (1) and as illustrated in Fig. 1. � Îk = Î spek − Î calck = ÎGk − ÎLk − ∑ m∈�k Îkm = 0 (1) The variables ÎGk , ÎLk , and Îkm , shown in (1), represent, respectively, the phasors of the currents generated and con- sumed at bus k and the current between buses k and m. The set of all neighboring buses of bus k is denoted by �k . Equation (1) can be reformulated according to the active and reactive powers specified in the conventional power flow equations, Pspe k and Qspe k , together with the voltage phasor at bus k, V̂k , resulting in (2). � Îk = Pspe k − j Qspe k V̂k − ∑ m∈�k Îkm = 0 (2) Equations (3) and (4) define, respectively, the magnitude and angle of the voltage at bus k, Vk and θk , and the current 123 124 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 phasor Îkm . V̂k = Vk (cos θk + j sin θk) (3) ∑ m∈�k Îkm = ∑ m∈K Vm (Gkm + j Bkm) (cos θm + j sin θm) = Î calck (4) In (4), the conductance and the susceptance between buses k andm are, respectively,Gkm and Bkm . The set�k represents all the buses m connected to bus k, while K represents the set formed from the union of the set �k and the bus k itself. Replacing (3) and (4) in (2) and separating into real,�Irk , and imaginary, �Iik , parts, the components of the current residues are obtained for bus k, as shown in (5) and (6). �Irk = 1 Vk ( Pspe k cos θk + Qspe k sin θk ) − ∑ m∈K Vm (Gkm cos θm − Bkm sin θm) = 0 (5) �Iik = 1 Vk ( Pspe k sin θk − Qspe k cos θk ) − ∑ m∈K Vm (Gkm sin θm + Bkm sin θk) = 0 (6) The matrix formulation (7) is derived by using the Newton-Raphson method in (5) and (6). It is a linearized system that is utilized to calculate the algebraic variables. [ �Ir �Ii ] ︸ ︷︷ ︸ �I = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ J411pf︷ ︸︸ ︷ J412pf︷ ︸︸ ︷ ∂�Ir ∂θ ∂�Ir ∂V ∂�Ii ∂θ ∂�Ii ∂V ︸ ︷︷ ︸ J421pf ︸ ︷︷ ︸ J422pf ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ [ �θ �V ] ︸ ︷︷ ︸ �y (7) 2.2 Current Residue Equations The residues of the active, �Pk , and reactive, �Qk , powers injected at bus k can be expressed as a function of the residues of the real and imaginary components, �Irk and�Iik , of the current injection at bus k, as shown in (8) and (9). �Pk = �Pspe k − �Pcalc k = Vk (�Irk cos θk + �Iik sin θk) (8) �Qk = �Qspe k − �Qcalc k = Vk (�Irk sin θk − �Iik cos θk) (9) (a) (b) Fig. 2 a UPFC’s model; b representation of the series voltage source as a current source In (8) and (9), Pcalc k and Qcalc k are the active and reactive power injections calculated at bus k. In addition, the manipu- lation of (8) and (9) allows to determine (10) and (11). These equations represent the components of the current residues as a function of the active and reactive power residues, which are used in the conventional formulation of the power flow problem. �Irk = �Pk cos θk+�Qk sin θk Vk = 0 (10) �Iik = �Pk sin θk−�Qk cos θk Vk = 0 (11) 2.3 The UPFC’s Current InjectionModel The UPFC is a FACTS that consists of two voltage source converters (VSCs), one shunt and one in series with the trans- mission line, which are connected to the power system via coupling transformers and generate three-phase sinusoidal voltages at the network frequency with controllable ampli- tude and phase angle (Hingorani, 2000). The converters are connected by a direct current link and it has a capacitor bank capable of supplying reactive power to the power system or controlling the voltage of the device’s common installation bus. The UPFC’s current injection model is achieved by mod- ifying the power injection model proposed by Noroozian et al. (1997), as shown in Fig. 2a. In this model, the equivalent circuit of the UPFC connected to the power system is rep- resented by an ideal voltage source, V̂s , connected between buses k and m, in series with a susceptance, bkm , that models the coupling transformer, and a shunt ideal current source, Îsh . To calculate the current injections at each bus, the series voltage source, shown in Fig. 2a and in (12), is converted into a current source, Îs , as illustrated in Fig. 2b and in (13). V̂s = r Vk (θk + ϕ) (12) Îs = −r Vkbkm [(cosϕ sin θk + sin ϕ cos θk) + j (sin ϕ sin θk − cosϕ cos θk)] (13) 123 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 125 (a) (b) Fig. 3 a Transformation of the series current source; b phasor diagram of the UPFC The current source Îs can be represented by using two shunt current sources, as shown in Fig. 3a, in which Î upfck is the current injected by the UPFC into bus k, while Î upfcm is the current injected into bus m. Figure 3b presents a pha- sor diagram (Huang et al., 2000), in which the voltage V̂s is decomposed into two components, one in phase (Vq ) and another in quadrature (Vp) with the voltage V̂k , of the com- mon bus of the installation of the device. The parameter r = V 2 p +V 2 q Vk and the angular phase is ϕ = arctan ( Vp Vq ) . By using (12) and (13) and by analyzing Fig. 3a and b, (14−17) can be obtained. cI upfcrk = Vm Vk bkm ( Vp cos θkm + Vq sin θkm ) cos θk + (Vqbkm + Iq ) sin θk (14) I upfcik = Vm Vk bkm ( Vp cos θkm + Vq sin θkm ) sin θk − (Vqbkm + Iq ) cos θk (15) I upfcrm = −bkm ( Vp cos θk + Vq sin θk ) (16) I upfcim = −bkm ( Vp sin θk − Vq cos θk ) (17) Fig. 4 The UPFC’s current injection model (a) (b) (c) Fig. 5 Control system structure of the UPFC In (14−17), I upfcrk and I upfcik are the real and imaginary components of the current injection at bus k, while I upfcrm and I upfcim are the real and imaginary components of the current injection at bus m. From (14−17) it is possible to obtain a current injection model for the UPFC (see Fig. 4), this representation is appro- priate for both small-signal stability studies and for static analysis. 2.4 Configuration of the UPFC’s Control System For the UPFC to perform the control of the power flows, it is necessary to use PI controllers (Fortes et al., 2016) as illustrated in Fig. 5. This control structure is used tomodulate the controlled variables of the voltage source converters VS1 and VS2, Vp, Vq , and Ip. In Fig. 5, the parameters of the controllers are the gains K u 1 , K u 2 , and K u 3 (p.u.), the time constants T u 1 , T u 2 , and T u 3 (s) and the time constants T pod m and Tm (s), which represent the 123 126 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 Fig. 6 Power balance at the final bus in which the UPFC is installed inherent delay of the control device (Hingorani, 2000). The supplementary signal from the POD controller, V pod sup , is used to modulate the quadrature component, Vp, of the converter VSC1. The complex power controlled by the UPFC, S̄ctrl m = Pctrl m + j Qctrl m , can be obtained by inspection from Fig. 6, by calculating the nodal power balance at bus m, as shown in (18). S̄ctrl m = S̄upfc m − S̄mk (18) In (18), S̄upfc m and S̄mk represent, respectively, the complex power injected by the UPFC at bus m and the complex power flow from bus m to bus k. The decomposition of the complex power flow (18) in its real and imaginary components makes it possible to determine the active, Pctrl m , and reactive, Qctrl m , power flows controlled by the UPFC. By analyzing Fig. 5, by inspection, the dynamic perfor- mance of the UPFC control structure is obtained, which is represented by (19−24). V̇p = K u 1 Tm ( P ref m − Pctrl m )+ 1 T pod m ( X1 − Vp − V pod sup ) (19) Ẋ1 = K u 1 T u 1 ( P ref m − Pctrl m ) (20) V̇q = K u 2 Tm ( Qref m − Qctrl m )+ 1 Tm ( X2 + Vq ) (21) Ẋ2 = K u 2 T u 2 ( Qref m − Qctrl m ) (22) İq = K u 3 Tm ( V ref k − Vk )+ 1 Tm ( X3 − Iq ) (23) Ẋ3 = K u 3 T u 3 ( V ref k − Vk ) (24) In (19−24), P ref m , Qref m , and V ref k are, respectively, the specified values of the active and reactive power flows on line km and the voltage magnitude at bus k. 2.5 Inclusion of the UPFC in the Current Injection Model In order to assess the UPFC’s performance, the respective equations that model the device must be included in the pro- posed formulation. For this to occur, it must be assumed that the state variables of the model are constant with respect to time, which facilitates their inclusion in the Newton– Raphson algorithm, according to the formulation proposed in Kopcak et al. (2007). As the purpose of this article is to model the power flow by current injections, the model should satisfy all current residues at the common installation buses of the UPFC, i.e., buses k and m shown in Fig. 4. This requirement is shown in (25) and (26). Îk = Î spek − Î calck + Î upfck (25) Îm = Î spem − Î calcm + Î upfcm (26) In (25) and (26), Î upfck and Î upfcm are the phasors repre- senting the current injections at buses k and m, in which the UPFC is installed, whereas, at the other buses of the power system, the residues of the components of the currents do not differ from the conventional power flow formulation. After linearizing the system formed by the equations resulting from the replacement in (25) of the current injec- tions of the UPFC and of the current residues defined, respectively, in (14–17) and (5–6) and by the set of dynamic equations (ẋu = fu(xu, y)) represented in (19–24), thematrix formulation by current injection for the UPFC is obtained, as shown in (27). (27) 123 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 127 (a) (b) Fig. 7 Control structure of a the PSSs and b the POD controller In (27), �xu = [�Vp �X1 �Vq �X2 �Iq �X3 ]t is the vector of the linearized state variables for the PI con- trollers of the UPFC, with the matrix J4u defined in (28). (28) 2.6 Dynamic Models for the PSS and POD Controllers The supplementary damping controllers used to damp the low-frequency oscillations in the power system are the power system stabilizers (PSSs) and the power oscillation damp- ing (POD). Their structures are presented in Fig. 7a and b, respectively. As it can be analyzed in Fig. 7a, the PSS is coupled to the AVR, which is represented by a gain Krk , a time constant Trk , the terminal voltage �Vk , the reference voltage �Vrefk , and the excitation voltage of the synchronous machine �E fd k . In Fig. 7b the POD controller is coupled to the control loop of the UPFC to modulate �V pod. In the structures shown inFig. 7a andb, there is a great sim- ilarity between the PSS and POD controllers, both differ only by the input and output signals used. The controllers have gains K pss and K pod, which are responsible for determining the amount of damping introduced by them; a washout block that functions as a high-pass filter determined by a time con- stant T pss w for thePSSand T pod w for thePOD,which is adjusted to allow the controller to act only in transient periods; and a phase compensation block represented by the time con- stants T pss 1 (T pod 1 ), T pss 2 (T pod 2 ), T pss 3 (T pod 3 ), and T pss 4 (T pod 4 ) which provides the appropriate phase advance characteristics to compensate the phase delay between the output of the con- trol loopof theAVRand the torqueproducedby the generator. Usually T pss 1 (T pod 1 ) = T pss 3 (T pod 3 ) and T pss 2 (T pod 2 ) = T pss 4 (T pod 4 ) (Kundur, 1994). For the input and output signals shown in Fig. 7a and b, the following configurations are adopted: the input signal used for the PSS is the angular speed, �ωk , of the rotor of the generator k, whereas for the POD it is the active power flow, �Pkm , on the transmission line adjacent to the installation of the UPFC–POD; the output signal for the PSS is the voltage �V pss sup added to the control loop of the AVR, as shown in Fig. 7a, while the output signal for the POD is the voltage V sup pod , added to the control loop of VSC1, as shown in Fig. 5a. An analysis of Fig. 7a and b allows to infer (29)− (32) and (33−36) representing, respectively, the dynamic behaviors of the PSS and POD controllers. �V̇1k = �ω̇k K pss k − 1 T pss ω �V1k (29) �V̇2k = 1 T pss 2 �V1k + T pss 1 T pss 2 �V̇1k − 1 T pss 2 �V2k (30) �V̇ pss supk = 1 T pss 4 �V2k + T pss 3 T pss 4 �V̇2k − 1 T pss 4 �V pss supk (31) �Ė fd k = Krk Trk ( �V pss supk + �V ref k − �Vk )− 1 Trk �E fd k (32) �Ẏ1 = − 1 Twp ( K pod�Pkm + �Y1 ) (33) �Ẏ2 = − 1 T pod 2 [( 1 − T pod 1 T pod 2 ) ( K pod�Pkm + �Y1 )− �Y2 ] (34) �Ẏ3 = − 1 T pod 4 {( 1 − T pod 3 T pod 4 ) [�Y2 + T pod 1 T pod 2 ( K pod�Pkm + �Y1 )] − �Y3 } (35) �V̇p = 1 T pod m { K u 1 ( �P ref m − �Pctrl m ) + (�X1 − �Vp ) − { �Y3 + T pod 3 T pod 4 [�Y2+ T pod 1 T pod 2 ( K pod�Pkm + �Y1 )]}} (36) 123 128 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 2.7 Current Sensitivity Model A linear model, called CSM, based on Kirchhoff’s current law is used to represent the power system (Fortes et al., 2018). An interesting feature of this model is the maintenance of all buses of the system in the model, what makes it possible the integration of FACTS–POD devices and PSS controllers in the network. The dynamics of a multimachine power system with ng generators, nb buses, np PSSs, and a UPFC–POD set is described by (37) − (44). [�xcsm] = [[�ω1 · · ·�ωng ] [ �δ1 · · · �δng ] [ �E ′ q1 · · · �E ′ qng ] [ �Efd1 · · ·�Efdng ]]t (37) [ �xpss ] = [[�V11 · · ·�V1np ] [ �V21 · · · �V2np ] [ �V pss sup1 · · · �V pss supnp ]]t (38) [ �xpod ] = [ �Y1 �Y2 �Y3 �V pod sup ]t (39) [�xT ] = [ �xg1 �xpss1 �xu �xpod · · · �xgng �xpssnp ]t (40) [�ucsm] = [[ �Pmec 1 · · · �Pmec ng ] [ �V r 1 · · · �V r ng ] [ �PL1 · · · �PLng ] [ �QL1 · · ·�QLng ]]t (41) [�uu] = [ P ref m P ref m Qref m Qref m V ref k V ref k ]t (42) [ �y ] = [[�θ1 · · · �θnb] [�V1 · · · �Vnb]] t (43) [ �ẋT 0 ] = ⎡ ⎢⎣ J1T J2T J3T J4pf + J4u︸ ︷︷ ︸ J4T ⎤ ⎥⎦ [ �xT �y ] + [ B1T B2u ] [ �uu ] (44) In (44), the submatrices J1T , J2T , J3T , and J4T associate the state variables with the algebraic variables of the system. The state variables are: the angular speed, �ω, the internal angle of the rotor, �δ, the internal quadrature voltage, �E ′ q , and field voltage of the generators, �Efd, the control param- eters of the converters VS1 and VS2 of the UPFC, �Vp, �Vq , and�Ip, the auxiliary variables,�X1,�X2, and�X3, the variables of the PSSs, �V 1k , �V 2k , and �V pss, and the variables of the POD controller, �Y1, �Y2, �Y3, and �V pod. The algebraic variables are the magnitude, �V , and the phase, �θ , of the voltages at the buses of the power sys- tem. The input variables are the mechanical power, �Pmec, the reference voltage of the AVR,�V r , the active,�PL , and reactive,�QL , loads, the active,�P ref m , and reactive,�Qref m , power references, and the voltage reference,�V ref k , of the PI controllers of the UPFC, present in the sub-matricesB1T and B2u . The state space representation is obtained by eliminating �y from the system defined in (44), which results in the state matrices A = J1T − J2T J4 −1 T J3T and the input B = B1T − J2T J4 −1 T B2u . 3 Parametrization Techniques for the Supplementary Damping Controllers The ABC algorithm and the FA, which were used to parametrize the PI and supplementary damping controllers, are described in depth in this section. 3.1 FA The FA was proposed by Yang in 2007 (Watanabe, 2009; Yang, 2008). The algorithm was conceived from the analysis of the frequency and intensity of the light emitted by these insects. In the FA, it is assumed that the initial population consists of N P candidate solutions to the optimization problem. Each solution describes the position of a firefly, which is repre- sented by a vector of dimension d containing the variables of the problem. Each solution zi = [zi1 zi2 · · · zid ] has a cor- responding objective function value F(zi ), i = 1, · · · , N P that corresponds to its quality. The brightness of eachfirefly is proportional to its corresponding objective function (quality) which, together with their attractiveness factor (β), dictates how strong the attraction of other members of the swarmwill be. Also, other constants such as the maximum attraction value (β0), the absorption coefficient (γ ), which determines the variation in attractiveness with increasing distance, and a randomization parameter (α) are also part of the FA and must be defined by the operator. Two key components of the FA must be specified before it can be implemented: the variation of light intensity and the formulation of the attractiveness factor. In the standard algorithm, the light intensity (I ) of a firefly used to repre- sent a solution zi is proportional to the value of its objective function. On the other hand, the luminous intensity I (ri j ) is defined according to (45). I (r) = I0 e −γ r2i j (45) In (45), I0 is used to indicate the source’s light intensity, and the light absorption is estimated using the absorption coefficient γ . The attractiveness factor β of the firefly is proportional to the luminous intensity of its flash, being, therefore, possible to represent it using (46). β = β0 e −γ r2i j (46) 123 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 129 In (46), β0 is the attractiveness for ri j = 0 and γ is the light absorption coefficient of the propagation medium. In the classical FA, the Euclidean distance ri j , between two fireflies, zi and z j , is expressed as shown in (47). ri j = ∥∥zi − z j ∥∥ = √√√√ d∑ k=1 ( zik − z jk )2 (47) In (47), d indicates the dimension of the problem. The movement of the i th firefly is directed to the brightest (most attractive) firefly j as shown in (48). zi = zi + β0 e −γ r2i j ( zi − z j )+ αεi (48) In (48), εi is a vector of random numbers generated using a Gaussian distribution. The firefly’s motions are made up of three terms: the current position of the i th firefly, the attrac- tion to the most attractive individual, and a random walk, consisting of a randomization parameter α (random number, considering a uniform distribution, in the interval [0, 1]). 3.2 ABC Algorithm Artificial bees are divided into three categories by the ABC algorithm: workers, observers, and explorers. Worker bees make up half of the colony, while observation bees make up the other half. Worker bees save knowledge about their food sources’ surroundings and transmit it on to observer bees, who prefer to choose the best sources among those supplied by workers. Then they focus their search on the food source they have chosen. The explorer bees are worker bees that quit their food sources at random and seek out new ones. The ABC algorithm’s stages are outlined below. 3.2.1 Population Initialization The initial population of potential solutions consists of SN food source locations. An initial food source, zi = [zi1 zi2 · · · zid ], is a randomly generated d-dimensional vec- tor that represents the optimization parameters, as shown in (49). zi j = zmin i j + εi j ( zmax i j − zmin i j ) (49) In (49) i = {1, 2, · · · , SN } and j = {1, 2, · · · , d}. On the other hand, zmin i j and zmax i j are, respectively, the lower and upper bounds of each optimization parameter and εi j is a random number generated in the range [0, 1]. Each of the SN food sources will be occupied by a worker bee and the value of F(zi ) (objective function) must be evaluated. 3.2.2 Bee Phase Initialization Using the search equation (50), each worker bee positioned at zi produces a new food source vi in the neighborhood of its present position. vi j = zi j + ( zi j − zk j ) (50) In (50), k ∈ {1, 2 · · · , SN } and j ∈ {1, 2, · · · , d} are indices randomly chosen, with k different from i , while i j is a random number between [−1, 1]. After determining vi , it will be evaluated and compared to zi . A greedy selection process is employed between old and new candidate solutions. If the objective function of vi is better than or equal to the objective function of zi , then vi will replace zi in the population. Otherwise, zi will be maintained. 3.2.3 Probabilistic Selection After the worker bees have completed their search, they tell the observation bees, who are in charge of the hive’s dancing area, of the best food sources they have discovered. From the information passed on, the observer bees memorize the places where these food sources, with the largest amounts of nectar, can be found. Then, all information presented by the worker bees is processed, and a food source is selected based on the probability associated with its amount of nectar, as presented in (51). pi = F(zi ) B N∑ n=1 F(zn) (51) In (51), the roulette-wheel selection method is used. Fur- thermore, it is possible to verify that the higher the value of F(zi ), the greater the probability that the i th food source is selected by the algorithm. 3.2.4 Observer Bees Phase According with the probability value pi , as presented in (51), a new food source zi can be selected. Once the food source is selected, a modification occurs in (50). If the amount of nectar found in the new food source (vi ) is greater than or equal to that obtained in the previous step (zi ), it will replace zi and will join the population as a new member. 3.2.5 Explorer Bees Phase If the newly selected food source represented by zi is not able to be improved after a pre-defined number of iterations, 123 130 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 Fig. 8 Region of interest for the eigenvalues this food source must be abandoned. If this happens, the corresponding bee will stop being a worker and will become an explorer, this will allow to randomly generate a new food source, as described in (49). 4 Problem Formulation For a power system to present a safe margin of operation for small-signal disturbances, the damping coefficients, ξi , of its low-frequency oscillatory modes must be sufficiently high for any operating point within its typical loading range. Therefore, the objective function F(zi ) should allow the opti- mization algorithm to conduct the search in the solution space of the problem and, consequently, determine an adjustment for the parameters of the supplementary damping controllers under different loading conditions. This ensures a minimum desired damping, ξdes, for any operating point within the specified range. Also, it must ensure that the frequencies of the oscillatory modes of interest,ωcalc, obtained at the end of the parametrization process, do not show significant differ- ences in comparison to those frequencies thatwere calculated without the use of controllers. In this sense, two objective functions, F1(zi ) and F2(zi ), will be defined, associated with the main objective function F(zi ), which will be responsi- ble, respectively, for the minimum desired damping and for the frequencies of the oscillatory modes of interest present in the power system. Thus, for each iteration of the optimization algorithm, the eigenvalues of the simulated system must be selected and allocated within a specific region in the complex plane, as shown in Fig. 8. From these considerations, at each iteration, two matrices will be defined, with dimensions q × p and n × p. The first one will consist of the q damping coefficients of the system and the second of the n eigenvalues of interests, λcalc, both at the p specified operation points. These matrices will be used Fig. 9 Representation of a solution proposal as input values to evaluate the functions F1(zi ) and F2(zi ), defined in (52) and (53). F1(zi ) = p∑ j=1 ∣∣∣ξdes − min(ξ j ) ∣∣∣ (52) F2(zi ) = p∑ j=1 n∑ i=1 ∣∣∣λdesi j − λcalci j ∣∣∣ (53) For the proposed optimization problem, the function F(zi ), composed of F1(zi ) and F2(zi ), is properly defined as shown in (54), in which η1 = η2 = 1 are weighting coef- ficients empirically chosen. minimize F(zi ) = η1F1(zi ) + η2F2(zi ) (54) A power system with ng generators equipped with np PSSs and a UPFC–POD was considered. The optimization algorithms must provide a solution with a set of gains and time constants for each PSS controller, for the POD, and for the PI controllers of the UPFC. In this way, the obtained solution is represented by a vector, zi , with the parameters of ns supplementary damping controllers, as shown in Fig. 9. 5 Test and Results This section presents the results obtained from the simu- lations carried out using the algorithms presented in Sect. 3 applied to the coordinated parametrization of the supple- mentary damping controllers and PI controllers of the UPFC considering different operational scenarios. The proposed models and optimization algorithms were implemented in MATLAB R2019a using a computer with a 3.20 GHz Intel® CoreTM i7-8700 processor and 32 GB of RAM. 5.1 Analysis of the New England System Static and dynamic analysis of the New England system (Fortes et al., 2016) are conducted. The single-line diagram of this system is shown in Fig. 10. The UPFC–POD set is allocated between buses 37 and 38 in one of the transmission lines that connect areas 1 and 2. This location is justified by its proximity to the region that has the worst voltage levels of the system. Also, the transmission line that connects buses 123 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 131 Fig. 10 Single-line diagram of the New England system 37 and 38 has a high inductive reactance compared to the other lines, which allows a greater margin of reactive com- pensation by the UPFC in addition to introducing additional damping to the interarea mode. The value of the coupling reactance of the transformer connecting the VSC1 to the transmission line (positioned between buses 37 and F1) is 0.01 p.u. The values of the parameters of the UPFC control structure (K u 1 , K u 2 , K u 3 , T u 1 , T u 2 , and T u 3 ) will be determined by the optimization algo- rithms and T pod m is 0.001 p.u. When the UPFC is allocated in the power system but does not actuate over the voltage and the power flows (a situation that will be called the base case), the variables related to the VSCs present values equal to zero. In thiswork, five different loading levels are defined (80%, 90%, 100%, 110%, and 120%) in relation to the specified nominal load level for this system. For this operating range, the voltage profile on the buses of the system is presented in Fig. 11. By analyzing Fig. 11a it is possible to infer that, in the condition of higher loading (120 %), the power system has voltage magnitudes at buses 11, 12, 13, 14, 15, 33, 34, 35, 36, and 37 outside the acceptable operating limits (0.95 – 1.05 p.u.). This violation can cause voltage instability if a major disturbance occurs. In this operating condition (with- out power flow control) the power flow on line F1 − 38 is −19.05 − j138.64 MVA. To solve the voltage problem, the UPFC was installed between buses 37 and F1 and, after its actuation in themanagement of the powerflowon line F1−38 (−75 − j150 MVA) and in the voltage magnitude at bus 37 (1.0 p.u.) we have a new voltage profile for the system, shown in Fig. 11b. By analyzing Fig. 11b it can be verified that the undervoltage problem at the marked buses in the high loading condition was corrected, and the voltage pro- file remains within the recommended range. In this operating condition, the control variables of the UPFC assume the val- (a) (b) Fig. 11 Voltage profile in the New England system for the specified loading range a without and b with the actuation of the UPFC ues of Vp = −0.097 p.u., Vq = −0.094 p.u., and Iq = 3.775 p.u. For the base case operating at nominal loading, the eigen- values of interest, obtained from the state matrix of the power system, are shown in Table 1, as well as the oscillation fre- quencies, ωni , and the associated damping coefficients, ξi . The eigenvalues shown in Table 1 indicate that the sys- tem has, for this point of operation, nine oscillatory modes, of which eight are local (L1–L8) and one is interarea (I1). Moreover, three of the local modes are unstable (L1–L3) while the other ones have low damping rates (less than 4 %). This analysis allows characterizing the system with an oscil- 123 132 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 Table 1 Dominant eigenvalues, undamped natural frequencies, and base case damping coefficients for nominal loading Modes Eigenvalues ξi (p.u.) ωni (Hz) L1 0.1550 ± j5.9610 −0.0260 0.9490 L2 0.1150 ± j6.3970 −0.0180 1.0183 L3 0.0749 ± j6.9225 −0.0108 1.1018 I1 −0.0068 ± j3.5036 0.0020 0.5576 L4 −0.1108 ± j6.5207 0.0170 1.0380 L5 −0.1880 ± j8.3228 0.0226 1.3249 L6 −0.2423 ± j8.3737 0.0289 1.3333 L7 −0.2116 ± j7.2181 0.0293 1.1493 L8 −0.2748 ± j8.1546 0.0337 1.2986 latory instability for the operating condition in the nominal loading. To correct the oscillatory instability verified in the power system, nine PSSs are installed at generators G1–G9 and a POD controller is installed at the UPFC. The local input signal adopted for the POD is the variation of the active power that flows through the line 37 − 34. 5.2 Evaluation of the Performances of theMethods To define the adjustment of the supplementary controllers, five operating points will be evaluated within the established loading range (80–120%) with the UPFC actuating and con- sidering a minimum damping of 15% (ξdes = 15%). For both algorithms the maximum number of iterations is equal to 1000, and the population size for both of them was 20. For the parametrization of the FA, the following values were considered: β0 = 2.0, α = 0.5, and γ = 1.0. In both algorithms, a set of constraints was imposed on the adjustment of time constants (seconds) and gains (p.u.), as shown in Table 2. To evaluate the performanceof theABCalgorithmandFA, 25 tests were performed for each of them, using the objec- tive function defined in (54). The data presented in Table 3 was obtained from two types of analysis. The first is spe- cific to the set of tests that produced adjustments that led the power system to the desired damping, i.e., that returned ξmin = ξdes ± 10%, which is equivalent to a minimum damping within the range of [14.85%, 15.15%]. The second includes all the tests and offers a general statistical overview of the results determined by each algorithm after 1000 itera- tions. The analysis of the data presented in Table 3 indicates that the adjustment obtained by the FA leads the power system to operate within the specified damping range in 20 of the 25 tests performed while the ABC algorithm was successful in only 14 tests. According to Table 3, when analyzing the statistical results of the set of minimum damping produced by all the tests (after 1000 iterations), it appears that the FA performs slightly better than theABCalgorithm. The average computational times of the ABC algorithm and FA to solve the problem considering ξmin = ξdes ± 10% were 0.38h and 0.17h, respectively. Figure 12 shows the average evolutions of the minimum damping coefficient of the power system, with a confidence interval of 95%, as a function of the iteration number, deter- mined by (a) the ABC algorithm and (b) the FA, respectively. Table 2 Bounds for the PI, PSSs, and POD control parameters Parameters T pod 1 T pod 2 K pod T u 1 T u 3 K u 1 K u 3 T pss 1k T pss 2k K pss k Bounds Minimum 0.01 0.25 0.10 0.001 0.01 0.001 1.00 0.50 0.01 1.00 Maximum 0.25 0.50 0.50 0.100 1.00 0.500 10.0 1.50 0.50 15.00 Table 3 Performance comparison of algorithms for 25 trials Minimum Comparative Method ABC FA damping (ξmin) performance criteria ξdes ± 10% 1 Number of solutions with ξmin = ξdes ± 10% 14 20 2 Number of iterations Minimum 359 21 Mean 381 176 Maximum 899 981 Any 3 Damping coefficient (p.u.) Best 0.1503 0.1500 Worst 0.1317 0.1320 Mean 0.1457 0.1478 Median 0.1490 0.1500 Standard deviation 0.0056 0.0047 123 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 133 (a) (b) Fig. 12 Average evolution of the damping coefficients with a the ABC algorithm, and b with the FA It corroborates the conclusions previously obtained about the necessity of a smaller number of iterations, on average, for the convergence of the FA. Figure 13 shows, for each of the five loading levels evalu- ated, the average minimum damping and its respective error (represented by the standard deviation), considering the set of 25 tests performed (after 1000 iterations) and the overall average damping of these five levels (dashed line). When analyzing Fig. 13, it is reasonable to conclude that the FA performed better than the ABC algorithm, since its adjustments generated better minimum damping for most of the loading levels considered. Besides, the averageminimum (a) (b) Fig. 13 Range of the minimum damping coefficients for different load- ing scenarios with a the ABC algorithm, and b with the FA damping of the five loading levels analyzed (dashed line) was closer to the desired damping (ξdes). To define which of the optimization algorithms is themost efficient (under the conditions established in this work) for the task of determining suitable adjustment parameters of the PSSs, UPFC–POD, and PI controllers for the New England test system, by specifying minimum damping (considering the five loading levels), for low-frequency oscillatorymodes, four criteria were used (the first two can be seen in Table 3 and the third in Fig. 13: (1) higher number of adjustments that guarantee minimum damping within the range between 14.85% and 15.15%; (2) damping coefficients closer to the 123 134 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 Table 4 Gains (p.u.) and time constants (s) for the PSSs parametrized with the FA Parameters PSS G1 PSS G2 PSS G3 PSS G4 PSS G5 PSS G6 PSS G7 PSS G8 PSS G9 T pss 1 = T pss 3 1.0173 0.9374 0.7131 0.8955 0.5632 0.8197 0.8117 0.6674 0.8410 T pss 2 = T pss 4 0.0337 0.1227 0.1517 0.0140 0.2587 0.0977 0.2837 0.0848 0.4081 K pss 10.5685 5.3601 4.2186 8.9316 4.5757 9.8772 1.0005 4.5640 7.7345 Table 5 Gains (p.u.) and time constants (s) for the PI and POD controllers parametrized with the FA Parameters of the UPFC–POD Parameters of the PIs T pod 1 = T pod 3 T pod 2 = T pod 4 K pod T u 1 = T u 2 T u 3 K u 1 = K u 2 K u 3 T pod m 0.1457 0.4551 0.3787 0.0597 0.0128 0.0206 5.6975 0.0010 Table 6 Dominant eigenvalues, undamped natural frequencies, and damping coefficients for the base case under nominal loading Modes Eigenvalues ξi (p.u.) ωni (Hz) L1 −0.9971 ± j5.9137 0.1663 0.9545 L2 −1.1030 ± j6.3485 0.1712 1.0255 L3 −1.0934 ± j6.9435 0.1555 1.1187 I1 −0.5808 ± j3.4849 0.1644 0.5623 L4 −1.2572 ± j6.6991 0.1845 1.0848 L5 −1.2987 ± j8.2684 0.1552 1.3321 L6 −1.2815 ± j8.4328 0.1502 1.3575 L7 −1.1366 ± j7.4300 0.1512 1.1963 L8 −1.6010 ± j7.9791 0.1967 1.2952 specified values (considering the 25 tests), i.e., with the best average and the smallest standard deviation; and (3) a global average of the minimum damping for the specified loading levels closest to ξdes. The FA performed better than the ABC algorithm in all the three considered criteria. Thus, it is possi- ble to conclude that the FA has a better performance than the ABC algorithm, being, therefore, accredited as a tool for the parametrization of the supplementary damping controllers. 5.3 Small-Signal Stability Analysis After verifying that the FA presents better performance than the ABC algorithm, the set of parameters used for the adjust- ment of the controllers was obtained in the test that showed the least dispersion for the values of the minimum damping (for the five operating points established), after 1000 itera- tions. The adjusted parameters of the PSSs, UPFC–POD, and PI controllers provided by the FA for the chosen test scenario are shown in Tables 4 and 5. For the nominal load level and after parametrization of the supplementary damping controllers, it is possible to observe from Table 6 that the power system, previously unstable (see Table 1), starts to operate stably within the damping range specified. The region of allocation in the complex plane of the eigenvalues of interest for the two cases studied (Fig. 14) reinforces this conclusion. The highmargin of stability in the face of small-signal dis- turbances provided by the proposed solution can be observed when the mechanical power of the generator G1 undergoes a (step) disturbance of 0.05 p.u. In this situation, the response of the variation in the angular speed of the generator G1 against the reference generator (G2) can be seen in Fig. 15. To evaluate the minimum damping of the power system (with the parametrization of the supplementary controllers according to Tables 4 and 5), in operating conditions different from those considered by the FA in the optimization process, but within the evaluated range (80–120% of loading for the base case), 17 operating points were considered, as it can be seen in Fig. 16. By analyzing Fig. 16, it is possible to conclude that the minimum damping of the power system remains practically unchanged, and around what was specified, even when con- sidering the load variation. On the other hand, there is a clear modification in the range when it comes to the maximum damping of the eigenvalues of interest, which is logical given that there is no control over its value. Moreover, the obtained parametrization provides high damping in all cases, demonstrating the effectiveness and robustness of the solution obtained by the FA. 6 Conclusions This article evaluated the performance of the artificial bee colony (ABC) algorithm and the firefly algorithm (FA) for conducting a coordinated parametrization of the proportional-integral (PI) controllers of the unified power flow controller (UPFC) and supplementary damping con- trollers. A current injection model for the UPFC was also presented, which can be utilized in both static and dynamic studies in power systems. Static analysis evaluated the performance of the proposed model for the UPFC installed in the New England system. 123 Journal of Control, Automation and Electrical Systems (2023) 34:121–136 135 Fig. 14 Location of the eigenvalues of interest after the parametrization of the PSSs, PI, and UPFC–POD controllers Fig. 15 Angular speed variation �ω1 − �ω2 Fig. 16 Minimumandmaximumdamping of the eigenvalues of interest as a function of the loading of the power system It was possible to verify, from the simulations, significant improvements in the voltage profile and an increase in the voltage stability margin of the power system. It was also concluded that the UPFC actuated effectively in the control of both the active and reactive power flows in the respective installation buses. Regarding the dynamic analysis performed, the two optimization algorithms had their performance compared through statistical indicators. The objective was to carry out a robust parametrization of the PI and supplementary damping controllers, considering different loading scenarios. From the obtained results, it was possible to conclude that the FA was more efficient than the ABC algorithm according to the cri- teria evaluated during the tests performed, showing that the system remained stable, even after loading variation, with damping greater than the specified in the project, demonstrat- ing the robustness of the parametrization provided by the FA. Therefore, it is possible to accredit the FA as a powerful tool in the study of small-signal stability in electric power sys- tems. Future works will include the modeling of intermittent renewable generation, such as wind and photovoltaic, in the formulation of the problem. Declarations Conflict of interest The authors declare that they have no known com- peting financial interests or personal relationships that could have appeared to influence the work reported in this paper. Ethical Approval The authors declare that: (i) this material is the authors’ own original work, which has not been previously published elsewhere; (ii) the paper is not currently being considered for publi- cation elsewhere; (ii) the paper reflects the authors’ own research and analysis in a truthful and complete manner; (iv) the results are appropri- ately placed in the context of prior and existing research; (v) all sources used are properly disclosed; (vi) all authors have been personally and actively involved in substantial work leading to the paper, and will take public responsibility for its content. References Anderson, P., & Fouad, A. A. (2003).Power system control and stability (2nd ed.). Piscataway: Wiley-IEEE Press. A John Wiley & Sons. DeMello, F. P., & Concordia, C. 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