A hybrid algorithm to tune power oscillation dampersfor FACTS devices in power systems

M.F. Castoldi a, D.S. Sanches a, M.R. Mansour b, N.G. Bretas b, R.A. Ramos b,n

a Universidade Federal Tecnologica do Parana, Cornelio Procopio, PR, Brazilb Escola de Engenharia de Sao Carlos, Universidade de Sao Paulo, Sao Carlos, SP, Brazil

a r t i c l e i n f o

Article history:Received 28 September 2012Accepted 4 November 2013Available online 6 December 2013

Keywords:Power system stabilizersPower system stabilityFACTS devicesEvolutionary algorithmsGradient descent

a b s t r a c t

The interaction between electrical and mechanical torques in the synchronous machines connected tobulk power transmission systems gives rise to electromechanical oscillations which, depending on theoperating conditions and type of disturbance, may be poorly damped or even unstable. Recently, acombination of power system stabilizers (PSSs) and power electronic devices known as FACTS (flexiblealternating current transmission systems) has been recognized as one of the most effective alternativesto deal with the problem. Tuning such a combination of controllers, however, is a challenging task evenfor a very skilled engineer, due to the large number of parameters to be adjusted under several operatingconditions. This paper proposes a hybrid method, based on a combination of evolutionary computation(performing a global search) and optimization techniques (performing a local search) that is capable ofadequately tuning these controllers, in a fast and reliable manner, with minimum intervention from thehuman designer. The results show that the proposed approach provides fast, reliable and robust tuning ofPSSs and FACTS devices for a problem in which both local and inter-area modes are targeted.

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1. Introduction

Electric power systems have undergone several transforma-tions in the last years, with a strong focus on more efficient energyconsumption (Palensky & Dietrich, 2011; Yang, Barria, & Green,2011) at the distribution level.

Some transformations have been seen at the generation andtransmission level as well, most of them with the objective toboost production efficiency by the introduction of competition.However, from the control viewpoint, most of the typical struc-tures that control the dynamic response of the system to pertur-bations remain unchanged.

These perturbations may have several different origins (e.g., asudden load increase or a short circuit in a transmission line)and can induce electromechanical oscillations in the powersystem, since the angular speed of the generators oscillates dueto sudden imbalances in their electrical and mechanical torques(Kundur, 1994). To mitigate the impact of such oscillations, themost common type of controller used is the PSS (power systemstabilizer) (Kundur, 1994). In some cases, however, the use of PSSsis not sufficient to guarantee a satisfactory level for a minimumdamping, as will be explained later.

Recently, combinations of PSSs and power electronics devicesknown as FACTS (flexible alternating current transmission systems)have been shown to be an effective alternative to enhance thedamping of electromechanical oscillations in power systems(Hingorani & Gyugyi, 2000). Among themost promising FACTS devicesto perform oscillation damping control is the thyristor controlledseries capacitor (TCSC) (Del Rosso, Canizares, & Dona, 2003), due to itseffectiveness for power flow control in series-connected circuitelements such as transmission lines and to its relatively lower costwhen compared to other high-end technologies.

In TCSCs (as well as in other types of FACTS devices), thefunction of oscillation damping control is performed by a supple-mentary controller know as a power oscillation damper (POD)(Rogers, 2000). The POD acts over the FACTS device much like thePSS acts over the automatic voltage regulator (AVR) in a synchro-nous machine: it modulates its reference value during the tran-sient period to ensure a well-damped response to a disturbance,with its action vanishing after steady-state conditions arereached again.

These controllers are implemented in digital control panels(during the commissioning stage), and their control actions areusually produced via software using microprocessors in digitalcontrol platforms. However, since the sampling frequency (usuallymore than 1 kHz) is much faster than the frequencies of theoscillations that must be controlled (which lie within the rangeof 0.1–3.0 Hz), the design of these controllers can be done entirelyin the continuous-time domain, with the resulting transfer-functions being discretized for implementation.

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n Principal corresponding author. Tel.: þ55 16 3373 9348;fax: þ55 16 3373 9372.

E-mail addresses: ramos@sc.usp.br, rodrigo.ramos@ieee.org,rodrigo.ramos@pq.cnpq.br (R.A. Ramos).

Control Engineering Practice 24 (2014) 25–32

At the middle part of the frequency range mentioned in theprevious paragraph (0.7 to 2.0 Hz) lies the type of electromecha-nical mode in which a single plants oscillates against the remain-ing of the system (Rogers, 2000). This type of oscillation is knownas a local mode, and usually can be dealt with very effectively byPSSs due to their relatively smaller complexity when compared toanother type of oscillation: the inter-area mode. Inter-area modesarise in bulk interconnected power systems when a group ofgenerators, in one area of the system, swings against anothergroup, in a different portion of the same system, and are oftenfound within the range of 0.1–0.7 Hz (Rogers, 2000). This type ofoscillation is much more complex due to the several factors thatplay a role in it: the structure of the transmission system, theoperating conditions and even the interactions among the severalgenerators involved and their respective controlling devices.

When inter-area oscillations arise, as stated before, one of themost effective alternatives to ensure a well-damped response ofthe system to a disturbance is the joint use of PSSs for AVRs of thesynchronous generators and PODs for the FACTS devices. However,since these controllers have to act simultaneously and in acoordinated manner, the problem of tuning their respectiveparameters becomes a challenge for even a highly skilled andexperienced engineer. The main issue in this case is the fact thatthe designer has to keep track of multiple parameters at once, andhumans have notorious difficulties to deal with these types ofmulti-dimensional search problems.

Even having to face these difficulties, the industry still relies onhuman expertise to perform the aforementioned tuning. Classicaltrial-and-error approaches for controller design are still employedfor PSS and POD tuning in utilities and independent systemoperators (ISOs) all over the world. Among the most usedtechniques in these trial-and-error processes are the selectivemodal analysis (Verghese, Perez-Arriaga, & Schweppe, 1982), theresidue analysis (Pagola, Perez-Arriaga, & Verghese, 1989) and theinduced torque coefficients (Pourbeik, Gibbard, & Vowles, 2002).However, the difficulties in handling a large number of operatingconditions and several modes of oscillation while keeping track ofmultiple controller parameters still place a significant burden onthe designer, which usually takes days or weeks to completeits task.

The academy has attempted to deal with this problem in avariety of ways. Approaches relying on H mixed-sensitivity theory(Chaudhuri & Pal, 2004), regional pole placement using linearmatrix inequalities (de Oliveira, Kuiava, Ramos, & Bretas, 2009;Ramos, Castoldi, Rodrigues, Borges, & Bretas, 2009) and geneticalgorithms (Do Bomfim, Taranto, & Falcao, 2000) are examples ofthe many approaches that have been proposed, but none of themhas already reached the status of the technique of choice for PSSand POD tuning by the industry worldwide, at least to theknowledge of the authors.

The cited tuning algorithms use search approaches based oneither local optimization methods or global optimization methodsto pursue their objective. The local optimization methods have thespeed as their main advantage, but they can present convergenceissues or get stuck in a local minimum point (Luenberger, 2003).On the other hand, global optimization methods tend to be slowdue to their many strategies to avoid getting stuck into localminimum points (Luenberger, 2003).

Based on this observation, this work proposes a controllertuning method that uses a hybrid structure mixing the twopreviously mentioned approaches, i.e., a global search methodcombined with a local search method. The method works byinitially setting up a minimum damping ratio to be achieved for alloscillation modes in all operating conditions of interest. Thisminimum damping ratio is defined as the goal of the searchprocedure, and is a user-defined input value. In the sequence, a

global search is performed by an evolutionary algorithm (EA),which starts with an initial set of candidate tunings for thecontroller parameters and works on this set in order to maximizethe resulting minimum damping ratio, until it becomes larger thanthe defined goal.

Due to the previously mentioned reasons, a pure globaloptimization approach could be too slow and, therefore, in ourprocedure the EA is stopped if a threshold (which is smaller thanthe design goal, yet close to it) is reached on the minimumdamping provided by a particular set of tunings provided by theEA. This is an indication that the EA has approached a promisingconvergence basin, and that the design goal can be reached withinthis basin by a local search approach.

At this point the algorithm switches to a gradient descentmethod, using the tuning that provided the best result from theprevious EA and working on it until the design goal is reached. If itis not reached, this is an indication that the local minimumassociated with the present convergence basin is smaller thanthe design goal, and the process has to be restarted with adifferent set of initial tunings. However, from the experiencegained with the application of this proposed hybrid approach,the need for restarting can be seen as a rare event, with the hybridalgorithm providing a satisfactory result in the vast majority of thetests that were carried out.

The paper is structured as follows: Section 2 depicts theformulation of tuning problem in terms of the controller para-meters and Section 3 presents our proposal for a hybrid algorithmthat tunes these controller parameters; the results of the applica-tion of the proposed algorithm to the New England/New Yorkbenchmark model are presented in Section 4, and Section 5finishes the paper with some concluding remarks.

2. Formulation of the tuning problem

The problem of parameter tuning of controllers is typical inlarge-scale industrial systems (Wu, Buttazzo, Bini, & Cervin, 2010).In the power system industry, the modeling process for designingPSSs and PODs is usually based on a set of nonlinear differential-algebraic equations in the form

_~x ¼ f ð ~x; ~u; z; λÞ ð1Þ

0¼ hð ~x; ~u; z; λÞ ð2Þ

~y ¼ gð ~x; ~u; z; λÞ ð3Þwhere ~xARn is the system state, ~uARp is the control input, ~yARq

is the measured output (which can be used for feedback), zARm isa vector of algebraic variables representing the transmission net-work coupling among the states of different generators, and λARl

is a vector of parameters, representing the load levels and otherquantities defining the system operating condition.

The algebraic constraints (2) can be eliminated from (1) to (3)using a linearization of this set of equations, followed by thesubstitution of the linearized algebraic constraints resulting from(2) into the linearized equations resulting from (1) and (3). The setof equations resulting from this process has the form

_xj ¼ AjxjþBjuj ð4Þ

yj ¼ CjxjþDjuj ð5Þwhere xjARn represents a deviation from an equilibrium value of~xj with respect to (1)–(3), obtained for a particular value of theparameter vector λ. Similarly, ujARp and yjARq represent devia-tions from ~uj and ~yj, respectively. In the industry, it is a typicalpractice to select a number of operating conditions, definingdifferent equilibria for ~xj, in such a way that these conditions are

M.F. Castoldi et al. / Control Engineering Practice 24 (2014) 25–3226

representative of the whole set of operating conditions that thesystem may experience, and use these operating conditions toobtain L linearized models in the form of (4) and (5).

To implement both PSSs and PODs for FACTS devices, phasecompensation is the typical control approach used by industrypractitioners. The block diagram of such a structure is shown inFig. 1.

The subscript i in all controller parameters of Fig. 1 denotes thatthis controller is added to the i-th FACTS in the case of a POD andto the ith AVR in the case of a PSS. The middle block in Fig. 1 isknown as a washout filter and guarantees that the controller gainis zero under steady-state conditions. This block possesses aderivative action represented by the term sTtwi, which can beincorporated into the plant to simplify the design formulation.Note that this does not imply that an ideal derivative signal mustbe measured in this plant, since this term can be combined withone of the poles of the controller after the design is carried out,and the result can be implemented as a typical washout block, asexplained in Ramos, Martins, and Bretas (2005). To do so, we candefine a new system output vector yj as

yj ¼ _y ¼ CjxjþDjuj ð6Þ

where C j ¼ CjþAj and Dj ¼ CjBj are matrices with dimensionsdetermined by Aj, Bj and Cj. It is important to remember that Dj

can be a non-zero matrix for some types of FACTS devices due tothe sensibility of the active power flow to variations in the FACTSstate variable.

Considering a controller with two lead–lag blocks (whichcorresponds to n¼2), the transfer function corresponding to theblock diagram in Fig. 1 can be put in the state-space form

_xci ¼�αi 0 0

γi�αiβi �γi 0

βiγi�αiβ2i γi�βiγi �γi

2664

3775xciþ

1βi

β2i

2664

3775yji ð7Þ

uji ¼ ½0 0 Ki�xci ð8Þwhere

αi ¼1Twi

; βi ¼T1i

T2i; γi ¼

1T2i

ð9Þ

In (7) and (8), xciARnci is a vector composed of the statevariables of ith controller. Eqs. (7) and (8) can be written in a morecompact form

_xci ¼ AcixciþBciyci ð10Þ

uji ¼ Ccixci ð11ÞSince we have one of set of Eqs. (10) and (11) for each

controller, we can lump all these nc sets of equations into a singleone, given by

_~xc ¼ ~Ac ~xcþ ~Bcyj ¼Ac1 … 0⋯ ⋱ ⋯0 … Acnc

264

375xcþ

Bc1 … 0⋯ ⋱ ⋯0 … Bcnc

264

375yi ð12Þ

~uj ¼ ~Cc ~xc ¼Cc1 … 0⋯ ⋱ ⋯0 … Ccnc

264

375xc ð13Þ

The final closed-loop models of the controlled power system canbe obtained from the combination of the open-loop system

models given by (4) and (6) with the controller models given by(12) and (13), which can be written in the form _~x j ¼ ~Aj ~xj, where~xj ¼ ½xj ~xc�T and

~Aj ¼Cj

~BjCc

~BcC j~Acþ ~BcDjC c

24

35 ð14Þ

for j¼ 1;…;L. From (14), it becomes clear that the problem oftuning the controllers parameters consists in finding a matrixtriplet ð ~Ac; ~Bc; ~CcÞ ensuring that matrices ~Aj, j¼ 1;…;L, fulfill somedesired performance criterion. The criterion that is most widelyaccepted by the industry, and used in this work, states that theperformance of closed-loop non-linear system can be consideredas satisfactory if all eigenvalues of all matrices ~Aj, j¼ 1;…;L,present a damping ratio greater than a certain pre-specifiedminimum value ξmin.

In this work we first use an EA to tune the controllerparameters until these controllers provide a certain initial targetdamping (close to the final desired minimum damping) to alleigenvalues of the matrices ~Aj; j¼ 1;…;L. After this initial targetdamping is reached, the parameters of the best solution given bythe EA are passed on (as initial conditions) to the second searchmethod used in this work, the gradient descent. The gradientdescent algorithm preserves the phase tunings provided by the EAand adjusts the controller gains to obtain the final desiredminimum damping. This proposed approach, together with itscorresponding algorithm, is explained in detail in the remaining ofthe paper.

3. The proposed combination of search methods

3.1. Evolutionary algorithms

Evolutionary algorithms are search and optimization methodsbased on the evolution mechanisms of the living beings (Eiben &Smith, 2008). There are several types of algorithms that can beclassified as EAs, the most common of them being the geneticalgorithms (Goldberg, 1989) and their corresponding variants(Tang, Yin, Kwong, Ng, & Man, 2011 is an example). In this work,the EA applied is a form of the genetic algorithm, but it will bereferred to as simply EA since this broader terminology is well-suited to avoid confusion with other methods.

Generally, genetic-type EAs work in the following way(Goldberg, 1989): an initial population of chromosomes is gener-ated, where each chromosome represents a possible solution forthe problem. This population is evaluated and each chromosomereceives a fitness value (according to the objective function),which represents the quality of its solution for the problem. Ingeneral, the most able chromosomes are selected for the nextgeneration and the less able chromosomes are discarded.

The selection method must prioritize chromosomes withhigher fitness value, but with no damage to genetic diversity ofthe population. After the selection, a part of the chromosomes canbe subject to modifications through crossover and mutationoperators, generating offspring. It is necessary to specify mutationand crossover rates that will define the probability of chromo-somes to receive such operations (Eiben & Smith, 2008). Thisprocess is repeated until a satisfactory solution is found or somestopping criterion is reached.

There are some important aspects to consider in the searchwith EAs to get a good performance and to adequately cover theregion of interest within the search space. One of these aspects isthe codification of the chromosome because, in general, eachchromosome encodes a solution for the problem and must bewritten in terms of the decision variables of the objective function

Fig. 1. Typical PSS or POD block diagram.

M.F. Castoldi et al. / Control Engineering Practice 24 (2014) 25–32 27

(Eiben & Smith, 2008; Goldberg, 1989). Other important aspectsare population size, genetic operators of crossover and mutationand their respective rates (Eiben & Smith, 2008; Goldberg, 1989).

In our work, the decision variables are the parameters α, β, γand K of each controller. Therefore, we propose an encoding forthe chromosomes so that each gene represents one of theseparameters. The parameter α is not considered, since its value isrelated to the washout time constant and can be arbitrarily fixedby the designer, within a pre-defined range (typically 3–20 s,Larsen & Swann, 1981), without degrading the quality of thesolution. Fig. 2 illustrates an example of the proposed encodingfor the chromosomes.

To obtain a solution with a set of parameters that is practicallyimplementable, it is necessary to include constraints on theparameters of the controllers. The range of 3–20 s for the washouttime constant gives

αi ¼ αi_const ; 0:05oαi_consto0:33 ð15Þ

where αi_const is a constant value set to αi, i¼ 1;…;nc.To determine the parameters βi, it is considered that these

parameters are relative to the maximum phase compensationφi_max that each controller can provide for their respective inputsignals. This relation is given by Ogata (2010)

βi ¼T1i

T2i¼ 1þ sin ðφmax_iÞ1� sin ðφmax_iÞ

ð16Þ

for i¼ 1;…;nc. Since the maximum feasible phase compensationfor each block of the controller is defined by practical constraintsas being no bigger than 701, the range of feasible values for theparameter βi is

0oβio32 ð17Þ

With fixed values of αi and βi, the phase compensation that willbe given to the input signal for the ith controller will bedetermined by the parameter γi. The highest value for the para-meter (γi_max) is calculated by Ogata (2010)

γi_max ¼ωk

ffiffið

pβiÞ ð18Þ

for i¼ 1;…;nc, where ωk is the oscillation frequency of the modeof interest λk. Since our interest damps both local and inter-areamodes, and we have that generally ωkr9 rad=s for local modes(which are the highest frequency modes in the range of interest),we have chosen the following range for the parameter γi:

0oγio40 ð19Þ

The static gain of the controller Ki is the only remainingparameter of (7) and (8) for which a range must be defined. Again,we relied on the industry practice to define

0oKio20 ð20Þ

for i¼ 1;…;nc, given that larger gains typically induce saturationof the controller output even for small perturbations.

With the previously defined ranges for the parameters, thefeasible solutions for the problem of tuning of the controllers arereduced to a certain region. Thus, the application of the EAproposed in this work can now be explained in detail.

First, an initial population is generated randomly. The indivi-duals for the mating pool are selected based on the tournamentselection scheme (Eiben & Smith, 2008). Three candidates areselected at random from the population and the best individualbased on the objective function is placed in the mating pool. The

tournament selection is done repeatedly until the mating pool getsfilled.

After that it is necessary to define the crossover operator tomatch the characteristics of individuals. Initially, two chromo-somes P1 and P2 are selected (these are considered as the parentchromosomes). Then one of the controllers encoded in P1 and P2 isselected, and its respective parameters are switched betweenthem, thus generating two new offspring chromosomes F1 andF2. Thus, through this combination it is expected to obtain idealvalues for parameters each controller. Fig. 3 illustrates an exampleof crossover application for the proposed method.

The mutation is applied to one chromosome, in which a certaincontroller is randomly selected and its parameters K, β and γ alsohave their values randomly changed within the range (18)–(20). Itis important to highlight that the number of individuals who sufferthe mutation will be determined by the mutation rate. Fig. 4 showsan example of the mutation used in this paper.

In this work, all operating points considered in the design areanalyzed simultaneously. Thus, each chromosome will be consid-ered at all operation points in parallel and the fitness value isobtained according to the lowest damping provided by thischromosome among all operating points analyzed.

To calculate the fitness value, the following objective functionwas used:

f ðKÞ ¼ �minðminðdampð ~AjðKÞÞÞÞ ð21Þ

for i¼ 1;…;L. This function finds the least damped mode amongall the j matrices representing the operating points considered inthe design. Thus, as the EA iterates, smaller values of this objectivefunction are generated. It is important to remark that movingalong the minimization of (21) corresponds to the maximization ofthe damping of the least damped mode (hence the negative sign inthe objective function). The firstmin in the objective function findsthe minimum damping of each eigenvalue of the closed-loopmatrix ~Aj and the second determines the smallest damping amongall those found by first min.

3.2. The gradient descent method

The gradient descent method, also called the method ofsteepest descent, is a first-order optimization algorithm to findthe nearest local minimum of an objective function f 0ðÞ, where f 0 :

Rn-R is a locally differentiable function. Basically, this methodstarts at an initial solution sk and, iteratively, minimizes thefunction f by moving from sk to skþ1 along the line extendingfrom sk in the direction of �∇f 0ðskÞ, the local downhill gradient(Luenberger, 2003).

In this work, the initial condition s0 for the gradient descentmethod is given by the output of the EA, which will be the fittestchromosome in the last population generated by the algorithm. Inaddition, the phases of the controllers are fixed (tuned by EA),while the gains are re-adjusted by the gradient descent method.

Consider the system in closed-loop given by (14), in thegradient descent method the matrices Aj, Bj, Cj, ~Ac and ~Bc , forj¼ 1;…;L, are fixed. Only the matrix ~Cc varies with the gain of the

Fig. 2. Chromosome representation.

Fig. 3. The proposed crossover operation.

M.F. Castoldi et al. / Control Engineering Practice 24 (2014) 25–3228

designed controller, as seen in (22)

~Cc ¼ KCcl ð22ÞIn (22), ~Cc is the output matrix, composed of all the controller

output gains, K is a static gain matrix with diagonal structure andCcl is the output matrix of the controllers, without their staticgains (i.e., with its entries set to 1).

The problem objective is again to move in the direction thatmaximizes the minimum damping ratio in the operating pointwith the least damping, until some satisfactory value of minimumdamping is reached. To do so, the same objective function (21) wasused, and the gradient descent method also moves towardsminimizing this objective function value, but stops when the finaldesired minimum damping is reached.

3.3. The proposed hybrid algorithm

As previously mentioned, both search and optimization meth-ods presented in Sections 3.1 and 3.2 present their inherentadvantages and disadvantages. The EA is capable of global opti-mization, but has slow convergence properties near local optima.The gradient descent is very well-suited for suboptimal localsearch, but cannot perform global optimization and is also slownear local optima.

For our problem, however, we found that a hybridization ofboth methods results in a very efficient approach to search for anadequate tuning of PSS and POD parameters. Fig. 5 summarizesthis hybrid approach, highlighted by the thick lines. The thin linespresent the method used for comparison, which is a purely EAapproach searching until the final desired minimum damping isreached.

The proposed method starts with an initial population for theEA, randomly generated, and applies the crossover and mutation

operators and the tournament selection until an intermediateminimum damping ratio ξ01 is achieved for all operating pointsconsidered in the design. Once this criterion is fulfilled, thealgorithm takes the best individual from the EA (the one withthe best fitness value, i.e., the smallest damping ratio) and passes iton to the gradient descent method.

In the sequence, the gradient descent iterates until the finaldesired minimum damping ξ02 is reached for all operating pointsof interest. In case the local optimum is not sufficient to fulfill thedesired final criterion, the algorithm is restarted with new randomvalues and the global EA search can try to find a new convergencebasin in which the corresponding local optimum is adequate, insuch a way that it has a surrounding region in which there aresolutions to the search problem under consideration. In our tests,however, there was no case in which a restart of the algorithmwasneeded.

4. Example of application of the proposed algorithm

In order to demonstrate the efficiency of the proposed algo-rithm, some tests were carried out on the benchmark NewEngland/New York interconnected system shown in Fig. 6.All the generators were described by a sixth-order model (Pal &Chaudhuri, 2005) and equipped with a first-order model of statictype AVR, with gains of 50 p.u/p.u. and a time constant of 0.01 s.The transmission system was modeled as a passive circuit and thesystem loads as constant impedances.

Usually, PSSs are tuned aiming to damp local modes, althoughsometime such tunings are also effective for inter-area mode.Nevertheless, in cases where inter-area modes are not properlydamped it is necessary to retune some PSSs of the system and/orto place FACTS devices, with their respective PODs, in strategicpoints of the system. This paper will assume that the secondoption was chosen, and thyristor controlled series capacitors(TCSCs) were selected to perform this task.

Typical simplifications (Rogers, 2000) are used to model theTCSC in this example. The TCSC input signal is the desiredreactance for the device, which compensates the line to generatethe desired power flow under stead-state conditions. This inputsignal is modulated by the POD output, and the net effect of thewhole TCSC dynamic behavior over the equivalent reactance of thedevice was modeled by a first-order linear block, shown in Fig. 7.

After a careful mode controllability/observability analysis (viaresidues and participation factors) (Pagola et al., 1989), 2 TCSCswith their respective PODs and 11 PSSs were inserted in the powersystem. The PSSs were inserted on generators G2, G3, G5, G7, G8,G9, G10, G11, G12, G14 and G15 aiming purely to provide asatisfactory damp for the local modes of these generators. Thefirst TCSC was installed between areas #1 and #2, in one of thelines that connects buses #60 and #61. The second was installedbetween areas #2 and #5, in the line connecting buses #18 and#49. The steady-state compensation level of both TCSCs corre-sponds to 30% of the reactance of its respective lines.

The tests in this paper considered an increase of 30% in the totalsystem load and a reduction of 25% of the total system load incomparison with base case (Pal & Chaudhuri, 2005). Together withthese 3 operating points, 15 other points were also taken intoaccount, consisting in variations of the first 3 ones previouslydescribed. These points were obtained with the following mod-ifications: (a) removing the line between buses #23 and #24;(b) removing the line between buses #30 and #31; (c) removingthe line between buses #39 and # 45; (d) removing the linebetween buses #57 and #60; and (e) removing the load off bus#67. It is important to remark that these contingencies wereconsidered in separate to take an N-1 criterion into account. The

Fig. 4. The proposed mutation operation.

Fig. 5. The proposed hybrid algorithm (thick lines) and the pure EA approach (thinlines) used for comparison.

M.F. Castoldi et al. / Control Engineering Practice 24 (2014) 25–32 29

open-loop system is highly unstable for all the considered opera-tion points, therefore requiring a simultaneous tuning of the 13controllers in order to stabilize the system in all operatingconditions of interest.

All the simulations were performed in a computer with a Core2 Quad Q6600 2.4 GHz processor and 4 GB DDR2 RAM memory.Both our proposed algorithm and the pure EA approach weretested in this same computer. At first, 11 PSSs and 2 PODs weresimultaneously tuned by using a global optimization methodbased purely on the EA. In this case, a population of 400individuals was used and the EA worked with crossover andmutation rates of 0.8 and 0.2, respectively (these values weredetermined by preliminary trials). The best individual of the initialpopulation provided a minimum damping ratio of 4.29% for thepoorest damped oscillation mode among all the operation pointsof interest. After 1811 s (approximately 30.2 min), the EA obtaineda controller which provided a 6.07% damping ratio for the leastdamped oscillation mode (6% was the target). Table 1 shows thevalues of the tuned controller parameters, for a controller struc-ture as shown in Fig. 1, with n¼2.

After the pure EA approach was concluded (to generate resultsfor the comparison that will be presented later on), the hybridapproach proposed in this paper was applied to the same problem.The EA in the hybrid algorithm started with a population of 400individuals, under the same conditions of the previous test. Atarget minimum damping of 5% was set, and the best individualresulting from the EA in the hybrid algorithm surpassed this targetafter an interval of 1154 s (approximately 19.2 min), reaching5.65% for the least damped mode. The parameters of this indivi-dual are shown in Table 2.

The individual in Table 2 was forwarded to the local searchbased on gradient descent of our hybrid approach, which startedretuning its gains, with a new target of 6% minimum damping.After 23 s a set of controller gains providing a damping ratio of6.07% for the least damped mode was found. The gain values

obtained by the local search method in our hybrid approach arepresented in Table 3. Therefore, the final tuning of the PSSs andPODs of the system resulting from our proposed hybrid approachcorresponds to the time constants in Table 2 combined with thegains in Table 3. It is important to notice that this set of tunedparameters fulfills the 6% target minimum damping just as well asthe one in Table 1 does, but the latter was obtained in less than20 min while the former took more than 30 min to be reached.

Nonlinear simulations were carried out under conditions thatwere considered in the design (in order to check the effectivenessof the solution found by the proposed algorithm), as well as under

Fig. 6. New England/New York benchmark test system.

Fig. 7. TCSC dynamic model used in this paper.

Table 1Controller parameters tuned only by EA.

Controller Gain T1 T2

POD 1 (18–69) 0.0608 0.2375 0.5991POD 2 (70–61) 33.303 0.1501 13.385PSS 1 (2) 48.478 13.835 0.2217PSS 2 (3) 68.966 0.2543 0.0751PSS 3 (5) 88.135 0.1766 0.0509PSS 4 (7) 97.274 0.5275 0.1189PSS 5 (8) 60.783 0.1455 0.0506PSS 6 (9) 94.507 0.5517 0.0763PSS 7 (10) 51.856 0.1721 0.0601PSS 8 (11) 75.339 0.1693 0.0524PSS 9 (12) 77.193 0.3846 0.0901PSS 10 (14) 95.791 0.1833 0.0565PSS 11 (15) 38.157 0.1885 0.0535

Table 2Controller parameters tuned only by the EA in the hybrid algorithm.

Controller Gain T1 T2

POD 1 (18–69) 0.0912 0.2603 0.5137POD 2 (70–61) 34.673 0.0871 11.481PSS 1 (2) 48.478 13.835 0.2217PSS 2 (3) 68.966 0.2543 0.0751PSS 3 (5) 38.127 0.3225 0.0533PSS 4 (7) 63.399 0.5138 0.0773PSS 5 (8) 35.880 0.5165 0.1479PSS 6 (9) 95.574 0.4020 0.0628PSS 7 (10) 75.376 0.3206 0.0598PSS 8 (11) 75.339 0.1693 0.0524PSS 9 (12) 77.193 0.3846 0.0901PSS 10 (14) 95.791 0.1833 0.0565PSS 11 (15) 38.157 0.1885 0.0535

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conditions that were not considered in the design (to verify therobustness of this solution). All the results of these simulationswere quite satisfactory, and to save space only the ones corre-sponding to a simulation that demonstrate the robustness of theobtained tuning are shown in the sequence. The operating condi-tion used in this simulation consists in a 15% load reduction (withrespect to the base case) and the removal of the line connectingbuses #30 and #31. A short circuit was applied to the line betweenbuses #47 and #48 at t¼2 s and removed after 16 ms of its

application (the short duration of the fault is intended to char-acterize it as a small perturbation).

Fig. 8 shows the response of the frequency of generators 1 and10, together with the active power flows on the lines in which theTCSCs were installed. These results show that the controllerstuned by the proposed hybrid algorithm are effective to dampthe electromechanical oscillations caused by small perturbationsand robust to variations in the operating conditions. Moreover, theproposed hybrid algorithm provides results faster than themethod based exclusively on EA, as expected.

5. Conclusion

This work proposed a hybrid search method to tune parametersof PSSs and PODs, in order to take advantage of the bestcharacteristics of global and local search approaches for thisparticular problem. For the global search, an EA was proposed insuch a way that its chromosomes encode the controller para-meters and, after a suitable number of generations, the resultingpopulation consists of individuals that are close to a solution to theproposed problem. The algorithm then switches to the localsearch, performed by a gradient descent method, which re-adjusts the controller gains in order to fulfill the target criterion,which is a minimum damping for all operating points consideredin the design.

The results in this paper show that the proposed approach isfaster than a pure EA approach, given that it takes advantage of the

Table 3Controller gains re-tuned by gradient descentmethod in the hybrid algorithm.

Controller Gain

POD 1 (18–69) 0.1808POD 2 (70–61) 34.672PSS 1 (2) 48.478PSS 2 (3) 68.966PSS 3 (5) 38.128PSS 4 (7) 63.401PSS 5 (8) 35.880PSS 6 (9) 95.579PSS 7 (10) 75.376PSS 8 (11) 75.338PSS 9 (12) 77.193PSS 10 (14) 95.818PSS 11 (15) 38.157

Fig. 8. Operating condition with the load of the base case reduced by 15% and without the line between buses #30 and 31.

M.F. Castoldi et al. / Control Engineering Practice 24 (2014) 25–32 31

faster convergence of the local search near a suboptimal solutionto the problem. It is also worth mentioning that the typicalindustry approach to solving similar problems is to rely on ahuman designer, who can take days to perform the designsreported in this paper. In this sense, the productivity gain for theindustry is evident, given that the intervention of the humandesigner in the proposed algorithm is very small, so that theemployee can dedicate its time to other tasks and only check thefeasibility of implementation of the resulting controller when theproposed algorithm finishes designing it.

Among the next directions of this research, we foresee the useof parallel computation as a promising possibility to speed up evenmore this hybrid approach. Given that both the EA and thegradient descent method involve tasks that can be performedsimultaneously, it is possible to take advantage of the multi-corearchitecture of modern processors.

Acknowledgement

The authors thank the support of FAPESP to this research, underGrant no. 2007/00062-5.

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