Adaptive Particle Swarm Optimization for Simultaneous Designof UPFC Damping Controllers

⇑ Corresponding author.E-mail address: mahdiyeh_eslami@yahoo.com (M. Eslami).

Mahdiyeh Eslami a,⇑, Hussain Shareef b, Mohd Raihan Taha c, Mohammad Khajehzadeh d

a Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Kerman, Iranb Department of Electrical Engineering, National University of Malaysia, Bangi 43600, Selangor, Malaysiac Department of Civil and Structural Engineering, National University of Malaysia, Bangi 43600, Selangor, Malaysiad Department of Civil Engineering, Anar Branch, Islamic Azad University, Anar, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 June 2012Received in revised form 28 July 2013Accepted 19 November 2013

Keywords:PSOAcceleration coefficientsLow-frequency oscillationUnified power flow controller

An adaptive particle swarm optimization based on nonlinear time-varying acceleration coefficients(NTVAC-PSO) is proposed for solving global optimization problems and damping of power system oscil-lations. The new method aims to control the global exploration ability of the original PSO algorithm andto increase its convergence rate with an acceptable solution in less iteration. A set of 10 well-knownbenchmark optimization problems is utilized to validate the performance of the NTVAC-PSO as a globaloptimization algorithm and to compare with similar methods. The numerical experiments show that theproposed algorithm leads to a significantly more accurate final solution for a variety of benchmark testfunctions faster. In addition, the simultaneous coordinated design of unified power flow controller-baseddamping controllers is presented to illustrate the feasibility and effectiveness of the new method. Theperformance of the proposed algorithm is compared with other methods through eigenvalue analysisand nonlinear time-domain simulation. The simulation studies show that the controllers designed usingNTVAC-PSO performed better than controllers designed by other methods. Moreover, experimentalresults confirm superior performance of the new method compared with other methods.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction device with the capability of regulating power flow in transmission

With the increasing electric power demand, power systems arebound to stress conditions, which result in undesirable systemvoltage and frequency. To provide a secure operation for powersystems, damping of power-system oscillations has received agreat deal of attention in power-system stability studies. For manyyears, power system stabilizers (PSSs) have been the choice of con-trol to damp out oscillations and to offset the negative damping ofautomatic voltage regulators. However, during some operatingconditions, this device may not produce enough damping, particu-larly in inter-area modes. Hence, the Flexible AC Transmission Sys-tems (FACTSs) technology emerges as an interesting approach tohelp alleviate several power system-operating difficulties, such asinter-area oscillations and controlling voltages at critical buses.This technology has very promising candidates for the power sys-tem-damping enhancement. Using a feedback supplementary con-trol, in addition to the FACTS device primary control, was observedto improve considerably system damping and voltage profile,which is advantageous over PSSs [1–5].

Among the FACTS devices, the unified power flow controller(UPFC) is one of the most important and most comprehensive

lines by its series converter and input voltage by its shunt con-verter [6]. The application of the UPFC to the interconnected powersystems provides a significant damping effect on tie-line poweroscillation through its supplementary control. Therefore, a fixedparameter controller based on the classical control theory is notcertainly suitable for the UPFC damping control design. Thus, aflexible controller needs to be developed. Several approaches havebeen proposed for damping controller design, such as root locusand sensitivity analysis, pole placement, and robust control. Thesetechniques have several drawbacks as the control law depends on alinearized machine model, while the control parameters are ad-justed to certain nominal operating conditions. Moreover, parame-ters of the controllers are not valid as the system conditions changeto nonlinear in case of large disturbances.

The optimal design problem of damping controllers is a com-plex optimization problem with the presence of multiple localminima. Hence, extensive studies have been conducted to employdifferent optimization methods for the solution. Population based,cooperative, and competitive stochastic search algorithms havebeen very popular in recent years in computational intelligence,such as in genetic algorithm [7,8], particle swarm optimization(PSO) [9], differential evolution [10], and gravitational searchalgorithm [11], are successfully implemented to solve simple andcomplex problems efficiently and effectively. Most of the

population-based search approaches are inspired by evolution asseen in nature. PSO was developed from the simulation of socialbehavior, which is prone to premature convergence [12]. PSO is astochastic population-based algorithm motivated by intelligentcollective behavior of some animals. The most important advanta-ges of the PSO are in its ease of implementation and few parame-ters for adjustment. In PSO, encouraging individuals to wanderthrough the entire search space, without clustering around localoptima, during the early stages of the optimization (exploration)is desirable. In addition, during the latter stages, enhancing conver-gence in the global optima is very important to find the optimumsolution efficiently (exploitation). Generally, the performance ofthe standard PSO depends almost entirely on its parameters.

Therefore, an adaptive PSO proposes to design a UPFC-basedpower oscillation damping (POD) controller in this study. To en-hance the performance and to provide good balance between glo-bal exploration and local exploitation abilities of the originalalgorithm, the PSO with nonlinear time-varying acceleration coef-ficients (NTVAC-PSO) is proposed. With a large cognitive parame-ter and small social parameter at the beginning, particles areallowed to move around the search space instead of moving to-ward the population best. However, a small cognitive parameterand a large social parameter allow the particles to converge inthe global optimum in the latter part of the optimization. A setof 10 well-known benchmark optimization problems is utilizedto validate the performance of the proposed NTVAC-PSO as a globaloptimization algorithm and to facilitate comparison with similarmethods. The numerical experiments show that the proposedalgorithm lead to a significantly more accurate final solution fora variety of benchmark test functions faster. In addition, the simul-taneous coordinated design of the UPFC POD controllers is pre-sented to illustrate the feasibility and the effectiveness of thenew method. The experimental analysis is tested on a two-areafour-machine system under different operating conditions. The re-sults demonstrate that the controllers designed by the NTVAC-PSOperform better than those by other methods in oscillations damp-ing, which perform and enhance greatly the dynamic stability.Moreover, the results confirm superior performance of the newmethod compared with other methods.

2. Proposed NTVAC-PSO approach

2.1. Review of PSO method

PSO is a population-based optimization technique introducedby Kennedy and Eberhart [13] to solve the unconstrained optimi-zation problem. In a PSO system, multiple candidate solutionscoexist and collaborate simultaneously. Each solution, which iscalled a particle, flies in the problem search space looking to landon the optimal position. A particle, during the generations, adjustsits position according to its own experience as well as the experi-ence of neighboring particles. The PSO system combines local andglobal search methods, and attempts to balance exploration andexploitation. The new velocity and position of a particle will be up-dated according to the following equation [14]:

Vi½kþ 1� ¼ w� Vi½k� þ c1 � r1 � ðpbesti½k� � Xi½k�Þ þ c2 � r2

� ðgbest½k� � Xi½k�Þ ; i ¼ 1;2; . . . ;N ð1Þ

Xi½kþ 1� ¼ Xi½k� þ Vi½kþ 1� ð2Þ

where N is the number of particles in the swarm, k is the number ofthe current iteration, w is an inertia weight, r1 and r2 are random vari-ables between 0 and 1, c1 and c2 are acceleration coefficients. Vi and Xi

are the velocity and position of the particle i, respectively. pbest rep-resents the local best position of particle i. gbest represents the glo-

bal-best position of all particles. Shi and Eberhart [15] introduced atime-varying inertia weight PSO (TVIW-PSO) with the linearlydecreasing inertia weight over the generations according to Eq. (3).

w ¼ wmax �ðwmax �wminÞ

kmax� k ð3Þ

where wmin and wmax are the initial and final values of the inertiaweight, respectively. kmax is the maximum number of iterations.Generally, the performance of the standard PSO depends on itsparameters. The algorithm is usually trapped in the local optimain early iterations and suffers from premature convergence whenfacing a complex optimization problem. Several efforts have beenmade to improve PSO to overcome the problem satisfactorily[16,9,17–21]. Recently, a time-varying acceleration coefficient PSO(TVAC-PSO) is proposed to reduces the cognitive component and in-creases the social component of acceleration coefficient, c1 and c2,with time [22]. In this approach, the acceleration coefficients arechanged according to the following equations:

c1 ¼ ðc1f � c1iÞ �k

kmaxþ c1i ð4Þ

c2 ¼ ðc2f � c2iÞ �k

kmaxþ c2i ð5Þ

2.2. NTVAC-PSO method

In PSO, encouraging individuals to wander through the entiresearch space, without clustering around local optima, during theearly stages of the optimization is desirable. During the latterstages, convergence in the global optima must be enhanced to findthe optimum solution efficiently.

This research introduces an NTVAC-PSO to enhance the perfor-mance and to provide good balance between global explorationand local exploitation abilities of the original algorithm. In the pro-posed method, the acceleration coefficients change according tothe following equations:

c1 ¼ ðc1i � c1f Þ � exp½�ð4k=kmaxÞ2� þ c1f ð6Þ

c2 ¼ ðc2i � c2f Þ � exp½�ð4k=kmaxÞ2� þ c2f ð7Þ

where c1i and c2i are the initial values of the acceleration coefficientc1 and c2, respectively. Similarly, c1f and c2f are the final values of theacceleration coefficient c1 and c2, respectively. The velocity updat-ing equation of NTVAC-PSO can be expressed as:

Vi½kþ 1� ¼ w� Vi½k� þ ðc1i � c1f Þ � exp½�ð4k=kmaxÞ2� þ c1f

� ��r1 � ðpbesti½k� � Xi½k�Þþððc2i � c2f Þ � exp½�ð4k=kmaxÞ2� þ c2f Þ�r2 � ðgbest½k� � Xi½k�Þ

ð8Þ

All parameters in the above equations were defined previously. InEq. (8), the first part, known as the cognitive component, encour-ages the particles to move toward their own best positions. The sec-ond part is known as the social component, which represents thecollaborative effect of the particles in finding the global optimalsolution. Figs. 1 and 2 present the differences between constant, lin-ear, and nonlinear acceleration coefficients for c1 and c2,respectively.

As seen in these figures, the proposed method reduces the cog-nitive component and increases the social component by changingthe acceleration coefficients c1 and c2 with time. The NTVAC-PSOprovides a larger value for cognitive component and smaller valuefor social component at the beginning, which allows particles to

Fig. 1. Variation of deferent type of cognitive coefficient with iterations.

Fig. 2. Variation of deferent type of social coefficient with iterations.

move around the search space instead of moving toward the pop-ulation best. In the latter part of the optimization, the new algo-rithm provides a smaller cognitive component and a larger socialcomponent that allows the particles to converge in the globaloptimum.

3. Mathematical model of power system with UPFC

The UPFC is a versatile FACTS controller, which has a wide rangeof control functions for the improvement of power system perfor-mance [23]. Fig. 3 shows an n-machine power system equippedwith a UPFC. The UPFC consists of an excitation transformer (ET),a boosting transformer (BT), two three-phase GTO based voltagesource converters (VSCs), and a DC link capacitors. The four inputcontrol signals to the UPFC are mB, dE, dB, and mE, where mE is theexcitation amplitude modulation ratio, mB is the boosting ampli-tude modulation ratio, dE is the excitation phase angle, and dB isthe boosting phase angle. These parameters are considered UPFCcontrol inputs to provide synchronized power compensation in aseries line without external voltage source.

3.1. Nonlinear dynamic model of the UPFC

The dynamic model of the UPFC is required to study the effect ofthe UPFC for enhancing the small-signal stability of the power sys-tem. By applying Park’s transformation and ignoring the resistanceand transients of the UPFC transformers, UPFC can be modeled asfollows [23,24]:

dvdc

dt¼ 3mE

4Cdc½cos dE sin dE�

iEd

iEq

� �þ 3mB

4Cdc½cos dB sin dB�

iBd

iBq

� �ð9Þ

mEtd

mEtq

� �¼

0 �xE

xE 0

� �iEd

iEq

� �þ

mE cos dEmdc2

mE sin dEmdc2

" #;

mBtd

mBtq

� �¼

0 �xB

xB 0

� �iBd

iBq

� �þ

mB cos dBmdc2

mB sin dBmdc2

" # ð10Þ

Eq. (10) can be written as:

VEt ¼ jxEIE þ VE; VBt ¼ jxBIE2 þ VB ð11Þ

where

IE ¼ iEd þ jiEq IE2 ¼ iBd þ jiBq

VE ¼ mEvDC2 ðcos dE þ j sin dEÞ ¼ mEvDC

2 ejdE ;

VB ¼ mBvDC2 ðcos dB þ j sin dBÞ ¼ mBvDC

2 ejdB

ð12Þ

where vEt, iE, vBt, and iB are the excitation voltage, excitation current,boosting voltage, and boosting current, respectively. Cdc is the DClink capacitance and Vdc is the DC link voltage. Lastly, xE is the ETreactance and xB is the BT reactance.

3.2. Nonlinear model of power system

The standard industry approach to power system modeling forcontroller design is based on a set of nonlinear differential alge-braic equations in the form:

_x ¼ fðx;uÞy ¼ gðx;uÞ

ð13Þ

where x and u are the state vector and input vector, respectively; yis the output vector; and f and g are the nonlinear functions. Thecomplex nonlinear model related to an n-machine interconnectedpower system, can be described by a set of differential–algebraicequations by assembling the models for each generator, load, andother devices, such as controls in the system, and connecting themvia the network algebraic equations. The resultant dynamic equa-tions are as follows:

_d ¼ xbðx� 1Þ_x ¼M�1ðTM � TE � Dðx� 1ÞÞ_E0q ¼ T0�1

D0 ðEFD � E0q þ ðXD � X0DÞIDÞ_EFD ¼ ðKAðVRef � VT þ UPSSÞ � EFDÞ=TA

TE ¼ IQ VTQ þ IDVTD

VTD ¼ XQ IQ ; VTQ ¼ _E0q � X0DID

ð14Þ

where xb is the synchronous speed, while d and x are the angle andspeed of the rotor, respectively. M is the machine inertia constant,and D is the damping coefficient. EFD is the generator field voltage,and T0D is the open-circuit field time constant. KA and TA are the gainand time constant of excitation system and VRef is the referencevoltage. The output current of the ith generator can be expressedon di–qi axes as:

Ii ¼ Idi þ jIqi ¼Xn

k¼1

Yik E0qiejð90�þdikÞ þ ðXqk � X 0dkÞIqkejdik

h ið15Þ

Fig. 3. An n-machine power system installed with a UPFC.

where Y is the system admittance matrix of the power system whenonly the n generator nodes are left.

3.3. Linearized model of power system with UPFC

Since the early 1970s, linear analysis techniques have been usedto study the dynamic behavior of power systems. A linearizedmodel of the n-machine power system installed with the PSS andUPFC-based controllers can be obtained by linearizing the nonlin-ear model around an operating point of the power system. Thesetechniques provide valuable insights in the dynamic characteristicsof a power system, which cannot be easily attained from time-do-main simulations alone. The linearized equations of n-machinepower system based stabilizers are derived as:

D _d ¼ x0DxD _x ¼M�1ð�DTE � DDxÞD _E0q ¼ T0�1

D0 �DE0q � ðXD � X0DÞDID þ DEFD

h iD _EFD ¼ ð�DEFD þ KAðDVRef � DVT þ DUPSSÞÞT�1

A

DTE ¼ DIQ_E0q0 þ IQ0D _E0q þ DIQ ðXQ � X0DÞID0 þ DIDðXQ � X0DÞIQ 0

DVTD ¼ XQ DIQ ; DVTQ ¼ D _E0q � X0DDID

ð16Þ

where

DTe ¼K1DdþK2DE0qþKpdDvDCþKpeDmEþKpdeDdEþKpbDmBþKpdbDdB

DEq ¼K4DdþK3DE0qþKqdDvDCþKqeDmEþKqdeDdEþKqbDmBþKqdbDdB

DVt ¼K5DdþK6DE0qþKvdDvDCþKveDmEþKvdeDdEþKvbDmBþKvdbDdB

D _vDC ¼K7DdþK8DE0q�K9DvDCþKceDmEþKcdeDdEþKcbDmBþKcdbDdB

ð17Þ

where the linearization constants K1–K9, Kpd, Kpe, Kpde, Kpb, Kpdb, Kqd,Kqe, Kqde, Kqb, Kqdb, KVd, KVe, KVde, KVb, KVdb, Kce, Kcde, Kcb, Kcdb are func-tions of the system parameters and initial operating conditions.DUPSS is the output signal vector of the PSS in the system. Instate-space representation, these equations can be arranged in com-pact form as:

D _X ¼ ADX þ BDU ð18Þ

where DX and DU are the state and control vector, respectively. Thestructure of the matrices A and B is:

D _d

DxD _E0qD _Efd

DvDC

26666664

37777775¼

0 x0I 0 0 0�M�1K1 �M�1D �M�1K2 0 �M�1Kpd

�T0�1do K4 0 �T0�1

do K3 T0�1do �T0�1

do Kqd

�T�1A KAK5 0 �T�1

A KAK6 T�1A �T�1

A KAKvd

K7 0 K8 0 �K9

26666664

37777775

Dd

DxDE0qDEfd

DvDC

26666664

37777775

þ

0 0 0 0�M�1Kpe �M�1Kpde �M�1Kpb �M�1Kpdb

�T0�1do Kqe �T0�1

do Kqde �T0�1do Kqb �T0�1

do Kqdb

�T�1A KAKve �T�1

A KAKvde �T�1A KAKvb �T�1

A KAKvdb

Kce Kcde Kcb Kcdb

26666664

37777775

DmE

DdE

DmB

DdB

26664

37775

ð19Þ

3.4. UPFC based damping controllers

The damping controller is designed to produce an electricaltorque in-phase with the speed deviation according to phasecompensation method to improve damping of the system oscilla-tions. The four control parameters of the UPFC can be modulatedto produce the damping torque as follows:

Fig. 5. UPFC with dc voltage regulator and damping controller.

Active power regulator:

dB ¼1

1þ TdBs

� �Kpp þ

Kpi

s

� �ðPRef � PÞ ð20Þ

Reactive power regulator:

mB ¼1

1þ TmBs

� �Kqp þ

Kqi

s

� �ðQRef � QÞ ð21Þ

AC voltage regulator:

mE ¼1

1þ TmEs

� �Kvp þ

Kvi

s

� �ðVRef � VÞ ð22Þ

DC voltage regulator:

dE ¼1

1þ TdEs

� �Kdcp þ

Kdci

s

� �ðvDCref � VDCÞ ð23Þ

The structure of the UPFC controller is shown in Fig. 4, where u canbe mB, dE, dB, and mE. To maintain the power flow between the seriesand shunt converters, a DC voltage regulator must be incorporated.The DC voltage is controlled by modulating the phase angle of theET voltage dE.

3.5. Optimal design of the damping controllers

To overcome the low frequency oscillation problem, supplemen-tal control action is applied to the generator excitation in the form ofPSS or UPFC device as POD controller. The four main control param-eters of the UPFC can be modulated to produce the appropriatedamping torque, which is illustrated in Fig. 4. In the current study,two controllers are attached to the UPFC to provide damping tothe poorly damped local and inter-area oscillation modes. The con-trollers can be superimposed on the normal control functions of theUPFC. The first controller can be attached to the function of AC volt-age regulation (mE) to damp the inter-area oscillation mode, whilethe second controller can be attached to the DC voltage regulation(dE) to damp the local oscillation modes. The dE damping controllerto be considered is shown in Fig. 5. The DC voltage regulator is a PIcontroller. The mE damping controller is almost the same as the dE

damping controller, where AC voltage regulation is a PI controller.Two controllers (means the UPFC is added to the two controllers)superimposed on the functions of the AC and DC voltage regulationsof UPFC were designed in coordination and based on the linearizedmodel of the power system. The POD controller has a structure sim-ilar to that of the PSS. The controller contains three main blocks,which are gain, washout filter, and two-block (lead/lag) phase com-pensators. The controller provides an electrical torque in phase withthe speed deviation to improve the damping of power system oscil-lations. The washout time Tw should have a value of 1–20 s definedto the electromechanical oscillation modes. In this study, 10 s is as-signed as the time constant (Tw), whereas the parameters of the PODcontroller such as K, T1, and T4 have yet to be determined. Theparameters of the damping controllers are obtained by using theNTVAC-PSO algorithm.

The main objective of the optimization technique is to improve thedynamic stability of the power system against disturbances at

Fig. 4. UPFC with lead-lag controller.

different loading conditions. The aim can be achieved by suitably tun-ing the damping controller parameters. The supplementary dampingstabilizer (lead/lag type) can be described mathematically as:

VðsÞ ¼ KðsÞyðsÞ ð24Þ

where K(s) is the transfer function of the supplementary dampingstabilizer, y(s) is the measurement signal, and V(s) is the output sig-nal from the supplementary damping stabilizer, which will provideadditional damping if modes are moved to the left. Such controlstructure can be represented by a linear equation set, in state spaceform, given by Eq. (25):

D _Xc ¼ AcDXc þ BcDU ð25Þ

where DXc is the controller state vector. Eq. (18) describes a linearmodel of the power system extracted around a certain operatingpoint. The closed loop description of the controlled system formedby the feedback connection of (18) and (25) may be represented by[11]:

D _Xcl ¼ AclDXcl; Acl ¼A BCc

BcC Ac

� �

DXcl ¼DX

DXc

� � ð26Þ

where DXcl is the state vector of the closed loop system. Acl repre-sents the state matrices of the closed loop system, while Ac, Bc,and Cc are the matrix variables to be determined by the design pro-cedure. The goal of the controller design is to place the eigenvalues(k) of matrix Acl in the left half of the complex plane. In the currentstudy, the damping ratio (n) and damping factor (r) for the oscilla-tion modes of the closed loop polytopic system are used as perfor-mance index in the design stage, since such performance index isuniversally accepted in power systems as an indicative of small sig-nal stability margin.

J ¼XNP

j¼1

XriPr0

ðr0 � rijÞ2 þ aXNP

j¼1

XfiPf0

ðf0 � fijÞ2; f ðxÞ ¼min J ð27Þ

The value of a is a weight for combining both damping factorsand ratios. r0 and f0 are the constant value of the expected damp-ing factor and ratio, respectively. The goal of the optimization pro-cess is to minimize J to realize appropriate damping for all modes,including electromechanical mode. The process can be done byexploring the search space of admissible control parameters, whichenhance the system damping characteristics. J is searched withinthe limited range of controller parameters:

minimize J

Kmini 6 Ki 6 Kmax

i

Tmin1i 6 T1i 6 Tmax

1i

Tmin2i 6 T2i 6 Tmax

2i

Tmin3i 6 T3i 6 Tmax

3i

Tmin4i 6 T4i 6 Tmax

4i

ð28Þ

The proposed approach employs the NTVAC-PSO to solve thisoptimization problem and to search for an optimal set of controllerparameters. The inequality constraints presented in Eq. (28) shouldbe considered for the optimization of controller parameters usingthe penalty method. The typical ranges of the optimized parame-ters are 0.01–100 for K and 0.001–2 for T. Therefore, the finalobjective function for optimal tuning of UPFC controllers usingNTVAC-PSO can be formulated as:

FðXÞ ¼ f ðXÞ þ rXp

i¼1

max f0; giðXÞgl ð29Þ

where r is the penalty factor, l is the power of the penalty function,g(X) is inequality constraints of Eq. (28), and p is total number ofinequality constraints. To acquire an optimal combination, thisstudy employs NTVAC-PSO to improve optimization synthesis and

Table 1Standard benchmark functions.

Function name Test function

Sphere F1ðxÞ ¼Pn

i¼1x2i

Schwefel F2ðxÞ ¼Pn

i¼1jxij þQn

i¼1jxijRotated hyper-elliposid

F3ðxÞ ¼Pn

i¼1Pi

j¼1xj

� �2

Schwefel F4ðxÞ ¼maxfjxij; 1 6 i 6 ngRosenbrock F5ðxÞ ¼

Pn�1i¼1 100ðxiþ1 � x2

i Þ2 þ ðxi � 1Þ2

h iStep F6ðxÞ ¼

Pni¼1ð½xi þ 0:5�Þ2

Noisy quadric F7ðxÞ ¼Pn

i¼1ix4i þ random½0;1Þ

Ackly F8ðxÞ ¼ �20 exp �0:2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Pni¼1x2

i

q� �� exp 1

n

Pni¼1 cos 2p

Griewank F9ðxÞ ¼ 1

4000

Pni¼1x2

i �Qn

i¼1 cos xiffiip� �

þ 1

Penalized F10ðxÞ ¼ pn f10 sin2ðpy1Þ þ

Pni¼1ðyi � 1Þ2½1þ 10 sin2ð3py

yi ¼ 1þ xiþ14

Table 2Minimization result of benchmark functions in Table 1. All results are the statistic results

Function Method Worst Mean

F1 NTVAC-PSO 2.0282 � 10�22 3.4174 � 10�23

TVAC-PSO 0.0974 0.0285TVIW-PSO 58.0518 39.3355

F2 NTVAC-PSO 1.0677 � 10�4 1.8053 � 10�5

TVAC-PSO 2.4874 1.7497TVIW-PSO 3.5307 2.8011

F3 NTVAC-PSO 4.7977 � 10�6 8.3252 � 10�7

TVAC-PSO 275.7041 177.9406TVIW-PSO 1.2575 � 103 763.9087

F4 NTVAC-PSO 1.2732 � 10�4 4.3247 � 10�5

TVAC-PSO 2.3864 1.9109TVIW-PSO 6.2625 5.7686

F5 NTVAC-PSO 10.2126 8.7987TVAC-PSO 122.8628 56.7178TVIW-PSO 1.7072 � 103 1.3463 � 103

F6 NTVAC-PSO 0 0TVAC-PSO 8.8234 � 10�23 3.6723 � 10�23

TVIW-PSO 0.1518 0.040F7 NTVAC-PSO 0.0010 7.2827 � 10�4

TVAC-PSO 0.0113 0.0073TVIW-PSO 0.0243 0.0211

F8 NTVAC-PSO 7.3275 � 10�13 2.9695 � 10�13

TVAC-PSO 0.0029 5.8438 � 10�4

TVIW-PSO 3.1159 2.6818F9 NTVAC-PSO 0.0123 0.0057

TVAC-PSO 0.0991 0.0390TVIW-PSO 1.5225 1.3428

F10 NTVAC-PSO 3.1935 � 10�24 5.4718 � 10�25

TVAC-PSO 0.1052 0.0217TVIW-PSO 0.7326 0.4067

to find the global optimum value of fitness function. The proposedapproach employs NTVAC-PSO to solve this optimization problemand to search for the optimal set of UPFC controller parameters.

4. Model verification

To demonstrate, compare, and analyze the effectiveness andperformance of our proposed NTVAC-PSO, testing the algorithmon 10 representative standard benchmark functions would beappropriate. These functions were extensively used to evaluatethe performance of optimization algorithms. Table 1 summarizesthe functions, the dimension (n), the admissible range of the vari-able (S # Rn), and the optimum value of the function (fopt). All thefunctions are to be minimized. The first seven functions (f1–f7) areunimodal functions, whereas the next three functions (f8�f10) are

n S fopt

30 [�100,100]n 0

30 [�10,10]n 030 [�100,100]n 0

30 [�100,100]n 030 [�30,30]n 0

30 [�100, 100]n 0

30 [�1.28,1.28]n 0

xiþ 20þ e 30 [�32, 32]n 0

30 [�600, 600]n 0

iþ1Þ� þ ðyn � 1Þ2g þPn

i¼1uðxi;10;100;4Þ 30 [�50,50]n 0

over 30 independent runs with kmax = 2000.

Median Best Std.

1.8357 � 10�25 1.1395 � 10�27 8.2624 � 10�23

0.0140 0.0043 0.038736.5584 24.2693 12.29063.5666 � 10�9 1.0586 � 10�9 4.3467 � 10�5

1.5864 1.1825 0.51093.0393 1.8640 0.66781.0919 � 10�8 1.9245 � 10�9 1.9436 � 10�6

179.9212 67.5181 83.1411683.6017 496.3461 295.86361.6975 � 10�5 4.8271 � 10�6 5.1699 � 10�5

1.9399 1.1273 0.49235.8570 5.2472 0.38068.9555 6.9689 1.328729.6613 22.7500 45.20221.4152 � 103 978.3462 280.96960 0 02.2212 � 10�23 4.7785 � 10�27 4.3049 � 10�23

0.0103 0.0034 0.06316.8749 � 10�4 5.0236 � 10�4 2.3789 � 10�4

0.0062 0.0037 0.00300.0202 0.0184 0.00242.0339 � 10�13 6.4837 � 10�14 2.5797 � 10�13

5.6079 � 10�6 2.6073 � 10�8 0.00132.7155 2.3045 0.31430.0049 0 0.00640.0334 0.0049 0.03791.3225 1.2184 0.11652.8202 � 10�27 3.2403 � 10�29 1.2968 � 10�24

3.5650e�004 1.2181e�005 0.04670.3893 0.1476 0.2418

Fig. 6. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F1 with n = 30.

Fig. 7. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F2 with n = 30.

Fig. 8. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F3 with n = 30.

Fig. 9. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F4 with n = 30.

multimodal optimization problems with a considerable amount oflocal minima.

The proposed method was applied to the minimization of thesebenchmark functions and the results were compared with TVAC-PSO and TVIW-PSO. For both NTVAC-PSO and TVAC-PSO, the valuesof c1i and c2f are considered 2.5; c1f and c2i are considered 0.5.Moreover, in TVIW-PSO, c1 = c2 = 2 and inertia factor (w) aredecreasing linearly from 0.9 to 0.4 for all algorithms. In all cases,population size is set to 50 (N = 50). The dimension is 30 (n = 30)and the maximum iteration (krmax) is 2000 for functions in Table 1.For the comparisons to be fair, all algorithms used the same num-ber of function evaluations. The algorithms were simulated 30times, and the results were recorded. From the recorded simulatedresults, statistical analyses were carried out. Table 2 summarizesthese results. For each method, the worst, mean, median, best,and standard deviation from the 30 independent runs were calcu-lated and compared.

The standard deviation and mean are the quality measures ofstatistical analyses. To prove the consistency and stability on theoptimal solution, standard deviations and means of the fitness va-lue were obtained during the 30 simulated runs. If the standard

deviation is very small, the solutions obtained in each of the 30runs were more or less equal, and fewer differences between thesolutions over each run existed. Therefore, if an algorithm providedvery low standard deviation, the algorithm can be concluded asconsistent. The results of Table 2 indicates that the mean fitnessevaluated by NTVAC-PSO in 2000 iterations were lower for allfunctions than those values computed by TVAC-PSO and TVIW-PSO. In addition, the NTVAC-PSO provided a very low standarddeviation for all benchmark functions among other methods.Hence, the NTVAC-PSO generated better optimal solutions and out-performed other methods considered in this paper. Figs. 6–15demonstrate the convergence rate comparison among the pro-posed NTVAC-PSO, TVAC-PSO, and TVIW-PSO on the 10 benchmarkfunctions. In these figures, the representative variations of themean best fitness in the form of logarithm values over the numberof iterations are depicted.

Figs. 6–15 show that the varying curves of fitness values usingNTVAC-PSO descend faster than the curves using TVAC-PSO andTVIW-PSO. At the same time, the best solution obtained byNTVAC-PSO was more optimal than the other solutions. Moreover,as shown in Figs. 6–9 and 15, the resulting history for minimization

Fig. 10. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F5 with n = 30.

Fig. 11. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F6 with n = 30.

Fig. 12. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F7 with n = 30.

Fig. 13. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F8 with n = 30.

Fig. 14. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F9 with n = 30.

Fig. 15. Performance comparison of NTVAC-PSO, TVAC-PSO and TVIW-PSO forminimization of F10 with n = 30.

Fig. 16. Flowchart of the NTVAC-PSO used for the optimization of controllers parameters.

Fig. 17. Multi-machine power system with UPFC.

of functions F1, F2, F3, F4, and F10 converged very quickly byTVAC-PSO and TVIW-PSO, but did not improve after the initialconvergence. This finding indicated that these algorithms fell intoa local trap, and were unable to escape. In other words, afterconvergence, the algorithms lost the ability to explore and becameinactive. From the testing results of 10 functions, NTVAC-PSOsignificantly surpassed other methods (e.g. TVAC-PSO and TVIW-PSO) in all selected test functions.

5. Practical applications

In the previous section, the efficiency and robustness of theproposed strategy, as a global optimization method, were verifiedby considering a number of benchmark problems. This sectioninvestigates the applicability and the effectiveness of the proposedmethod to design controller parameters. Fig. 16 presents theimplementation flowchart of the proposed NTVAC-PSO for

Table 3Result of coordinated design of two controllers attached to the UPFC.

Controller parameters TVIW-PSO based mE TVIW-PSO based dE TVAC-PSO based mE TVAC-PSO based dE NTVAC-PSO based mE NTVAC-PSO based dE

T1 0.483 0.715 0.470 0.260 0.310 0.370T2 0.012 0.043 0.380 0.093 0.057 0.043T3 0.529 0.092 0.140 0.280 0.120 0.450T4 0.062 0.016 0.036 0.260 0.068 0.059K 84.49 49.56 64.76 39.65 37.25 28.36

Table 4Oscillation modes in the example power system of Fig. 17 with UPFC.

Oscillation modes Frequency Damping ratio

Without PSSs 0.04438 ± j4.031 0.641 0.011TVIW-PSO �1.0208 ± j4.899 0.779 0.203TVAC-PSO �1.3775 ± j5.778 0.918 0.232NTVAC-PSO �1.9531 ± j6.153 0.979 0.301

Fig. 18. Local mode of oscillation (w1-w2).

Fig. 19. Local mode of oscillation (w3-w4).

Fig. 20. Inter-area mode of oscillation (w2-w3).

Fig. 21. Inter-area mode of oscillation (w1-w3).

evaluating the optimal controller parameters. In the followingcases, the maximum iteration number (kmax) is set to 100, andother parameters considered are specified in the previous section.The two-area–four-machine power system was used to demon-strate the design of damping controllers.

5.1. Two-area–four-machine system

Fig. 17 shows the two-area–four-machine system used to dem-onstrate the damping controller design. The system, introduced in

Fig. 22. Local mode of oscillation (w1-w2).

Fig. 23. Local mode of oscillation (w3-w4).

Fig. 24. Inter-area mode of oscillation (w2-w3).

Fig. 25. Inter-area mode of oscillation (w1-w3).

Fig. 26. Local mode of oscillation (w1-w2).

Fig. 27. Local mode of oscillation (w3-w4).

Fig. 28. Inter-area mode of oscillation (w2-w3).

Fig. 29. Inter-area mode of oscillation (w1-w3).

[25], is a benchmark system for inter-area oscillation studies. TwoPSSs are supposed to be simultaneously designed for Machines 2and 3 [17]. The sub-transient model for the generators and theIEEE-type DC1 and DC2 excitation systems are applied to Machines1 and 4. The IEEE-type ST3 compound source rectifier exciter mod-el is applied to Machines 2 and 3. The system data and the defini-tion of the small-signal stability are adopted from [26]. The simplemodel shows the fundamental electromechanical oscillations thatare inherent in interconnected power systems. In the system, thereare three oscillation modes. The two local modes (difference inspeed of Gen.1 and Gen.2 and difference in speed of Gen.3 andGen.4) and inter-area mode (difference in speed of Gen.2 andGen.3 and difference in speed of Gen.1 and Gen.3) are poorlydamped without damping controllers. A UPFC is installed betweenbuses 7 and 10 with the primary functions of active power andreactive power control as well as voltage regulation implementedby four PI controllers.

The proposed approach uses the NTVAC-PSO to solve this opti-mization problem and to search for the optimal set of controllersparameters. This procedure has a robust stabilizer and can satisfac-torily operate over a wide range of operating conditions. The valuesof r0 and f0 are set as �1.5 and 0.3, respectively. Note that NTVAC-PSO algorithm should be executed several times before selectingthe optimal set of controller parameters. The final values of theoptimized parameters with objective function, J, are given inTable 3. The obtained parameters by the algorithms are placed inthe study system. Table 4 presents the close-loop eigenvalue withminimum damping ratio obtained for all operating conditions afterthe application of various optimization methods in the system. Forthe system without stabilizer, it can be observed that some of themodes are weakly damped and for some operating conditions, thesystem is unstable. It is clear that the system stability is greatly en-hanced by proposed controllers. After the optimized coordinatedtuning of controllers, all the electro-mechanical modes are welldamped. The TVAC-PSO and TVIW-PSO cannot move the worsteigenvalue to the D-shape sector, whereas the NTVAC-PSO can.This feature greatly enhances the system stability and improvesthe damping characteristic of the electro-mechanical mode. There-fore, the use of nonlinear time-varying concept in PSO is an effec-tive approach to enhance the global searching capability and toimprove the performance stability.

To assess the effectiveness and robustness of the proposed con-trollers, time-domain simulation is carried out for various fault dis-turbances. Taking generator G3 as the reference and the pre-disturbance operating condition in units P1 = 0.4444, Q1 = 0.2056,P2 = 0.6667, Q2 = 0.2611, P3 = 1.5767, Q3 = 0.2186, P4 = 0.3333, andQ4 = 0.2244, the response of the network to different disturbancesis presented to establish the superiority of the proposed controllersover other methods. The following case studies are undertaken forevaluating the performance of the proposed controller.

Case 1: A three-phase fault of 100 ms duration is simulated inthe middle of a line connecting bus 7 and 8. Figs. 18–21 presentthe local and inter-area mode of the oscillations. These figuresshow the comparison of the NTVAC-PSO based UPFC controllerwith the TVAC-PSO and TVIW-PSO based UPFC controller. Theperformance of NTVAC-PSO is quite prominent compared withother optimization methods. These figures show that designedcontrollers using TVAC-PSO and TVIW-PSO provide a gooddamping for the study system, but the one designed byNTVAC-PSO performs better.Case 2: A three-phase fault of 100 ms duration is simulatedat the middle of one of the transmission lines connectingbus 9 and 10 with the same operating condition as Case 1.The inter-area and local modes of oscillations are shown inFigs. 22–25. The performance of the proposed method is found

to be significant in damping both the inter-area and local modeof oscillations. These figures show that the controllers designedby NTVAC-PSO method perform better and has a better featurecomparing to those designed TVAC-PSO and TVIW-PSO.Case 3: To validate the effectiveness of the proposed method ina multi-machine environment, the operating conditions (perunit) of the power network are changed to P1 = 0.5556,Q1 = 0.2056, P2 = 0.5556, Q2 = 0.2611, P3 = 1.3739, Q3 = 0.1502,P4 = 0.5556, and Q4 = 0.2244. The three-phase fault of Case 2 isinitiated for a 100 ms duration. Figs. 26–29 present the inter-area and local mode of oscillations for the transient disturbance.These responses establish the highly effective performance ofthe proposed controllers. Comparison of the results of thepresented method, TVAC-PSO and TVIW-PSO demonstrate thesuperior successful functioning of the NTVAC-PSO in dampingoscillations and enhance greatly the dynamic stability of thepower system.

These time-domain simulations are consistent with the resultsof the eigenvalue analysis. These figures show that the controllersdesigned using the proposed method performed better and had

better features compared with controllers designed by othermethods.

It is observed that, with the new method, the oscillations aredamped. These time domain simulations agree well with the re-sults of eigenvalue analysis. These figures show that the controllersdesigned by NTVAC-PSO method perform better and has a betterfeature comparing to those designed TVAC-PSO and TVIW-PSO.

6. Conclusions

This paper introduces a new adaptive PSO based on nonlinearacceleration coefficients. In the new approach, the NTVAC en-hances the performance and provides good balance between globalexploration and local exploitation abilities of the original algo-rithm. The proposed modification is easy to implement withoutadditional computational complexity. The performance of the pro-posed algorithm as a global optimization technique is evaluated byusing a set of 10 well-known unimodal/multimodal benchmarkfunctions. Compared with the findings obtained by using othermethods, the results obtained by the NTVAC-PSO algorithm per-formed excellently in terms of accuracy, convergence rate, stabil-ity, and robustness. The new method is also implemented to theoptimal design of UPFC controllers for oscillation damping. Com-paring the results of the proposed method, the TVAC-PSO andTVIW-PSO revealed that the NTVAC-PSO has superior and success-ful performance in oscillation damping and greatly enhances thedynamic stability of the power system.

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