Electrical Power and Energy Systems 85 (2017) 50–66

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

A power control strategy to improve power system stability in thepresence of wind farms using FACTS devices and predictive control

http://dx.doi.org/10.1016/j.ijepes.2016.08.0020142-0615/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: M.Darabian@znu.ac.ir (M. Darabian), ajalilvand@znu.ac.ir

(A. Jalilvand).

Mohsen Darabian ⇑, Abolfazl JalilvandDepartment of Electrical Engineering, University of Zanjan, Zanjan, Iran

a r t i c l e i n f o

Article history:Received 29 February 2016Received in revised form 14 June 2016Accepted 6 August 2016Available online 24 August 2016

Keywords:Power system stabilityPredictive controlStatic Synchronous Series Compensator(SSSC)Super Capacitor Energy Storage System(SCESS)Doubly Fed Induction Generator (DFIG)Wind turbine

a b s t r a c t

The main objective of this paper that distinguishes it from other similar articles is to employ predictivecontrol strategy to improve the stability of power systems (4- machines and 10-machine) in presence ofwind farms based on Doubly Fed Induction Generator (DFIG), using Static Synchronous SeriesCompensator (SSSC) and Super Capacitor Energy Storage System (SCESS). In this paper, SCESS is usedto control the active power in the Grid Side Convertor (GSC) and SSSC is employed to reduce low fre-quency oscillations. The proposed strategy based on the predictive control can be simultaneously usedto control the active and reactive power of the Rotor Side Convertor (RSC) as well as damping controllerdesign for SCESS and SSSC. A function is used in the predictive control strategy to reduce computationalcomplexity in selecting the input paths of Laguerre functions. Moreover, the sampling time is reduced bymeans of employing the exponential data weighting. Simulation results for the function-based predictivecontrol using disturbance scenario in the field of non-linear time are compared with the other two meth-ods, model-based predictive control and classic model (without using the predictive control). The effec-tiveness of the proposed strategy in improving stability is confirmed through simulation result.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Motivation and approach

Due to world population growth today, diminishing fossil fuelssources and concerns about environmental pollution, the use ofrenewable energy resources has drawn more and more attentionof researchers. Among the renewable energy sources, wind energyis one of the most popular type of energy to produce electricitythroughout the world. But because of the fluctuating nature ofwind power, the use of this energy causes the electricity producedby wind farms to be oscillated in electrical grids. Hence, the exis-tence of such fluctuations from the stability and power qualitypoint of view is non-negligible in a power system [1,2]. This canbe considered in all types of variable and constant speed wind tur-bines. In this regard, the use of Energy Storage System (ESS) andFlexible AC Transmission Systems (FACTS) devices can be very use-ful as a compensator to reduce oscillations and increase dampingin power systems [3,4]. Extensive studies have been done in thefield of power system stability in the presence of wind farms andcompensator devices. However, finding a method that can lead to

the development of industry and technology requires previousstudies and employing their advantages and disadvantages. There-fore, in this paper that its main objective is to use the predictivecontrol strategy to improve the stability of a hybrid power systemincluding wind farms, synchronous generators, energy storage sys-tems and FACTS devices, the most recent studies in each field areneeded to be evaluated.

1.2. Review of previous publications

Due to low-cost and direct control of active and reactive power,doubly fed induction generator is considered as one of the mostcommon types of variable speed wind turbine [5]. The rotor of thisgenerator is connected to the grid through a back-to-back bi-directional converter and its stator is connected directly to the grid.A three loop controller including GSC, RSC and a control loop for DClink capacitor for connecting these two converters is used to con-trol the bi-directional converter. Several studies have been madein recent years to stabilize the DFIG control systems, some ofwhich include sensitivity analysis [6], small signal stability analy-sis [6–8], eigenvalues analysis [9], approaches based on optimizingcontrol parameters in each of DFIG converters [10–12], a statefeedback [13] and robust control H2/H1 [14]. In theoptimization-based methods, the parameters of PI controller in

Nomenclature

Pxt extracted power from the wind turbine (W)qxt air density (kg/m3)Kxt swept area of blades (m2)Vxt wind speed (m/s);Dxt performance coefficient of bladesbxt blade pitch anglekxt tip speed ratiod1 –d9 constantsRb blade radius (m)xb angular velocity of blade (rad/s)Lss self-inductance of statorLrr self-inductance of rotorLmm mutual inductanceRs stator resistanceRr rotor resistanceids stator current in d-axisiqs stator current in q-axisidr rotor current in d-axisvds rotor current in q-axisvdr stator voltage in d-axisvqs stator voltage in q-axisvdr rotor voltage in d-axisvqr rotor voltage in q-axisHt inertia constant of wind turbineHg inertia constant of generatorxt angular speed of wind turbinexr angular speed of rotor of generatorTxt mechanical torque of wind turbineTtg shaft torqueText electrical torque of wind turbineKt damping coefficient of turbineKg damping coefficient of generatorKtg inertia constant of wind turbineLtg inertia constant of generatorPdc active power of the DC linkPrw active power of the rotor-side converterPgw active power of the grid-side converter

Cdc capacity of the DC link capacitorVdc voltage of the DC link capacitorZq1 & Zi1 PI controller coefficients for regulating the reactive

powerZq2 & Zi2 voltage of the DC link capacitorZq3 & Zi3 PI controller coefficients for regulating the reactive

poweridrw_ref current control in d-axis for RSCiqrw_ref current control in q-axis for RSCQsw_ref reference reactive powerxrw_ref reference speedZbg & Zig coefficients of the PI controller for regulating the voltage

of DC link capacitorZpb & Zpi coefficients of the PI controller for regulating the cur-

rent of GSCiqgw_ref reference current control in q-axis for GSCVdc_ref reference voltage of the DC link capacitorZbb & Zib coefficients of the PI controllersbb delay time constant for blade pitch angle controlPgw power of wind turbine measured for the blade pitch

controlPgw_ref reference power of wind turbine for the blade pitch con-

trolx(k) state vector of MPCu(k) input vector of MPCe(k) disturbance vector of MPCy(k) output vector of MPCk sampling instantGk weighting matrix of the cost functionSk weighting matrix of control action in the cost functiony0(n + k) prediction vector of the output signalyref (n + k) reference path of system’s futureDu(n + k) action control vectormc turns ratio of the coupling transformer of SSSCZinv modulation index of SSSCVdc_sssc DC capacitor voltage of SSSCbs phase angle of the injected voltage of SSSC

M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66 51

each of the induction generator converters are optimized by intel-ligent algorithms to apply the output of the controller with thelowest error to the relevant converter. PI controllers core problemis their dependence on operating point as well as their sensitivityto change in system conditions. Hence, the neural network andfuzzy logic based algorithms have been used in some studies tosolve this problemwhich have their own complexity and disadvan-tages as well [15,16]. Another method for reducing oscillations ofthe DFIG output power is to use Super Capacitor Energy StorageSystem [17,18]. Given that the use of power electronic devises isreduced in energy storage systems, in addition to reducing the costin case of designing a proper controller, an output power with thelowest oscillation will be achieved by DFIG [19]. In these systems,if the wind turbine undergoes oscillations as a result of drasticchange in wind, the power required by the system can be easilycompensated through a DC-DC converter [20]. In addition, Fly-wheel Energy Storage System (FESS) [21,22], and SuperconductingMagnetic Energy Storage (SMES) can be noted as other energy stor-age systems used at AC bus to compensate reactive power [23–25].One of the methods to reduce low frequency oscillation (LFO) inthe power system stability studies is to design damping controllerfor power system stabilizer (PSS) or FACTS devices. The SSSC is anew generation of FACTS devices which is connected in series withtransmission lines and leads the power flowing in transmission

lines to be converted from capacitive to inductive [26,27]. There-fore, the regional and inter-regional oscillations can be mitigatedthrough a proper controller design for SSSC and PSS. Differentmethods have been proposed to design a suitable controller forSSSC [28–33]. Wavelet neural adaptive method to design PID con-troller [28], nonlinear robust control method to improve dampingand transient stability [29], methods based on intelligent algo-rithms to optimize lead-lag controller parameters in the designof coordinated PSS-SSSC [30,31], damping controller design forSSSC in the presence of wind farms using adaptive-network-based fuzzy inference system (ANFIS) considering time delay [32]and modal analysis [33] are some of the approaches to increasestability of power systems using this type of FACTS devices. Theabove-mentioned techniques are able to provide a desiredresponse to system in case of the detailed design. But in case ofmultiple disturbances in a power system and lack of certainty ina wind system, a method is required to guarantee stability underall conditions. In other words, this method should be able to applythe constraints of the system as well as simultaneously control ofmultiple parameters with the lowest output error so as to ensurethe stability of the system. The predictive control is presented asa very powerful control tool for meeting the above requirements.Model Predictive Control (MPC) is assumed as one of the advancedcontrol methods in the field of industrial operations and research

52 M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66

activities [34]. In this method, by employing the process model andminimizing an objective function, reasonable control signals areachieved to design the control loop parameters.

The selection of appropriate control signals for Power SystemStabilizer (PSS) [35], the design of damping controller for a HVDCsystem [36], the control of active and reactive power in a wind tur-bine modeled in the state space [37,38], the control of active andreactive power in a micro-grid to coordinate wind turbine and abattery considering wind speed and network consumption [39]and the prediction of out power of a micro-grid to control voltageand active power in a 33-buses distribution system [40] are someof MPC applications in solving problems related to power systemstability. Different predictive control strategies have been pro-posed in various references, all of which are based on the modelpredictive. It has been tried in these methods to employ somemathematical equations and intelligent algorithms to improvethe process of solving the predictive model problems. In thisregard, the non-linear predictive model with Offset-Free to designStatic Var Compensator (SVC) [41], robust predictive control todesign control signals of FACTS devices to improve transient stabil-ity [42], the non-linear predictive control based on Takagi–Sugeno

Vw 5

IG

ac /dc /ac

Vw 10

IG

ac /dc /ac

V

Vw 2

IG

ac /dc /ac

Vw 7

IG

ac /dc /ac

V

Vw 1

IG

ac /dc /ac

Vw 6

IG

ac /dc /ac

V

Z 2,3

Z 4,5

Z 7,8

Z 9,10

Z 1,2 Z 6,7

Z 1

Z 6

0.69/33 KV

0.69/33 KV

0.69/33 KV 0.69/33 KV

0.69/33 KV

0.69/33 KV

G 1

G 2

SSSC

20/230 KV

20/230 KV

1

2

5

7

86

TL 1

25 KM 10 KM

110 KM

110 KM

Load

1 C7

PCC (A )

Fig. 1. The model of four-machine power system

fuzzy model [43], distributed model predictive control [44] andfunctional predictive control [45,46] are included as moreadvanced methods for predictive model which are required toimplement the advanced mathematical equations in the field ofcontrol engineering.

1.3. Contributions

In this paper, functional model predictive control is used toenhance stability of the four machine power system with approachof the active and reactive power control in a wind turbine. Activepower control in grid-side convertor is carried out using a dampingcontroller in the SCESS structure. This structure is designed in sucha way that in addition to balancing DC voltage link, have the abilityto control the active power in GSC. The active and reactive powercontrol in the rotor-side converter is conducted with the approachof selecting the proper control signals for the RSC inverter. More-over, in order to mitigate low frequency oscillations, a dampingcontroller is used in the SCESS structure to control line power flowand increase the power system damping.

w 15

IG

ac /dc /ac

Vw 20

IG

ac /dc /ac

w 12

IG

ac /dc /ac

Vw 17

IG

ac /dc /ac

w 11

IG

ac /dc /ac

Vw 16

IG

ac /dc /ac

Z 12,13

Z 14,15

Z 17,18

Z 19,20

Z 11,12 Z 16,17

Z 11

Z 16

0.69/33 KV

0.69/33 KV

0.69/33 KV

0.69/33 KV

0.69/33 KV

0.69/33 KV

33/230 KV

G 3

G 4

20/230 KV

230/20 KV 3

4

9

10

12

11

TL 2

25 KM10 KM

110 KM

110 KM

Load

2

C8

PCC (B)

100 MW - Wind Farme

connected to the wind farm and SSSC-SCESS.

M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66 53

1.4. Paper structure

This paper is set out as follow: Section 2 presents the powersystem modeling including synchronous generator dynamic equa-tions, the grid and rotor-side converters equations for DFIG and thedamping controller design equations for SCESS and SSSC, Section 3presents the modeling of predictive control strategy and its imple-mentation on the power system, Section 4 presents the simulationresults and finally the conclusion is presented in Section 5.

2. Modeling of power system elements

2.1. Synchronous generator model

In this section, the equations of the test system shown in Fig. 1are described. In this paper, the two-axis model is employed toanalyze the dynamic equations of the synchronous generator[47]. In the two-axis model, the impact of sub-transient reactanceis ignored and only the synchronous generator transient reactanceis used in the modeling. The excitation system including AVR-PSSis type IEEE-1 and its relevant parameters are given in AppendixA (Table 1). The two-axis model dynamic equations for the ith syn-chronous generator can be stated as follow:

dE0qi

dt¼ 1

T 0doi

½�E0qi þ Efdi þ ðXdi � X0

diÞ�Idi ð1Þ

dE0di

dt¼ 1

T 0qoi

½�E0di � ðXqi � X0

qiÞIqi� ð2Þ

ddidt

¼ xi �xs ð3Þ

dxi

dt¼ xs

2Hi½Tmi � ½IdiE0

di þ IqiE0qi � ðX0

qi � X0diÞIdiIqi� � Diðxi �xsÞ� ð4Þ

Table 1Employed system parameters.

Generator G1 G2 G3 G4 Ex

Multi-machine power systemRated MW 719 700 700 719 KA

Rated MVAR 185 235 176 202 TAVbase kV 20 20 20 20 KE

Xd (p.u.) 1.8 1.8 1.8 1.8 TEX0

d (p.u.) 0.3 0.3 0.3 0.3 KF

Xq (p.u.) 1.7 1.7 1.7 1.7 TFX0

q (p.u.) 0.55 0.55 0.55 0.55 AX

s0do 8 8 8 8 BXs0qo 0.05 0.05 0.05 0.05 –H (s) 6.5 6.175 6.175 6.5 –xs (p.u.) 1 1 1 1 –

DFIG based wind turbine (100 MW)P = 5 MW V = 0.69 KV Rs = 0.042 p.u. Rr = 0.005 p.u.Xtg = 0.55 p.u. Kt = 0.5 p.u. Ktg = 2.5 p.u. Ltg = 0.93 p.u.Zq2 = 8.6 Zi2 = 3.87 Zq3 = 15 Zi3 = 9.2bwt_min = 0� bwt_max = 30� Zbb = 1.11 d1 = 0.22d6 = 6.161 d7 = 11.89 d8 = �12.95 d9 = 0.088

SSSC (±50 MVAR)S = 50 MVA V = 161 kV F = 60 HZ R = 0.01 L = 0.02

Transmision linesRTL1 = RTL2 = 0.00005 p.u. XTL1 = XTL2 = 0.0012 p.u.Z1 = Z6 = Z11 = Z16 = 0.0738 + j0.1050 Ω Z1,2 = Z6,7 = Z11,12 = Z16,17 =Z4,5 = Z9,10 = Z14,15 = Z19,20 =0.3658 + j0.1591X –SCESSC = 20 mF Cees = 20 F Lees = 50

2.2. Doubly fed induction generator model

In the DFIG based wind turbine system, the GSC and RSC areconnected to each other through a DC link and in back-to-backform as seen in Fig. 2. The DC link duty is to keep the balancebetween two convertors. In this paper, the vector technique is usedto model the synchronous generator to control the active and reac-tive powers [6]. But due to the space constraints, the electricalequations of synchronous generator, the mechanical and aerody-namic equations of wind turbine are ignored. But the equationsof RSC and GSC controllers are fully expressed due to the controlof active and reactive powers as the main subject of this paper.The more comprehensive information regarding the modeling ofrelevant controllers for DFIG is given in [6,9].

2.3. Mathematical model of RSC

The main duties of the RSC are to control the output active andreactive output power of DFIG, extract the maximum power fromwind and provide the reactive power required by the inductiongenerator. In this controller, active power and voltage are con-trolled by vqrw and vdrw components, respectively. As seen inFig. 3, voltage control is done through reactive power control bymeasuring Qsw and reactive power control is carried out throughmeasuring the turbine speed xrw. Comparing these control signalswith reference signals by a PI controller for each of the parame-ters, the reference signals (iqrw_ref, idrw_ref) are produced. Thereference signals in d-q axis are compared with the currentreference signal to produce the error signal and then after passingthrough two PI controllers, v⁄qrw and v⁄drw signals are obtained.These reference signals after amplifying by two other componentsof current signals produce vqrw, vdrw signals in order to be sentto PWM. And finally, a proper pulse is sent by PWM to the inverterto apply switching [9]. The RSC equations can be stated asfollow:

citer Exciter 1 Exciter 2 Exciter 3 Exciter 4

20 20 20 20(s) 0.2 0.2 0.2 0.2

1 1 1 1(s) 0.314 0.314 0.314 0.314

0.063 0.063 0.063 0.063(s) 0.35 0.35 0.35 0.35

0.015 0.015 0.015 0.0159.6 9.6 9.6 9.6– – – –– – – –– – – –

Cdc = 0.01 F Lmm = 2.9 p.u. Lrr = 3.056 p.u. Lss = 3.071 p.u.Ht = 0.05 p.u. Hg = 10.2 p.u. Zq1 = 15 Zi1 = 9.2Zbg = 17.35 Zig = 10.43 Zpb = 12 Zib = 8.53d2 = 116 d3 = 0.954 d4 = 0.18 d5 = 0.955– – – –

Vdc = 40 kV Cdc = 175 lF Zb_sssc = 0.0015 Zi_sssc = 0.15

0.0771 + j0.1148X Z3,4 = Z8,9 = Z13,14 = Z18,19 = 0.2756 + j0.1558X–

mH VSC = 2 kV Vdc = 4 kV

++

+

+-- sZ

Z igbg +

sZ

Z pipb +

Xqgwi

dgwV*dgwVrefdgwi _refdcV _

dcV

GB DFIG

DC

DC

Supercapacitor

Bank

Ces

s

LGrid

GSC

Cdc

VdcgwV

gwi

refdcV _

rwiRSC

swQrefswQ _

WT Controller

tωβ

tgTwtT

refrw _ω

rwω

rwV

sP sQ

gwP gwQrwS

Vw

Fig. 2. Block diagram of DFIG-based wind turbine.

sZZ 1i

1q +refrw _ω

+-rwω

refqrwi _

+ - sZZ 2i

2q +*qrwv

-

mmrs Lww )( −

rrrs Lww )( −

dswi

qrwi

rrrs Lww )( −drwi

qrwv+

sZZ 3i

3q ++-

refdrwi _

+-

sZZ 2i

2q +*drwv +

- drwv

+

mmrs Lww )( −qswi

refswQ _

swQ

+

Fig. 3. Block diagram of the rotor-side converter.

54 M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66

_u1 ¼ xrw ref �xrw

iqrw ref ¼ Zq1ðxrw ref �xrwÞ þ Zi1u1

_u2 ¼ iqrw�ref � iqrw ¼ Zq1ðxrw ref �xrwÞ þ Zi1u1 � iqrw

_u3 ¼ Qsw ref � Qsw

idrw�ref ¼ Zq3ðQsw ref � QswÞ þ Zi3u3

_u4 ¼ idrw�ref � idrw ¼ Zq3ðQsw ref � QswÞ þ Zi3u3 � idrw

vqrw ¼ Zq2ðZq1ðxrw ref �xrwÞ þ Zi1u1 � iqrwÞ þ Zi2u2

þsrwxsLmmidsw þ srwxsLrriqrw

vdrw ¼ Zq2ðZq3ðQsw ref � QswÞ þ Zi3u3 � idrwÞþZi2u4 � srwxsLmmiqsw � srwxsLrridrw

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

ð5Þ

++

- -

sZZ pi

pb +

Tg

TgXdgwi

qgwV*qgwVrefqgwi _

Fig. 4. Control block diagram of the grid-side converter.

2.4. Mathematical model of GSC

The main duties of the GSC are to keep the DC voltage Link in aconstant value and control the reactive power of the network. The

detailed block diagram of the GSC is shown in Fig. 4. According tothis figure, comparing the reference signal of DC link (vdc_ref) withthe measured value (vdc) by a PI controller, the current referencesignal in the d-axis is obtained. The current reference signals(iqgw_reg, idgw_reg) are compared with the measured values and thenafter passing through two PI controllers, v⁄dgw and v⁄qgw signals aregenerated. This control signal after combining with the current sig-nals generate vqgw, vdgw signals in order to be sent to PWM. Andfinally, proper pulses provided by PWM are sent to the inverter toapply switching [9]. The GSC equations can be stated as follow:

_u5 ¼Vdc�ref �Vdc

idgw�ref ¼�ZbgDVdc þZigu5

_u6 ¼ idgw�ref � idgw ¼�ZbgDVdc þZigu5� idgw_u7 ¼vqgw�ref � iqgwDvdgw ¼ Zpg _u6þZpiu6 ¼ Zpgð�ZbgDVdc þZigu5� idgwÞþZpiu6

Dvqgw ¼ Zpg _u7þZpiu7 ¼ Zpgðiqgw�ref � iqgwÞþZpiu6

8>>>>>>>><>>>>>>>>:

ð6Þ

2.5. Controller of SCEES convertor

In this paper, a SCEES is used to guarantee the maintenance ofthe DC link voltage. In addition, a damping controller is designedin the SCEES system to control the reactive power of the systemunder network changes. This convertor is composed of a capacitorbank and a dual switch DC/DC converter which is connected to the

+

-

refgwP _

gwP

+-

-+

-*_ SCdcV

SCdcV _

maxCV

minCV

gwPΔgwP′Δ

sγΔ

K1

K2

DC LinkLess

Super Capacitor

Bank-+SCV

K2K1

S1

w

w

ST1ST+S

KKC D+

CSVP

PST1K

+

Gate Signal Generator

Fig. 5. Block diagram for designing the damping controller of the SCESS.

M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66 55

DFIG through the DC link (see Fig. 5). This convertor can be used inboost and buck modes depending on the switches K1 and K2. WhenK1 is open, the converter operates in boost mode and in case of K2

being open, the convertor operates in buck mode. When K1 is close(buck mode), based on the ratio of the capacitor bank voltage to theDC link voltage, the contribution of switch K1 (when switch K1 ison) can be obtained as follow (S1):

S1 ¼ Vsc

Vdcð7Þ

Moreover, when K2 is close (boost mode), S2 is defined as 1-S1.The ratio of the capacitor bank voltage to the DC link voltage isconsidered as 0.5 in this paper which means S1 = 0.5. The SCEESequations can be formulated as follow:

DPgwdt ¼ Pgw ref � Pgw

DP0gwdt ¼ ðPgw ref � PgwÞ � DP0gw

Tw

dVCSdt ¼ KC ðPgw ref � PgwÞ � DP0gw

Tw

� �þ KDDP

0gw

dDcsdt ¼ 1

TP½KPðV�

dc SC � Vdc SC � VCSÞ � Dcs�

8>>>>>><>>>>>>:

ð8Þ

Voltage Source Inverte

dcC

+ -VseVm

Fig. 6. Circuit struct

2.6. Mathematical model of SSSC

The SSSC can be used as a reactive power compensator based onvoltage source inverter connected in series with transmission line.In other words, this compensator can operate in two modes: thecapacitive and inductive. The structure of SSSC including a voltagesource inverter, a series transformer, a capacitor of capacity Cdc anda control block is demonstrated in Fig. 6. The SSSC series injectedvoltage in d-q axis can be stated as follow using synchronous ref-erence frame:

vds ¼ mcZinvVdc sssc CosðbsÞvqs ¼ mcZinvVdc sssc SinðbsÞ

�ð9Þ

Dynamic equation of DC link capacitor to preserve the balanceof power in dc and ac sides is expressed as follow:

Vdc sssc ¼ 1Cdc

mcZinvðid cos bs þ iq sin bsÞ �Vdc sssc

Rdc

� �ð10Þ

The block diagram of SSSC for the capacitive mode along withdamping controller is depicted in Fig. 7. By adding a control signalbased on XF, the transmitting power in the line can be controlled toreduce low frequency oscillations (see Fig. 7). The SSSC dampingcontroller equations can be expressed as follows:

DPLdt ¼ PL ref � PL

DP0Ldt ¼ ðPL ref � PLÞ � DP0L

Tw

dXFdt ¼ KF ðPL ref � PLÞ � DP0L

Tw

� �þ KADP

0L

8>>><>>>:

ð11Þ

3. Predictive control

In the problems requiring the prediction of system’s futurebehavior, the model-based predictive control is a powerful tech-nique [48,49]. The information predicted by this method is usedto obtain the optimal point based on the criteria of each specificproblem. As the basis of this method is on the process model,therefore, the predicted inputs and outputs can also be used forstate estimation of the process. The new measurements of the pro-cess model sampled in each time instant are injected into the con-trol loop, and on this basis, the predictive horizon is forecasted. Themerit of this strategy is that in each sampling interval, a con-strained optimization problem is solved. The limitations and everychange in the process constraints can be applied to the system asthe error signal. One of the appealing features of the predictivecontrol which has distinguished it from the other control methodsis that a series of control variables with a given length, i.e. the pre-dictive horizon is calculated for the future behavior the system.Fig. 8 shows the block diagram of this control method. Based on

r Controller

Xref

Vn

ure of an SSSC.

Magnitude and Phase Angle Clculator

d-q Transformation

diqi

Phase Loked Loop (PLL)

i

mv

Gate Patern Logic

2π−

maxFXminFX

FX

invZ

1+

-ssscdcV _

S

ZZ

ssscissscb

__ +

I

-

+ π

-+

++ +

++ +

VSI

refLP _

LPS1

w

w

ST1ST

+LPΔLP ′Δ

SK

K AF +

refX

θ

irθ

iθ vθ seβ

seα

Vdc_refVsc_ref

Fig. 7. Block diagram for designing the damping controller of the SSSC.

Optimizer

ConstraintsCost Function

Predictor

Plant

)(nyref

)(ny

)(nε

UΔ

)( knu +Δ

)(ny

)(ry

)(ˆ kny +

)( knyref +

)( kn +ε

++

Fig. 8. Controller structure of the model predictive control.

56 M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66

this figure, the problem constraints, the objective function, and theoutput of the prediction system can be applied to the optimizationsystem in order to obtain an appropriate output for the system. Asin this paper the predictive control strategy in the multi-objectiveform is used, therefore, the utilized model is in the state space toaccurately follow the desired objectives. Thus, the equations ofthe MPC are represented in the discrete state space as (12):

xðkþ 1Þ ¼ AzxðkÞ þ BzuðkÞ þ EzdðkÞyðkÞ ¼ CzxðkÞ

�ð12Þ

The objective function is selected in a way that the future out-puts are able to track the reference signal in the prediction horizon,and the required control action is low as possible as more. There-fore, in order to attain the desired objectives, the objective functionof the predictive control can be described as (13):

FfitðnÞ ¼Xma

k¼1

Gkðy0ðnþ kÞ � yref ðnþ kÞÞ2 þXmb

k¼1

SkDuðnþ kÞ2 ð13Þ

According to the above relation, the prediction vector, which isconsidered for the system’s output, is defined as a 1 �ma matrix inwhich ma is called the prediction horizon; Also, Du is a 1 �mb

matrix in which mb is named the control horizon.

3.1. The considered constraints in the model predictive control

The following constraints are considered in solving of the prob-lem by the predictive control:

� Limitation on the amplitude and variations of the input;� Limitation on the state variables;� Limitation on the output variables.

The above constraints can be mathematically described as (14):

umin 6 uðnþ kÞ 6 umax;Dumin 6 Duðnþ kÞ 6 Dumax

xmin 6 xðnþ kÞ 6 xmax;Dxmin 6 Dxðnþ kÞ 6 Dxmax

ymin 6 yðnþ kÞ 6 ymax;Dymin 6 Dyðnþ kÞ 6 Dymax

8><>: ð14Þ

The optimal input control sequence is given by solving theobjective function (13) with system constraint (14).

3.2. Functional model predictive control (FMPC)

3.2.1. Laguerre-based model predictive controlIn the conventional MPC, the future control signal is considered

as a vector of forward shift operator with length of mb.

M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66 57

DU ¼ ½DuðnÞ; . . . ;Dbðnþ kÞ; . . . ;Dbðnþmb � 1Þ� ð15Þ

where mb unknown control variables are achieved in the optimiza-tion procedure. If the controlled system is a multivariable system,then this procedure should be done for all inputs, which need largecomputational burden. Therefore, MPC may not be fast enough tobe used as a real-time optimal control for such systems. The prob-lem will be worse when large prediction horizon is needed toachieve high closed-loop performance. A solution to this drawbackis using functional MPC. In the functional MPC, future input isassumed to be a linear combination of a few simple base functions.In principle, these could be any appropriate functions. However inpractice, a polynomial basis is usually used [50]. This approximationof input trajectory can be more accurate by proper selection of basefunction. Using FMPC, the term used in the optimization procedurecan be reduced to a fraction of that required by classical MPC.Therefore, the computational load will be reduced largely.

In this paper, orthonormal basis Laguerre function is used formodeling input trajectory. Laguerre polynomial is one of the mostpopular orthonormal base functions, which has extensive applica-tions in system identification [51]. The z-transform of g-thLaguerre function is given by (16):

Cg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� b2

pz� b

z�1 � b

1� bz�1

� g�1

;0 6 b 6 1 ð16Þ

In this transform, b is the pole of the power system; if 0 < b < 1,the system will be stable.

Optimizer

Constraints

-

)(_ rQ refsw

+

-+

-+

+

+

)(rV drw

)(rVqrw

)(rSγΔ

)(rX F

-

-)(riω

)(_ rrefrwω

)(_ rP refgw

)(_ rP refL

)( rsω

)(_ 1rQ refsw +

-+ )( 1rV drw +

-+ )( 1rV qrw +

)(_ 1rrefrw +ω

)(_ 1rP refL +

-+

)( 1rs +ω)( 1ri +ω

)( 1rS +Δ γ-+

-+

)(_ 1rP refgw +

)( 1rX F +

Fig. 9. Block diagram of th

Now, each input control signal can be described using theLaguerre functions as below:

Duðnþ kÞ �Xmg¼1

ag � f gðkÞ ð17Þ

In the above relation, fg is the transposed form of the Laguerrefunctions defined in Eq. (16), and ag is named the parameter vector.In practical applications, the value of m is considered to be lowerthan 10. Choosing larger values for m will increase the input pathsprediction for the Laguerre functions.

3.2.2. Exponentially weighted model predictive controlClosed-loop performance of MPC depends on the magnitude of

prediction horizon ma. Generally, by increasing the magnitude ofprediction horizon, the closed-loop performance will be improved.However, practically, selection of large prediction horizon is lim-ited by numerical issue, particularly in the process with high sam-pling rate. One approach to overcome this drawback is to useexponential data weighting in model predictive control [52].

To design discrete model predictive control with exponentialdata weighting, input, state, and output vectors are changed inthe following way (18):

DUT ¼ ½q�0DuðnÞ; . . . ;q�ðmb�1ÞDuðnþmb � 1Þ�XT ¼ ½q�1xðnþ 1Þ; . . . ;q�maxðnþmaÞ�YT ¼ ½q�1yðnþ 1Þ; . . . ;q�mayðnþmaÞ�

8><>: ð18Þ

Objective Function

)(rQ sw

)(rrwω )(rVqrw

)(rV drw

)(rSγΔ

)( rX F

)(riω

)(rPgw

)( rPL

)( riδ

)(

1r

Qsw

+

)(

1r

rw+

ω

)(

1r

P gw

+

)(

1r

P L+

)(

1r

i+

δ

)(rVdrw

)(rVqrw

)(rSγΔ

)(rX F

)(riω

Pow

er s

ysyt

em

Predictor

e proposed strategy.

(B)(A)

(D)(C)

(F)(E)

(H)(G)

0 1 2 3 4 5 6 7 8 9 10

-2

-1

0

1

2

x 10-3

Time [sec]

Δω13

[p.

u]

Case 1Case 2Case 3

0 1 2 3 4 5 6 7 8 9 10

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

Time [sec]

Δω24

[p.

u]

Case 1Case 2Case 3

0 1 2 3 4 5 6 7 8 9 10

20

25

30

35

Time [sec]

Δδ13

[de

g]

Case 1Case 2Case 3

0 1 2 3 4 5 6 7 8 9 10

20

22

24

26

28

30

32

34

36

Time [sec]

Δδ24

[de

g]

Case 1Case 2Case 3

0 1 2 3 4 5 6 7 8 9 10

0.95

1

1.05

1.1

Time [sec]

VG

1 [p.

u]

Case 1Case 2Case 3

0 1 2 3 4 5 6 7 8 9 10

0.95

1

1.05

1.1

Time [sec]

VG

3 [p.

u]•

Case 1Case 2Case 3

0 1 2 3 4 5 6 7 8 9 10

3

3.5

4

4.5

5

5.5

Time [sec]

Act

ive

Pow

er o

f B

us 8

[p.

u]

Case 1Case 2Case 3

0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

Time [sec]

Rea

ctiv

e P

ower

of B

us 8

[p.

u]

Case 1Case 2Case 3

Fig. 10. The response of time-domain simulation of three-phase short circuit fault in four-machine power system.

58 M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66

In the above equation, the symbol q has been used for represent-ing the adjustment of parameters in exponential weight. The value ofq is chosen to be larger than 1. Therefore, the new equations of theutilized model in the state space can be explained as (19):

xðnþ 1Þ ¼ AxðnÞ þ BDuðnÞyðnÞ ¼ CxðnÞ

(ð19Þ

Substituting the following relations in the above equationresults in the new objective function of (21):

A ¼ Aq; B ¼ B

q; C ¼ C

qð20Þ

FfitðnÞ ¼Xma

k¼1

Gkðyðnþ kÞ � kref ðnþ kÞÞ2 þXmb

k¼1

SkDuðnþ kÞ2 ð21Þ

Also, the constraints of (14) are modified to (22):

q�Zumin 6 uðnþkÞ6q�Zumax;q�ZDumin 6DuðnþkÞ6q�ZDumax

q�Zxmin 6 xðnþkÞ6q�Zxmax;q�ZDxmin 6DxðnþkÞ6q�ZDxmax

q�Zymin 6 kðnþkÞ6q�Zymax;q�ZDymin 6q�ZDyðnþkÞ6q�ZDymax

8><>:

ð22ÞAfter solving Eq. (21), the input path should be rewritten as

(23):

(J)(I)

(L)(K)

(N)(M)

(P)(O)

0 1 2 3 4 5 6 7 8 9 10

0.2

0.3

0.4

0.5

0.6

Time [sec]

0 1 2 3 4 5 6 7 8 9 10Time [sec]

0 1 2 3 4 5 6 7 8 9 10Time [sec]

0 1 2 3 4 5 6 7 8 9 10Time [sec]

0 1 2 3 4 5 6 7 8 9 10Time [sec]

Pg

1 [p.

u]

Case 1Case 2Case 3

0 1 2 3 4 5 6 7 8 9 100.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time [sec]

Qs 1 [

p.u]

Case 1Case 2Case 3

0.4

0.6

0.8

1

1.2

PW

ind

Farm

[p.

u]

Case 1Case 2Case 3

0.4

0.5

0.6

0.7

0.8

0.9

Q W

ind

Farm

[p.

u]

Case 1Case 2Case 3

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Volta

ge o

f DC

Lin

k [p

.u]

Case 1Case 2Case 3

0

0.5

1

1.5

2

V W

F1 [

p.u]

Case 1Case 2Case 3

0

0.5

1

1.5

2

VP

CC

A [

p.u]

Case 1Case 2Case 3

0.4698

0.4699

0.47

0.4701

0.4702

0.4703

Vdr

w

Case 1Case 2Case 3

0 1 2 3 4 5 6 7 8 9 10Time [sec]

0 1 2 3 4 5 6 7 8 9 10Time [sec]

Fig. 10 (continued)

M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66 59

DUT ¼ ½b0DuðkÞ; . . . ; bðmb�1ÞDuðkþmb � 1Þ� ð23ÞIn brief, the sequence of solving the problem of functional pre-

dictive control can be followed as the below stages:

� Assigning a proper value for q;� Substituting the matrices (A,B,C) and the variables (U,X,Y) inEqs. (20) and (21);

� Applying the constraints in the objective function according tothe characteristics of the problem using Eqs. (22) and (23);

� Implementing the optimization procedure for the objectivefunction based on the Laguerre functions, and calculating thecoefficients of this function;

� Processing the input control signals chosen by the Laguerrefunctions using Eq. (17).

� Sorting the inputs according to (23), and applying it to the con-sidered system.

The functional MPC differs from the classical MPC in somecases; The Laguerre function and the exponentially weights repre-sented in (16) and (17) are employed to produce the initial controlinput sequence Duðnþ kÞ in case of functional MPC. Then, by min-

imizing the cost function FfitðnÞ described by (21) the optimal con-trol trajectory is achieved by means of the initial control inputsequence. The computational burden to acquire the optimal

(R)(Q)

0 1 2 3 4 5 6 7 8 9 10Time [sec]

0 1 2 3 4 5 6 7 8 9 10Time [sec]

0.3898

0.3899

0.39

0.3901

0.3902

0.3903

Vqr

w

Case 1Case 2Case 3

0.6

0.62

0.64

0.66

0.68

Vγ s

Case 1Case 2Case 3

(S)

0 1 2 3 4 5 6 7 8 9 100.45

0.5

0.55

0.6

0.65

Time (sec)

XF

Case 1Case 2Case 3

Fig. 10 (continued)

60 M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66

control trajectory is reduced employing the initial control inputsequence with suitable weighting factors, Gk and Sk. Whereas, incase of classical MPC, minimizing the cost function FfitðnÞ describedin (13) results in the optimal control trajectory Duðnþ kÞ directly.In order to minimize the cost function FfitðnÞ in this case, more cal-culations are required to obtain the optimal control trajectory.

3.3. Implementation of the proposed strategy for the system understudy

In order to implement the predictive strategy on the systemunder study, the relations outlined in the strategy should beadapted to the power system model (as seen in Fig. 9). Therefore,it is essential to specify the network dynamic equations includingsynchronous generators, wind farm, SSSC and SCESS in state space.The state space equations can be represented as follow:

_X ¼ AX þ BU þ ER

Y ¼ CX þ DU

(ð24Þ

where X is the system state vector and is defined in this paper asfollow:

X ¼ ½XSG;XRSC ;XGSC ;XSCESS;XSSSC �T ;XSG ¼ ½E0

qi; E0di; di;xi�;XRSC ¼ ½u1; u2; u3; u4�

XGSC ¼ ½u5;u6;u7�;XSCESS ¼ ½DPgw;DP0gw;VCS;Dcs�;

XSSSC ¼ ½DPL;DP0L;XF �

8>>>><>>>>:where D is the variable value (D = 0), U is the input vector of predic-tive control (U = [di, xrw_ref, Qsw_ref, Pgw_ref, PL_ref]T) and R is the dis-turbance vector (R = 0). In order to modify the optimum responseobtained from the predictive control, constraints should be definedwithin a legal range. In this paper, the constraints are defined as fol-low to achieve the desired objectives:� Active and reactive powers control of DFIG with the approach ofselecting proper reference vectors for the RSC (Vdrw_min 6 Vdrw -6 Vdrw_max, Vqrw_min 6 Vqrw 6 Vqrw_max).

� Active power control of GSC with the approach of selectingproper output for damping controller in super capacitor energystorage system (VcS_min 6 Vcs 6 Vcs_max).

� Line power flow control with the approach of selecting properoutput for damping controller in SSSC (XF_min 6 XF 6 XF_max).

In general, the min/max ranges of the above-mentioned defini-tions are expressed as follow (25):

vdrw min

vqrw min

Vcs min

XF min

xi min

26666664

37777775¼ 0 6 u 6 1 ¼

vdrw max

vqrw max

Vcs max

XF max

xi max

26666664

37777775

ð25Þ

The constraints applied to control signals slow down the simu-lation speed whereas this problem does not happen in case of statevectors. Hence, the constraints applied to state vectors within alegal min/max range are stated as follow (26):

E0qi min

E0di min

u3 min

u5 min

266664

377775 ¼

0000

26664

37775 6 x 6

0:30:20:70:4

26664

37775 ¼

E0di max

E0di max

u3 max

u5 max

26664

37775 ð26Þ

In addition, the predictive control strategy parameters used in thesimulation are set as b = 0.24, m = 7, q = 1.08, ma = 200, mb = 5 forFMPC and mb = 80 for MPC. The coefficients in weight matrixes areselected as G = 0.14� Imb � mb and S = 1� Ima � ma, respectively aswell.The sampling time for the predictive controller is assumed as 0.03 s.

4. Simulation result

4.1. Four-machine two-area power system

As a larger system, a two area four machine power systemequipped with a wind farm and a SSSC is employed to evaluate

Table 2The eigen values of four-machine power system (in rad/s).

Operation points Variables PI MPC FMPC

xr1. . .5 = 1.09 p.u. Dx13 �0.43 ± 2.92j �1.18 ± 1.98j �1.32 ± 1.97jVx1. . .5 = 12 m/s Dd24 �0.31 ± 3.12j �1.19 ± 1.99j �1.41 ± 1.98jxr6. . .10 = 1.03 p.u. VG3 �0.47 ± 3. 25j �1.16 ± 2.19j �1.46 ± 2.19Vx6. . .10 = 11.5 m/s PBus-8 �0.33 ± 2.94j �1.31 ± 2.32j �1.65 ± 2.34jxr11. . .15 = 1 p.u. QBus-8 �0.44 ± 2.89j �1.22 ± 2.14j �1.68 ± 2.16jVx11. . .15 = 11 m/s Pg1 �0.38 ± 2.99j �1.35 ± 2.17j �1.75 ± 2.17jxr16. . .20 = 0.994 p.u. VWT-1 �0.31 ± 3.09j �1.21 ± 2.15j �1.54 ± 2.14jVx16. . .20 = 10.5 m/s QWT-3 �0.48 ± 3.31j �1.27 ± 2.23j �1.68 ± 2.23j

xr1. . .5 = 0.810 p.u. Dx12 �0.48 ± 2.89j �1.31 ± 2.22j �1.54 ± 2.25jVx1. . .5 = 8.5 m/s Dd13 �0.23 ± 3.03j �1.41 ± 2.19j �1.78 ± 2.19jxr6. . .10 = 0.925 p.u. VG4 �0.31 ± 3.23j �1.38 ± 2.16j �1.52 ± 2.17jVx6. . .10 = 9.5 m/s PBus-8 �0.39 ± 3.17j �1.44 ± 2.14j �1.79 ± 2.14jxr11. . .15 = 1 p.u. QBus-8 �0.26 ± 3.14j �1.27 ± 2.11j �1.69 ± 2.11jVx11. . .15 = 11 m/s Pg6 �0.13 ± 3.54j �1.49 ± 2.26j �1.96 ± 2.25jxr16. . .20 = 1.09 p.u. VWT-6 �0.18 ± 3.87j �1.53 ± 2.31j �2.04 ± 2.29jVx16. . .20 = 12 m/s QWT-9 0.16 ± 3.65j �1.48 ± 2.37j �1.93 ± 2.36j

xr1. . .5 = 0.729 p.u. Dx34 �0.36 ± 3.96j �1.16 ± 1.95j �1.59 ± 1.96jVx1. . .5 = 8 m/s VG3 �0.31 ± 3.73j �1.11 ± 1.97j �1.61 ± 1.97jxr6. . .10 = 0.875 p.u. PBus-8 �0.28 ± 4.76j �1.27 ± 2.21j �1.85 ± 2.23jVx6. . .10 = 9 m/s QBus-8 �0.12 ± 5.87j �1.17 ± 2.41j �1.87 ± 2.38jxr11. . .15 = 0.975 p.u. Pg11 �0.08 ± 5.76j �1.09 ± 2.01j �1.54 ± 2.02jVx11. . .15 = 10 m/s Pg17 �0.03 ± 5.98j �1.15 ± 2.12j �1.78 ± 2.12jxr16. . .20 = 1 p.u. QWT-12 �0.02 ± 6.11j �1.18 ± 2.23j �1.76 ± 2.24jVx16. . .20 = 11 m/s QWT-17 �0.05 ± 5.87j �1.10 ± 2.11j �1.44 ± 2.11j

8G

1G

3G

5G

9G

7G

1

39

31

10

8

25

26

28 29

9

2416

1514 21

6

22

23

7

19

20

5

43

30

12

11 13

35

2

3736

34

33

32 17

2718

6G

2G

10G

4G

SSSC

WF 1

WF 2

Fig. 11. Single-line diagram of New-England system with wind turbines and SSSC.

M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66 61

the proposed scheme. The wind farm consists of 20 wind turbinesbased on DFIG with capacity of 5 MW for each unit which is trans-ferred through a 0.69/33 kV HV transformer. The total power pro-duced by these turbines is 100 MW and is transferred to the power

system through a 33/230 kV HV transformer. It is worthmentioning that each of these turbines are equipped with SCESSand wind speed is considered as of 12 m per second for each ofthem. As seen in Fig. 1, a SSSC is used between buses 8 and 9 to

62 M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66

mitigate low frequency oscillation between two areas. The moredetailed data of the system is given in Appendix A. A three-phasefault at t = 2 s is applied to Bus number 10 and cleared after0.2 s.

Three different scenarios are considered as follow and resultsare shown in Fig. 10(A)–(I):

(A)

(C)

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

2 x 10-3

Time (sec)

Δω1-

5 [rad

/ se

c]

PIMPCFMPC

-4

-2

0

2

4

6

8

10

Pow

er fl

ow b

etw

een

buse

s 1-

39 [p

.u.]

PIMPCFMPC

(E)

(G)

4.5

4.6

4.7

4.8

4.9

5

5.1

5.2

Rea

ctiv

e po

wer

of W

F1

[p.u

.] PIMPCFMPC

0.7

0.75

0.8

0.85

0.9

0.95

Vdr

w

PIMPCFMPC

0 2 4 6 8 10 12 14 16 18 20Time (sec)

0 2 4 6 8 10 12 14 16 18 20Time (sec)

0 2 4 6 8 10 12 14 16 18 20Time (sec)

Fig. 12. The response of time-dom

Case1: PI controller in absence of SCESS-SSSC compensators.Case2: MPC controller in presence of SCESS-SSSC compensators.Case3: FMPC controller in presence of SCESS-SSSC compensators.

The variations in speed deviation for G1–3 and G2–4 are shown inFig. 10(A) and (B), respectively. In addition, the variations in rotor

(B)

(D)

0 2 4 6 8 10 12 14 16 18 20Time (sec)

0 2 4 6 8 10 12 14 16 18 20Time (sec)

-2

-1.5

-1

-0.5

0

0.5

1

1.5 x 10-4

Δω2-

4 [rad

/ se

c]

PIMPCFMPC

-0.5

0

0.5

1

1.5

2

2.5

3P

ower

flow

bet

wee

n bu

ses

26-2

8 [p

.u]

PIMPCFMPC

(F)

(H)

0.996

0.998

1

1.002

1.004

1.006

1.008

1.01

PC

C v

olta

ge o

f WF

1 [p

.u.]

PIMPCFMPC

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Vqr

w

PIMPCFMPC

0 2 4 6 8 10 12 14 16 18 20Time (sec)

0 2 4 6 8 10 12 14 16 18 20Time (sec)

ain simulation of scenario I.

(J)(I)

0.55

0.555

0.56

0.565

Vγ s

PIMPCFMPC

0.675

0.68

0.685

0.69

0.695

XF

PIMPCFMPC

0 2 4 6 8 10 12 14 16 18 20Time (sec)

0 2 4 6 8 10 12 14 16 18 20Time (sec)

Fig. 12 (continued)

0 5 10 15 20 25 30 35 400

2

4

6

8

10

Time (sec)

Plo

ad [

p.u.

]

Fig. 13. The response of the load changes.

M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66 63

angle for G1–3 and G2–4 are depicted in Fig. 10(C) and (D), respec-tively. The better damping characteristics can be seen in the thirdcase 3 compared to two other Cases. The terminal voltage of G1and G3 are demonstrated in Fig. 10(E) and (F), respectively. Asobserved in these figures, the FMPC scheme is not only very suc-cessful in designing the damping controller, but also has significantimpact on improving the dynamic stability of the power system.The variations in active and reactive power flown from bus 8 areshown in Fig. 10(G) and (H), respectively. The results of activepower control (Pgi) on the GSC and reactive power control (QSi)on the RSC for wind turbine 1 are shown in Fig. 10(I) and (J),respectively. The responses of whole wind farm active and reactivepower are illustrated in Fig.10(K) and (L). According to these fig-ures, better performance by the wind farm and consequently bet-ter improvement in the power system stability are acquiredthrough designing the damping controller for SSSC. The responseof the DC link voltage and the terminal voltage for wind turbine1 are shown in Fig.10(M) and (N), respectively. Moreover, Fig. 10(O) illustrates the response of voltage variations for bus PCC (A).Fig. 10(P) and (Q) respectively show rotor-side converter switchingsignals on axis d and q. Furthermore, the input damping signals forSCESS and SSSC through the predictive controllers and PI controllerare illustrated in Fig. 10(R) and (S), respectively.

Based on these figures, the average settling time for Cases 1–3 isreached as 9.963, 5.221 and 3.171 s, respectively. Also, the simula-tion time of the FMPC, MPC, and PI controller is 768.246, 892.324and 741.987 s, respectively. Thus, it can be said that the functionalmodel predictive controller is faster than the classic predictivemodel, but, it is a bit slower than the classic model (conventionalPI controller). However, this low speed can be ignored due to theadvantages of the FMPC in controlling the active and reactive

powers of the wind turbine’s converters and damping of the oscil-lations. In order to verify the effectiveness of the proposed strategy,the eigenvalues for each part of the system are tabulated in Table 2under different wind speed for the turbines. As seen in this table,the proposed strategy has robust and successful performance indamping oscillations even under different wind speeds, whereasin the conventional method there is the risk of instability withoutusing the predictive controllers and compensator devices. Thisarises from the fact that by increasing or decreasing the windspeed, the imaginary parts of the eigenvalues are increasing (beingmore stable) and their real parts are constant or decreasing (beingless stable). This means that oscillations damping is decreasing(being less stable) and can trigger a system instability problem.

4.2. Ten-machine thirty-nine-bus power system

In this section, Ten-Machine New-England Power System isemployed to evaluate the robustness of the proposed method.The detailed data of this system can be found in [53]. For this pur-pose, two different scenarios based on three-phase short-circuitfault and changes in the system are investigated. The proposedmethods for PI, MPC and FMPC in presence of the SSSC-SCESS arecompared together in both cases. The single-line diagram of thesystem is shown in Fig. 11.

4.2.1. Scenario I: three-phase short circuit faultIn this scenario a wind farm of 36 MVA equipped with SCESS is

connected to bus 14. A three phase fault are applied at t = 2 sbetween buses 3 and 4 and cleared after 0.1 s. The speed of thewind farm is also increased from 11 m per second to 12 m per sec-ond at t = 2 s with step-size of 0.25. The speed deviations of G1–5

and G2–4 are depicted in Fig. 12(A) and (B), respectively. The muchbetter damping characteristics using FMPC is acquired for theactive power of lines 1–39 and 26–28 as shown in Fig. 12(C) and (D). As seen in Fig. 12(E) and (F), the reactive power ofthe wind farm and Vpcc are effectively controlled through the pre-dictive strategy and has better performance in damping LFOs com-pared to PI controller. Rotor-side converter switching signals onaxis d and q are demonstrated in Fig. 12(G) and (H), respectively.Moreover, the controlled damping signals for SCESS and SSSC usingthe predictive strategy and PI controller are shown in Fig. 12(I) and (J), respectively. According to these figures, in addition toimproving overshot and undershot, the settling time is alsosignificantly decreased using FMPC controller. The average settlingtime for FMPC, MPC and PI controllers are obtained as 7.112, 11.24and 19.985, respectively.

(B)(A)

0 5 10 15 20 25 30 35 40-2

-1

0

1

2

3 x 10-3

Time (sec)

Δω9-

10 [r

ad /

sec]

PIMPCFMPC

0 5 10 15 20 25 30 35 40-1.5

-1

-0.5

0

0.5

1

1.5 x 10-3

Time (sec)

Δω6-

8 [rad

/ se

c]

PIMPCFMPC

(D)(C)

(F)(E)

(H)(G)

0 5 10 15 20 25 30 35 40

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (sec)

0 5 10 15 20 25 30 35 40

Time (sec)

Pow

er fl

ow b

etw

een

buse

s 16

-19

[p.u

.]

PIMPCFMPC

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

Time (sec)

Pow

er fl

ow b

etw

een

buse

s 16

-17

[p.u

.]

PIMPCFMPC

3

4

5

6

7

8

Rea

ctiv

e po

wer

of W

F2

[p.u

.] PIMPCFMPC

0.997

0.998

0.999

1

1.001

1.002

1.003

PC

C V

olta

ge o

f WF

2 [p

.u.]

PIMPCdata3

0.55

0.6

0.65

0.7

0.75

Vdr

w

PIMPCFMPC

0.65

0.7

0.75

0.8

Vqr

w

PIMPCFMPC

0 5 10 15 20 25 30 35 40

Time (sec)0 5 10 15 20 25 30 35 40

Time (sec)

0 5 10 15 20 25 30 35 40

Time (sec)

Fig. 14. The response of time-domain simulation of scenario II.

64 M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66

4.2.2. Scenario II: changes in the system load in presence of two windfarms

In this scenario two wind farms with a total rated power outputof 72 MVA (36 MVA for each one) connected to buses 14 and 16 areconsidered. The proposed controller performance under rapidchange in load connected to bus 16 is evaluated through followingpattern: the load is increased from 3.5 (p.u) to 9.5 (p.u) at t = 5 sand then decreased from 9.5 (p.u.) To 1.5 at t = 12 s and once again

increased from 1.5 (p.u.) to 3.5 (p.u.) at t = 25 s. Fig. 13 shows thedetails of these changes. It should be noted that by increasingthe load, the wind speed for both wind turbines is also increased(from 11 m per second to 12 m per second) and decreasing the loadresults in reducing the wind speed from 12 m per second to 11 mper second. The speed deviations of G9–10 and G6–8 in terms of theload changes are shown in Fig. 14(A) and (B), respectively. As seenin these figures, the predictive controllers are substantially able to

(J)(I)

0

0.2

0.4

0.6

0.8

1

Vγ s

PIMPCdata3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

XF

PIMPCFMPC

0 5 10 15 20 25 30 35 40

Time (sec)0 5 10 15 20 25 30 35 40

Time (sec)

Fig. 14 (continued)

Table 3The performance of the proposed controllers in terms of objective function and computational time.

Case study Scenarios ma,mb Computational time periteration

Cost function

– – MPC FMPC MPC FMPC MPC FMPC

10 Machine I 200,80 200,5 4.22 0.46 0.162 0.144II 200,80 200,5 5.87 0.51 0.167 0.146

4 Machine – 200,80 200,5 4.11 0.41 0.158 0.134

M. Darabian, A. Jalilvand / Electrical Power and Energy Systems 85 (2017) 50–66 65

respond to the load changes and mitigate the resultant deviations.The active power variations of lines 16–19 and 16–17 are shown inFig. 14(C) and (D), respectively. As seen in these figures, bydecreasing the load at t = 5 s, the power flown between lines is ini-tially increased and then by increasing the load at t = 12 s thepower is declined. Finally, decreasing the load at t = 25 s resultsin increasing the power flown between lines. The reactive powervariations and Vpcc related to the second wind farm are depictedin Fig. 14(E) and (F), respectively. Fig. 12(G) and (H) respectivelyshows rotor-side converter switching signals on axis d and q. Fur-thermore, the input damping signals for SCESS and SSSC throughthe predictive controllers and PI controller are illustrated inFig. 14(I) and (J), respectively. According to obtain results from thisscenario, the robustness of the predictive controllers is confirmedunder rapid power changes compared to PI controller and it is alsoverified that FMPC has much effective ability in damping oscilla-tions in comparison to MPC and PI.

4.3. Computational aspects of the method

In this section the performance of FMPC and MPC controllers interms of computational time and objective functions are evaluated.As seen in the table, the value of objective function for FMPC is farless than MPC for both power systems. Since the objective functionis defined as the difference between input and output signals, itcan be concluded that the least value in this case represents theoptimal performance of the controllers. Given that the unknownvariables in FMPC are 16 times less than MPC, therefore the com-putational time for each iteration of FMPC is much less than thatof MPC as shown in Table 3. This reduction in the computationaltime can be considered as a benefit for FMPC controller.

5. Conclusion

In this paper FMPC strategy is proposed as an analytical method toenhancepower systemstability and is comparedwithmodel-basedpre-dictive controller and classic controller (without using predictive con-trol). In this technique the Laguerre function is used to reducecomputationaleffort inselecting inputpathsandalso inorder todecreasesampling time in the prediction horizon the exponential data weightingis employed. A four-machine and ten-machine power systems are

employed to evaluate the effectiveness of the proposed FMPS scheme.The tested power systems include wind turbines and SCESS and SSSCcompensators. Themain duty of the SCESS is to control the active powerof GSC in the wind turbines. Moreover, SSSC is used to reduce low fre-quencyoscillations throughdesigning adampingcontroller (usingFMPCandMPCmethods) for each of the compensators. In addition, the activeand reactive power control are carried out on the RSC to acquire appro-priatecontrol signals (vqrw,vdrw) to improve thepowersystemstability.Athree phase fault is applied to both power systems inMATLAB/Simulinkenvironment to run simulations. It is verified through simulation resultsthat the FMPC and MPC strategies have better abilities in damping lowfrequency oscillations as well as active and reactive power control com-pared to the conventional method. Furthermore, 20 wind turbines areused in the four-machinepowersystemasawind farmandtheeigenval-ues under different wind speeds are extracted and evaluated. The moredesirable eigenvalues and consequently better stability enhancement isacquired through the FMPC strategy compared to other Cases.

Appendix A

See Table 1.

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