Electrical Power and Energy Systems 43 (2012) 859–867

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Electrical Power and Energy Systems

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

Model predictive based load frequency control_design concerning wind turbines

Tarek Hassan Mohamed b,⇑, Jorge Morel c, Hassan Bevrani a, Takashi Hiyama c

a Dept. of Electrical & Computer Engineering, University of Kurdistan, Sanandaj, Iranb Faculty of Energy, Aswan University, Egyptc Dept. of Electrical & Computer Engineering, Kumamoto University, Kumamoto, Japan

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

Article history:Received 28 October 2011Received in revised form 18 April 2012Accepted 5 June 2012

Keywords:Load frequency controlIntegral controlModel predictive controlWind turbine

0142-0615/$ - see front matter � 2012 Published byhttp://dx.doi.org/10.1016/j.ijepes.2012.06.032

⇑ Corresponding author.E-mail addresses: tarekfoe@yahoo.com (T.H. M

(H. Bevrani).

This paper presents a load frequency control (LFC) design using the model predictive control (MPC) tech-nique in a multi-area power system in the presence of wind turbines. In the studied system, each localarea controller is designed independently such that stability of the overall closed loop system is guaran-teed. A frequency response model of multi-area power system including wind turbines is introduced. Aphysical constraints of the governors and turbines are considered in the model. The model was employedin the MPC structures. Digital simulations for a two area power system are provided to validate the effec-tiveness of the proposed scheme. The results show that, with the proposed MPC technique, the overallclosed loop system performance demonstrated robustness in the face of uncertainties due to governorsand turbines parameters variation and load disturbances. Also, it was denoted that wind turbine has apositive effect on the total response of the system. A performance comparison between the proposed con-troller with and without the participation of the wind turbines and a classical integral control scheme iscarried out for confirming the superiority of the proposed MPC technique in the presence of the partic-ipation of the wind turbines.

� 2012 Published by Elsevier Ltd.rson

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1. Introduction

In fact, the LFC becomes an important function of power systemoperation where the main objective is to regulate the output powerof each generator at prescribed levels while keeping the frequencyfluctuations within pre-specified limits. Different types of control-lers have been proposed in literature for the load frequency controldesign. The proportional integral PI controller, is widely employedin the LFC application [1]. But this type is considered as a fixedparameters controller which designed at nominal operating pointsand may no longer be suitable in all operating conditions. For thisreason, adaptive gain scheduling approaches have been proposedfor LFC synthesis [2–4].

Robust adaptive control schemes have been developed to dealwith changes in system parameters. Fuzzy logic controllers havebeen used in many reports for LFC design in a two area power sys-tem [5], with and without nonlinearities. The applications of artifi-cial neural network, genetic algorithms, and optimal control to LFChave been reported in [6,7]. In their findings it is observed that thetransient response is oscillatory and it seems some other eleganttechniques are needed to achieve a desirable performance. On

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Elsevier Ltd.

ohamed), bervrani@uok.ac.ir

the other hand, the MPC appears to be an efficient strategy tocontrol many applications in industry, it has many advantagessuch as fast response, robustness against load disturbance andparameters uncertainty.

Its straightforward design procedure is considered as a majoradvantage of the MPC. Given a model of the system, only an objec-tive function incorporating the control objectives needs to be setup. Additional physical constraints can be easily dealt with by add-ing them as inequality constraints, whereas soft constraints can beaccounted for in the objective function using large penalties. More-over, the MPC is well adapted to different physical setups and it al-lows for a unified approach [8–10].

Recently, a few attempts studied the idea of wind turbines inthe issue of LFC. Variable speed wind turbines (VSWTs), the mostutilize type of modern WTs, is partially or totally decoupled fromthe power network due to the power electronic converters limitingtheir capacity to provide primary frequency support to the net-work in case of disturbances. The inertial response of WTs is dis-cussed in detail in [11,12].

A detailed background of frequency responses, including pri-mary and secondary responses, are given in [11]. A detailed com-parison is made about fixed-speed wind turbines (FSWTs) anddoubly fed induction generator (DFIG) type wind turbines (WTs)through detailed simulations, the potential of FSWTs to contributeto the frequency response. On the other hand, these simulationsalso show that the DFIG-based WTs have negligible contribution

Table 1Parameters for Fig. 1 [14].

X2 X3 T1

1Rr

LmLss

L0ws Rs

860 T.H. Mohamed et al. / Electrical Power and Energy Systems 43 (2012) 859–867

r

to frequency responses and, hence it sounds that an additional con-trol loop is necessary.

In [12], it is reported that full converter (FC) type WTs are com-pletely decoupled from the power grid and no contribution is givento the frequency regulation. In addition, it is pointed out that theDFIG-type WTs have some small contribution to the powernetwork.

In [13], and [15], fast response and robustness against parame-ter uncertainties and load changes can be obtained using MPCcontroller for both single and multi-area load frequency controlapplication respectively, but without the participation of WTs.

While in [16], an attempt of study the effect of merging thewind turbines in the power system controlled by model predictiveload frequency control method, a positive effect on the system re-sponse could be noted, but that only for a single area.

This paper studies the effect of merging the wind turbines onthe system frequency response of multi area power system. In thispaper, each local area includes an aggregated wind turbine model(which consists of 200 wind turbine units) beside the main gener-ation unit. The MPC technique law produces its optimal output de-rived from a quadratic cost function minimization based on thedynamic model of the single area power system.The techniquecalculates the optimal control signal while respecting the givenconstrains over the output frequency deviation and the loadchange. The effects of the physical constraints such as generationrate constraint (GRC) and speed governor dead band [1] are consid-ered. The power system with the proposed MPC technique hasbeen tested through the effect of uncertainties due to governorand turbine parameters variation and load disturbance using com-puter simulation. A comparison has been made between the MPC(with and without wind turbine participation) and the traditionalintegral controller confirming the superiority of the proposedMPC technique and showing the positive effect of the WT partici-pation on the total system performance. Also, the simulation re-sults proved that the proposed controller can be successfullyapplied to the application of power system load frequency control.

The rest of the paper is organized as follows: A simplified windturbine model is presented in Section 2. The description of thedynamics of the power system is given in Section 3. General con-sideration about MPC and its cost function are presented in Section4. The implementation scheme of a multi area power systemtogether with the MPC technique is described in Section 5. Simula-tion results and general remarks are presented in Section 6. Finally,the paper is concluded in Section 7.

2. Simplified wind turbine model for frequency studies

Fig. 1 shows a simplified model of DFIG based wind turbine forfrequency response [14]. This simplified model can be described bythe following equations:

_iqr ¼ �1T1

� �iqr þ

X2

T1

� �Vqr ð1Þ

Autho

r Pe

Fig. 1. Simplified model of DFIG based wind turbine.

_w ¼ � X3

2Ht

� �iqr þ

12Ht

� �Tm ð2Þ

Pe ¼ wX3iqr ð3Þ

and for linearization, Eq. (3) can be rewritten as:

Pe ¼ woptX3iqr ð4Þ

and

Te ¼ iqs ¼ �Lm

LSSiqr ð5Þ

where the parameters are defined as follows: wopt is the operatingpoint of the rotational speed, Te the electromagnetic torque, Tm

the mechanical power change, w the rotational speed, Pe the activepower of wind turbine, iqr the q-axis component of the rotor cur-rent, Vqr the q-axis component of the rotor voltage, Ht the equivalentinertia constant of wind turbine.

Table 1 shows the detailed expressions of the main parametersutilized for the simplified model of Fig. 1.where

L0 ¼ Lrr þL2

m

Lss

" #; Lss ¼ Ls þ Lm; Lrr ¼ Lrs þ Lm

and, Lm is the magnetizing inductance, Rr and Rs the rotor and statorresistances, respectively, Lr and Ls the rotor and stator leakageinductances, respectively, Lrr and Lss the rotor and stator self-induc-tances, respectively, ws is the synchronous speed.

3. System dynamics

A multi-area power system comprises areas that are intercon-nected by tie-lines. The trend of frequency measured in each con-trol area is an indicator of the trend of the mismatch power in theinterconnection and not in the control area alone. The LFC systemin each control area of an interconnected (multi-area) power sys-tem should control the interchange power with the other controlareas as well as its local frequency. Therefore, the dynamic LFC sys-tem model must take into account the tie-line power signal. Forthis purpose, consider Fig. 2, which shows a power system withN-control areas [1].

In this section, a frequency response model for any area-i of Npower system control areas with an aggregated generator unit ineach area is described [1].

The overall generator–load dynamic relationship between theincremental mismatch power (DPmi � DPLi) and the frequencydeviation (Dfi) can be express

D _f i ¼1

2Hi

� �DPmi �

12Hi

� �DPLi �

Di

2Hi

� �Dfi �

12Hi

� �DPtie;i ð6Þ

the dynamic of the governor can be expressed as:

D _Pmi ¼1Tti

� �DPgi �

1Tti

� �DPmi ð7Þ

the dynamic of the turbine can be expressed as:

D _Pgi ¼1

Tgi

� �DPci �

1RiTgi

� �Dfi �

1Tgi

� �DPgi ð8Þ

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Fig. 2. Dynamic model of a control area in an interconnected environment.

T.H. Mohamed et al. / Electrical Power and Energy Systems 43 (2012) 859–867 861

s

py

the total tie-line power change between area-i and the other areascan be calculated as:

D _Ptie;i ¼ 2pXN

j ¼ 1j–i

TijDfi �XN

j ¼ 1j–i

TijDfj

2666664

3777775 ð9Þ

In a multi-area power system, in addition to regulating area fre-quency, the supplementary control should maintain the net inter-change power with neighbouring areas at scheduled values. This isgenerally accomplished by adding a tie-line flow deviation to thefrequency deviation in the supplementary feedback loop. A suit-able linear combination of frequency and tie-line power changesfor area i, is known as the area control error (ACE),

ACEi ¼ DPtie;i þ BiDfi ð10Þ

Eqs. (6)–(9) in additionally to Eqs. (1) and (2) represent the fre-quency response model for N power system control areas withone generator unit and wind turbine unit in each area and can becombined in the following state space model:

thor

Per

D _Pgi

D _Pmi

D _f i

D _Ptie;i

D _iqr;i

D _wi

26666666664

37777777775¼

� 1Tgi

0 � 1Ri �Tgi

0 0 01

Tti� 1

Tti0 0 0 0

0 12Hi

� D2Hi

� 12Hi

� X3 :wopt

2Hi0

0 0 2pXN

j ¼ 1j–1

Tij 0 0

0 0 0 0 � 1T1

0

0 0 0 0 � X32Ht

0

2666666666666666664

3777777777777777775

DPgi

DPmi

Dfi

DPtie;i

Diqr;i

Dwi

2666666664

3777777775þ

y ¼0 0 Bi 1 0 00 0 0 0 0 1

� �DPg

DPm

DfDPtie;i

Diqr

Dw

2666666664

3777777775

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

Au

where Pgi is the governor output change of area i. Pmi the mechanicalpower change of area i. fi the frequency deviation of area i. PLi theload change of area i. Pci the supplementary control action of areai. yi the system output of area i. Hi the equivalent inertia constantof area i. Di the equivalent damping coefficient of area i. Ri the speeddroop characteristic of area i. Tgi, Tti the governor and turbine timeconstants of area i. ACEi the control error of area i. Bi the a frequencybias factor of area i. Tij the tie-line synchronizing coefficient witharea j. DPtie,i the total tie-line power change between area i andthe other areas. Dmi the control area interface, Dv i ¼

PNj ¼ 1j–i

TijDfj

This state space model will be applied to the controller of thewind turbine unit, while the state space model used in the powersystem area controller will not include the dynamics of the windturbine, and can be expressed as:

D _Pgi

D _Pmi

D _f i

D _Ptie;i

26664

37775¼

� 1Tgi

0 � 1Ri Tgi

01

Tti� 1

Tti0 0

0 12Hi

� Di2Hi

� 12Hi

0 0 2pXN

j¼ 1j–1

Tij 0

266666666664

377777777775

DPgi

DPmi

Dfi

DPtie;i

2664

3775þ

0 00 0� 1

2Hi0

0 �2p

26664

37775 DPLi

DVi

� �þ

1Tgi

00

264

375DPei

y¼ ACEi ¼ ½0 0 Bi 1�

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

9>>>>>>>>>>>>=>>>>>>>>>>>>;ð12Þ

onal

Co

1Tgi

0

0 00 00 00 X2

T1

0 0

26666666664

37777777775

DPci

DVqr

� �þ

0 0 00 0 0� 1

2Hi0 0

0 �2p 00 0 00 0 1

2Ht

26666666664

37777777775

DPLi

DVi

DTm

264

375

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

ð11Þ

Model

ModelPast

Pastoutputs

controls

Optimizer

Free responceReference trajectory

Forcedresponse

Totalresponse

Futurecontrols

Cost function J

Constraints

_+

+ +

Futureerrors

Fig. 3. A simple structure of the MPC controller.

Table 2Parameters and data of a practical single area power system.

D (pu/Hz) H (pu s) R (Hz/pu) Tg (s) Tt (s)

0.015 0.08335 3.00 0.08 0.4

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4. Model predictive control

The MPC has proved to efficiently control a wide range of appli-cations in industry such as chemical process, petrol industry, elec-tromechanical systems and many other applications. The MPCscheme is based on an explicit use of a prediction model of the sys-tem response to obtain the control actions by minimizing an objec-tive function. Optimization objectives include minimization of thedifference between the predicted and reference response, and thecontrol effort subjected to prescribed constraints. The effectivenessof the MPC is demonstrated to be equivalent to the optimal control.It displays its main strength in its computational expediency,

Fig. 4. The block diagram of a two area power system

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real-time applications, intrinsic compensation for time delays,treatment of constraints, and potential for future extensions ofthe methodology. At each control interval, the first input in theoptimal sequence is sent into the plant, and the entire calculationis repeated at subsequent control intervals. The purpose of takingnew measurements at each time step is to compensate for unmea-sured disturbances and model inaccuracy, both of which causethe system output to be different from the one predicted by themodel [8,9].

Fig. 3 shows a simple structure of the MPC controller. An inter-nal model is used to predict the future plant outputs based on thepast and current values of the inputs and outputs and on the pro-posed optimal future control actions. The prediction has two maincomponents: the free response which being expected behavior ofthe output assuming zero future control actions, and the forcedresponse which being the additional component of the outputresponse due to the candidate set of future controls. For a linearsystems, the total prediction can be calculated by summing bothof free and forced responses. The optimizer is used to calculatethe best set of future control action by minimizing a cost function(J). The optimization is subject to constraints on both manipulatedand controlled variables [10].al

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including the proposed MPC controller and WT.

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T.H. Mohamed et al. / Electrical Power and Energy Systems 43 (2012) 859–867 863

The general objective is to tighten the future output error tozero, with a minimum input effort. The cost function to be mini-mized is generally a weighted sum of square predicted errorsand square future control values, e.g. in the Generalized PredictiveControl (GPC):

JðN1;N2;NuÞ ¼XN2

j¼N1

bðjÞ½yðkþ jjkÞ �wðkþ jÞ�2 þXNu

j¼1

kðjÞ½uðkþ j� 1Þ�2 ð13Þ

where N1, N2 are the lower and upper prediction horizons over theoutput, Nu is the control horizon, bðjÞ; kðjÞ are weighting factors. Thecontrol horizon permits to decrease the number of calculated futurecontrol according to the relation: Du(k + j) for J P Nu. The w(k + j)represents the reference trajectory over the future horizon N.

Constraints over the control signal, the outputs and the controlsignal changing can be added to the cost function as follows:

umin 6 uðkÞ 6 umax

Dumin 6 DuðkÞ 6 Dumax

ymin 6 yðkÞ 6 ymax

ð14Þ

Solution of Eq. (13) gives the optimal sequence of control signalover the horizon N while respecting the given constraints of Eq.(14).

The MPC have many advantages, in particularly it can pilot a bigvariety of process, being simple to apply in the case of multivari-able system, can compensate the effect of pure delay by the predic-tion, inducing the anticipate effect in the closed loop, being asimple technique of control to be applied and also offer optimalsolution while respecting the given constraints. On the other hand,this type of restructure required the knowledge of model for thesystem, and in the presence of constraints it becomes a relatively

Table 3Wind turbine parameters and operating point [14].

Operating point (mw) Wind speed (m/s) Rotational speed (m/s) R

247 11 1.17 0

Xm is the magnetizing reactance, (Pwt)Base = 400 MVA, (wr)Base = 1.15 (rad/s).

95 100 105 110 115-0.06

-0.04

-0.02

0(a)

Δf 1

(Hz)

95 100 105 110 115

-0.04

-0.02

0

0.02(b)

(c)

Δf 2

(Hz)

95 100 105 110 115-15

-10

-5

0

5x 10-3

Tim

ΔP

tie(p

u)

Fig. 5. Power system response for case1 of Scenario 1: with (MPC with WT participatiocurve) and conventional controller (dotted curve).

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more complex regulator than a simple conventional controllersuch as a proportional integral differential (PID), and it takes moretime for on-line calculations

5. System configuration

The block diagram of a simplified frequency response model fora two area power system with aggregated unit including the pro-posed MPC controller is shown in Fig. 4. Each area consists of theoverall rotating mass and load and an aggregated generator unitincluding one nonlinear turbine with GRC, one governor withdead-band constraint [1], and aggregated wind turbine model(which consists of 200 wind turbine units of 2 MW rated VSWTs),as each local area controller can be designed independently. On theother hand, the frequency deviation and wind turbine rotationalspeed are used as feedback for the closed loop control system. Bothof the reference and measured area control error ACEi, (ACEre-

f,i = 0 HZ) and both of the reference and measured wind rotationalspeed (wref,i = wopt) are fed to the model predictive controller MPCi,2

of the wind turbine unit in order to obtain the signal DVqr,i (q-axiscomponent of the deviation of the rotor voltage, which fed to thewind turbine), and the second controller MPCi,1 is fed by only bythe reference and measured area control error ACEi, (ACEref,i = 0 HZ)in order to obtain the supplementary control action DPci which isadded to the negative frequency feedback signal to give the signalwhich feeds the governor giving the governor valve position whichsupplies the turbine to give the mechanical power change DPmi

which is affected by the load change DPLi, the tie-line powerchange DPtie,i and the active power change of wind turbine DPe

(which multiplied by a gain equal to 0.5 to make a correlationna

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r (pu) Rs (pu) Xlr (pu) Xls (pu) Xm (pu) Ht (pu)

.00552 0.00491 0.1 0.09273 3.9654 4.5

120 125 130 135 140

120 125 130 135 140

120 125 130 135 140

e, s

n) (bold and solid curve) and with (MPC without WT participation) (solid and thin

o

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between power system and wind turbine bases) giving the input ofthe rotating mass and load block to provide the actual frequencydeviation Df. In addition, the tie-line flow deviation is added tothe frequency deviation in the supplementary feedback loop togive the area control error ACEi.

6. Results and discussions

Computer simulations have been carried out in order to validatethe effectiveness of the proposed scheme. The Matlab/Simulinksoftware package has been used for this purpose. A practicalmulti area power system with the nominal parameters [1] listedin Table 2, is considered.

The simulation studies are carried out for the proposed control-ler with generation rate constraint (GRC) of 10% pu per minute. The

Fig. 7. Power system response forcase 2 of Scenario 1, with (MPC with WT participationcurve) and conventional controller (dotted curve).

Fig. 6. Wind turbine responses in the first case, (a) rotationa

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maximum value of dead band for governor is specified as 0.05%.The parameters of the MPC controller are set as follows:

prediction horizon = 10,control horizon = 2,weights on manipulated variables = 0,weights on manipulated variable rates = 0.1,weights on the output signals = 1, andsampling interval = 0.1 s.

Constraints are imposed over the control action, and frequencydeviation are considered as follows :

Max. control action = 0.25 pu.Min. control action = �0.25 pu.Max. frequency deviation = 0.25 pu.

) (thick and solid curve) and with (MPC without WT participation) (solid and thin

l speed w, and (b) power change of wind turbine DPe1.

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T.H. Mohamed et al. / Electrical Power and Energy Systems 43 (2012) 859–867 865

Min. frequency deviation = �0.25 pu.

The wind turbine used consists of 200 units of 2 MW rated var-iable speed wind turbine VSWTs, the wind turbine parameters andoperating point are indicated in Table 3.

6.1. Scenario 1

6.1.1. Case 1The system performance with the proposed MPC controllers in

case of wind turbine participation at nominal parameters is testedand compared with the system performances with both MPC con-trollers without wind turbine participation using a conventionalintegrator (K(s) = �0.3/s) in the presence of a step load change

Table 4Parameters and data of a practical three control area power system.

Area K(s) D (pu/Hz) 2H (pu s)

Area-1 �0.3/s 0.015 0.1667

Area-2 �0.2/s 0.016 0.2017

Area-3 �0.4/s 0.015 0.1247

(Pe)Base = 800 MVA.

Fig. 9. Three-control area power

100 100.5 101 101.5 102 10

0.01

0.02(a)

ΔP c2

(pu)

100 100.5 101 101.5 102-5

0

5x 10

-4

(b)

Tim

V qr,2

(pu)

Fig. 8. System control signals: (a) out

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s

(DPL is assumed to be 0.02 pu at t = 100 s). Fig. 5 shows the simu-lation results of the MPC with WT participation, MPC without WTparticipation (as in [15]) and only conventional integrator systems.The results from the top to the bottom are: the frequency devia-tions for Area 1, the frequency deviations for Area 2, and thetie-line power change DPtie,i. It has been noticed that with theproposed MPC controller with the participation of WT, the systemis more stable and fast compared to the system with conventionalcontroller and merging the wind turbine leaded to the enhance-ment of the performance of the MPC controller. Fig. 6 shows bothrotational speed (at the top) and power change of wind turbine (atthe bottom). As shown in this figure, the change in the wind tur-bine electrical power can cooperate to enhance the overall systemresponse performance.

R (Hz/pu) Tg (s) Tt (s) Tij

3.00 0.08 0.40 T12 = 0.20T13 = 0.25

2.73 0.06 0.44 T21 = 0.20T23 = 0.15

2.82 0.07 0.3 T31 = 0.25T32 = 0.15

system with wind turbines.

02.5 103 103.5 104 104.5 105

102.5 103 103.5 104 104.5 105

e,s

put of MPC3, (b) output of MPC4.

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Fig. 10. Power system response to scenario B with MPC with participation of wind turbines (thick and solid curve), only MPC (solid and thin curve) controller, andconventional controller (dotted curve).

866 T.H. Mohamed et al. / Electrical Power and Energy Systems 43 (2012) 859–867

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6.1.2. Case 2The robustness of the proposed system against parameters

uncertainty is validated. In this case, both of the governor and tur-bine time constants are increased to Tg1 = 0.105 s (ffi31% change),Tt1 = 0.785 s (ffi95% change), Tg2 = 0.105 s (ffi66% change) andTt2 = 0.6 s (ffi38% change), respectively (this can happen in caseof off line change of the practical turbine and governor, while thecontroller keeps the nominal values of these parts). Fig. 7 depictsthe system response with MPC with WT participation, MPC with-out WT participation and only integrator in this case of study.The load change is assumed to be as described in the first case. Ithas been shown that, with the traditional controller, the systembecomes unstable while with the MPC controller, the system re-sponse is more convenient. Also, the figure indicates that the pres-ence of wind turbine leads to enhance the system performancewith MPC controller, significantly. While Fig. 8 shows the controlsignals of the controllers MPC3 and MPC4 of Area 2.ut

hor P

e

A

6.2. Scenario 2

To illustrate the behavior of LFC system with the proposeddecentralized MPC controller in the presence of wind turbine in amulti-area power system, consider three identical interconnectedcontrol areas as shown in Fig. 9. The simulation parameters [1]are given in Table 4, while the wind turbine parameters are thesame for all areas and are listed in Table 3. The system is testedat a simultaneous 0.02-pu load step disturbance in control area-2and against wide range of parameters uncertainty is validated. Inthis case, the governor and turbine time constants of each areaare increased to Tg1 = 0.105 s (ffi31% change), Tt1 = 0.785 s (ffi95%change), Tg2 = 0.105 s (ffi66% change) and Tt2 = 0.6 s (ffi38%change), Tg3 = 0.15 s (ffi100% change) and Tt3 = 0.7 s (ffi100%change), respectively. Fig. 10 depicts the response of both the pro-posed MPC controllers in the presence and without the participa-tion of the wind turbine (as in [15]) and that response of theclassical integrator controller (with gains listed in Table 4) in the

case of above uncertainty. The results from the top to the bottomare: frequency deviation in area-1, frequency deviation in area-3,tie-line power change in area-1, and tie-line power change inarea-3. From the comparisons in Fig. 10, that even at this severecondition of uncertainties, system with MPC controllers keeps sta-ble, and it is appeared that wind turbine affected positively on thesystem response.

7. Conclusions

This paper studies the merging of wind turbines in a multi areapower system controlled by a robust load frequency control basedon the model predictive control technique.

Digital simulations have been carried out in order to validatethe effectiveness of the proposed scheme. The proposed controllerhas been tested for several mismatched parameters and loaddisturbance.

A performance comparison between the proposed MPC in thepresence of wind turbine and MPC without wind turbine and con-ventional integral controllers is carried out. The simulation resultsdemonstrate that the closed-loop system with MPC controller isrobust against the parameter perturbation of the system and hasdesirable performance in comparison of classical integral controldesign in all of the performed test scenarios. Also, it was denotedthat wind turbine has a positive impact on the total response ofthe system. For future work, to solve the problem of the complexoptimization problem procedure required in MPC design, simula-tion results can be used to train a neural network to give the samerobust response with ability of practical implementation.

Appendix A. A

Fig. 1 shows a simplified model of DFIG based wind turbine(WT) for frequency response, this model is investigated from thedetailed model [17], which can be described As:

Voltage equations

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T.H. Mohamed et al. / Electrical Power and Energy Systems 43 (2012) 859–867 867

s

Vds ¼ RSids � wqs þ1

wS

ddt

wds ðA:1Þ

Vqs ¼ RSiqs � wds þ1

wS

ddt

wqs ðA:2Þ

Vdr ¼ Rridr � swqs þ1

wS

ddt

wdr ðA:3Þ

Vqr ¼ Rriqr þ swdr þ1

wS

ddt

wqr ðA:4Þ

Flux equations

wds ¼ Lssids þ Lmidr ðA:5Þ

wqs ¼ Lssiqs þ Lmiqr ðA:6Þ

wdr ¼ Lrridr þ Lmids ðA:7Þ

wqr ¼ Lrriqr þ Lmiqs ðA:8Þ

where

Lss ¼ Ls þ Lm

and

Lrr ¼ Lr þ Lm

Torque equation

Te ¼ wds � iqs � wqs � ids ¼ wdr � iqr � wqr � idr ðA:9Þ

Manipulating (A.1), (A.2), (A.3), (A.4), (A.5), (A.6), (A.7), (A.8) thefollowing equations can be derived:

_idr ¼�wSRr

Lrridr þ s�ws � iqr þ

ws

Lrr�Vdr þ s�ws �

Lm

Lrr� iqs �

Lm

Lrr� _ids ðA:10Þ

_iqr ¼�wSRr

Lrriqr � s�ws � idr þ

ws

Lrr�Vqr � s�ws �

Lm

Lrr� ids �

Lm

Lrr� _iqs ðA:11Þ

For vector control of the DFIG, the d-axis was chosen such that itcoincides with the maximum of the stator flux, therefore,wds = 1 pu and wqs = 1 pu. Substituting in (A.6), the following equa-tion can be obtained:

iqs ¼ �Lm

LSSiqr ðA:12Þ

then Eq. (A.11) can be simplified and converted to S domain as:

S 1� L2m

LSSLrr

" #iqr ¼ �

wSRr

Lrriqr þ

ws

LrrVdr ðA:13Þ

Autho

r Per

and

iqr ¼1Rr� 1½1þ ST1�

Vqr ðA:14Þ

substituting, wds = 1 pu and wqs = 1 pu in (A.9) and the using (A.12),the following equation can be obtained for the electromagnetictorque:

Te ¼ iqs ¼ �Lm

LSSiqr ðA:15Þ

While the mechanical equation can described as:

pe ¼ Tewr ðA:16Þ

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