Research ArticleModel Predictive Control for Load Frequency Control withWind Turbines

Yi Zhang12 Xiangjie Liu2 and Yujia Yan2

1Department of Electrical Engineering North China University of Science and Technology Tangshan 063000 China2The State Key Laboratory of Alternate Electrical Power Systemwith Renewable Energy Sources North China Electric PowerUniversityBeijing 102206 China

Correspondence should be addressed to Yi Zhang zhangyizhouzhao163com

Received 23 August 2015 Accepted 12 October 2015

Academic Editor Onur Toker

Copyright copy 2015 Yi Zhang et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Reliable load frequency (LFC) control is crucial to the operation and design of modern electric power systems Considering theLFC problem of a four-area interconnected power system with wind turbines this paper presents a distributed model predictivecontrol (DMPC) based on coordination schemeThe proposed algorithm solves a series of local optimization problems tominimizea performance objective for each control area The scheme incorporates the two critical nonlinear constraints for example thegeneration rate constraint (GRC) and the valve limit into convex optimization problems Furthermore the algorithm reducesthe impact on the randomness and intermittence of wind turbine effectively A performance comparison between the proposedcontroller with and that without the participation of the wind turbines is carried out Good performance is obtained in the presenceof power system nonlinearities due to the governors and turbines constraints and load change disturbances

1 Introduction

Wind energy is considered as a promising and encoringrenewable energy alternative for power generation owing toenvironmental and economical benefits The world marketof wind installation set a new record in the year of 2014and reached a total size of 51 GW [1] Nowadays due to theinterconnection of more distributed generators especiallywind turbines that are committed to grid operation electricpower system has become more complicated than ever

Power system LFC incorporating WTGs can be a quitechallengeable issue The output power of WTGs varies withwind speed fluctuation [2] This wind power fluctuationimposed additional power imbalance to the power systemand may cause frequency deviation from the nominal value[3] Significant frequency deviations may lead to the discon-nection of some loads and generations and even can leadto whole power system oscillations Previous studies [4ndash6]provide extensive overviews of the primary and secondaryfrequency control strategies of power systems with windpower plants

LFC secondary frequency control has been performedby integrating the area control error (ACE) which acts on theload reference settings of the governors LFC tasks are main-taining tie-line power flow and system frequency close tonominal value for themultiarea interconnected power system[7] As a fundamental characteristic of electric power opera-tions frequency of the systemdeviates from its nominal valuedue to generation-demand imbalance Conventional gener-ators in which the turbine rotational speed is nearly constantprovide inertia and governor response against frequencydeviations however the speed of a wind turbine is not syn-chronous with the grid and is usually controlled to trackthe maximum power point It implies that the wind turbineswill have less time to react to the power imbalance probablyresulting in lager frequency deviations

Thus it is thus necessary to establish the optimal profile ofthe WTGs power surge in coordination with the characteris-tics of conventional plants to achieve a more economical andreliable operation of power system With the large amount ofrealistic constraints for example generation rate constraints

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2015 Article ID 282740 17 pageshttpdxdoiorg1011552015282740

2 Journal of Control Science and Engineering

(GRCs) in the conventional units the pitch angle and gener-ator torque constraints in WTGs the LFC becomes a large-scale distributed multiconstraints optimization problem

Recently a few attempts studied the idea of wind turbinesin the issue of LFC [8ndash10] Two types of wind farm modelsare derived and demonstrated to portray the capability ofset-point tracking under automatic generation control (AGC)[8] This inference leads to the development of a simplifiedwind farm model that is specially designed for the set-pointcontrol in the power system study However the durabilityof inertia effect depends on the allowable rotor speed rangeAn adaptive fuzzy logic structure was used to propose a newLFC scheme in the interconnected large-scale power systemin the presence of wind turbines [9]The performance againstsudden load change and wind power fluctuations in differentwind power penetration rates is confirmed by simulationA flatness-based method to control frequency and powerflow for multiarea power system with wind turbine is pre-sented in [10] And practical constraints such as generatorramping rates of wind turbine generator can be consideredin designing the controllers As abovementioned referencethe control schemes are designed for each area to maintainthe frequency at nominal value and to keep power flowsnear scheduled values However local controller in each areadoes not work cooperatively towards satisfying systemwidecontrol objectives In addition the control scheme [8ndash10]mentioned above could yield unsatisfactory performancesince the effects of nonlinearities such as generation rate con-straint and generation ramping rate were not considered

Model predictive control (MPC) also called recedinghorizon control was originally developed to be an effectivemethod for processing industrial control It transforms thecontrol problem into a finite horizon optimal control problemthat can also satisfy multivariable constraints on the inputoutput and state variables In the power industry MPC hasbeen successfully used in controlling power plant steam-boiler generation processes [11ndash13] In power system con-trol MPC was first developed to be an economic-orientedLFC [14] which generates the control action based on theopen-loop optimization method over a finite horizon MPChas subsequently been developed to realize the constrainedoptimal algorithm for LFC problem In [15] the constrainthandling ability of MPC is employed to effectively accountfor the generation rate constraints (GRCs) but without theanalysis of closed-loop stability and robustness RecentlyMPC has been successfully used in LFC design of multiareapower system with wind turbines [16] However each areacontroller is designed independently and the communicationbetween the local controllers is not considered On the otherhand with the size and capacity of wind farms increasing inrecent years traditional centralized MPCs encounter manydifficulties due to limitations in exchanging information withlarge-scale geographically extensive control areas In order todeal with these issues advanced distributed control strategieshave to be investigated and implemented

Developing decentralizeddistributed LFC structures canbe an effective way of solving this problemThe decentralizedmodel predictive control scheme for the LFC of multiareainterconnected power system is presented in [17] However

the local controller does not consider generation rate con-straint that is only imposed on the turbine in the simulationIn the distributed MPC (DMPC) the benefits from usinga decentralized structure are partially preserved and theplant-wide performance and stability are improved throughcoordination [18 19] In [20] feasible cooperation-basedMPCmethod is used in distributed LFC instead of centralizedMPC It is noted that the range of load change used in thecases is very large and inappropriate for the LFC issue

This paper studies the effect of merging the wind turbineson the system frequency of multiarea power system Thefirst control area includes an aggregated wind turbine model(which consists of 60 wind turbine units) beside the thermalpower plant According to the distributed LFC structurethe dynamics model of the four-area interconnected powersystem is established In our scheme the overall power systemis decomposed into four areas and each area has its ownlocal MPC controller These areas-based MPCs exchangetheir measurement and predictions by communication andincorporate the information from other controllers intotheir local objective so as to coordinate with each otherThe controllers calculate the optimal control signal whilerespecting constraints over the wind turbines output fre-quency deviation and the load change Not only do theeffects of the physical constraints conclude generating rateconstraints (GRCs) and the limit of governor position inconventional power plant but also thewind speed constraintsin wind turbines are considered Comparisons of responseto step load change computational burden and robustnesshave been made between DMPC centralized MPC anddecentralizedMPCThe results confirm the superiority of theproposed DMPC technique

The remainder of the paper is organized as follows Mod-eling of wind turbines participation in LFC is presented inSection 2 and the proposed DMPC algorithm is presented inSection 3 Section 4 presents the application of the algorithmin a four-area interconnected power systemThe conclusionsare presented in Section 5

2 Distributed Model of Hybrid Power System

Figure 1 illustrates the interconnected power system con-sisting of four control areas connected by tie-lines whichconsists of thermal power plant variable speed wind turbines(VSWTs) and hydro power plant In area 1 wind turbineis taken into consideration as it can provide a new solutionto the contradiction between economic development andenvironment pollution Area 4 is the thermal power plantwhile area 2 and area 3 are hydro power plants

Detailed compositions of each area are shown in Figures2ndash4 In addition area 1 includes an aggregated wind turbinemodel which consists of 40 VSWT units while the capacityof thermal plant is 600MW The variables and parametersare listed in Table 1 In each control area a change in localdemand (load) alters the nominal frequency The DMPCin each control area 119894 manipulates the load reference set-pointΔ119875ref 119894 to drive the frequency deviationsΔ119891119894 and tie-linepower flow deviations Δ119875tie119894119895 to zero

Journal of Control Science and Engineering 3

Table 1 Power system variables and parameter

Parametervariable Description Unit120596119903

Angular velocity of rotor rads120596119892

Angular velocity of high speed shaft and generator rads119879119892

Generator reaction torque Nm119879119903

Aerodynamic torque Nm119870119904

Total stiffness on low speed shaft Nmrad119869119903

Inertia of the rotor (low speed shaft and gearbox) Kgm2

119869119892

Inertia of the rotor (high speed shaft and gearbox) Kgm2

119899gear Exchange ratio Null120578gear Efficiency of the gear box 120591120573

Actuator time constant s119870120573

Actuator gain HzpuMWV119898

Wind speed ms120579ref Pitch demand rad120579 Pitch angle rad119875ref119890

Power demand puMW119875119890

The output of wind turbine puMWΔ119891

119894

(119905) Frequency deviation HzΔ119875

119892119894

(119905) Generator output power deviation puMWΔ119883

119892119894

(119905) Governor valve position deviation puΔ119883

119892ℎ119894

(119905) Governor valve servomotor position deviation puΔ119875

119905119894119890119894

(119905) Tie-line active power deviation puMWΔ119875

119889119894

(119905) Load disturbance puMW119870119875119894

Power system gain HzpuMW119870119903119894

Reheat turbine gain HzpuMW119879119875119894

Power system time constant s119879119903119894

Reheat turbine time constant s119879119882119894

Water starting time s119879119894119894

119879119877119894

Hydro governor time constants s119879119866119894

Thermal governor time constant s119879119879119894

Turbine time constant s119870119878119894119895

Interconnection gain between control areas puMW119870119861119894

Frequency bias factor puMWHz119877119894

Speed drop due to governor action HzpuMWACE

119894

Area control error puMW

P12tie P23

tie

P14tie P34

tie

P24tie

P31tie

Control area 1Wind turbine

Thermal power plant

Control area 2

Hydro power plant

Control area 3

Hydro power plant

Control area 4

Thermal power plant

Figure 1 The four-area interconnected hybrid power system

21Wind TurbineModel Awind turbine is an installation forconverting kinetic energy extracted from wind to electricalenergy Figure 5 illustrates the basicmodel structure of awindturbine and the interactions between the different dynamiccomponents in the model The whole wind turbine canbe divided into four subsystems aerodynamics subsystemmechanical subsystem electrical and actuator subsystem[21]

The linearization model for the variable speed wind tur-bine in Figure 6 can be represented by

Δ120576

= Δ120596119903

minus1

119899gearΔ120596

119892

(1a)

Δ119903

= minus119870119904

119869119903

Δ120593120576

+1

119869119903

Δ119879119903

(1b)

4 Journal of Control Science and Engineering

KBi

ACEi

Σ MPCi

ΔPci

1

Ri

Σ1

1 + sTGi

ΔXgi

1 + sKrTri1 + sTri

ΔPri

1

1 + sTTi

ΔPgi

Σ

ΔPdi

KPi

1 + sTPi

Δfi

ΔPeΔPi

sumj

Ksij(Δfi minus Δfj)

2120587

s

Wind turbines

ΔPtiei

[Δ120579ref ΔTgref]

Figure 2 Block diagram of a thermal power plant and wind turbines (119894 = 1)

KBi

ACEi

Σ Σ

ΣMPCi

ΔPci

1

Ri

1

1 + sTGi

ΔXgiΔXghi ΔPgi

ΔPdi

KP2

1 + sTP2

Δfi

sumj

Ksij(Δfi minus Δfj)

2120587

s

1 + sTRi1 + sTii

1 minus sTWi

1 + 05sTWi

ΔPtiei

Figure 3 Block diagram of a hydro power plant (119894 = 2 3)

KBi

ACEi

Σ Σ

ΣMPCi

ΔPci

1

Ri

1

1 + sTGi

ΔXgi

1 + sKrTri1 + sTri

1

1 + sTTi

ΔPgi

ΔPdi

KPi

1 + sTPi

Δfi

sumj

Ksij(Δfi minus Δfj)2120587

s

ΔPtiei

Figure 4 Block diagram of a thermal power plant (119894 = 4)

mTr

120596r 120596g

ΔPe

Pe

120579120579ref Tgref

Prefe

Rotor

Driver train

Generator

Actuator

Power electronics

Power

Controller

Figure 5 Diagram of a variable speed wind turbine

Journal of Control Science and Engineering 5

m

ΔPe

minusminus

1

Ts + 1

1

s

ΔTgΔ120579

Δ120579ref

ΔTgref

Generatormodel

Wind speed

MPCWind turbine

model

Wind turbines

Figure 6 Diagram of wind power plant in area 1

Δ119892

=

120578gear119870119904

119899gearΔ120593

120576

minus1

119869119903

Δ119879119892

(1c)

Δ 120579 = minus1

120591120573

Δ120579 +

119870120573

120591120573

Δ120579ref (1d)

The generator reaction torque 119879119892

and the reference pitchangle 120579ref are used as indicator of the input of VSWT as119906119890

= [Δ120573ref Δ119879119892]119879

isin 1198772 Moreover 120578 is the efficiency of the

generator and 120596119892

and 119879119892

are used as indicator of the outputpower as 119875

119890

= 120578120596119892

119879119892

isin 1198771 where 120596

119892

is the angular velocityof generator shaft A generalized representation of the state-space model of the variable speed turbine can be described as

119890

(119905) = 119860 (V119898

) 119909119890

(119905) + 1198611

(V119898

) 120596 (119905) + 1198612

119906119890

(119905) (2a)

119911119890

(119905) = 119862119909119890

(119905) + 1198631

120596 (119905) + 1198632

119906119890

(119905) (2b)

with

119860 (V119898

) =

[[[[[[[[[[[[[

[

0 1 minus1

119899gear0

minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 minus1

120591120579

]]]]]]]]]]]]]

]

1198611

(V119898

) =

[[[[[[[[

[

0

1

119869119903

120597119879119903

120597V

10038161003816100381610038161003816100381610038161003816op

0

0

]]]]]]]]

]

1198612

=

[[[[[[[[[

[

0 0

0 0

0 minus1

119869119892

119870120573

120591120573

0

]]]]]]]]]

]

119862 = [

0 0 1 0

0 0 0 0]

1198631

= [

0

0]

1198632

= [

0 0

0 1]

119909119890

= [Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

119906119890

= [Δ120579ref Δ119879119892ref]119879

119911119890

= [Δ120596119892

Δ119879119892]119879

119910119890

= 119875119890

= 120578120596119892

119879119892

(3)

22 Four-Area Power System with Wind Turbine Denotingthat the control area 119894 (119894 = 1 2 3 4) is to be interconnectedwith the control area 119895 119895 = 119894 through a tie-line a linear con-tinuous time-varyingmodel of control area 119894 can bewritten as

119894

= 119860119894119894

119909119894

+ 119861119894119894

119906119894

+ 119865119894119894

119889119894

+sum

119894 =119895

(119860119894119895

119909119895

+ 119861119894119895

119906119895

+ 119865119894119895

119889119895

)

119910119894

= 119862119894119894

119909119894

(4)

where 119909119894

isin 119877119899 119906

119894

isin 119877119898 119889

119894

isin 119877119896 and 119910

119894

isin 119877119897 are the state

vector the control signal vector the disturbance vector andthe vector of output of control area 119894 respectively 119909

119895

isin 119877119901

119906119895

isin 119877119902 and 119889

119895

isin 119877119904 are the state vector the control signal

vector and the disturbance vector of neighbor controlarea respectively Matrices 119860

119894119894

119861119894119894

119862119894119894

and 119865119894119894

representappropriate systemmatrices of control area 119894 and119860

119894119895

119861119894119895

and119865119894119895

represent the matrices of interaction variables betweenarea 119894 and area 119895 Tie-line power for area 119894 is represented by

Δ119875tie119894 =4

sum

119895=1

119895 =119894

Δ119875119894119895

tie =4

sum

119895=1

119895 =119894

119870119904119894119895

(Δ119891119894

minus Δ119891119895

)

Δ119875119894119895

tie = minusΔ119875119895119894

tie

(5)

6 Journal of Control Science and Engineering

The state disturbance and output vectors for area 119894 aredefined by

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

(119894 = 1)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119883119892ℎ119894]119879

(119894 = 2 3)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119875119903119894

(119905)]119879

(119894 = 4)

119889119894

= Δ119875119889119894

(119894 = 1 2 3 4)

1199061

= [Δ1198751198881

Δ120579ref Δ119879119892]119879

119910119894

= ACE119894

= [119870119861119894

Δ119891119894

+ Δ119875tie119894] (119894 = 1 2 3 4)

(6)

The state control and disturbance matrices for area 1 areas follows

11986011

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus1

1198791198751

minus1198701198751

1198791198751

1198701198751

1198791198751

0 0 0 0 0

sum

119895

119870119904119894119895

0 0 0 0 0 0 0

0 0 minus1

1198791198791

01

1198791198791

0 0 0

1

1198791198661

0 0 minus1

1198791198661

0 0 0 0

0 0 0 0 0 1 minus1

119899gear0

0 0 0 0 minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

0 0 0 0

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 0 0 0 0 minus1

120591120579

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986111

=

[[[[[[[

[

0 0 01

1198791198661

0 0 0 0

0 0 0 0 0 0 0

119870120573

120591120579

0 0 0 0 0 0 minus1

119869119892

0

]]]]]]]

]

119879

11986511

=[[[

[

minus1198701198751

1198791198751

0 0 0 0 0 0 0

0 0 0 0 01

119869119903

120597119879119903

120597V119898

10038161003816100381610038161003816100381610038161003816op0 0

]]]

]

119879

11986211

= [1198701198871

1 0 0 0 0 0 0]

(7)

However for hydro plants in areas 2 and 3 they are asfollows

11986022

= 11986033

=

[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

2120572 0 minus2

119879119882119894

2120581 2120573

minus120572 0 0 minus1

1198792119894

minus120573

minus1

1198791119894

119877119894

0 0 0 minus1

1198791119894

]]]]]]]]]]]]]]]

]

11986122

= 11986133

= [0 0 minus2119877119894

120572 119877119894

1205721

1198791119894

]

119879

11986222

= 11986233

= [119870119861119894

1 0 0 0]

11986522

= 11986533

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(8)

where 120572 = 119879119877119894

1198791119894

1198792119894

119877119894

120573 = (119879119877119894

minus1198791119894

)1198791119894

1198792119894

and 120581 = (1198792119894

+

119879119882119894

)1198792119894

119879119882119894

Journal of Control Science and Engineering 7

However for thermal power plants in area 4 they are asfollows

11986044

=

[[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

0 0 minus1

119879119879119894

01

119879119879119894

minus1

119879119866119894

119877119894

0 0 minus1

119879119866119894

0

minus119870119903119894

119879119866119894

119877119894

0 01

119879119903119894

minus119870119903119894

119879119866119894

minus1

119879119903119894

]]]]]]]]]]]]]]]]

]

11986144

= [0 0 01

119879119866119894

0]

119879

11986244

= [119870119861119894

1 0 0 0]

11986544

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(9)

The interactionmatrices between the four control areas are asfollows

119860119894119895

=

[[[[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]]]]

]

(119894 = 1 119895 = 2 3 4)

119860119894119895

=

[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]

]

(119894 = 1 119895 = 2 3 119894 = 119895)

119860119894119895

=

[[[[[

[

0 0 0 0 0

minus119870119878119894119895

0 0 0 0

0 0 0 0 0

0 0 0 0 0

]]]]]

]

(119894 = 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 08times4

119865119894119895

= 08times2

(119894 = 1 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 05times1

119865119894119895

= 05times1

(119894 = 119895 = 2 3 4 119894 = 119895)

(10)

TheGRCs for the thermal plants are |Δ119892119894

| le 00017 puMWs and the hydro units are |Δ

119892119894

| le 0045 puMWs In addi-tion the load disturbance is constrained to |Δ

119889119894| le 03

3 Distributed Model Predictive Controller

31 Distributed Model Predictive Controller The block dia-gram of the DMPC scheme for a four-area interconnectedpower system is illustrated in Figure 7 Though there existslarge amount of variables in the interconnected powersystem the 30 state variables expressed in (1a) (1b) (1c)and (1d) concerning the frequency the generator outputpower the governor valve (servomotor) position the tie-lineactive power the wind power and the 4 load disturbanceΔ119875

119889119894

are crucial to LFC problem They can be measured orestimated directly by the local controller The DMPC in eacharea exchange control information through the power linecommunication which is a sole networking technology withhigh reliability that can provide high speed communicationto power grids applications [22]

Distributed MPC The partitioned discrete-time model forcontrol area 119894 of the continuous-time four-area intercon-nected power system ((1a) (1b) (1c) and (1d)) can beexpressed as follows

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

+sum

119894 =119895

(119860119894119895

119909119895

(119896) + 119861119894119895

119906119895

(119896) + 119865119894119895

119889119895

(119896))

119910119894

(119896) = 119862119894119894

119909119894

(119896)

(11)

where 119860119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

represent the discretenewmatrices obtained from original matrices in (4) based onthe Zero-Order Hold (ZOH) method

Assume that the state variables 119909119894

(119896) and the disturbance119863119894

can be measured or estimated directly by the controllerin area 119894 at sampling time 119896 Optimizations and exchange ofvariables are termed iterate The iteration number is denotedby 119901

For DMPC the optimal state-input trajectory (119909119894

119906119894

) foreach area 119894 119894 = 1 2 3 4 at iterate 119901 is obtained as the solutionto the optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (12)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896) + 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)] (13)

8 Journal of Control Science and Engineering

MPC 1 MPC 2

MPC 3MPC 4

Communication network

Thermal plantwind turbines

Hydro power plant

Thermal power plant

Hydro power plant

Figure 7 Block diagram of DMPC for power system with wind turbines

Subject to 10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00017 119894 = 1 4 (14a)

10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00045 119894 = 2 3 (14b)

10038171003817100381710038171199091198944 (119896 + 119899 | 119896)10038171003817100381710038172 le 120590119894 119894 = 1 2 3 4 (14c)

For notational convenience we drop the 119896 dependence of119909119894

(119896) 119906119894

(119896) 119894 = 1 2 3 4 It is shown in [20] that each 119909119894

canbe expressed as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119892119894119895

119909119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(15)

with

119909119894

= [119909119894

(119896 + 1 | 119896)119879

119909119894

(119896 + 2 | 119896)119879

sdot sdot sdot 119909119894

(119896 + 119873119901

| 119896)119879

]

119879

119906119894

= [119906119894

(119896 | 119896)119879

119906119894

(119896 + 1 | 119896)119879

sdot sdot sdot 119906119894

(119896 + 119873119888

minus 1 | 119896)119879

]119879

(16)

Let119873119888

denote the control horizon and let119873119901

denote thepredictive horizon 119909

119894

is no more a vector but a matrix after

iteration obtained from original equation (4)Thematrices in(15) have detailed expressions as follows

119864119894119894

=

[[[[[[[

[

119861119894119894

0 sdot sdot sdot 0

119860119894119894

119861119894119894

119861119894119894

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119894

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119864119894119895

=

[[[[[[[

[

119861119894119895

0 sdot sdot sdot 0

119860119894119894

119861119894119895

119861119894119895

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119895

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119891119894119894

=

[[[[[[[

[

119860119894119894

119860119894119894

119860119894119894

119860119873minus1

119894119894

119860119894119894

]]]]]]]

]

Journal of Control Science and Engineering 9

119891119894119895

=

[[[[[[[

[

119860119894119895

119860119894119894

119860119894119895

119860119873minus1

119894119894

119860119894119895

]]]]]]]

]

120573119894119894

=

[[[[[[[

[

119865119894119894

119860119894119894

119865119894119894

119860119873minus1

119894119894

119865119894119894

]]]]]]]

]

120573119894119895

=

[[[[[[[

[

119865119894119895

119860119894119894

119865119894119895

119860119873minus1

119894119894

119865119894119895

]]]]]]]

]

119892119894119895

=

[[[[[[[[[[

[

0 0 0 sdot sdot sdot 0

119860119894119895

0 0 sdot sdot sdot 0

119860119894119894

119860119894119895

119860119894119895

0 sdot sdot sdot 0

sdot sdot sdot

119860119873minus2

119894119894

119860119894119895

119860119873minus3

119894119894

119860119894119895

sdot sdot sdot 119860119894119895

0

]]]]]]]]]]

]

(17)

where 119864119894119894

119891119894119894

120573119894119894

119864119894119895

119891119894119895

120573119894119895

and 119892119894119895

are the new matricesobtained from 119860

119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

after iterationCombining the models in (15) gives the following system

of equations

Λ119909 = 120576 + 120583119909 (119896) + 120601119889 (119896) (18)

with

Λ =

[[[[[

[

119868 minus11989212

minus11989213

minus11989214

minus11989221

119868 minus11989223

minus11989224

minus11989231

minus11989232

119868 minus11989234

minus11989241

minus11989242

minus11989243

119868

]]]]]

]

120576 =

[[[[[[

[

11986411

11986412

11986413

11986414

11986421

11986422

11986423

11986424

11986431

11986432

11986433

11986434

11986441

11986442

11986443

11986444

]]]]]]

]

120583 =

[[[[[[

[

11989111

11989112

11989113

11989114

11989121

11989122

11989123

11989124

11989131

11989132

11989133

11989134

11989141

11989142

11989143

11989144

]]]]]]

]

120601 =

[[[[[[

[

12057311

12057312

12057313

12057314

12057321

12057322

12057323

12057324

12057331

12057332

12057333

12057334

12057341

12057342

12057343

12057344

]]]]]]

]

119909 = [1199091

1199092

1199093

1199094]119879

= [1199061

1199062

1199063

1199064]119879

(19)

Since matrix Λ is invertible we can write it as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(20)

in which

119864119894119895

= Λminus1

120576

119891119894119895

= Λminus1

120583

120573119894119895

= Λminus1

120601

(21)

To do so we eliminate the unknownmatrix 119909119895

because wehave knowledge of 119909

119895

(119896) since it is just a vector at time 119896In the distributed MPC algorithm for subsystem 119894 the

control signal 119880119894

is designed at each time interval 119896 ge 0 Bysolving the following optimization problem denoted by 119869

119894

itis usually defined as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ (120574119894

+ Γ119894

+sum

119894 =119895

119867119894119895

119906119895

)

119879

119906119894

(22)

in which

Q119894

= diag119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119876119894

120596119894

119876119894

)

R119894

= diag119873119888

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119877119894

120596119894

119877119894

)

Φ119894

= R119894

+ 119864119879

119894119894

Q119894

119864119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

120574119894

= 119864119879

119894119894

Q119894

119892119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119892119895119894

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

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RotatingMachinery

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Submit your manuscripts athttpwwwhindawicom

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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DistributedSensor Networks

International Journal of

2 Journal of Control Science and Engineering

(GRCs) in the conventional units the pitch angle and gener-ator torque constraints in WTGs the LFC becomes a large-scale distributed multiconstraints optimization problem

Recently a few attempts studied the idea of wind turbinesin the issue of LFC [8ndash10] Two types of wind farm modelsare derived and demonstrated to portray the capability ofset-point tracking under automatic generation control (AGC)[8] This inference leads to the development of a simplifiedwind farm model that is specially designed for the set-pointcontrol in the power system study However the durabilityof inertia effect depends on the allowable rotor speed rangeAn adaptive fuzzy logic structure was used to propose a newLFC scheme in the interconnected large-scale power systemin the presence of wind turbines [9]The performance againstsudden load change and wind power fluctuations in differentwind power penetration rates is confirmed by simulationA flatness-based method to control frequency and powerflow for multiarea power system with wind turbine is pre-sented in [10] And practical constraints such as generatorramping rates of wind turbine generator can be consideredin designing the controllers As abovementioned referencethe control schemes are designed for each area to maintainthe frequency at nominal value and to keep power flowsnear scheduled values However local controller in each areadoes not work cooperatively towards satisfying systemwidecontrol objectives In addition the control scheme [8ndash10]mentioned above could yield unsatisfactory performancesince the effects of nonlinearities such as generation rate con-straint and generation ramping rate were not considered

Model predictive control (MPC) also called recedinghorizon control was originally developed to be an effectivemethod for processing industrial control It transforms thecontrol problem into a finite horizon optimal control problemthat can also satisfy multivariable constraints on the inputoutput and state variables In the power industry MPC hasbeen successfully used in controlling power plant steam-boiler generation processes [11ndash13] In power system con-trol MPC was first developed to be an economic-orientedLFC [14] which generates the control action based on theopen-loop optimization method over a finite horizon MPChas subsequently been developed to realize the constrainedoptimal algorithm for LFC problem In [15] the constrainthandling ability of MPC is employed to effectively accountfor the generation rate constraints (GRCs) but without theanalysis of closed-loop stability and robustness RecentlyMPC has been successfully used in LFC design of multiareapower system with wind turbines [16] However each areacontroller is designed independently and the communicationbetween the local controllers is not considered On the otherhand with the size and capacity of wind farms increasing inrecent years traditional centralized MPCs encounter manydifficulties due to limitations in exchanging information withlarge-scale geographically extensive control areas In order todeal with these issues advanced distributed control strategieshave to be investigated and implemented

Developing decentralizeddistributed LFC structures canbe an effective way of solving this problemThe decentralizedmodel predictive control scheme for the LFC of multiareainterconnected power system is presented in [17] However

the local controller does not consider generation rate con-straint that is only imposed on the turbine in the simulationIn the distributed MPC (DMPC) the benefits from usinga decentralized structure are partially preserved and theplant-wide performance and stability are improved throughcoordination [18 19] In [20] feasible cooperation-basedMPCmethod is used in distributed LFC instead of centralizedMPC It is noted that the range of load change used in thecases is very large and inappropriate for the LFC issue

This paper studies the effect of merging the wind turbineson the system frequency of multiarea power system Thefirst control area includes an aggregated wind turbine model(which consists of 60 wind turbine units) beside the thermalpower plant According to the distributed LFC structurethe dynamics model of the four-area interconnected powersystem is established In our scheme the overall power systemis decomposed into four areas and each area has its ownlocal MPC controller These areas-based MPCs exchangetheir measurement and predictions by communication andincorporate the information from other controllers intotheir local objective so as to coordinate with each otherThe controllers calculate the optimal control signal whilerespecting constraints over the wind turbines output fre-quency deviation and the load change Not only do theeffects of the physical constraints conclude generating rateconstraints (GRCs) and the limit of governor position inconventional power plant but also thewind speed constraintsin wind turbines are considered Comparisons of responseto step load change computational burden and robustnesshave been made between DMPC centralized MPC anddecentralizedMPCThe results confirm the superiority of theproposed DMPC technique

The remainder of the paper is organized as follows Mod-eling of wind turbines participation in LFC is presented inSection 2 and the proposed DMPC algorithm is presented inSection 3 Section 4 presents the application of the algorithmin a four-area interconnected power systemThe conclusionsare presented in Section 5

2 Distributed Model of Hybrid Power System

Figure 1 illustrates the interconnected power system con-sisting of four control areas connected by tie-lines whichconsists of thermal power plant variable speed wind turbines(VSWTs) and hydro power plant In area 1 wind turbineis taken into consideration as it can provide a new solutionto the contradiction between economic development andenvironment pollution Area 4 is the thermal power plantwhile area 2 and area 3 are hydro power plants

Detailed compositions of each area are shown in Figures2ndash4 In addition area 1 includes an aggregated wind turbinemodel which consists of 40 VSWT units while the capacityof thermal plant is 600MW The variables and parametersare listed in Table 1 In each control area a change in localdemand (load) alters the nominal frequency The DMPCin each control area 119894 manipulates the load reference set-pointΔ119875ref 119894 to drive the frequency deviationsΔ119891119894 and tie-linepower flow deviations Δ119875tie119894119895 to zero

Journal of Control Science and Engineering 3

Table 1 Power system variables and parameter

Parametervariable Description Unit120596119903

Angular velocity of rotor rads120596119892

Angular velocity of high speed shaft and generator rads119879119892

Generator reaction torque Nm119879119903

Aerodynamic torque Nm119870119904

Total stiffness on low speed shaft Nmrad119869119903

Inertia of the rotor (low speed shaft and gearbox) Kgm2

119869119892

Inertia of the rotor (high speed shaft and gearbox) Kgm2

119899gear Exchange ratio Null120578gear Efficiency of the gear box 120591120573

Actuator time constant s119870120573

Actuator gain HzpuMWV119898

Wind speed ms120579ref Pitch demand rad120579 Pitch angle rad119875ref119890

Power demand puMW119875119890

The output of wind turbine puMWΔ119891

119894

(119905) Frequency deviation HzΔ119875

119892119894

(119905) Generator output power deviation puMWΔ119883

119892119894

(119905) Governor valve position deviation puΔ119883

119892ℎ119894

(119905) Governor valve servomotor position deviation puΔ119875

119905119894119890119894

(119905) Tie-line active power deviation puMWΔ119875

119889119894

(119905) Load disturbance puMW119870119875119894

Power system gain HzpuMW119870119903119894

Reheat turbine gain HzpuMW119879119875119894

Power system time constant s119879119903119894

Reheat turbine time constant s119879119882119894

Water starting time s119879119894119894

119879119877119894

Hydro governor time constants s119879119866119894

Thermal governor time constant s119879119879119894

Turbine time constant s119870119878119894119895

Interconnection gain between control areas puMW119870119861119894

Frequency bias factor puMWHz119877119894

Speed drop due to governor action HzpuMWACE

119894

Area control error puMW

P12tie P23

tie

P14tie P34

tie

P24tie

P31tie

Control area 1Wind turbine

Thermal power plant

Control area 2

Hydro power plant

Control area 3

Hydro power plant

Control area 4

Thermal power plant

Figure 1 The four-area interconnected hybrid power system

21Wind TurbineModel Awind turbine is an installation forconverting kinetic energy extracted from wind to electricalenergy Figure 5 illustrates the basicmodel structure of awindturbine and the interactions between the different dynamiccomponents in the model The whole wind turbine canbe divided into four subsystems aerodynamics subsystemmechanical subsystem electrical and actuator subsystem[21]

The linearization model for the variable speed wind tur-bine in Figure 6 can be represented by

Δ120576

= Δ120596119903

minus1

119899gearΔ120596

119892

(1a)

Δ119903

= minus119870119904

119869119903

Δ120593120576

+1

119869119903

Δ119879119903

(1b)

4 Journal of Control Science and Engineering

KBi

ACEi

Σ MPCi

ΔPci

1

Ri

Σ1

1 + sTGi

ΔXgi

1 + sKrTri1 + sTri

ΔPri

1

1 + sTTi

ΔPgi

Σ

ΔPdi

KPi

1 + sTPi

Δfi

ΔPeΔPi

sumj

Ksij(Δfi minus Δfj)

2120587

s

Wind turbines

ΔPtiei

[Δ120579ref ΔTgref]

Figure 2 Block diagram of a thermal power plant and wind turbines (119894 = 1)

KBi

ACEi

Σ Σ

ΣMPCi

ΔPci

1

Ri

1

1 + sTGi

ΔXgiΔXghi ΔPgi

ΔPdi

KP2

1 + sTP2

Δfi

sumj

Ksij(Δfi minus Δfj)

2120587

s

1 + sTRi1 + sTii

1 minus sTWi

1 + 05sTWi

ΔPtiei

Figure 3 Block diagram of a hydro power plant (119894 = 2 3)

KBi

ACEi

Σ Σ

ΣMPCi

ΔPci

1

Ri

1

1 + sTGi

ΔXgi

1 + sKrTri1 + sTri

1

1 + sTTi

ΔPgi

ΔPdi

KPi

1 + sTPi

Δfi

sumj

Ksij(Δfi minus Δfj)2120587

s

ΔPtiei

Figure 4 Block diagram of a thermal power plant (119894 = 4)

mTr

120596r 120596g

ΔPe

Pe

120579120579ref Tgref

Prefe

Rotor

Driver train

Generator

Actuator

Power electronics

Power

Controller

Figure 5 Diagram of a variable speed wind turbine

Journal of Control Science and Engineering 5

m

ΔPe

minusminus

1

Ts + 1

1

s

ΔTgΔ120579

Δ120579ref

ΔTgref

Generatormodel

Wind speed

MPCWind turbine

model

Wind turbines

Figure 6 Diagram of wind power plant in area 1

Δ119892

=

120578gear119870119904

119899gearΔ120593

120576

minus1

119869119903

Δ119879119892

(1c)

Δ 120579 = minus1

120591120573

Δ120579 +

119870120573

120591120573

Δ120579ref (1d)

The generator reaction torque 119879119892

and the reference pitchangle 120579ref are used as indicator of the input of VSWT as119906119890

= [Δ120573ref Δ119879119892]119879

isin 1198772 Moreover 120578 is the efficiency of the

generator and 120596119892

and 119879119892

are used as indicator of the outputpower as 119875

119890

= 120578120596119892

119879119892

isin 1198771 where 120596

119892

is the angular velocityof generator shaft A generalized representation of the state-space model of the variable speed turbine can be described as

119890

(119905) = 119860 (V119898

) 119909119890

(119905) + 1198611

(V119898

) 120596 (119905) + 1198612

119906119890

(119905) (2a)

119911119890

(119905) = 119862119909119890

(119905) + 1198631

120596 (119905) + 1198632

119906119890

(119905) (2b)

with

119860 (V119898

) =

[[[[[[[[[[[[[

[

0 1 minus1

119899gear0

minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 minus1

120591120579

]]]]]]]]]]]]]

]

1198611

(V119898

) =

[[[[[[[[

[

0

1

119869119903

120597119879119903

120597V

10038161003816100381610038161003816100381610038161003816op

0

0

]]]]]]]]

]

1198612

=

[[[[[[[[[

[

0 0

0 0

0 minus1

119869119892

119870120573

120591120573

0

]]]]]]]]]

]

119862 = [

0 0 1 0

0 0 0 0]

1198631

= [

0

0]

1198632

= [

0 0

0 1]

119909119890

= [Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

119906119890

= [Δ120579ref Δ119879119892ref]119879

119911119890

= [Δ120596119892

Δ119879119892]119879

119910119890

= 119875119890

= 120578120596119892

119879119892

(3)

22 Four-Area Power System with Wind Turbine Denotingthat the control area 119894 (119894 = 1 2 3 4) is to be interconnectedwith the control area 119895 119895 = 119894 through a tie-line a linear con-tinuous time-varyingmodel of control area 119894 can bewritten as

119894

= 119860119894119894

119909119894

+ 119861119894119894

119906119894

+ 119865119894119894

119889119894

+sum

119894 =119895

(119860119894119895

119909119895

+ 119861119894119895

119906119895

+ 119865119894119895

119889119895

)

119910119894

= 119862119894119894

119909119894

(4)

where 119909119894

isin 119877119899 119906

119894

isin 119877119898 119889

119894

isin 119877119896 and 119910

119894

isin 119877119897 are the state

vector the control signal vector the disturbance vector andthe vector of output of control area 119894 respectively 119909

119895

isin 119877119901

119906119895

isin 119877119902 and 119889

119895

isin 119877119904 are the state vector the control signal

vector and the disturbance vector of neighbor controlarea respectively Matrices 119860

119894119894

119861119894119894

119862119894119894

and 119865119894119894

representappropriate systemmatrices of control area 119894 and119860

119894119895

119861119894119895

and119865119894119895

represent the matrices of interaction variables betweenarea 119894 and area 119895 Tie-line power for area 119894 is represented by

Δ119875tie119894 =4

sum

119895=1

119895 =119894

Δ119875119894119895

tie =4

sum

119895=1

119895 =119894

119870119904119894119895

(Δ119891119894

minus Δ119891119895

)

Δ119875119894119895

tie = minusΔ119875119895119894

tie

(5)

6 Journal of Control Science and Engineering

The state disturbance and output vectors for area 119894 aredefined by

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

(119894 = 1)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119883119892ℎ119894]119879

(119894 = 2 3)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119875119903119894

(119905)]119879

(119894 = 4)

119889119894

= Δ119875119889119894

(119894 = 1 2 3 4)

1199061

= [Δ1198751198881

Δ120579ref Δ119879119892]119879

119910119894

= ACE119894

= [119870119861119894

Δ119891119894

+ Δ119875tie119894] (119894 = 1 2 3 4)

(6)

The state control and disturbance matrices for area 1 areas follows

11986011

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus1

1198791198751

minus1198701198751

1198791198751

1198701198751

1198791198751

0 0 0 0 0

sum

119895

119870119904119894119895

0 0 0 0 0 0 0

0 0 minus1

1198791198791

01

1198791198791

0 0 0

1

1198791198661

0 0 minus1

1198791198661

0 0 0 0

0 0 0 0 0 1 minus1

119899gear0

0 0 0 0 minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

0 0 0 0

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 0 0 0 0 minus1

120591120579

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986111

=

[[[[[[[

[

0 0 01

1198791198661

0 0 0 0

0 0 0 0 0 0 0

119870120573

120591120579

0 0 0 0 0 0 minus1

119869119892

0

]]]]]]]

]

119879

11986511

=[[[

[

minus1198701198751

1198791198751

0 0 0 0 0 0 0

0 0 0 0 01

119869119903

120597119879119903

120597V119898

10038161003816100381610038161003816100381610038161003816op0 0

]]]

]

119879

11986211

= [1198701198871

1 0 0 0 0 0 0]

(7)

However for hydro plants in areas 2 and 3 they are asfollows

11986022

= 11986033

=

[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

2120572 0 minus2

119879119882119894

2120581 2120573

minus120572 0 0 minus1

1198792119894

minus120573

minus1

1198791119894

119877119894

0 0 0 minus1

1198791119894

]]]]]]]]]]]]]]]

]

11986122

= 11986133

= [0 0 minus2119877119894

120572 119877119894

1205721

1198791119894

]

119879

11986222

= 11986233

= [119870119861119894

1 0 0 0]

11986522

= 11986533

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(8)

where 120572 = 119879119877119894

1198791119894

1198792119894

119877119894

120573 = (119879119877119894

minus1198791119894

)1198791119894

1198792119894

and 120581 = (1198792119894

+

119879119882119894

)1198792119894

119879119882119894

Journal of Control Science and Engineering 7

However for thermal power plants in area 4 they are asfollows

11986044

=

[[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

0 0 minus1

119879119879119894

01

119879119879119894

minus1

119879119866119894

119877119894

0 0 minus1

119879119866119894

0

minus119870119903119894

119879119866119894

119877119894

0 01

119879119903119894

minus119870119903119894

119879119866119894

minus1

119879119903119894

]]]]]]]]]]]]]]]]

]

11986144

= [0 0 01

119879119866119894

0]

119879

11986244

= [119870119861119894

1 0 0 0]

11986544

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(9)

The interactionmatrices between the four control areas are asfollows

119860119894119895

=

[[[[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]]]]

]

(119894 = 1 119895 = 2 3 4)

119860119894119895

=

[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]

]

(119894 = 1 119895 = 2 3 119894 = 119895)

119860119894119895

=

[[[[[

[

0 0 0 0 0

minus119870119878119894119895

0 0 0 0

0 0 0 0 0

0 0 0 0 0

]]]]]

]

(119894 = 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 08times4

119865119894119895

= 08times2

(119894 = 1 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 05times1

119865119894119895

= 05times1

(119894 = 119895 = 2 3 4 119894 = 119895)

(10)

TheGRCs for the thermal plants are |Δ119892119894

| le 00017 puMWs and the hydro units are |Δ

119892119894

| le 0045 puMWs In addi-tion the load disturbance is constrained to |Δ

119889119894| le 03

3 Distributed Model Predictive Controller

31 Distributed Model Predictive Controller The block dia-gram of the DMPC scheme for a four-area interconnectedpower system is illustrated in Figure 7 Though there existslarge amount of variables in the interconnected powersystem the 30 state variables expressed in (1a) (1b) (1c)and (1d) concerning the frequency the generator outputpower the governor valve (servomotor) position the tie-lineactive power the wind power and the 4 load disturbanceΔ119875

119889119894

are crucial to LFC problem They can be measured orestimated directly by the local controller The DMPC in eacharea exchange control information through the power linecommunication which is a sole networking technology withhigh reliability that can provide high speed communicationto power grids applications [22]

Distributed MPC The partitioned discrete-time model forcontrol area 119894 of the continuous-time four-area intercon-nected power system ((1a) (1b) (1c) and (1d)) can beexpressed as follows

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

+sum

119894 =119895

(119860119894119895

119909119895

(119896) + 119861119894119895

119906119895

(119896) + 119865119894119895

119889119895

(119896))

119910119894

(119896) = 119862119894119894

119909119894

(119896)

(11)

where 119860119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

represent the discretenewmatrices obtained from original matrices in (4) based onthe Zero-Order Hold (ZOH) method

Assume that the state variables 119909119894

(119896) and the disturbance119863119894

can be measured or estimated directly by the controllerin area 119894 at sampling time 119896 Optimizations and exchange ofvariables are termed iterate The iteration number is denotedby 119901

For DMPC the optimal state-input trajectory (119909119894

119906119894

) foreach area 119894 119894 = 1 2 3 4 at iterate 119901 is obtained as the solutionto the optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (12)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896) + 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)] (13)

8 Journal of Control Science and Engineering

MPC 1 MPC 2

MPC 3MPC 4

Communication network

Thermal plantwind turbines

Hydro power plant

Thermal power plant

Hydro power plant

Figure 7 Block diagram of DMPC for power system with wind turbines

Subject to 10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00017 119894 = 1 4 (14a)

10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00045 119894 = 2 3 (14b)

10038171003817100381710038171199091198944 (119896 + 119899 | 119896)10038171003817100381710038172 le 120590119894 119894 = 1 2 3 4 (14c)

For notational convenience we drop the 119896 dependence of119909119894

(119896) 119906119894

(119896) 119894 = 1 2 3 4 It is shown in [20] that each 119909119894

canbe expressed as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119892119894119895

119909119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(15)

with

119909119894

= [119909119894

(119896 + 1 | 119896)119879

119909119894

(119896 + 2 | 119896)119879

sdot sdot sdot 119909119894

(119896 + 119873119901

| 119896)119879

]

119879

119906119894

= [119906119894

(119896 | 119896)119879

119906119894

(119896 + 1 | 119896)119879

sdot sdot sdot 119906119894

(119896 + 119873119888

minus 1 | 119896)119879

]119879

(16)

Let119873119888

denote the control horizon and let119873119901

denote thepredictive horizon 119909

119894

is no more a vector but a matrix after

iteration obtained from original equation (4)Thematrices in(15) have detailed expressions as follows

119864119894119894

=

[[[[[[[

[

119861119894119894

0 sdot sdot sdot 0

119860119894119894

119861119894119894

119861119894119894

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119894

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119864119894119895

=

[[[[[[[

[

119861119894119895

0 sdot sdot sdot 0

119860119894119894

119861119894119895

119861119894119895

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119895

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119891119894119894

=

[[[[[[[

[

119860119894119894

119860119894119894

119860119894119894

119860119873minus1

119894119894

119860119894119894

]]]]]]]

]

Journal of Control Science and Engineering 9

119891119894119895

=

[[[[[[[

[

119860119894119895

119860119894119894

119860119894119895

119860119873minus1

119894119894

119860119894119895

]]]]]]]

]

120573119894119894

=

[[[[[[[

[

119865119894119894

119860119894119894

119865119894119894

119860119873minus1

119894119894

119865119894119894

]]]]]]]

]

120573119894119895

=

[[[[[[[

[

119865119894119895

119860119894119894

119865119894119895

119860119873minus1

119894119894

119865119894119895

]]]]]]]

]

119892119894119895

=

[[[[[[[[[[

[

0 0 0 sdot sdot sdot 0

119860119894119895

0 0 sdot sdot sdot 0

119860119894119894

119860119894119895

119860119894119895

0 sdot sdot sdot 0

sdot sdot sdot

119860119873minus2

119894119894

119860119894119895

119860119873minus3

119894119894

119860119894119895

sdot sdot sdot 119860119894119895

0

]]]]]]]]]]

]

(17)

where 119864119894119894

119891119894119894

120573119894119894

119864119894119895

119891119894119895

120573119894119895

and 119892119894119895

are the new matricesobtained from 119860

119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

after iterationCombining the models in (15) gives the following system

of equations

Λ119909 = 120576 + 120583119909 (119896) + 120601119889 (119896) (18)

with

Λ =

[[[[[

[

119868 minus11989212

minus11989213

minus11989214

minus11989221

119868 minus11989223

minus11989224

minus11989231

minus11989232

119868 minus11989234

minus11989241

minus11989242

minus11989243

119868

]]]]]

]

120576 =

[[[[[[

[

11986411

11986412

11986413

11986414

11986421

11986422

11986423

11986424

11986431

11986432

11986433

11986434

11986441

11986442

11986443

11986444

]]]]]]

]

120583 =

[[[[[[

[

11989111

11989112

11989113

11989114

11989121

11989122

11989123

11989124

11989131

11989132

11989133

11989134

11989141

11989142

11989143

11989144

]]]]]]

]

120601 =

[[[[[[

[

12057311

12057312

12057313

12057314

12057321

12057322

12057323

12057324

12057331

12057332

12057333

12057334

12057341

12057342

12057343

12057344

]]]]]]

]

119909 = [1199091

1199092

1199093

1199094]119879

= [1199061

1199062

1199063

1199064]119879

(19)

Since matrix Λ is invertible we can write it as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(20)

in which

119864119894119895

= Λminus1

120576

119891119894119895

= Λminus1

120583

120573119894119895

= Λminus1

120601

(21)

To do so we eliminate the unknownmatrix 119909119895

because wehave knowledge of 119909

119895

(119896) since it is just a vector at time 119896In the distributed MPC algorithm for subsystem 119894 the

control signal 119880119894

is designed at each time interval 119896 ge 0 Bysolving the following optimization problem denoted by 119869

119894

itis usually defined as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ (120574119894

+ Γ119894

+sum

119894 =119895

119867119894119895

119906119895

)

119879

119906119894

(22)

in which

Q119894

= diag119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119876119894

120596119894

119876119894

)

R119894

= diag119873119888

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119877119894

120596119894

119877119894

)

Φ119894

= R119894

+ 119864119879

119894119894

Q119894

119864119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

120574119894

= 119864119879

119894119894

Q119894

119892119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119892119895119894

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

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International Journal of

Journal of Control Science and Engineering 3

Table 1 Power system variables and parameter

Parametervariable Description Unit120596119903

Angular velocity of rotor rads120596119892

Angular velocity of high speed shaft and generator rads119879119892

Generator reaction torque Nm119879119903

Aerodynamic torque Nm119870119904

Total stiffness on low speed shaft Nmrad119869119903

Inertia of the rotor (low speed shaft and gearbox) Kgm2

119869119892

Inertia of the rotor (high speed shaft and gearbox) Kgm2

119899gear Exchange ratio Null120578gear Efficiency of the gear box 120591120573

Actuator time constant s119870120573

Actuator gain HzpuMWV119898

Wind speed ms120579ref Pitch demand rad120579 Pitch angle rad119875ref119890

Power demand puMW119875119890

The output of wind turbine puMWΔ119891

119894

(119905) Frequency deviation HzΔ119875

119892119894

(119905) Generator output power deviation puMWΔ119883

119892119894

(119905) Governor valve position deviation puΔ119883

119892ℎ119894

(119905) Governor valve servomotor position deviation puΔ119875

119905119894119890119894

(119905) Tie-line active power deviation puMWΔ119875

119889119894

(119905) Load disturbance puMW119870119875119894

Power system gain HzpuMW119870119903119894

Reheat turbine gain HzpuMW119879119875119894

Power system time constant s119879119903119894

Reheat turbine time constant s119879119882119894

Water starting time s119879119894119894

119879119877119894

Hydro governor time constants s119879119866119894

Thermal governor time constant s119879119879119894

Turbine time constant s119870119878119894119895

Interconnection gain between control areas puMW119870119861119894

Frequency bias factor puMWHz119877119894

Speed drop due to governor action HzpuMWACE

119894

Area control error puMW

P12tie P23

tie

P14tie P34

tie

P24tie

P31tie

Control area 1Wind turbine

Thermal power plant

Control area 2

Hydro power plant

Control area 3

Hydro power plant

Control area 4

Thermal power plant

Figure 1 The four-area interconnected hybrid power system

21Wind TurbineModel Awind turbine is an installation forconverting kinetic energy extracted from wind to electricalenergy Figure 5 illustrates the basicmodel structure of awindturbine and the interactions between the different dynamiccomponents in the model The whole wind turbine canbe divided into four subsystems aerodynamics subsystemmechanical subsystem electrical and actuator subsystem[21]

The linearization model for the variable speed wind tur-bine in Figure 6 can be represented by

Δ120576

= Δ120596119903

minus1

119899gearΔ120596

119892

(1a)

Δ119903

= minus119870119904

119869119903

Δ120593120576

+1

119869119903

Δ119879119903

(1b)

4 Journal of Control Science and Engineering

KBi

ACEi

Σ MPCi

ΔPci

1

Ri

Σ1

1 + sTGi

ΔXgi

1 + sKrTri1 + sTri

ΔPri

1

1 + sTTi

ΔPgi

Σ

ΔPdi

KPi

1 + sTPi

Δfi

ΔPeΔPi

sumj

Ksij(Δfi minus Δfj)

2120587

s

Wind turbines

ΔPtiei

[Δ120579ref ΔTgref]

Figure 2 Block diagram of a thermal power plant and wind turbines (119894 = 1)

KBi

ACEi

Σ Σ

ΣMPCi

ΔPci

1

Ri

1

1 + sTGi

ΔXgiΔXghi ΔPgi

ΔPdi

KP2

1 + sTP2

Δfi

sumj

Ksij(Δfi minus Δfj)

2120587

s

1 + sTRi1 + sTii

1 minus sTWi

1 + 05sTWi

ΔPtiei

Figure 3 Block diagram of a hydro power plant (119894 = 2 3)

KBi

ACEi

Σ Σ

ΣMPCi

ΔPci

1

Ri

1

1 + sTGi

ΔXgi

1 + sKrTri1 + sTri

1

1 + sTTi

ΔPgi

ΔPdi

KPi

1 + sTPi

Δfi

sumj

Ksij(Δfi minus Δfj)2120587

s

ΔPtiei

Figure 4 Block diagram of a thermal power plant (119894 = 4)

mTr

120596r 120596g

ΔPe

Pe

120579120579ref Tgref

Prefe

Rotor

Driver train

Generator

Actuator

Power electronics

Power

Controller

Figure 5 Diagram of a variable speed wind turbine

Journal of Control Science and Engineering 5

m

ΔPe

minusminus

1

Ts + 1

1

s

ΔTgΔ120579

Δ120579ref

ΔTgref

Generatormodel

Wind speed

MPCWind turbine

model

Wind turbines

Figure 6 Diagram of wind power plant in area 1

Δ119892

=

120578gear119870119904

119899gearΔ120593

120576

minus1

119869119903

Δ119879119892

(1c)

Δ 120579 = minus1

120591120573

Δ120579 +

119870120573

120591120573

Δ120579ref (1d)

The generator reaction torque 119879119892

and the reference pitchangle 120579ref are used as indicator of the input of VSWT as119906119890

= [Δ120573ref Δ119879119892]119879

isin 1198772 Moreover 120578 is the efficiency of the

generator and 120596119892

and 119879119892

are used as indicator of the outputpower as 119875

119890

= 120578120596119892

119879119892

isin 1198771 where 120596

119892

is the angular velocityof generator shaft A generalized representation of the state-space model of the variable speed turbine can be described as

119890

(119905) = 119860 (V119898

) 119909119890

(119905) + 1198611

(V119898

) 120596 (119905) + 1198612

119906119890

(119905) (2a)

119911119890

(119905) = 119862119909119890

(119905) + 1198631

120596 (119905) + 1198632

119906119890

(119905) (2b)

with

119860 (V119898

) =

[[[[[[[[[[[[[

[

0 1 minus1

119899gear0

minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 minus1

120591120579

]]]]]]]]]]]]]

]

1198611

(V119898

) =

[[[[[[[[

[

0

1

119869119903

120597119879119903

120597V

10038161003816100381610038161003816100381610038161003816op

0

0

]]]]]]]]

]

1198612

=

[[[[[[[[[

[

0 0

0 0

0 minus1

119869119892

119870120573

120591120573

0

]]]]]]]]]

]

119862 = [

0 0 1 0

0 0 0 0]

1198631

= [

0

0]

1198632

= [

0 0

0 1]

119909119890

= [Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

119906119890

= [Δ120579ref Δ119879119892ref]119879

119911119890

= [Δ120596119892

Δ119879119892]119879

119910119890

= 119875119890

= 120578120596119892

119879119892

(3)

22 Four-Area Power System with Wind Turbine Denotingthat the control area 119894 (119894 = 1 2 3 4) is to be interconnectedwith the control area 119895 119895 = 119894 through a tie-line a linear con-tinuous time-varyingmodel of control area 119894 can bewritten as

119894

= 119860119894119894

119909119894

+ 119861119894119894

119906119894

+ 119865119894119894

119889119894

+sum

119894 =119895

(119860119894119895

119909119895

+ 119861119894119895

119906119895

+ 119865119894119895

119889119895

)

119910119894

= 119862119894119894

119909119894

(4)

where 119909119894

isin 119877119899 119906

119894

isin 119877119898 119889

119894

isin 119877119896 and 119910

119894

isin 119877119897 are the state

vector the control signal vector the disturbance vector andthe vector of output of control area 119894 respectively 119909

119895

isin 119877119901

119906119895

isin 119877119902 and 119889

119895

isin 119877119904 are the state vector the control signal

vector and the disturbance vector of neighbor controlarea respectively Matrices 119860

119894119894

119861119894119894

119862119894119894

and 119865119894119894

representappropriate systemmatrices of control area 119894 and119860

119894119895

119861119894119895

and119865119894119895

represent the matrices of interaction variables betweenarea 119894 and area 119895 Tie-line power for area 119894 is represented by

Δ119875tie119894 =4

sum

119895=1

119895 =119894

Δ119875119894119895

tie =4

sum

119895=1

119895 =119894

119870119904119894119895

(Δ119891119894

minus Δ119891119895

)

Δ119875119894119895

tie = minusΔ119875119895119894

tie

(5)

6 Journal of Control Science and Engineering

The state disturbance and output vectors for area 119894 aredefined by

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

(119894 = 1)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119883119892ℎ119894]119879

(119894 = 2 3)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119875119903119894

(119905)]119879

(119894 = 4)

119889119894

= Δ119875119889119894

(119894 = 1 2 3 4)

1199061

= [Δ1198751198881

Δ120579ref Δ119879119892]119879

119910119894

= ACE119894

= [119870119861119894

Δ119891119894

+ Δ119875tie119894] (119894 = 1 2 3 4)

(6)

The state control and disturbance matrices for area 1 areas follows

11986011

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus1

1198791198751

minus1198701198751

1198791198751

1198701198751

1198791198751

0 0 0 0 0

sum

119895

119870119904119894119895

0 0 0 0 0 0 0

0 0 minus1

1198791198791

01

1198791198791

0 0 0

1

1198791198661

0 0 minus1

1198791198661

0 0 0 0

0 0 0 0 0 1 minus1

119899gear0

0 0 0 0 minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

0 0 0 0

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 0 0 0 0 minus1

120591120579

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986111

=

[[[[[[[

[

0 0 01

1198791198661

0 0 0 0

0 0 0 0 0 0 0

119870120573

120591120579

0 0 0 0 0 0 minus1

119869119892

0

]]]]]]]

]

119879

11986511

=[[[

[

minus1198701198751

1198791198751

0 0 0 0 0 0 0

0 0 0 0 01

119869119903

120597119879119903

120597V119898

10038161003816100381610038161003816100381610038161003816op0 0

]]]

]

119879

11986211

= [1198701198871

1 0 0 0 0 0 0]

(7)

However for hydro plants in areas 2 and 3 they are asfollows

11986022

= 11986033

=

[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

2120572 0 minus2

119879119882119894

2120581 2120573

minus120572 0 0 minus1

1198792119894

minus120573

minus1

1198791119894

119877119894

0 0 0 minus1

1198791119894

]]]]]]]]]]]]]]]

]

11986122

= 11986133

= [0 0 minus2119877119894

120572 119877119894

1205721

1198791119894

]

119879

11986222

= 11986233

= [119870119861119894

1 0 0 0]

11986522

= 11986533

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(8)

where 120572 = 119879119877119894

1198791119894

1198792119894

119877119894

120573 = (119879119877119894

minus1198791119894

)1198791119894

1198792119894

and 120581 = (1198792119894

+

119879119882119894

)1198792119894

119879119882119894

Journal of Control Science and Engineering 7

However for thermal power plants in area 4 they are asfollows

11986044

=

[[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

0 0 minus1

119879119879119894

01

119879119879119894

minus1

119879119866119894

119877119894

0 0 minus1

119879119866119894

0

minus119870119903119894

119879119866119894

119877119894

0 01

119879119903119894

minus119870119903119894

119879119866119894

minus1

119879119903119894

]]]]]]]]]]]]]]]]

]

11986144

= [0 0 01

119879119866119894

0]

119879

11986244

= [119870119861119894

1 0 0 0]

11986544

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(9)

The interactionmatrices between the four control areas are asfollows

119860119894119895

=

[[[[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]]]]

]

(119894 = 1 119895 = 2 3 4)

119860119894119895

=

[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]

]

(119894 = 1 119895 = 2 3 119894 = 119895)

119860119894119895

=

[[[[[

[

0 0 0 0 0

minus119870119878119894119895

0 0 0 0

0 0 0 0 0

0 0 0 0 0

]]]]]

]

(119894 = 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 08times4

119865119894119895

= 08times2

(119894 = 1 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 05times1

119865119894119895

= 05times1

(119894 = 119895 = 2 3 4 119894 = 119895)

(10)

TheGRCs for the thermal plants are |Δ119892119894

| le 00017 puMWs and the hydro units are |Δ

119892119894

| le 0045 puMWs In addi-tion the load disturbance is constrained to |Δ

119889119894| le 03

3 Distributed Model Predictive Controller

31 Distributed Model Predictive Controller The block dia-gram of the DMPC scheme for a four-area interconnectedpower system is illustrated in Figure 7 Though there existslarge amount of variables in the interconnected powersystem the 30 state variables expressed in (1a) (1b) (1c)and (1d) concerning the frequency the generator outputpower the governor valve (servomotor) position the tie-lineactive power the wind power and the 4 load disturbanceΔ119875

119889119894

are crucial to LFC problem They can be measured orestimated directly by the local controller The DMPC in eacharea exchange control information through the power linecommunication which is a sole networking technology withhigh reliability that can provide high speed communicationto power grids applications [22]

Distributed MPC The partitioned discrete-time model forcontrol area 119894 of the continuous-time four-area intercon-nected power system ((1a) (1b) (1c) and (1d)) can beexpressed as follows

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

+sum

119894 =119895

(119860119894119895

119909119895

(119896) + 119861119894119895

119906119895

(119896) + 119865119894119895

119889119895

(119896))

119910119894

(119896) = 119862119894119894

119909119894

(119896)

(11)

where 119860119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

represent the discretenewmatrices obtained from original matrices in (4) based onthe Zero-Order Hold (ZOH) method

Assume that the state variables 119909119894

(119896) and the disturbance119863119894

can be measured or estimated directly by the controllerin area 119894 at sampling time 119896 Optimizations and exchange ofvariables are termed iterate The iteration number is denotedby 119901

For DMPC the optimal state-input trajectory (119909119894

119906119894

) foreach area 119894 119894 = 1 2 3 4 at iterate 119901 is obtained as the solutionto the optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (12)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896) + 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)] (13)

8 Journal of Control Science and Engineering

MPC 1 MPC 2

MPC 3MPC 4

Communication network

Thermal plantwind turbines

Hydro power plant

Thermal power plant

Hydro power plant

Figure 7 Block diagram of DMPC for power system with wind turbines

Subject to 10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00017 119894 = 1 4 (14a)

10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00045 119894 = 2 3 (14b)

10038171003817100381710038171199091198944 (119896 + 119899 | 119896)10038171003817100381710038172 le 120590119894 119894 = 1 2 3 4 (14c)

For notational convenience we drop the 119896 dependence of119909119894

(119896) 119906119894

(119896) 119894 = 1 2 3 4 It is shown in [20] that each 119909119894

canbe expressed as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119892119894119895

119909119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(15)

with

119909119894

= [119909119894

(119896 + 1 | 119896)119879

119909119894

(119896 + 2 | 119896)119879

sdot sdot sdot 119909119894

(119896 + 119873119901

| 119896)119879

]

119879

119906119894

= [119906119894

(119896 | 119896)119879

119906119894

(119896 + 1 | 119896)119879

sdot sdot sdot 119906119894

(119896 + 119873119888

minus 1 | 119896)119879

]119879

(16)

Let119873119888

denote the control horizon and let119873119901

denote thepredictive horizon 119909

119894

is no more a vector but a matrix after

iteration obtained from original equation (4)Thematrices in(15) have detailed expressions as follows

119864119894119894

=

[[[[[[[

[

119861119894119894

0 sdot sdot sdot 0

119860119894119894

119861119894119894

119861119894119894

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119894

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119864119894119895

=

[[[[[[[

[

119861119894119895

0 sdot sdot sdot 0

119860119894119894

119861119894119895

119861119894119895

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119895

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119891119894119894

=

[[[[[[[

[

119860119894119894

119860119894119894

119860119894119894

119860119873minus1

119894119894

119860119894119894

]]]]]]]

]

Journal of Control Science and Engineering 9

119891119894119895

=

[[[[[[[

[

119860119894119895

119860119894119894

119860119894119895

119860119873minus1

119894119894

119860119894119895

]]]]]]]

]

120573119894119894

=

[[[[[[[

[

119865119894119894

119860119894119894

119865119894119894

119860119873minus1

119894119894

119865119894119894

]]]]]]]

]

120573119894119895

=

[[[[[[[

[

119865119894119895

119860119894119894

119865119894119895

119860119873minus1

119894119894

119865119894119895

]]]]]]]

]

119892119894119895

=

[[[[[[[[[[

[

0 0 0 sdot sdot sdot 0

119860119894119895

0 0 sdot sdot sdot 0

119860119894119894

119860119894119895

119860119894119895

0 sdot sdot sdot 0

sdot sdot sdot

119860119873minus2

119894119894

119860119894119895

119860119873minus3

119894119894

119860119894119895

sdot sdot sdot 119860119894119895

0

]]]]]]]]]]

]

(17)

where 119864119894119894

119891119894119894

120573119894119894

119864119894119895

119891119894119895

120573119894119895

and 119892119894119895

are the new matricesobtained from 119860

119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

after iterationCombining the models in (15) gives the following system

of equations

Λ119909 = 120576 + 120583119909 (119896) + 120601119889 (119896) (18)

with

Λ =

[[[[[

[

119868 minus11989212

minus11989213

minus11989214

minus11989221

119868 minus11989223

minus11989224

minus11989231

minus11989232

119868 minus11989234

minus11989241

minus11989242

minus11989243

119868

]]]]]

]

120576 =

[[[[[[

[

11986411

11986412

11986413

11986414

11986421

11986422

11986423

11986424

11986431

11986432

11986433

11986434

11986441

11986442

11986443

11986444

]]]]]]

]

120583 =

[[[[[[

[

11989111

11989112

11989113

11989114

11989121

11989122

11989123

11989124

11989131

11989132

11989133

11989134

11989141

11989142

11989143

11989144

]]]]]]

]

120601 =

[[[[[[

[

12057311

12057312

12057313

12057314

12057321

12057322

12057323

12057324

12057331

12057332

12057333

12057334

12057341

12057342

12057343

12057344

]]]]]]

]

119909 = [1199091

1199092

1199093

1199094]119879

= [1199061

1199062

1199063

1199064]119879

(19)

Since matrix Λ is invertible we can write it as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(20)

in which

119864119894119895

= Λminus1

120576

119891119894119895

= Λminus1

120583

120573119894119895

= Λminus1

120601

(21)

To do so we eliminate the unknownmatrix 119909119895

because wehave knowledge of 119909

119895

(119896) since it is just a vector at time 119896In the distributed MPC algorithm for subsystem 119894 the

control signal 119880119894

is designed at each time interval 119896 ge 0 Bysolving the following optimization problem denoted by 119869

119894

itis usually defined as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ (120574119894

+ Γ119894

+sum

119894 =119895

119867119894119895

119906119895

)

119879

119906119894

(22)

in which

Q119894

= diag119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119876119894

120596119894

119876119894

)

R119894

= diag119873119888

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119877119894

120596119894

119877119894

)

Φ119894

= R119894

+ 119864119879

119894119894

Q119894

119864119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

120574119894

= 119864119879

119894119894

Q119894

119892119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119892119895119894

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Journal of Control Science and Engineering

KBi

ACEi

Σ MPCi

ΔPci

1

Ri

Σ1

1 + sTGi

ΔXgi

1 + sKrTri1 + sTri

ΔPri

1

1 + sTTi

ΔPgi

Σ

ΔPdi

KPi

1 + sTPi

Δfi

ΔPeΔPi

sumj

Ksij(Δfi minus Δfj)

2120587

s

Wind turbines

ΔPtiei

[Δ120579ref ΔTgref]

Figure 2 Block diagram of a thermal power plant and wind turbines (119894 = 1)

KBi

ACEi

Σ Σ

ΣMPCi

ΔPci

1

Ri

1

1 + sTGi

ΔXgiΔXghi ΔPgi

ΔPdi

KP2

1 + sTP2

Δfi

sumj

Ksij(Δfi minus Δfj)

2120587

s

1 + sTRi1 + sTii

1 minus sTWi

1 + 05sTWi

ΔPtiei

Figure 3 Block diagram of a hydro power plant (119894 = 2 3)

KBi

ACEi

Σ Σ

ΣMPCi

ΔPci

1

Ri

1

1 + sTGi

ΔXgi

1 + sKrTri1 + sTri

1

1 + sTTi

ΔPgi

ΔPdi

KPi

1 + sTPi

Δfi

sumj

Ksij(Δfi minus Δfj)2120587

s

ΔPtiei

Figure 4 Block diagram of a thermal power plant (119894 = 4)

mTr

120596r 120596g

ΔPe

Pe

120579120579ref Tgref

Prefe

Rotor

Driver train

Generator

Actuator

Power electronics

Power

Controller

Figure 5 Diagram of a variable speed wind turbine

Journal of Control Science and Engineering 5

m

ΔPe

minusminus

1

Ts + 1

1

s

ΔTgΔ120579

Δ120579ref

ΔTgref

Generatormodel

Wind speed

MPCWind turbine

model

Wind turbines

Figure 6 Diagram of wind power plant in area 1

Δ119892

=

120578gear119870119904

119899gearΔ120593

120576

minus1

119869119903

Δ119879119892

(1c)

Δ 120579 = minus1

120591120573

Δ120579 +

119870120573

120591120573

Δ120579ref (1d)

The generator reaction torque 119879119892

and the reference pitchangle 120579ref are used as indicator of the input of VSWT as119906119890

= [Δ120573ref Δ119879119892]119879

isin 1198772 Moreover 120578 is the efficiency of the

generator and 120596119892

and 119879119892

are used as indicator of the outputpower as 119875

119890

= 120578120596119892

119879119892

isin 1198771 where 120596

119892

is the angular velocityof generator shaft A generalized representation of the state-space model of the variable speed turbine can be described as

119890

(119905) = 119860 (V119898

) 119909119890

(119905) + 1198611

(V119898

) 120596 (119905) + 1198612

119906119890

(119905) (2a)

119911119890

(119905) = 119862119909119890

(119905) + 1198631

120596 (119905) + 1198632

119906119890

(119905) (2b)

with

119860 (V119898

) =

[[[[[[[[[[[[[

[

0 1 minus1

119899gear0

minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 minus1

120591120579

]]]]]]]]]]]]]

]

1198611

(V119898

) =

[[[[[[[[

[

0

1

119869119903

120597119879119903

120597V

10038161003816100381610038161003816100381610038161003816op

0

0

]]]]]]]]

]

1198612

=

[[[[[[[[[

[

0 0

0 0

0 minus1

119869119892

119870120573

120591120573

0

]]]]]]]]]

]

119862 = [

0 0 1 0

0 0 0 0]

1198631

= [

0

0]

1198632

= [

0 0

0 1]

119909119890

= [Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

119906119890

= [Δ120579ref Δ119879119892ref]119879

119911119890

= [Δ120596119892

Δ119879119892]119879

119910119890

= 119875119890

= 120578120596119892

119879119892

(3)

22 Four-Area Power System with Wind Turbine Denotingthat the control area 119894 (119894 = 1 2 3 4) is to be interconnectedwith the control area 119895 119895 = 119894 through a tie-line a linear con-tinuous time-varyingmodel of control area 119894 can bewritten as

119894

= 119860119894119894

119909119894

+ 119861119894119894

119906119894

+ 119865119894119894

119889119894

+sum

119894 =119895

(119860119894119895

119909119895

+ 119861119894119895

119906119895

+ 119865119894119895

119889119895

)

119910119894

= 119862119894119894

119909119894

(4)

where 119909119894

isin 119877119899 119906

119894

isin 119877119898 119889

119894

isin 119877119896 and 119910

119894

isin 119877119897 are the state

vector the control signal vector the disturbance vector andthe vector of output of control area 119894 respectively 119909

119895

isin 119877119901

119906119895

isin 119877119902 and 119889

119895

isin 119877119904 are the state vector the control signal

vector and the disturbance vector of neighbor controlarea respectively Matrices 119860

119894119894

119861119894119894

119862119894119894

and 119865119894119894

representappropriate systemmatrices of control area 119894 and119860

119894119895

119861119894119895

and119865119894119895

represent the matrices of interaction variables betweenarea 119894 and area 119895 Tie-line power for area 119894 is represented by

Δ119875tie119894 =4

sum

119895=1

119895 =119894

Δ119875119894119895

tie =4

sum

119895=1

119895 =119894

119870119904119894119895

(Δ119891119894

minus Δ119891119895

)

Δ119875119894119895

tie = minusΔ119875119895119894

tie

(5)

6 Journal of Control Science and Engineering

The state disturbance and output vectors for area 119894 aredefined by

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

(119894 = 1)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119883119892ℎ119894]119879

(119894 = 2 3)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119875119903119894

(119905)]119879

(119894 = 4)

119889119894

= Δ119875119889119894

(119894 = 1 2 3 4)

1199061

= [Δ1198751198881

Δ120579ref Δ119879119892]119879

119910119894

= ACE119894

= [119870119861119894

Δ119891119894

+ Δ119875tie119894] (119894 = 1 2 3 4)

(6)

The state control and disturbance matrices for area 1 areas follows

11986011

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus1

1198791198751

minus1198701198751

1198791198751

1198701198751

1198791198751

0 0 0 0 0

sum

119895

119870119904119894119895

0 0 0 0 0 0 0

0 0 minus1

1198791198791

01

1198791198791

0 0 0

1

1198791198661

0 0 minus1

1198791198661

0 0 0 0

0 0 0 0 0 1 minus1

119899gear0

0 0 0 0 minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

0 0 0 0

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 0 0 0 0 minus1

120591120579

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986111

=

[[[[[[[

[

0 0 01

1198791198661

0 0 0 0

0 0 0 0 0 0 0

119870120573

120591120579

0 0 0 0 0 0 minus1

119869119892

0

]]]]]]]

]

119879

11986511

=[[[

[

minus1198701198751

1198791198751

0 0 0 0 0 0 0

0 0 0 0 01

119869119903

120597119879119903

120597V119898

10038161003816100381610038161003816100381610038161003816op0 0

]]]

]

119879

11986211

= [1198701198871

1 0 0 0 0 0 0]

(7)

However for hydro plants in areas 2 and 3 they are asfollows

11986022

= 11986033

=

[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

2120572 0 minus2

119879119882119894

2120581 2120573

minus120572 0 0 minus1

1198792119894

minus120573

minus1

1198791119894

119877119894

0 0 0 minus1

1198791119894

]]]]]]]]]]]]]]]

]

11986122

= 11986133

= [0 0 minus2119877119894

120572 119877119894

1205721

1198791119894

]

119879

11986222

= 11986233

= [119870119861119894

1 0 0 0]

11986522

= 11986533

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(8)

where 120572 = 119879119877119894

1198791119894

1198792119894

119877119894

120573 = (119879119877119894

minus1198791119894

)1198791119894

1198792119894

and 120581 = (1198792119894

+

119879119882119894

)1198792119894

119879119882119894

Journal of Control Science and Engineering 7

However for thermal power plants in area 4 they are asfollows

11986044

=

[[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

0 0 minus1

119879119879119894

01

119879119879119894

minus1

119879119866119894

119877119894

0 0 minus1

119879119866119894

0

minus119870119903119894

119879119866119894

119877119894

0 01

119879119903119894

minus119870119903119894

119879119866119894

minus1

119879119903119894

]]]]]]]]]]]]]]]]

]

11986144

= [0 0 01

119879119866119894

0]

119879

11986244

= [119870119861119894

1 0 0 0]

11986544

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(9)

The interactionmatrices between the four control areas are asfollows

119860119894119895

=

[[[[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]]]]

]

(119894 = 1 119895 = 2 3 4)

119860119894119895

=

[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]

]

(119894 = 1 119895 = 2 3 119894 = 119895)

119860119894119895

=

[[[[[

[

0 0 0 0 0

minus119870119878119894119895

0 0 0 0

0 0 0 0 0

0 0 0 0 0

]]]]]

]

(119894 = 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 08times4

119865119894119895

= 08times2

(119894 = 1 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 05times1

119865119894119895

= 05times1

(119894 = 119895 = 2 3 4 119894 = 119895)

(10)

TheGRCs for the thermal plants are |Δ119892119894

| le 00017 puMWs and the hydro units are |Δ

119892119894

| le 0045 puMWs In addi-tion the load disturbance is constrained to |Δ

119889119894| le 03

3 Distributed Model Predictive Controller

31 Distributed Model Predictive Controller The block dia-gram of the DMPC scheme for a four-area interconnectedpower system is illustrated in Figure 7 Though there existslarge amount of variables in the interconnected powersystem the 30 state variables expressed in (1a) (1b) (1c)and (1d) concerning the frequency the generator outputpower the governor valve (servomotor) position the tie-lineactive power the wind power and the 4 load disturbanceΔ119875

119889119894

are crucial to LFC problem They can be measured orestimated directly by the local controller The DMPC in eacharea exchange control information through the power linecommunication which is a sole networking technology withhigh reliability that can provide high speed communicationto power grids applications [22]

Distributed MPC The partitioned discrete-time model forcontrol area 119894 of the continuous-time four-area intercon-nected power system ((1a) (1b) (1c) and (1d)) can beexpressed as follows

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

+sum

119894 =119895

(119860119894119895

119909119895

(119896) + 119861119894119895

119906119895

(119896) + 119865119894119895

119889119895

(119896))

119910119894

(119896) = 119862119894119894

119909119894

(119896)

(11)

where 119860119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

represent the discretenewmatrices obtained from original matrices in (4) based onthe Zero-Order Hold (ZOH) method

Assume that the state variables 119909119894

(119896) and the disturbance119863119894

can be measured or estimated directly by the controllerin area 119894 at sampling time 119896 Optimizations and exchange ofvariables are termed iterate The iteration number is denotedby 119901

For DMPC the optimal state-input trajectory (119909119894

119906119894

) foreach area 119894 119894 = 1 2 3 4 at iterate 119901 is obtained as the solutionto the optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (12)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896) + 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)] (13)

8 Journal of Control Science and Engineering

MPC 1 MPC 2

MPC 3MPC 4

Communication network

Thermal plantwind turbines

Hydro power plant

Thermal power plant

Hydro power plant

Figure 7 Block diagram of DMPC for power system with wind turbines

Subject to 10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00017 119894 = 1 4 (14a)

10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00045 119894 = 2 3 (14b)

10038171003817100381710038171199091198944 (119896 + 119899 | 119896)10038171003817100381710038172 le 120590119894 119894 = 1 2 3 4 (14c)

For notational convenience we drop the 119896 dependence of119909119894

(119896) 119906119894

(119896) 119894 = 1 2 3 4 It is shown in [20] that each 119909119894

canbe expressed as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119892119894119895

119909119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(15)

with

119909119894

= [119909119894

(119896 + 1 | 119896)119879

119909119894

(119896 + 2 | 119896)119879

sdot sdot sdot 119909119894

(119896 + 119873119901

| 119896)119879

]

119879

119906119894

= [119906119894

(119896 | 119896)119879

119906119894

(119896 + 1 | 119896)119879

sdot sdot sdot 119906119894

(119896 + 119873119888

minus 1 | 119896)119879

]119879

(16)

Let119873119888

denote the control horizon and let119873119901

denote thepredictive horizon 119909

119894

is no more a vector but a matrix after

iteration obtained from original equation (4)Thematrices in(15) have detailed expressions as follows

119864119894119894

=

[[[[[[[

[

119861119894119894

0 sdot sdot sdot 0

119860119894119894

119861119894119894

119861119894119894

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119894

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119864119894119895

=

[[[[[[[

[

119861119894119895

0 sdot sdot sdot 0

119860119894119894

119861119894119895

119861119894119895

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119895

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119891119894119894

=

[[[[[[[

[

119860119894119894

119860119894119894

119860119894119894

119860119873minus1

119894119894

119860119894119894

]]]]]]]

]

Journal of Control Science and Engineering 9

119891119894119895

=

[[[[[[[

[

119860119894119895

119860119894119894

119860119894119895

119860119873minus1

119894119894

119860119894119895

]]]]]]]

]

120573119894119894

=

[[[[[[[

[

119865119894119894

119860119894119894

119865119894119894

119860119873minus1

119894119894

119865119894119894

]]]]]]]

]

120573119894119895

=

[[[[[[[

[

119865119894119895

119860119894119894

119865119894119895

119860119873minus1

119894119894

119865119894119895

]]]]]]]

]

119892119894119895

=

[[[[[[[[[[

[

0 0 0 sdot sdot sdot 0

119860119894119895

0 0 sdot sdot sdot 0

119860119894119894

119860119894119895

119860119894119895

0 sdot sdot sdot 0

sdot sdot sdot

119860119873minus2

119894119894

119860119894119895

119860119873minus3

119894119894

119860119894119895

sdot sdot sdot 119860119894119895

0

]]]]]]]]]]

]

(17)

where 119864119894119894

119891119894119894

120573119894119894

119864119894119895

119891119894119895

120573119894119895

and 119892119894119895

are the new matricesobtained from 119860

119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

after iterationCombining the models in (15) gives the following system

of equations

Λ119909 = 120576 + 120583119909 (119896) + 120601119889 (119896) (18)

with

Λ =

[[[[[

[

119868 minus11989212

minus11989213

minus11989214

minus11989221

119868 minus11989223

minus11989224

minus11989231

minus11989232

119868 minus11989234

minus11989241

minus11989242

minus11989243

119868

]]]]]

]

120576 =

[[[[[[

[

11986411

11986412

11986413

11986414

11986421

11986422

11986423

11986424

11986431

11986432

11986433

11986434

11986441

11986442

11986443

11986444

]]]]]]

]

120583 =

[[[[[[

[

11989111

11989112

11989113

11989114

11989121

11989122

11989123

11989124

11989131

11989132

11989133

11989134

11989141

11989142

11989143

11989144

]]]]]]

]

120601 =

[[[[[[

[

12057311

12057312

12057313

12057314

12057321

12057322

12057323

12057324

12057331

12057332

12057333

12057334

12057341

12057342

12057343

12057344

]]]]]]

]

119909 = [1199091

1199092

1199093

1199094]119879

= [1199061

1199062

1199063

1199064]119879

(19)

Since matrix Λ is invertible we can write it as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(20)

in which

119864119894119895

= Λminus1

120576

119891119894119895

= Λminus1

120583

120573119894119895

= Λminus1

120601

(21)

To do so we eliminate the unknownmatrix 119909119895

because wehave knowledge of 119909

119895

(119896) since it is just a vector at time 119896In the distributed MPC algorithm for subsystem 119894 the

control signal 119880119894

is designed at each time interval 119896 ge 0 Bysolving the following optimization problem denoted by 119869

119894

itis usually defined as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ (120574119894

+ Γ119894

+sum

119894 =119895

119867119894119895

119906119895

)

119879

119906119894

(22)

in which

Q119894

= diag119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119876119894

120596119894

119876119894

)

R119894

= diag119873119888

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119877119894

120596119894

119877119894

)

Φ119894

= R119894

+ 119864119879

119894119894

Q119894

119864119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

120574119894

= 119864119879

119894119894

Q119894

119892119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119892119895119894

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 5

m

ΔPe

minusminus

1

Ts + 1

1

s

ΔTgΔ120579

Δ120579ref

ΔTgref

Generatormodel

Wind speed

MPCWind turbine

model

Wind turbines

Figure 6 Diagram of wind power plant in area 1

Δ119892

=

120578gear119870119904

119899gearΔ120593

120576

minus1

119869119903

Δ119879119892

(1c)

Δ 120579 = minus1

120591120573

Δ120579 +

119870120573

120591120573

Δ120579ref (1d)

The generator reaction torque 119879119892

and the reference pitchangle 120579ref are used as indicator of the input of VSWT as119906119890

= [Δ120573ref Δ119879119892]119879

isin 1198772 Moreover 120578 is the efficiency of the

generator and 120596119892

and 119879119892

are used as indicator of the outputpower as 119875

119890

= 120578120596119892

119879119892

isin 1198771 where 120596

119892

is the angular velocityof generator shaft A generalized representation of the state-space model of the variable speed turbine can be described as

119890

(119905) = 119860 (V119898

) 119909119890

(119905) + 1198611

(V119898

) 120596 (119905) + 1198612

119906119890

(119905) (2a)

119911119890

(119905) = 119862119909119890

(119905) + 1198631

120596 (119905) + 1198632

119906119890

(119905) (2b)

with

119860 (V119898

) =

[[[[[[[[[[[[[

[

0 1 minus1

119899gear0

minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 minus1

120591120579

]]]]]]]]]]]]]

]

1198611

(V119898

) =

[[[[[[[[

[

0

1

119869119903

120597119879119903

120597V

10038161003816100381610038161003816100381610038161003816op

0

0

]]]]]]]]

]

1198612

=

[[[[[[[[[

[

0 0

0 0

0 minus1

119869119892

119870120573

120591120573

0

]]]]]]]]]

]

119862 = [

0 0 1 0

0 0 0 0]

1198631

= [

0

0]

1198632

= [

0 0

0 1]

119909119890

= [Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

119906119890

= [Δ120579ref Δ119879119892ref]119879

119911119890

= [Δ120596119892

Δ119879119892]119879

119910119890

= 119875119890

= 120578120596119892

119879119892

(3)

22 Four-Area Power System with Wind Turbine Denotingthat the control area 119894 (119894 = 1 2 3 4) is to be interconnectedwith the control area 119895 119895 = 119894 through a tie-line a linear con-tinuous time-varyingmodel of control area 119894 can bewritten as

119894

= 119860119894119894

119909119894

+ 119861119894119894

119906119894

+ 119865119894119894

119889119894

+sum

119894 =119895

(119860119894119895

119909119895

+ 119861119894119895

119906119895

+ 119865119894119895

119889119895

)

119910119894

= 119862119894119894

119909119894

(4)

where 119909119894

isin 119877119899 119906

119894

isin 119877119898 119889

119894

isin 119877119896 and 119910

119894

isin 119877119897 are the state

vector the control signal vector the disturbance vector andthe vector of output of control area 119894 respectively 119909

119895

isin 119877119901

119906119895

isin 119877119902 and 119889

119895

isin 119877119904 are the state vector the control signal

vector and the disturbance vector of neighbor controlarea respectively Matrices 119860

119894119894

119861119894119894

119862119894119894

and 119865119894119894

representappropriate systemmatrices of control area 119894 and119860

119894119895

119861119894119895

and119865119894119895

represent the matrices of interaction variables betweenarea 119894 and area 119895 Tie-line power for area 119894 is represented by

Δ119875tie119894 =4

sum

119895=1

119895 =119894

Δ119875119894119895

tie =4

sum

119895=1

119895 =119894

119870119904119894119895

(Δ119891119894

minus Δ119891119895

)

Δ119875119894119895

tie = minusΔ119875119895119894

tie

(5)

6 Journal of Control Science and Engineering

The state disturbance and output vectors for area 119894 aredefined by

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

(119894 = 1)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119883119892ℎ119894]119879

(119894 = 2 3)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119875119903119894

(119905)]119879

(119894 = 4)

119889119894

= Δ119875119889119894

(119894 = 1 2 3 4)

1199061

= [Δ1198751198881

Δ120579ref Δ119879119892]119879

119910119894

= ACE119894

= [119870119861119894

Δ119891119894

+ Δ119875tie119894] (119894 = 1 2 3 4)

(6)

The state control and disturbance matrices for area 1 areas follows

11986011

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus1

1198791198751

minus1198701198751

1198791198751

1198701198751

1198791198751

0 0 0 0 0

sum

119895

119870119904119894119895

0 0 0 0 0 0 0

0 0 minus1

1198791198791

01

1198791198791

0 0 0

1

1198791198661

0 0 minus1

1198791198661

0 0 0 0

0 0 0 0 0 1 minus1

119899gear0

0 0 0 0 minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

0 0 0 0

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 0 0 0 0 minus1

120591120579

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986111

=

[[[[[[[

[

0 0 01

1198791198661

0 0 0 0

0 0 0 0 0 0 0

119870120573

120591120579

0 0 0 0 0 0 minus1

119869119892

0

]]]]]]]

]

119879

11986511

=[[[

[

minus1198701198751

1198791198751

0 0 0 0 0 0 0

0 0 0 0 01

119869119903

120597119879119903

120597V119898

10038161003816100381610038161003816100381610038161003816op0 0

]]]

]

119879

11986211

= [1198701198871

1 0 0 0 0 0 0]

(7)

However for hydro plants in areas 2 and 3 they are asfollows

11986022

= 11986033

=

[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

2120572 0 minus2

119879119882119894

2120581 2120573

minus120572 0 0 minus1

1198792119894

minus120573

minus1

1198791119894

119877119894

0 0 0 minus1

1198791119894

]]]]]]]]]]]]]]]

]

11986122

= 11986133

= [0 0 minus2119877119894

120572 119877119894

1205721

1198791119894

]

119879

11986222

= 11986233

= [119870119861119894

1 0 0 0]

11986522

= 11986533

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(8)

where 120572 = 119879119877119894

1198791119894

1198792119894

119877119894

120573 = (119879119877119894

minus1198791119894

)1198791119894

1198792119894

and 120581 = (1198792119894

+

119879119882119894

)1198792119894

119879119882119894

Journal of Control Science and Engineering 7

However for thermal power plants in area 4 they are asfollows

11986044

=

[[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

0 0 minus1

119879119879119894

01

119879119879119894

minus1

119879119866119894

119877119894

0 0 minus1

119879119866119894

0

minus119870119903119894

119879119866119894

119877119894

0 01

119879119903119894

minus119870119903119894

119879119866119894

minus1

119879119903119894

]]]]]]]]]]]]]]]]

]

11986144

= [0 0 01

119879119866119894

0]

119879

11986244

= [119870119861119894

1 0 0 0]

11986544

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(9)

The interactionmatrices between the four control areas are asfollows

119860119894119895

=

[[[[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]]]]

]

(119894 = 1 119895 = 2 3 4)

119860119894119895

=

[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]

]

(119894 = 1 119895 = 2 3 119894 = 119895)

119860119894119895

=

[[[[[

[

0 0 0 0 0

minus119870119878119894119895

0 0 0 0

0 0 0 0 0

0 0 0 0 0

]]]]]

]

(119894 = 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 08times4

119865119894119895

= 08times2

(119894 = 1 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 05times1

119865119894119895

= 05times1

(119894 = 119895 = 2 3 4 119894 = 119895)

(10)

TheGRCs for the thermal plants are |Δ119892119894

| le 00017 puMWs and the hydro units are |Δ

119892119894

| le 0045 puMWs In addi-tion the load disturbance is constrained to |Δ

119889119894| le 03

3 Distributed Model Predictive Controller

31 Distributed Model Predictive Controller The block dia-gram of the DMPC scheme for a four-area interconnectedpower system is illustrated in Figure 7 Though there existslarge amount of variables in the interconnected powersystem the 30 state variables expressed in (1a) (1b) (1c)and (1d) concerning the frequency the generator outputpower the governor valve (servomotor) position the tie-lineactive power the wind power and the 4 load disturbanceΔ119875

119889119894

are crucial to LFC problem They can be measured orestimated directly by the local controller The DMPC in eacharea exchange control information through the power linecommunication which is a sole networking technology withhigh reliability that can provide high speed communicationto power grids applications [22]

Distributed MPC The partitioned discrete-time model forcontrol area 119894 of the continuous-time four-area intercon-nected power system ((1a) (1b) (1c) and (1d)) can beexpressed as follows

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

+sum

119894 =119895

(119860119894119895

119909119895

(119896) + 119861119894119895

119906119895

(119896) + 119865119894119895

119889119895

(119896))

119910119894

(119896) = 119862119894119894

119909119894

(119896)

(11)

where 119860119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

represent the discretenewmatrices obtained from original matrices in (4) based onthe Zero-Order Hold (ZOH) method

Assume that the state variables 119909119894

(119896) and the disturbance119863119894

can be measured or estimated directly by the controllerin area 119894 at sampling time 119896 Optimizations and exchange ofvariables are termed iterate The iteration number is denotedby 119901

For DMPC the optimal state-input trajectory (119909119894

119906119894

) foreach area 119894 119894 = 1 2 3 4 at iterate 119901 is obtained as the solutionto the optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (12)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896) + 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)] (13)

8 Journal of Control Science and Engineering

MPC 1 MPC 2

MPC 3MPC 4

Communication network

Thermal plantwind turbines

Hydro power plant

Thermal power plant

Hydro power plant

Figure 7 Block diagram of DMPC for power system with wind turbines

Subject to 10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00017 119894 = 1 4 (14a)

10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00045 119894 = 2 3 (14b)

10038171003817100381710038171199091198944 (119896 + 119899 | 119896)10038171003817100381710038172 le 120590119894 119894 = 1 2 3 4 (14c)

For notational convenience we drop the 119896 dependence of119909119894

(119896) 119906119894

(119896) 119894 = 1 2 3 4 It is shown in [20] that each 119909119894

canbe expressed as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119892119894119895

119909119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(15)

with

119909119894

= [119909119894

(119896 + 1 | 119896)119879

119909119894

(119896 + 2 | 119896)119879

sdot sdot sdot 119909119894

(119896 + 119873119901

| 119896)119879

]

119879

119906119894

= [119906119894

(119896 | 119896)119879

119906119894

(119896 + 1 | 119896)119879

sdot sdot sdot 119906119894

(119896 + 119873119888

minus 1 | 119896)119879

]119879

(16)

Let119873119888

denote the control horizon and let119873119901

denote thepredictive horizon 119909

119894

is no more a vector but a matrix after

iteration obtained from original equation (4)Thematrices in(15) have detailed expressions as follows

119864119894119894

=

[[[[[[[

[

119861119894119894

0 sdot sdot sdot 0

119860119894119894

119861119894119894

119861119894119894

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119894

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119864119894119895

=

[[[[[[[

[

119861119894119895

0 sdot sdot sdot 0

119860119894119894

119861119894119895

119861119894119895

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119895

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119891119894119894

=

[[[[[[[

[

119860119894119894

119860119894119894

119860119894119894

119860119873minus1

119894119894

119860119894119894

]]]]]]]

]

Journal of Control Science and Engineering 9

119891119894119895

=

[[[[[[[

[

119860119894119895

119860119894119894

119860119894119895

119860119873minus1

119894119894

119860119894119895

]]]]]]]

]

120573119894119894

=

[[[[[[[

[

119865119894119894

119860119894119894

119865119894119894

119860119873minus1

119894119894

119865119894119894

]]]]]]]

]

120573119894119895

=

[[[[[[[

[

119865119894119895

119860119894119894

119865119894119895

119860119873minus1

119894119894

119865119894119895

]]]]]]]

]

119892119894119895

=

[[[[[[[[[[

[

0 0 0 sdot sdot sdot 0

119860119894119895

0 0 sdot sdot sdot 0

119860119894119894

119860119894119895

119860119894119895

0 sdot sdot sdot 0

sdot sdot sdot

119860119873minus2

119894119894

119860119894119895

119860119873minus3

119894119894

119860119894119895

sdot sdot sdot 119860119894119895

0

]]]]]]]]]]

]

(17)

where 119864119894119894

119891119894119894

120573119894119894

119864119894119895

119891119894119895

120573119894119895

and 119892119894119895

are the new matricesobtained from 119860

119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

after iterationCombining the models in (15) gives the following system

of equations

Λ119909 = 120576 + 120583119909 (119896) + 120601119889 (119896) (18)

with

Λ =

[[[[[

[

119868 minus11989212

minus11989213

minus11989214

minus11989221

119868 minus11989223

minus11989224

minus11989231

minus11989232

119868 minus11989234

minus11989241

minus11989242

minus11989243

119868

]]]]]

]

120576 =

[[[[[[

[

11986411

11986412

11986413

11986414

11986421

11986422

11986423

11986424

11986431

11986432

11986433

11986434

11986441

11986442

11986443

11986444

]]]]]]

]

120583 =

[[[[[[

[

11989111

11989112

11989113

11989114

11989121

11989122

11989123

11989124

11989131

11989132

11989133

11989134

11989141

11989142

11989143

11989144

]]]]]]

]

120601 =

[[[[[[

[

12057311

12057312

12057313

12057314

12057321

12057322

12057323

12057324

12057331

12057332

12057333

12057334

12057341

12057342

12057343

12057344

]]]]]]

]

119909 = [1199091

1199092

1199093

1199094]119879

= [1199061

1199062

1199063

1199064]119879

(19)

Since matrix Λ is invertible we can write it as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(20)

in which

119864119894119895

= Λminus1

120576

119891119894119895

= Λminus1

120583

120573119894119895

= Λminus1

120601

(21)

To do so we eliminate the unknownmatrix 119909119895

because wehave knowledge of 119909

119895

(119896) since it is just a vector at time 119896In the distributed MPC algorithm for subsystem 119894 the

control signal 119880119894

is designed at each time interval 119896 ge 0 Bysolving the following optimization problem denoted by 119869

119894

itis usually defined as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ (120574119894

+ Γ119894

+sum

119894 =119895

119867119894119895

119906119895

)

119879

119906119894

(22)

in which

Q119894

= diag119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119876119894

120596119894

119876119894

)

R119894

= diag119873119888

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119877119894

120596119894

119877119894

)

Φ119894

= R119894

+ 119864119879

119894119894

Q119894

119864119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

120574119894

= 119864119879

119894119894

Q119894

119892119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119892119895119894

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Control Science and Engineering

The state disturbance and output vectors for area 119894 aredefined by

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ120593120576

Δ120596119903

Δ120596119892

Δ120579]119879

(119894 = 1)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119883119892ℎ119894]119879

(119894 = 2 3)

119909119894

= [Δ119891119894

Δ119875tie119894 Δ119875119892119894 Δ119883119892119894

Δ119875119903119894

(119905)]119879

(119894 = 4)

119889119894

= Δ119875119889119894

(119894 = 1 2 3 4)

1199061

= [Δ1198751198881

Δ120579ref Δ119879119892]119879

119910119894

= ACE119894

= [119870119861119894

Δ119891119894

+ Δ119875tie119894] (119894 = 1 2 3 4)

(6)

The state control and disturbance matrices for area 1 areas follows

11986011

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

minus1

1198791198751

minus1198701198751

1198791198751

1198701198751

1198791198751

0 0 0 0 0

sum

119895

119870119904119894119895

0 0 0 0 0 0 0

0 0 minus1

1198791198791

01

1198791198791

0 0 0

1

1198791198661

0 0 minus1

1198791198661

0 0 0 0

0 0 0 0 0 1 minus1

119899gear0

0 0 0 0 minus119870119904

119869119903

1

119869119903

120597119879119903

120597120596119903

10038161003816100381610038161003816100381610038161003816op0

1

119869119903

120597119879119903

120597120579

10038161003816100381610038161003816100381610038161003816op

0 0 0 0

120578gear119870119904

119899gear1198691198920 0 0

0 0 0 0 0 0 0 minus1

120591120579

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

11986111

=

[[[[[[[

[

0 0 01

1198791198661

0 0 0 0

0 0 0 0 0 0 0

119870120573

120591120579

0 0 0 0 0 0 minus1

119869119892

0

]]]]]]]

]

119879

11986511

=[[[

[

minus1198701198751

1198791198751

0 0 0 0 0 0 0

0 0 0 0 01

119869119903

120597119879119903

120597V119898

10038161003816100381610038161003816100381610038161003816op0 0

]]]

]

119879

11986211

= [1198701198871

1 0 0 0 0 0 0]

(7)

However for hydro plants in areas 2 and 3 they are asfollows

11986022

= 11986033

=

[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

2120572 0 minus2

119879119882119894

2120581 2120573

minus120572 0 0 minus1

1198792119894

minus120573

minus1

1198791119894

119877119894

0 0 0 minus1

1198791119894

]]]]]]]]]]]]]]]

]

11986122

= 11986133

= [0 0 minus2119877119894

120572 119877119894

1205721

1198791119894

]

119879

11986222

= 11986233

= [119870119861119894

1 0 0 0]

11986522

= 11986533

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(8)

where 120572 = 119879119877119894

1198791119894

1198792119894

119877119894

120573 = (119879119877119894

minus1198791119894

)1198791119894

1198792119894

and 120581 = (1198792119894

+

119879119882119894

)1198792119894

119879119882119894

Journal of Control Science and Engineering 7

However for thermal power plants in area 4 they are asfollows

11986044

=

[[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

0 0 minus1

119879119879119894

01

119879119879119894

minus1

119879119866119894

119877119894

0 0 minus1

119879119866119894

0

minus119870119903119894

119879119866119894

119877119894

0 01

119879119903119894

minus119870119903119894

119879119866119894

minus1

119879119903119894

]]]]]]]]]]]]]]]]

]

11986144

= [0 0 01

119879119866119894

0]

119879

11986244

= [119870119861119894

1 0 0 0]

11986544

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(9)

The interactionmatrices between the four control areas are asfollows

119860119894119895

=

[[[[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]]]]

]

(119894 = 1 119895 = 2 3 4)

119860119894119895

=

[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]

]

(119894 = 1 119895 = 2 3 119894 = 119895)

119860119894119895

=

[[[[[

[

0 0 0 0 0

minus119870119878119894119895

0 0 0 0

0 0 0 0 0

0 0 0 0 0

]]]]]

]

(119894 = 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 08times4

119865119894119895

= 08times2

(119894 = 1 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 05times1

119865119894119895

= 05times1

(119894 = 119895 = 2 3 4 119894 = 119895)

(10)

TheGRCs for the thermal plants are |Δ119892119894

| le 00017 puMWs and the hydro units are |Δ

119892119894

| le 0045 puMWs In addi-tion the load disturbance is constrained to |Δ

119889119894| le 03

3 Distributed Model Predictive Controller

31 Distributed Model Predictive Controller The block dia-gram of the DMPC scheme for a four-area interconnectedpower system is illustrated in Figure 7 Though there existslarge amount of variables in the interconnected powersystem the 30 state variables expressed in (1a) (1b) (1c)and (1d) concerning the frequency the generator outputpower the governor valve (servomotor) position the tie-lineactive power the wind power and the 4 load disturbanceΔ119875

119889119894

are crucial to LFC problem They can be measured orestimated directly by the local controller The DMPC in eacharea exchange control information through the power linecommunication which is a sole networking technology withhigh reliability that can provide high speed communicationto power grids applications [22]

Distributed MPC The partitioned discrete-time model forcontrol area 119894 of the continuous-time four-area intercon-nected power system ((1a) (1b) (1c) and (1d)) can beexpressed as follows

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

+sum

119894 =119895

(119860119894119895

119909119895

(119896) + 119861119894119895

119906119895

(119896) + 119865119894119895

119889119895

(119896))

119910119894

(119896) = 119862119894119894

119909119894

(119896)

(11)

where 119860119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

represent the discretenewmatrices obtained from original matrices in (4) based onthe Zero-Order Hold (ZOH) method

Assume that the state variables 119909119894

(119896) and the disturbance119863119894

can be measured or estimated directly by the controllerin area 119894 at sampling time 119896 Optimizations and exchange ofvariables are termed iterate The iteration number is denotedby 119901

For DMPC the optimal state-input trajectory (119909119894

119906119894

) foreach area 119894 119894 = 1 2 3 4 at iterate 119901 is obtained as the solutionto the optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (12)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896) + 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)] (13)

8 Journal of Control Science and Engineering

MPC 1 MPC 2

MPC 3MPC 4

Communication network

Thermal plantwind turbines

Hydro power plant

Thermal power plant

Hydro power plant

Figure 7 Block diagram of DMPC for power system with wind turbines

Subject to 10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00017 119894 = 1 4 (14a)

10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00045 119894 = 2 3 (14b)

10038171003817100381710038171199091198944 (119896 + 119899 | 119896)10038171003817100381710038172 le 120590119894 119894 = 1 2 3 4 (14c)

For notational convenience we drop the 119896 dependence of119909119894

(119896) 119906119894

(119896) 119894 = 1 2 3 4 It is shown in [20] that each 119909119894

canbe expressed as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119892119894119895

119909119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(15)

with

119909119894

= [119909119894

(119896 + 1 | 119896)119879

119909119894

(119896 + 2 | 119896)119879

sdot sdot sdot 119909119894

(119896 + 119873119901

| 119896)119879

]

119879

119906119894

= [119906119894

(119896 | 119896)119879

119906119894

(119896 + 1 | 119896)119879

sdot sdot sdot 119906119894

(119896 + 119873119888

minus 1 | 119896)119879

]119879

(16)

Let119873119888

denote the control horizon and let119873119901

denote thepredictive horizon 119909

119894

is no more a vector but a matrix after

iteration obtained from original equation (4)Thematrices in(15) have detailed expressions as follows

119864119894119894

=

[[[[[[[

[

119861119894119894

0 sdot sdot sdot 0

119860119894119894

119861119894119894

119861119894119894

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119894

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119864119894119895

=

[[[[[[[

[

119861119894119895

0 sdot sdot sdot 0

119860119894119894

119861119894119895

119861119894119895

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119895

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119891119894119894

=

[[[[[[[

[

119860119894119894

119860119894119894

119860119894119894

119860119873minus1

119894119894

119860119894119894

]]]]]]]

]

Journal of Control Science and Engineering 9

119891119894119895

=

[[[[[[[

[

119860119894119895

119860119894119894

119860119894119895

119860119873minus1

119894119894

119860119894119895

]]]]]]]

]

120573119894119894

=

[[[[[[[

[

119865119894119894

119860119894119894

119865119894119894

119860119873minus1

119894119894

119865119894119894

]]]]]]]

]

120573119894119895

=

[[[[[[[

[

119865119894119895

119860119894119894

119865119894119895

119860119873minus1

119894119894

119865119894119895

]]]]]]]

]

119892119894119895

=

[[[[[[[[[[

[

0 0 0 sdot sdot sdot 0

119860119894119895

0 0 sdot sdot sdot 0

119860119894119894

119860119894119895

119860119894119895

0 sdot sdot sdot 0

sdot sdot sdot

119860119873minus2

119894119894

119860119894119895

119860119873minus3

119894119894

119860119894119895

sdot sdot sdot 119860119894119895

0

]]]]]]]]]]

]

(17)

where 119864119894119894

119891119894119894

120573119894119894

119864119894119895

119891119894119895

120573119894119895

and 119892119894119895

are the new matricesobtained from 119860

119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

after iterationCombining the models in (15) gives the following system

of equations

Λ119909 = 120576 + 120583119909 (119896) + 120601119889 (119896) (18)

with

Λ =

[[[[[

[

119868 minus11989212

minus11989213

minus11989214

minus11989221

119868 minus11989223

minus11989224

minus11989231

minus11989232

119868 minus11989234

minus11989241

minus11989242

minus11989243

119868

]]]]]

]

120576 =

[[[[[[

[

11986411

11986412

11986413

11986414

11986421

11986422

11986423

11986424

11986431

11986432

11986433

11986434

11986441

11986442

11986443

11986444

]]]]]]

]

120583 =

[[[[[[

[

11989111

11989112

11989113

11989114

11989121

11989122

11989123

11989124

11989131

11989132

11989133

11989134

11989141

11989142

11989143

11989144

]]]]]]

]

120601 =

[[[[[[

[

12057311

12057312

12057313

12057314

12057321

12057322

12057323

12057324

12057331

12057332

12057333

12057334

12057341

12057342

12057343

12057344

]]]]]]

]

119909 = [1199091

1199092

1199093

1199094]119879

= [1199061

1199062

1199063

1199064]119879

(19)

Since matrix Λ is invertible we can write it as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(20)

in which

119864119894119895

= Λminus1

120576

119891119894119895

= Λminus1

120583

120573119894119895

= Λminus1

120601

(21)

To do so we eliminate the unknownmatrix 119909119895

because wehave knowledge of 119909

119895

(119896) since it is just a vector at time 119896In the distributed MPC algorithm for subsystem 119894 the

control signal 119880119894

is designed at each time interval 119896 ge 0 Bysolving the following optimization problem denoted by 119869

119894

itis usually defined as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ (120574119894

+ Γ119894

+sum

119894 =119895

119867119894119895

119906119895

)

119879

119906119894

(22)

in which

Q119894

= diag119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119876119894

120596119894

119876119894

)

R119894

= diag119873119888

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119877119894

120596119894

119877119894

)

Φ119894

= R119894

+ 119864119879

119894119894

Q119894

119864119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

120574119894

= 119864119879

119894119894

Q119894

119892119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119892119895119894

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 7

However for thermal power plants in area 4 they are asfollows

11986044

=

[[[[[[[[[[[[[[[[

[

minus1

119879119875119894

minus119870119875119894

119879119875119894

119870119875119894

119879119875119894

0 0

sum

119895

119870119878119894119895

0 0 0 0

0 0 minus1

119879119879119894

01

119879119879119894

minus1

119879119866119894

119877119894

0 0 minus1

119879119866119894

0

minus119870119903119894

119879119866119894

119877119894

0 01

119879119903119894

minus119870119903119894

119879119866119894

minus1

119879119903119894

]]]]]]]]]]]]]]]]

]

11986144

= [0 0 01

119879119866119894

0]

119879

11986244

= [119870119861119894

1 0 0 0]

11986544

= [minus

119870119901119894

119879119901119894

0 0 0 0]

119879

(9)

The interactionmatrices between the four control areas are asfollows

119860119894119895

=

[[[[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]]]]

]

(119894 = 1 119895 = 2 3 4)

119860119894119895

=

[[[[[

[

0 0 0 0 0 0 0 0

minus119870119878119894119895

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]]]]]

]

(119894 = 1 119895 = 2 3 119894 = 119895)

119860119894119895

=

[[[[[

[

0 0 0 0 0

minus119870119878119894119895

0 0 0 0

0 0 0 0 0

0 0 0 0 0

]]]]]

]

(119894 = 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 08times4

119865119894119895

= 08times2

(119894 = 1 119895 = 2 3 4 119894 = 119895)

119861119894119895

= 05times1

119865119894119895

= 05times1

(119894 = 119895 = 2 3 4 119894 = 119895)

(10)

TheGRCs for the thermal plants are |Δ119892119894

| le 00017 puMWs and the hydro units are |Δ

119892119894

| le 0045 puMWs In addi-tion the load disturbance is constrained to |Δ

119889119894| le 03

3 Distributed Model Predictive Controller

31 Distributed Model Predictive Controller The block dia-gram of the DMPC scheme for a four-area interconnectedpower system is illustrated in Figure 7 Though there existslarge amount of variables in the interconnected powersystem the 30 state variables expressed in (1a) (1b) (1c)and (1d) concerning the frequency the generator outputpower the governor valve (servomotor) position the tie-lineactive power the wind power and the 4 load disturbanceΔ119875

119889119894

are crucial to LFC problem They can be measured orestimated directly by the local controller The DMPC in eacharea exchange control information through the power linecommunication which is a sole networking technology withhigh reliability that can provide high speed communicationto power grids applications [22]

Distributed MPC The partitioned discrete-time model forcontrol area 119894 of the continuous-time four-area intercon-nected power system ((1a) (1b) (1c) and (1d)) can beexpressed as follows

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

+sum

119894 =119895

(119860119894119895

119909119895

(119896) + 119861119894119895

119906119895

(119896) + 119865119894119895

119889119895

(119896))

119910119894

(119896) = 119862119894119894

119909119894

(119896)

(11)

where 119860119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

represent the discretenewmatrices obtained from original matrices in (4) based onthe Zero-Order Hold (ZOH) method

Assume that the state variables 119909119894

(119896) and the disturbance119863119894

can be measured or estimated directly by the controllerin area 119894 at sampling time 119896 Optimizations and exchange ofvariables are termed iterate The iteration number is denotedby 119901

For DMPC the optimal state-input trajectory (119909119894

119906119894

) foreach area 119894 119894 = 1 2 3 4 at iterate 119901 is obtained as the solutionto the optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (12)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896) + 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)] (13)

8 Journal of Control Science and Engineering

MPC 1 MPC 2

MPC 3MPC 4

Communication network

Thermal plantwind turbines

Hydro power plant

Thermal power plant

Hydro power plant

Figure 7 Block diagram of DMPC for power system with wind turbines

Subject to 10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00017 119894 = 1 4 (14a)

10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00045 119894 = 2 3 (14b)

10038171003817100381710038171199091198944 (119896 + 119899 | 119896)10038171003817100381710038172 le 120590119894 119894 = 1 2 3 4 (14c)

For notational convenience we drop the 119896 dependence of119909119894

(119896) 119906119894

(119896) 119894 = 1 2 3 4 It is shown in [20] that each 119909119894

canbe expressed as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119892119894119895

119909119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(15)

with

119909119894

= [119909119894

(119896 + 1 | 119896)119879

119909119894

(119896 + 2 | 119896)119879

sdot sdot sdot 119909119894

(119896 + 119873119901

| 119896)119879

]

119879

119906119894

= [119906119894

(119896 | 119896)119879

119906119894

(119896 + 1 | 119896)119879

sdot sdot sdot 119906119894

(119896 + 119873119888

minus 1 | 119896)119879

]119879

(16)

Let119873119888

denote the control horizon and let119873119901

denote thepredictive horizon 119909

119894

is no more a vector but a matrix after

iteration obtained from original equation (4)Thematrices in(15) have detailed expressions as follows

119864119894119894

=

[[[[[[[

[

119861119894119894

0 sdot sdot sdot 0

119860119894119894

119861119894119894

119861119894119894

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119894

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119864119894119895

=

[[[[[[[

[

119861119894119895

0 sdot sdot sdot 0

119860119894119894

119861119894119895

119861119894119895

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119895

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119891119894119894

=

[[[[[[[

[

119860119894119894

119860119894119894

119860119894119894

119860119873minus1

119894119894

119860119894119894

]]]]]]]

]

Journal of Control Science and Engineering 9

119891119894119895

=

[[[[[[[

[

119860119894119895

119860119894119894

119860119894119895

119860119873minus1

119894119894

119860119894119895

]]]]]]]

]

120573119894119894

=

[[[[[[[

[

119865119894119894

119860119894119894

119865119894119894

119860119873minus1

119894119894

119865119894119894

]]]]]]]

]

120573119894119895

=

[[[[[[[

[

119865119894119895

119860119894119894

119865119894119895

119860119873minus1

119894119894

119865119894119895

]]]]]]]

]

119892119894119895

=

[[[[[[[[[[

[

0 0 0 sdot sdot sdot 0

119860119894119895

0 0 sdot sdot sdot 0

119860119894119894

119860119894119895

119860119894119895

0 sdot sdot sdot 0

sdot sdot sdot

119860119873minus2

119894119894

119860119894119895

119860119873minus3

119894119894

119860119894119895

sdot sdot sdot 119860119894119895

0

]]]]]]]]]]

]

(17)

where 119864119894119894

119891119894119894

120573119894119894

119864119894119895

119891119894119895

120573119894119895

and 119892119894119895

are the new matricesobtained from 119860

119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

after iterationCombining the models in (15) gives the following system

of equations

Λ119909 = 120576 + 120583119909 (119896) + 120601119889 (119896) (18)

with

Λ =

[[[[[

[

119868 minus11989212

minus11989213

minus11989214

minus11989221

119868 minus11989223

minus11989224

minus11989231

minus11989232

119868 minus11989234

minus11989241

minus11989242

minus11989243

119868

]]]]]

]

120576 =

[[[[[[

[

11986411

11986412

11986413

11986414

11986421

11986422

11986423

11986424

11986431

11986432

11986433

11986434

11986441

11986442

11986443

11986444

]]]]]]

]

120583 =

[[[[[[

[

11989111

11989112

11989113

11989114

11989121

11989122

11989123

11989124

11989131

11989132

11989133

11989134

11989141

11989142

11989143

11989144

]]]]]]

]

120601 =

[[[[[[

[

12057311

12057312

12057313

12057314

12057321

12057322

12057323

12057324

12057331

12057332

12057333

12057334

12057341

12057342

12057343

12057344

]]]]]]

]

119909 = [1199091

1199092

1199093

1199094]119879

= [1199061

1199062

1199063

1199064]119879

(19)

Since matrix Λ is invertible we can write it as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(20)

in which

119864119894119895

= Λminus1

120576

119891119894119895

= Λminus1

120583

120573119894119895

= Λminus1

120601

(21)

To do so we eliminate the unknownmatrix 119909119895

because wehave knowledge of 119909

119895

(119896) since it is just a vector at time 119896In the distributed MPC algorithm for subsystem 119894 the

control signal 119880119894

is designed at each time interval 119896 ge 0 Bysolving the following optimization problem denoted by 119869

119894

itis usually defined as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ (120574119894

+ Γ119894

+sum

119894 =119895

119867119894119895

119906119895

)

119879

119906119894

(22)

in which

Q119894

= diag119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119876119894

120596119894

119876119894

)

R119894

= diag119873119888

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119877119894

120596119894

119877119894

)

Φ119894

= R119894

+ 119864119879

119894119894

Q119894

119864119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

120574119894

= 119864119879

119894119894

Q119894

119892119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119892119895119894

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

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DistributedSensor Networks

International Journal of

8 Journal of Control Science and Engineering

MPC 1 MPC 2

MPC 3MPC 4

Communication network

Thermal plantwind turbines

Hydro power plant

Thermal power plant

Hydro power plant

Figure 7 Block diagram of DMPC for power system with wind turbines

Subject to 10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00017 119894 = 1 4 (14a)

10038171003817100381710038171199091198943 (119896 + 119899 | 119896)10038171003817100381710038172le 00045 119894 = 2 3 (14b)

10038171003817100381710038171199091198944 (119896 + 119899 | 119896)10038171003817100381710038172 le 120590119894 119894 = 1 2 3 4 (14c)

For notational convenience we drop the 119896 dependence of119909119894

(119896) 119906119894

(119896) 119894 = 1 2 3 4 It is shown in [20] that each 119909119894

canbe expressed as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119892119894119895

119909119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(15)

with

119909119894

= [119909119894

(119896 + 1 | 119896)119879

119909119894

(119896 + 2 | 119896)119879

sdot sdot sdot 119909119894

(119896 + 119873119901

| 119896)119879

]

119879

119906119894

= [119906119894

(119896 | 119896)119879

119906119894

(119896 + 1 | 119896)119879

sdot sdot sdot 119906119894

(119896 + 119873119888

minus 1 | 119896)119879

]119879

(16)

Let119873119888

denote the control horizon and let119873119901

denote thepredictive horizon 119909

119894

is no more a vector but a matrix after

iteration obtained from original equation (4)Thematrices in(15) have detailed expressions as follows

119864119894119894

=

[[[[[[[

[

119861119894119894

0 sdot sdot sdot 0

119860119894119894

119861119894119894

119861119894119894

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119894

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119864119894119895

=

[[[[[[[

[

119861119894119895

0 sdot sdot sdot 0

119860119894119894

119861119894119895

119861119894119895

sdot sdot sdot 0

119860119873minus1

119894119894

119861119894119895

119860119873minus2

119894119894

sdot sdot sdot 0

]]]]]]]

]

119891119894119894

=

[[[[[[[

[

119860119894119894

119860119894119894

119860119894119894

119860119873minus1

119894119894

119860119894119894

]]]]]]]

]

Journal of Control Science and Engineering 9

119891119894119895

=

[[[[[[[

[

119860119894119895

119860119894119894

119860119894119895

119860119873minus1

119894119894

119860119894119895

]]]]]]]

]

120573119894119894

=

[[[[[[[

[

119865119894119894

119860119894119894

119865119894119894

119860119873minus1

119894119894

119865119894119894

]]]]]]]

]

120573119894119895

=

[[[[[[[

[

119865119894119895

119860119894119894

119865119894119895

119860119873minus1

119894119894

119865119894119895

]]]]]]]

]

119892119894119895

=

[[[[[[[[[[

[

0 0 0 sdot sdot sdot 0

119860119894119895

0 0 sdot sdot sdot 0

119860119894119894

119860119894119895

119860119894119895

0 sdot sdot sdot 0

sdot sdot sdot

119860119873minus2

119894119894

119860119894119895

119860119873minus3

119894119894

119860119894119895

sdot sdot sdot 119860119894119895

0

]]]]]]]]]]

]

(17)

where 119864119894119894

119891119894119894

120573119894119894

119864119894119895

119891119894119895

120573119894119895

and 119892119894119895

are the new matricesobtained from 119860

119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

after iterationCombining the models in (15) gives the following system

of equations

Λ119909 = 120576 + 120583119909 (119896) + 120601119889 (119896) (18)

with

Λ =

[[[[[

[

119868 minus11989212

minus11989213

minus11989214

minus11989221

119868 minus11989223

minus11989224

minus11989231

minus11989232

119868 minus11989234

minus11989241

minus11989242

minus11989243

119868

]]]]]

]

120576 =

[[[[[[

[

11986411

11986412

11986413

11986414

11986421

11986422

11986423

11986424

11986431

11986432

11986433

11986434

11986441

11986442

11986443

11986444

]]]]]]

]

120583 =

[[[[[[

[

11989111

11989112

11989113

11989114

11989121

11989122

11989123

11989124

11989131

11989132

11989133

11989134

11989141

11989142

11989143

11989144

]]]]]]

]

120601 =

[[[[[[

[

12057311

12057312

12057313

12057314

12057321

12057322

12057323

12057324

12057331

12057332

12057333

12057334

12057341

12057342

12057343

12057344

]]]]]]

]

119909 = [1199091

1199092

1199093

1199094]119879

= [1199061

1199062

1199063

1199064]119879

(19)

Since matrix Λ is invertible we can write it as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(20)

in which

119864119894119895

= Λminus1

120576

119891119894119895

= Λminus1

120583

120573119894119895

= Λminus1

120601

(21)

To do so we eliminate the unknownmatrix 119909119895

because wehave knowledge of 119909

119895

(119896) since it is just a vector at time 119896In the distributed MPC algorithm for subsystem 119894 the

control signal 119880119894

is designed at each time interval 119896 ge 0 Bysolving the following optimization problem denoted by 119869

119894

itis usually defined as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ (120574119894

+ Γ119894

+sum

119894 =119895

119867119894119895

119906119895

)

119879

119906119894

(22)

in which

Q119894

= diag119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119876119894

120596119894

119876119894

)

R119894

= diag119873119888

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119877119894

120596119894

119877119894

)

Φ119894

= R119894

+ 119864119879

119894119894

Q119894

119864119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

120574119894

= 119864119879

119894119894

Q119894

119892119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119892119895119894

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 9

119891119894119895

=

[[[[[[[

[

119860119894119895

119860119894119894

119860119894119895

119860119873minus1

119894119894

119860119894119895

]]]]]]]

]

120573119894119894

=

[[[[[[[

[

119865119894119894

119860119894119894

119865119894119894

119860119873minus1

119894119894

119865119894119894

]]]]]]]

]

120573119894119895

=

[[[[[[[

[

119865119894119895

119860119894119894

119865119894119895

119860119873minus1

119894119894

119865119894119895

]]]]]]]

]

119892119894119895

=

[[[[[[[[[[

[

0 0 0 sdot sdot sdot 0

119860119894119895

0 0 sdot sdot sdot 0

119860119894119894

119860119894119895

119860119894119895

0 sdot sdot sdot 0

sdot sdot sdot

119860119873minus2

119894119894

119860119894119895

119860119873minus3

119894119894

119860119894119895

sdot sdot sdot 119860119894119895

0

]]]]]]]]]]

]

(17)

where 119864119894119894

119891119894119894

120573119894119894

119864119894119895

119891119894119895

120573119894119895

and 119892119894119895

are the new matricesobtained from 119860

119894119894

119861119894119894

119862119894119894

119865119894119894

119860119894119895

119861119894119895

and 119865119894119895

after iterationCombining the models in (15) gives the following system

of equations

Λ119909 = 120576 + 120583119909 (119896) + 120601119889 (119896) (18)

with

Λ =

[[[[[

[

119868 minus11989212

minus11989213

minus11989214

minus11989221

119868 minus11989223

minus11989224

minus11989231

minus11989232

119868 minus11989234

minus11989241

minus11989242

minus11989243

119868

]]]]]

]

120576 =

[[[[[[

[

11986411

11986412

11986413

11986414

11986421

11986422

11986423

11986424

11986431

11986432

11986433

11986434

11986441

11986442

11986443

11986444

]]]]]]

]

120583 =

[[[[[[

[

11989111

11989112

11989113

11989114

11989121

11989122

11989123

11989124

11989131

11989132

11989133

11989134

11989141

11989142

11989143

11989144

]]]]]]

]

120601 =

[[[[[[

[

12057311

12057312

12057313

12057314

12057321

12057322

12057323

12057324

12057331

12057332

12057333

12057334

12057341

12057342

12057343

12057344

]]]]]]

]

119909 = [1199091

1199092

1199093

1199094]119879

= [1199061

1199062

1199063

1199064]119879

(19)

Since matrix Λ is invertible we can write it as

119909119894

= 119864119894119894

119906119894

+ 119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896)

+sum

119894 =119895

(119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896))

(20)

in which

119864119894119895

= Λminus1

120576

119891119894119895

= Λminus1

120583

120573119894119895

= Λminus1

120601

(21)

To do so we eliminate the unknownmatrix 119909119895

because wehave knowledge of 119909

119895

(119896) since it is just a vector at time 119896In the distributed MPC algorithm for subsystem 119894 the

control signal 119880119894

is designed at each time interval 119896 ge 0 Bysolving the following optimization problem denoted by 119869

119894

itis usually defined as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ (120574119894

+ Γ119894

+sum

119894 =119895

119867119894119895

119906119895

)

119879

119906119894

(22)

in which

Q119894

= diag119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119876119894

120596119894

119876119894

)

R119894

= diag119873119888

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞(120596

119894

119877119894

120596119894

119877119894

)

Φ119894

= R119894

+ 119864119879

119894119894

Q119894

119864119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

120574119894

= 119864119879

119894119894

Q119894

119892119894119894

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119892119895119894

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

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International Journal of

10 Journal of Control Science and Engineering

119892119894119894

= 119891119894119894

119909119894

(119896) +

4

sum

119895=1

119891119894119895

119909119895

(119896)

Γ119894

= 119864119879

119894119894

Q119894

120588119894

+

4

sum

119895=1

119864119879

119895119894

Q119895

120588119895

120588119894

= 120573119894119894

119889119894

(119896) +

4

sum

119895=1

120573119894119895

119889119895

(119896)

119867119894119895

= 119864119879

119894119894

Q119894

119864119894119895

+

4

sum

119895=1

119895 =119894

119864119879

119895119894

Q119895

119864119895119894

(23)

At time interval 119896 (22) is implemented based on thefuture states and manipulated variables The first input inthe optimal sequence is injected into the processes and theprocedure is repeated at subsequent time intervals

119876119894

ge 0 119877119894

ge 0 are symmetric weighting matrices and120596119894

gt 0sum4

119894=1

120596119894

= 1Define 120578

119894

= 120574119894

+ Γ119894

+ sum119895 =119894

119867119894119895

119906119895

Then (22) is rewritten as

119869119894

= min119906119894

1

2119906119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

(24)

32 Constraint Handling The two crucial nonlinearities forexample the GRCs and the valve position limits of thegovernor have been considered as the state constraints in thedesigned DMPC as shown in Figures 8 and 9

In power system the GRC can be expressed asΔ

119892

(119896)min le Δ119892(119896) le Δ119892(119896)max and then the constraintson Δ119875

119892

can be expressed as follows

119879 (Δ119892

(119896))min + Δ119875119892 (119896 minus 1) le Δ119875119892 (119896)

le 119879 (Δ119892

(119896))max + Δ119875119892 (119896 minus 1) (25)

Δ119875119892

= [Δ119875119892

(119896 + 1 | 119896) Δ119875119892

(119896 + 2 | 119896) sdot sdot sdot Δ119875119892

(119896 + 119873119901

| 119896)]119879

(26)

Since Δ119875119892119894

= 1198831198943

the state form can be expressed as

Δ119875119892

= 119878119894

119909119894

(27)

where 119878119894

= diag(119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞120596119894

119878119894119894

120596119894

119878119894119894

)When 119894 = 1 4 119878

119894119894

= [0 0 1 0 0] and when 119894 = 2 3119878119894119894

= [0 0 1 0 0] with (25) and (27) the constraints onΔ119875

119892

(119896) are expressed as119873119894

le 119878119894

119909119894

le 119872119894

Define

119873119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119873119894

119873119894

sdot sdot sdot 119873119894

]]]

]

119879

119872119894

=[[[

[

119873119901

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞119872

119894

119872119894

sdot sdot sdot 119872119894

]]]

]

119879

(28)

where119873119894

and119872119894

are obtained from (15)Consider the constraints on Δ119875

119892

(119896)

[

119878119894

119864119894119894

minus119878119894

119864119894119894

] 119906119894

le

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(29)

Define

Ψ119894

= [

119878119894

119864119894119894

minus119878119894

119864119894119894

]

Π119894

=

[[[[[[[[

[

119872119894

minus 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119895 =119894

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

minus119873119894

+ 119878119894

119891119894119894

119909119894

(119896) + 120573119894119894

119889119894

(119896) +sum

119894 =119895

[119864119894119895

119906119895

+ 119891119894119895

119909119895

(119896) + 120573119894119895

119889119895

(119896)]

]]]]]]]]

]

(30)

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 11

1

RiΔfi

ΔPgi1

sui

minus

minus+ +

1

1 + sTGi

ΔXgi(s) 1

TTi

GRC

Figure 8 Thermal power plant with GRC

1

Ri

Δfi

ΔPgiui

minus

+

1

1 + sT1i

ΔXghi(s) 1 + sTRi1 + sT2i

ΔXgi(s) 1 minus sTWi

1 + 05sTWi

GRC

Figure 9 Hydro power plant with GRC

Then distributedMPC algorithm (24) for multiple-inter-connected system can be transformed into the following opti-mization problem with GRC constraints

119869119894

=min119906119894

1

2119879

119894

Φ119894

119906119879

119894

+ 120578119879

119894

119906119894

Subject to Ψ119894

119906119894

le Π119894

(31)

33 The DMPC Algorithm

Step 1 (initialization) The constant matrices 119877119894

119877119895

and 119876119894

119876119895

at control interval 119896 = 0 are given Choose the specifiederror tolerance 120576

119894

Set iteration 119901 = 0

Step 2 (communication) The controller in each subsystem 119894

exchanges its previous predictions 119909119894

(119896) 119909119895

(119896) set 1199060119894

(119896) and1199060

119895

(119896) at initial instant

Step 3 (optimization and iteration)

While 119901 lt 119901max

119906lowast(119901)

119894

is solved by the optimal problem (31)

If 119906(119901)119894

minus 119906(119901minus1)

119894

le 120576119894

forall119894 isin 1 2 3 4

BreakEnd if

Exchange the solutions 119906119901119894

and 119906119901119895

and set 119901 = 119901 + 1

If 120576119894

= 0 forall119894 isin 1 2 3 4

BreakEnd if

End while

Step 4 (assignment and prediction) Send out 119906119894

(119896) = 119906119894

(119896)Otherwise 119906

119894

(119896) = 119906119894

(119896 minus 1) Predict the future states

Step 5 (implementation) Set 119896 = 119896 + 1 and repeat Step 1

4 Simulation Results

In this section the four-area power system stability is ana-lyzed and the performances of the proposed DMPC havebeen tested in case of wind turbines participation at nominalparameters The simulation of the proposed DMPC schemeis also verified by two cases The performance and theimplementation of the proposed DMPC are compared withother two types of typical LFC scheme

As comparison we design the centralized MPC anddecentralized MPC controller for four-area interconnectedpower system respectively The four-area interconnectedpower system can be described as

119909 (119896 + 1) = 119860119909 (119896) + 119861119906 (119896) + 119865119889 (119896)

119910 (119896 + 1) = 119862119909 (119896)

(32)

where

119860 =

[[[[[

[

11986011

11986012

11986013

11986014

11986021

11986022

11986023

11986024

11986031

11986032

11986033

11986034

11986041

11986042

11986043

11986044

]]]]]

]

119861 =

[[[[[

[

11986111

11986112

11986113

11986114

11986121

11986122

11986123

11986124

11986131

11986132

11986133

11986134

11986141

11986142

11986143

11986144

]]]]]

]

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Journal of Control Science and Engineering

119862 =

[[[[[

[

11986211

0 0 0

0 11986222

0 0

0 0 11986233

0

0 0 0 11986244

]]]]]

]

119865 =

[[[[[

[

11986511

0 0 0

0 11986522

0 0

0 0 11986533

0

0 0 0 11986544

]]]]]

]

119909 = [119909119879

1

119909119879

2

119909119879

3

119909119879

4

]119879

119906 = [119906119879

1

119906119879

2

119906119879

3

119906119879

4

]119879

119910 = [119910119879

1

119910119879

2

119910119879

3

119910119879

4

]119879

119889 = [119889119879

1

119889119879

2

119889119879

3

119889119879

4

]119879

(33)with constraints (12) (13) (14a) (14b) and (14c) for each con-trol area In centralizedMPC framework theMPC for overallsystem (32) solves the following optimization problem

min119906(119896+119899|119896)

119869 (119896) (34)

119869 (119896) =

119873

sum

119899=0

[119909119879

(119896 + 119899 | 119896)119876119909 (119896 + 119899 | 119896)

+ 119906119879

(119896 + 119899 | 119896) 119877119906 (119896 + 119899 | 119896)]

(35)

subject to (14a) (14b) and (14c)Theweightingmatrices119876 and119877 in objective function (35)

are chosen as 119877 = diag(1 1 1 1) and

119876 = diag(1000 0 0 1000 1000 0 0 1000 1000

0 0 1000 1000 0 0 1000) (36)

In the decentralized modeling framework it is assumedthat the interaction between the control areas is negligibleSubsequently the decentralized model for each control areais

119909119894

(119896 + 1) = 119860119894119894

119909119894

(119896) + 119861119894119894

119906119894

(119896) + 119865119894119894

119889119894

(119896)

119910119894

(119896 + 1) = 119862119894119894

119909119894

(119896)

(37)

with the system matrices and constraints (12) (13) (14a)(14b) and (14c) for each control area denoted as in Section 2In decentralized MPC framework each control area basedMPC solves the following optimization problem

min119906119894(119896+119899|119896)

119869119894

(119896) (38)

119869119894

(119896) =

119873

sum

119899=0

[119909119879

119894

(119896 + 119899 | 119896)119876119894

119909119894

(119896 + 119899 | 119896)

+ 119906119879

119894

(119896 + 119899 | 119896) 119877119894

119906119894

(119896 + 119899 | 119896)]

(39)

subject to (14a) (14b) and (14c)

The weighting matrices 119876119894

and 119877119894

in objective function(39) are chosen as 119877

1

= 1198772

= 1198773

= 1198774

= 1 and

1198761

= 1198762

= 1198763

= 1198764

= diag (1000 0 0 1000) (40)

Choose the prediction horizon of the centralized MPCdecentralized MPC and RDMPC to be 119873 = 15 choosethe control horizon to be 119873

119888

= 10 and choose the sampletime 119879

119904

= 01 and 120582 = 01 Consider GRC for the ther-mal power plants in area 1 and area 4 to be |Δ119894

119892

| le 119903 =

01 puMWmin = 00017 puMWs and GRC for the hydropower plants in area 2 and area 3 to be |Δ119894

119892

| le 119903 =

27 puMWmin = 0045 puMWs In addition area 1includes an aggregated wind turbine model which consists of30 wind turbine units of 2MW rated VSWTswhile the capac-ity of thermal plant is 600MW The wind turbine param-eters and operating points [23] are indicated as follows

Operating point 80MW wind speed 12ms

119879119892

= 37819Nm 120596119892

= 105 rads 120596119903

= 26869 rads

119870119904

= 7871198906Nmrad 119899gear = 1 287 120578gear = 975

119869119903

= 28675 kgm2 119869119892

= 545432 kgm2

1198773

= 33HzpuMW 1198774

= 3HzpuMW

The parameters for the thermal and hydro plants used in thesimulation are listed as follows

1198701198751

= 120HzpuMW 1198701198752

= 115HzpuMW

1198701198753

= 80HzpuMW 1198701198754

= 75HzpuMW

1198791198751

= 20 s 1198791198752

= 20 s 1198791198753

= 13 s 1198791198754

= 15 s

1198771

= 24HzpuMW 1198772

= 25HzpuMW

1198773

= 33HzpuMW 1198774

= 3HzpuMW

1198701198611

= 0425 puMWHz 1198701198612

= 0409 puMWHz

1198701198613

= 0316 puMWHz 1198701198614

= 0347 puMWHz

1198791198661

= 008 s 1198791198662

= 01 s 1198791198663

= 008 s 1198791198664

= 02 s

1198791198791

= 1198791198794

= 03 s 1198791199031

= 1198791199034

= 10 s 1198791198772

= 06 s

1198791198773

= 0513 s 11987922

= 5 s 11987923

= 10 s 1198791198822

= 1 s 1198791198823

=

2 s

11987011987812

= minus11987011987821

= 0545 puMW

11987011987823

= minus11987011987832

= 0444 puMW

11987011987813

= minus11987011987831

= 0545 puMW

11987011987814

= minus11987011987841

= 05 puMW

11987011987824

= minus11987011987842

= 0545 puMW

11987011987834

= minus11987011987843

= 0545 puMW

Case 1 (response to step load change without wind turbinesparticipation) Wind turbine is present but it does notprovide any power support in the event of grid frequencydeviation An event is simulated in which a system shown in

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 13Δf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 10 Response of frequency deviation to step load disturbance in Case 1 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

Table 2 Cost of the different strategies

Strategy Cost [20]Centralized MPC 010Decentralized MPC 0083Distributed MPC 0078

Figure 1 is subjected to step load disturbances as give in (41)at 119905 = 10 s Consider

Δ1198751198891

= Δ1198751198892

= Δ1198751198893

= Δ1198751198894

= 01 (41)

Figure 10 shows the simulation results of distributedMPC centralized MPC and decentralized MPC withoutwind turbine participation and only conventional integra-tor systems The relative performance of distributed MPCcentralized MPC and decentralized MPC rejecting the loaddisturbance in each area in Figure 10 is denoted by soliddotted and dashed lines respectively It has been noticedthat the closed-loop trajectory of distributed MPC obtainedby algorithm is little fast and almost indistinguishable fromthe closed-loop trajectory of centralized MPC It successfullyimproves the dynamic response of area frequencies comparedwith decentralized MPC

The control costs defined by [20] for different strategiesare listed in Table 2 It is obviously seen that the DMPCcontroller needs nearly as much CPU time as decentralizedMPC controller and significantly less CPU time than cen-tralized MPC controllers The proposed DMPC algorithmhas significant computational advantages when compared tocentralized MPC while achieving the best performance

Case 2 (response to step load change with wind turbinesparticipation) Wind turbine is present and it will provideactive power support in the event of grid frequency deviationAn event is simulated in which a system shown in Figure 1 issubjected to step load disturbances as give in (41) at 119905 = 10 sMean wind speed is assumed to be 17ms in area 1

In Figures 11 and 12 the behavior for the frequency ispresented for Case 2 where the wind turbines are partici-pating in load frequency control The results from top tothe bottom in Figure 11 are the frequency deviations for area1 to area 4 and in Figure 12 are six tie-lines power changeIn simulation it is obvious that both the DMPC and thecentralized MPC converge rapidly and drive the local fre-quency changes and tie-line power deviation to zero Thewind turbines that have participated in the interconnectedpower system do not affect the performance of the powersystem under distributed MPC and centralized MPC whilesatisfying all the physical constraints for example the GRCthe limit of the governors and load step change constraintsHowever with decentralized MPC the rapid convergencecannot be guaranteed in the presence of wind turbines in area1 This confirms the performance advantage of the proposeddistributed model predictive control algorithm

Figure 13 shows the dynamic response of active powerdeviation Δ119875

119890

and rotor speed 120596119892

of wind turbine whileparticipating in the load frequency controlWhen the controlis activated the frequency deviation becomes zero whichconsequently eliminated the additional active power devia-tion Δ119875

119890

and wind turbine is driven to operate again at theoptimal rotor speed 120596

119892

It may be noted here that an increasein power step on top of the converter further reduces the rotorspeed thereby transferring more kinetic power to reduce thefrequency dip As shown in this figure the distributed MPC

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Journal of Control Science and EngineeringΔf1

(Hz)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf2

(Hz)

minus006

minus004

minus002

0

002

5 10 15 20 25 30 35 40 45 500Time (s)

minus006

minus004

minus002

0

002

Δf3

(Hz)

minus006

minus004

minus002

0

002

Δf4

(Hz)

Figure 11 Response of frequency deviation to step load disturbance in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

times10minus3 times10minus3

times10minus3times10minus3

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

2

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

10

5 10 15 20 25 30 35 40 45 500Time (s)

times10minus4

5 10 15 20 25 30 35 40 45 500Time (s)

minus5

0

5

5 10 15 20 25 30 35 40 45 500Time (s)

minus1

minus05

0

05

1

15

minus1

minus05

0

05

1

15

ΔP

tie12

(pu

MW

)ΔP

tie14

(pu

MW

)ΔP

tie24

(pu

MW

)

ΔP

tie13

(pu

MW

)ΔP

tie23

(pu

MW

)ΔP

tie34

(pu

MW

)

Figure 12 Response of tie-line active power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) anddecentralized MPC (dashed line)

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 15

5 10 15 20 25 30 35 40 45 500Time (s)

040506070809

1ΔPe

(pu

MW

)

085

09

095

1

105

5 10 15 20 25 30 35 40 45 500Time (s)

120596g

(pu

)

Figure 13 Wind turbine response of electrical power and rotor speed in Case 2 distributed MPC (solid line) centralized MPC (dotted line)and decentralized MPC (dashed line)

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0

002

004

006

U1

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U2

5 10 15 20 25 30 35 40 45 500Time (s)

minus001

0

001

002

003

004

U3

minus002

0

002

004

006

008

U4

Figure 14 Control signal of distributed MPC in Case 2 Δ120579ref in area 1 (solid line) Δ119875119888119894

in four areas (dotted line) and Δ119879119892

in area 1 (dashedline)

in the presence of wind turbine has desirable performance incomparison to centralized MPC and decentralized MPC

The distributed MPC control actions as shown inFigure 14 Δ120579ref Δ119875119888119894 and Δ119879119892 in four areas are depicted assolid dotted and dashed line respectively Δ120579ref and Δ119879119892 arethe control signals of wind turbine in area 1 and Δ119875

119888119894

is thecontrol signal of traditional power plants in the four areasFigure 15 shows the generating outputs of traditional plants

5 Conclusions

In this paper a DMPC scheme is presented for the LFC of afour-area interconnected power system with wind turbinesThe state and input constraints including the valve positionlimit on the governor and the GRCs were incorporated intothe systemdesign In our scheme each control area has a localMPC controller in which the four controllers coordinated

with each other by exchanging their information Compar-isons of response to step load change and computationalburden have been made between DMPC centralized MPCand decentralized MPC The simulation results verified thereliability of the DMPC for achieving a performance that hasadvantages over the centralized MPC and distributed MPCin the presence of load changes Moreover the proposedDMPC scheme can guarantee a good performance underthe wind turbines participation in LFC Future work will bethe extension of the proposed DMPC to different renewableenergy contained LFC since the greater utilization of inter-mittent renewable resources will induce greater power flowfluctuations

Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

16 Journal of Control Science and Engineering

5 10 15 20 25 30 35 40 45 500Time (s)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg4

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012

ΔPg3

(pu

MW

)

5 10 15 20 25 30 35 40 45 500Time (s)

minus002

0002004006008

01012014

ΔPg2

(pu

MW

)

0

002

004

006

ΔPg1

(pu

MW

)

Figure 15 Response of generated power deviation in Case 2 distributed MPC (solid line) centralized MPC (dotted line) and decentralizedMPC (dashed line)

Acknowledgments

This project was supported by National Natural ScienceFoundation of China under Grants 60974051 and 61273144Natural Science Foundation of Beijing under Grant 4122071Scientific Technology Research and Development PlanProject of Tangshan under Grant 13130298b and ScientificTechnology Research andDevelopment Plan Project ofHebeiunder Grant z2014070

References

[1] Global Wind Energy Council Global Wind Report on AnnualMarket Global Wind Energy Council 2014

[2] H Bevrani F Daneshfar and R P Daneshmand ldquoIntelligentpower system frequency regulations concerning the integrationof wind power unitsrdquo in Wind Power Systems Applications ofComputational Intelligence L FWang C Singh and A KusiakEds Green Energy and Technology pp 407ndash437 SpringerBerlin Germany 2010

[3] X Yingcheng and T Nengling ldquoReview of contribution tofrequency control through variable speedwind turbinerdquoRenew-able Energy vol 36 no 6 pp 1671ndash1677 2011

[4] Y-Z Sun Z-S Zhang G-J Li and J Lin ldquoReview on frequencycontrol of power systems with wind power penetrationrdquo in Pro-ceedings of the International Conference on Power System Tech-nology pp 1ndash8 IEEE Hangzhou China October 2010

[5] S K Pandey S R Mohanty and N Kishor ldquoA literature surveyon load-frequency control for conventional and distributiongeneration power systemsrdquo Renewable and Sustainable EnergyReviews vol 25 pp 318ndash334 2013

[6] F Dıaz-Gonzalez M Hau A Sumper and O Gomis-BellmuntldquoParticipation of wind power plants in system frequency con-trol review of grid code requirements and control methodsrdquo

Renewable and Sustainable Energy Reviews vol 34 pp 551ndash5642014

[7] H ShayeghiHA Shayanfar andA Jalili ldquoLoad frequency con-trol strategies a state-of-the-art survey for the researcherrdquoEnergy Conversion andManagement vol 50 no 2 pp 344ndash3532009

[8] L-R Chang-Chien C-C Sun and Y-J Yeh ldquoModeling ofwind farm participation in AGCrdquo IEEE Transactions on PowerSystems vol 29 no 3 pp 1204ndash1211 2014

[9] H Bevrani and P R Daneshmand ldquoFuzzy logic-based load-frequency control concerning high penetration of wind tur-binesrdquo IEEE Systems Journal vol 6 no 1 pp 173ndash180 2012

[10] M H Variani and K Tomsovic ldquoDistributed automatic genera-tion control using flatness-based approach for high penetrationof wind generationrdquo IEEE Transactions on Power Systems vol28 no 3 pp 3002ndash3009 2013

[11] X J Liu P Guan and C W Chan ldquoNonlinear multivari-able power plant coordinate control by constrained predictiveschemerdquo IEEE Transactions on Control Systems Technology vol18 no 5 pp 1116ndash1125 2010

[12] X-J Liu and C W Chan ldquoNeuro-fuzzy generalized predictivecontrol of boiler steam temperaturerdquo IEEE Transactions onEnergy Conversion vol 21 no 4 pp 900ndash908 2006

[13] X J Liu and X B Kong ldquoNonlinear fuzzy model predictiveiterative learning control for drum-type boilerndashturbine systemrdquoJournal of Process Control vol 23 no 8 pp 1023ndash1040 2013

[14] D Rerkpreedapong N Atic and A Feliachi ldquoEconomy ori-ented model predictive load frequency controlrdquo in Proceedingsof the Large Engineering Systems Conference on Power Engineer-ing pp 12ndash16 IEEE Montreal Canada May 2003

[15] X Liu X Kong and X Deng ldquoPower system model predictiveload frequency controlrdquo in Proceedings of the American ControlConference (ACC rsquo12) pp 6602ndash6607 June 2012

[16] T H Mohamed J Morel H Bevrani and T Hiyama ldquoModelpredictive based load frequency control design concerning

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Control Science and Engineering 17

wind turbinesrdquo International Journal of Electrical Power ampEnergy Systems vol 43 no 1 pp 859ndash867 2012

[17] T H Mohamed H Bevrani A A Hassan and T HiyamaldquoDecentralized model predictive based load frequency controlin an interconnected power systemrdquo Energy Conversion andManagement vol 52 no 2 pp 1208ndash1214 2011

[18] Y Zheng S Li and H Qiu ldquoNetworked coordination-baseddistributed model predictive control for large-scale systemrdquoIEEE Transactions on Control Systems Technology vol 21 no 3pp 991ndash998 2013

[19] E Camponogara and H F Scherer ldquoDistributed optimizationfor model predictive control of linear dynamic networks withcontrol-input and output constraintsrdquo IEEE Transactions onAutomation Science and Engineering vol 8 no 1 pp 233ndash2422011

[20] A N Venkat I A Hiskens J B Rawlings and S J WrightldquoDistributed MPC strategies with application to power systemautomatic generation controlrdquo IEEE Transactions on ControlSystems Technology vol 16 no 6 pp 1192ndash1206 2008

[21] M Mirzaei N K Poulsen and H H Niemann ldquoRobust modelpredictive control of a wind turbinerdquo in Proceedings of the Amer-icanControl Conference (ACC rsquo12) pp 114ndash119 Toronto CanadaJune 2012

[22] M Yigit V C Gungor G Tuna M Rangoussi and E FadelldquoPower line communication technologies for smart grid appli-cations a review of advances and challengesrdquo Computer Net-works vol 70 pp 366ndash383 2014

[23] M Ma H Chen X Liu and F Allgower ldquoMoving horizon119867

infin control of variable speed wind turbines with actuator sat-urationrdquo IET Renewable Power Generation vol 8 no 5 article498 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of