research article robust adaptive reactive power control...

Research ArticleRobust Adaptive Reactive Power Control forDoubly Fed Induction Generator

Huabin Wen1 Yu Zeng2 Lei Wang3 Feng Yang4 and Y D Song3

1 School of Electrical Engineering Beijing Jiaotong University Beijing 100044 China2 School of Energy Science and Engineering UESTC Chengdu 611731 China3 School of Automation Chongqing University Chongqing 400044 China4 School of Automation Engineering UESTC Chengdu 611731 China

Correspondence should be addressed to Y D Song ydsongcqueducn

Received 3 February 2014 Accepted 19 March 2014 Published 20 May 2014

Academic Editor Peng Shi

Copyright copy 2014 Huabin Wen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The problem of reactive power control for mains-side inverter (MSI) in doubly fed induction generator (DFIG) is studied inthis paper To accommodate the modelling nonlinearities and inherent uncertainties a novel robust adaptive control algorithmfor MSI is proposed by utilizing Lyapunov theory that ensures asymptotic stability of the system under unpredictable externaldisturbances and significant parametric uncertainties The distinguishing benefit of the aforementioned scheme consists in itscapabilities to maintain satisfactory performance under varying operation conditions without the need for manually redesigningor reprogramming the control gains in contrast to the commonly used PIPID control Simulations are also built to confirm thecorrectness and benefits of the control scheme

1 Introduction

Doubly fed induction generator (DFIG) enjoys more notice-able advantages compared with other kinds of wind genera-tors [1] For example by keeping the rotor current frequencyat a constant level DFIG can produce nearly constant powerfrom the stator and by keeping an optimal tip-speed ratioDFIG is able to capture themaximumwind power at differentwind speeds [2] A wind power generation system equippedwith DFIG requires a converter with only one-third ofthe power rating leading to a less expensive system withreduced power loss [3] DFIG can also control reactive powerseparately from active power with a reasonable adoptionof orientation frame [4] Especially DFIG can stabilizethe power network voltage by providing some controllablereactive power thus improving power factor or voltagecharacteristics [5]

Researches on the blackout in Canada and Americain 2003 indicated that if reactive power was provided intime the cascaded outages of several power system devicesmight have been avoided Reactive power is closely related

to voltage level and power factor (pf) and terminal voltageTo prevent power network instability problem some powercompanies have proposed several standards which must bestrictly met when the wind generators connect to the system[6] Therefore reactive power control in DFIG for windturbines has become a research topic of theoretical andpractical importance that has attracted considerable attentionduring the past decade leading to a number of technicalresults on reactive power control of DFIG in wind turbinesBrekken and Mohan [7] deal with the harmonic componenton the frame with a low bandwidth filter A PI and a statespace based controller for reactive power are studied byMachmoum et al [8] The limitation of generation capabilityon both converters of DFIG is analysed in Engelhardt et al[9] Slootweg et al [10] study the voltage control scheme byreactive power compensation on theRSI without consideringreactive power generation ability of grid-side inverter (MSI)In Tapia et al [11] a similar problem is investigated in whichMSI contribution to voltage control is ignored It shouldbe noted that MSI can be the main reactive compensatoras a STATCOM as shown in Kayikci and Milanovic [12]

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 369349 8 pageshttpdxdoiorg1011552014369349

2 Abstract and Applied Analysis

DC-link

Control

Gearbox

LN

is

in

ir

iN

uN

CfRSI un

usPm

ur

Lrf

MSI

P120596

Figure 1 DFIG drive topology

β

120572120579N

d

q

120596N

uN

Figure 2 Voltage orientation

An interesting effort has been made on using both the RSIand the MSI to design reactive power regulator in DFIG[13] There are several other coordinated RSI and MSI basedvoltage-control methods suggested in the literature (see egAkhmatov [14] and Ackermann [15] and the references citedtherein)

While reactive power control of DFIG has been exten-sively studied during the past few years there are some openissues which have practical and theoretical importance inthis area For instance from a reliable operation and real-time implementation point of view currently there is nouniform framework for the design of a cost-effective andreliable method for reactive power adjustment As a matterof fact in most existing works either the control develop-ment and closed-loop system stability conditions are basedon largely oversimplified linear dynamics or the resultantcontrol algorithms are prohibitively too complex for real-timeimplementation One reason leading to such barrier is thefact that differential equations of DFIG are nonlinear andcomplex in nature To facilitate control design most existingmethods have carried on the tradition of using linear modelwithout fully recognizing modelling uncertainty externaldisturbance or implementation cost As such these controlmethods which heavily depend on linear model and precisesystem parameters seldom satisfactorily work in practice Itis interesting to notice that different applications of adaptivecontrol method have been studied in various fields such as inpower systems [16 17] and in robot controlling [18] Adaptivecontrol method can help to solve the above problems

This paper proposes a computationally inexpensive con-trol algorithm for controlling reactive power in DFIG forwind turbines The main interest in the mentioned methodis primarily motivated by some practical implementationsituations where algorithm cost-effectiveness appears to bethe prior concern Meanwhile there exist the possibilitiesthat the system parameters and dynamics are not alwaysfully available for the sake of some constraints A dynamicmodel which reflects the electrical connection effects of theMSI of DIFG wind turbine system is established in thispaper Inspired by the recent work on using core informationfor control design [19] a simple yet effective robust adap-tive control scheme is developed The superior features ofthe resultant control scheme consist in the significance indealing with unpredictable lumped disturbances and simplestructure In fact only little information of the parame-tersdynamics is necessary for the construction of the controlalgorithm Meanwhile complicated and painful trail-and-error process for control gains determination is no longerneeded These friendly advantages are favourable in practicalimplementations

2 System Topology and Dynamic Equations

Drive topologies of DFIG have been intensively studied in theliterature [13] Drive topology of DFIG containing the currentand power flow is shown in Figure 1 Wind power capturedfrom the wind turbine transfers into electric power throughthe gear box and the induction generator The inductiongenerator is quite special since it has a dual converter whichis made of electric devices such as IGBTs The size of theconverter is determined according to the desired speed rangeWith a proper control for the RSI and MSI the separatecontrol of reactive power from active power is achieved TheDFIG employs some inductors between the rotor terminalsand the RSI as filters In order to suppress harmonics outputfilters are also used in the DFIG

21 Voltage Orientation Considering the deep couplingnature between reactive and active power a well-chosen ori-entation can help to control the two variables independentlyIn this paper the coordinate system rotates synchronouslyalong with mains voltage vector By adopting the phase-locked loop (PLL) scheme [20] the mains voltage vector iswell tracked by 119889-axis in the frameThus we obtain 119906

119873= 119906119873119889

and 119906119873119902

= 0 Based on such voltage orientation currentcomponents on the 119902- and 119889-axis are considered separatelyas reactive component and active componentThus the 119902-axiscurrent component is responsible for controlling the reactivepower production which will be used in later discussionThisimplication is illustrated in Figure 2 and 120596

119873is the angular

speed of 119906119873

22 Generator Model In this work we follow the modellingmethods as described in Rabelo et al [13] The stator voltagefrequency is the same as the net frequency that is 120596

119873= 120596119878

and the slip frequency is determined by 120596119903= 119878120596119873 119906119904and

Abstract and Applied Analysis 3

Robust adaptive control

32minusqlowastn ilowastnq

uNd

Control plant GPqn

Gci

middot

GFqn

Qnqninq

Figure 3 Block diagram of the system

119906119903are obtained by using the synchronous rotating frame for

induction generator in Leonhard [2] Consider

1199061015840

119903= 1198771015840

1199031198941015840

119903+119889120595119903

119889119905+ 119895119878119908

119873120595119903

119906119904= 119877119904119894119904+119889120595119904

119889119905+ 119895119908119873120595119904

(1)

Flux linkage of the rotor 120595119903and flux linkage of the stator

120595119904are calculated as

120595119903= 1198711015840

1199031198941015840

119903+ 119871119898119894119904 120595

119904= 119871119904119894119904+ 1198711198981198941015840

119903 (2)

Electromagnetic torque of the drive system is establishedby

119879119890=3

2119899119871119898

119871119904

sdotI 1205951199041198941015840lowast

119903 (3)

where ldquolowastrdquo denotes the conjugate complex value and Isdot

denotes the imaginary part and 119899 represents pole pairsThe current components on the stator side with orthogo-

nal coordinate are written as

119894119904119889= minus

119871119898

119871119904

119894119903119889

119894119904119902=120595119904119902

119871119904

minus119871119898

119871119904

119894119903119902

(4)

where 119894119904119889is the stator current component on the 119889-axis and

119894119904119902is the stator current component on the 119902-axisUsing the flux linkage equations above and replacing the

stator current component the following equations can beobtained

119906119903119902= 119877119903119894119903119902+ 120590119871119903

119889119894119903119902

119889119905minus 119878119908119873120590119871119903119894119903119902

119906119903119889= 119877119903119894119903119889+ 120590119871119903

119889119894119903119889

119889119905+ 119878119908119873120590119871119903119894119903119902+ 119878119908119873

119871119898

119871119904

120595119904119902

(5)

where 120590 = 1 minus (1198711198982119871119904119871119903) The basic electrical formulas

above will be used to construct the inner rotor currentcontroller

23 LC Filter Model Considering the inverter synchronizedwith the mains voltage and ignoring the voltage drop on the119871119873 the MSI output current dynamics can be described as

follows

119906119873119902

= 119877119891119894119899119902+ 119871119891

119889119894119899119902

119889119905minus 119908119873119871119891119894119899119889+ 119906119899119902= 0 (6)

119906119873119889

= 119877119891119894119899119889+ 119871119891

119889119894119899119889

119889119905minus 119908119873119871119891119894119899119902+ 119906119899119889 (7)

The block diagram in Figure 1 together with the aboveequations indicated the mutual and internal relationship inthe drive topology of DFIG and the wind power generationsystem

3 Reactive Power Control Design

Considering the external disturbance acting on the systemreactive power at the MSI is governed by the dynamicequation as given in (8) that is

11990211989911987911198792+ 119902119899(1198791+ 1198792) + 119902119899+ ℎ (sdot) = 119870119902119899119894

lowast

119899119902 (8)

which can be shown in detail as follows

31 Current Inner Loop Control This is a cross-coupled 2DproblemThe transfer function of the plant 119866

119894(119904) is described

as

119866119894 (119904) =

119870119894

119904119879119894+ 1

(9)

where 119870119894= 1119877 and 119879

119894= 119871119877 The transfer functions

are identical on both 119889 and 119902 axes The time delay of signalpreconditioning and processing can be regarded as a smalltime constant 119879sum ≪ 119879

119894 Note that 119879sum is small and its

accurate value is normally difficult to obtain The first-ordertransfer function of such time constant part is described as

119866sum (119904) =1

119904119879sum + 1 (10)

By utilizing of a simple controller119866119877119894 the following open-

loop transfer function is established

119866119900119894 (119904) = 119866119877119894119866119894119866sum = 119870

119875119894

119904119879119868119894+ 1

119904119879119868119894

119870119894

119904119879119894+ 1

1

119904119879sum + 1 (11)

where 119870119875119894

is the differential parameter and 119879119868119894is the inte-

grated time


32 Output Current Control Output current control shouldtrack the input current without heavy overshoots A generalmethod to solve such problems is proposed in Follinger [21]To achieve a well damping factor 1radic2 we can set119870

119875119894and119879119868119894

as follows

119870119875119894=

119871

2119879sum 119879

119868119894= 119879119894 (12)

The closed-loop transfer function becomes

119866119888119894 (119904) =

119866119900119894

1 + 119866119900119894

=1

21198792sum1199042 + 2119879sum119904 + 1

(13)

Since 119879sum ≪ 119879119894 the square value in the 1199042 term can be

ignored Thus the closed-loop function of the current innercontrol is simplified and obtained as

119866119888119894 (119904) cong

1

2119879sum119904 + 1 (14)

At the mains-side inverter and under the voltage orienta-tion discussed above the reactive power is established by

119876119899=3

2I 119906119899119894lowast

119899 =

3

2(119906119873119902119894119899119889minus 119906119873119889119894119899119902) = minus

3

2119906119873119889119894119899119902 (15)

Before passing into the controller the data of the inputreactive power pass through a filter The filter is given by aone-order transfer function as described in Rabelo et al [13]Consider

119866119865119902119899

=119902119899

119876119899

=1

(119904119879119865119902119899

+ 1)

(16)

where 119902119899is the ultimate actual reactive power through the

filter and 119879119865119902119899

is the filter time constantWe take the current inner control in (13) into account

Thus the control plant becomes the following second-ordertransfer function

119866119875119902119899

= 119870119902119899119866119888119894119866119865119902119899

=119870119902119899

(1199042119879sum + 1) (119904119879119865119902119899

+ 1)

(17)

where

119870119902119899= minus(

3

2) 119906119873119889 (18)

The block diagramof the system adopting robust adaptivecontrol methods is shown in Figure 3 The primary objectiveis to build the reference current 119894lowast

119899119902which makes the actual

reactive power 119902119899regulate the reference one 119902lowast

119899asymptoti-

callyWe rewrite control plant equation (16) as

119902119899

119894lowast119899119902

=119870119899119902

(1199041198791+ 1) (119904119879

2+ 1)

(19)

where 1198791= 2119879sum 1198792 = 119879

119865119899119902 Using Laplace inverse trans-

form it follows that11990211989911987911198792+ 119902119899(1198791+ 1198792) + 119902119899= 119870119902119899119894lowast

119899119902 (20)

Taking the external disturbances acting on the systeminto account (20) becomes (8) where ℎ(sdot) is the externaldisturbances

33 Robust Adaptive Control for Mains-Side Reactive PowerIn this section a robust adaptive control for mains-side reac-tive power will be proposed To build a meaningful adaptivecontrol scheme two realistic assumptions are adopted

Assumption 1 Voltage amplitude at net connecting pointremains nonzero Thus with the voltage orientation and (6)119906119873119889

can be regarded as a positive known number so that119870119902119899

with respect to (18) is a negative known number

Remark 2 Assumption 1 imposed here rather standard inaddressing system stabilization is practical because the windpower generation system will be shut off if the voltage at thenet is extremely low thus the zero voltage does not occur forthe situation under consideration

To design the tracking controller we define the reactivepower tracking error as

119890 = 119902119899minus 119902lowast

119899 (21)

To simplify controller design we introduce 120576 and define 120576as

120576 = 120573119890 + 119890 (22)

where 120573 gt 0 is a designed constantApparently if 120576 converges to zero as time increases 119890 and

119890 also converge to zero which means if we can design acontroller that forces 120576 to converge to zero as time increasesthen problem will be solved

Taking derivative of (22) with respect to time we get

120576 = 120573 119890 + 119890 (23)

Substituting e with (21) and using 119902119899as given in (8) we

obtain

120576 =119870119902119899

11987911198792

119894lowast

119899119902+

1

11987911198792

[minusℎ (sdot) minus 119902119899 minus (1198791 + 1198792) 119902119899] + 120573 119890 minus 119902

lowast

119899

(24)

Or equivalently

120576 = 119861119906 + 119871 (sdot) (25)

where 119906 = minus119894lowast119899119902

119861 = minus119870119902119899

11987911198792

119871 (sdot) =1

11987911198792


lowast

119899

(26)

Obviously 119861 is positive because of the definition of119870119902119899as

given in (18) Note that

119871 (sdot) le

10038161003816100381610038161003816100381610038161003816

1

11987911198792

10038161003816100381610038161003816100381610038161003816[|ℎ (sdot)| +

10038161003816100381610038161199021198991003816100381610038161003816 +

1003816100381610038161003816(1198791 + 1198792)10038161003816100381610038161003816100381610038161003816 1199021198991003816100381610038161003816] +

1003816100381610038161003816120573 1198901003816100381610038161003816 +

1003816100381610038161003816 119902lowast

119899

1003816100381610038161003816

(27)


Assumption 3 In this assumption the characteristic of thedisturbance is discussed The external disturbance ℎ(sdot) isbounded which leads to the existence of an unknown con-stant such that

|ℎ (sdot)| le 1198861 lt infin (28)

Also the time constants 1198791and 119879

2 although unknown in

general are bounded so that10038161003816100381610038161003816100381610038161003816

1

11987911198792

10038161003816100381610038161003816100381610038161003816le 1198862lt infin

1003816100381610038161003816(1198791 + 1198792)1003816100381610038161003816 le 1198863 lt infin (29)

where 1198862and 1198863are some unknown positive constants

This assumption quite reasonable in practice allows forthe establishment of

|119871 (sdot)| le 119886120593 (119902119899) (30)

where

119886 = max (1198862sdot 1198861 1198862 1198862sdot 1198863 1) (31)

120593 (119902119899) = 1 +

10038161003816100381610038161199021198991003816100381610038161003816 +

1003816100381610038161003816 1199021198991003816100381610038161003816 + 120573 | 119890| +

1003816100381610038161003816 119902lowast

119899

1003816100381610038161003816 (32)

We design the input as

119906 = minus (1198700+ ) 120576 (33a)

where1198700gt 0 is a design constant and is updated as

=119886120593

|120576| (33b)

where 119886 is the estimation of 119886 and is updated as

119886 = |120576| 120593 (33c)

Theorem 4 For such system established by (8) under theassumptions if 119906 is calculated by (33a) (33b) and (33c)then the reactive power is ensured to track the desired oneasymptotically

Proof The result can be justified using the following Lya-punov function

119881 =1

21205762+

1

2119861min(119886 minus 119886119861min)

2 (34)

where 119861min is constant and 0 lt 119861min le 119861 for forall119861 isin 119871infin

Differentiating 119881 leads to

= 120576 120576 minus (119886 minus 119886119861min)119886 (35)

From (25) and (33a) we get

120576 = minus1198611198700120576 minus 119861120576 + 119871 (36)

Equation (35) becomes

= 120576 (minus1198611198700120576 minus 119861120576 + 119871) minus (119886 minus 119886119861min)

119886 (37)

Substituting with (33b) and 119886 with (33c)

= 120576119871 minus 11986111987001205762minus 119861119886120593 |120576| minus 120593 |120576| (119886 minus 119886119861min)

le |120576| |119871| minus 11986111987001205762minus 119861119886120593 |120576| minus 120593 |120576| (119886 minus 119886119861min)

= minus11986111987001205762+ (|120576| |119871| minus |120576| 120593119886) + (119861min119886120593 |120576| minus 119861119886120593 |120576|)

(38)

Because we have

|119871 (sdot)| le 119886120593 0 lt 119861min le 119861 (39)

we can get

le minus119861min11987001205762lt 0 (40)

As 119861min gt 0 it is readily shown from (40) that both 120576 and119886 are bounded Furthermore we can show that 120576 is boundedand thus 120576 is uniformly continuousTherefore using BarbalatLemma [22] the reality is obtained that 120576 converges to zeroasymptotically By utilizing (22) ultimately 119890 and 119890 convergeto zero asymptotically then the result is established

Remark 5 It should be noted that when the states get closerto zero the control schememight experience chatteringThishowever can be avoided by using the simple but classicmeans of replacing 119911|119911| with 119911(|119911| + 120591) where 120591 is smallMeanwhile in order to prevent the estimate 119886 from drifting(33c) can be modified to

119886 = minus1205901119886 + 1205902

12057621205932

|120576| 120593 + 120591 1205901gt 0 120590

2gt 0 (41a)

In this case we have the following ultimately uniformlybounded (UUB) tracking control result

Theorem 6 Also for system established by (8) under theassumptions if the following robust adaptive control algorithmis adopted

119906 = minus (1198700+ ) 120576 (41b)


=119886120593

|120576| + 120591 (41c)

where 119886 is the estimation of a and is updated by (33a) then thesystem is ensured to be UUB stable

Proof This theorem can be proved by utilizing of the meth-ods in Cai et al [23]

Remark 7 Instead of using PI controller as in Rabelo et al[13] a simple robust adaptive control scheme is developedhere in which one only needs to specify the parameters in aclear direction that is119870

0gt 0 and 120573 gt 0

Remark 8 The significance of the developed control schemeis twofold


(1) The control scheme developed here does not relyon the precise value for the time constants 119879

1 1198792

Also there is no need for analytical estimation ofthe unknown parameters 119886

1 1198862 1198863 Such fact can

sufficiently simplify the design procedure and imple-mentation of the proposed control algorithm

(2) As the parameter 119886 involved in the controller isupdated automatically via the algorithm and suchprocess is independent of operation conditions noredesign or reprogramming is needed during thesystem operation

4 Simulation Verification

Aiming at validating the correctness of the robust adaptivereactive control scheme simulations with MatlabSimulinkare presented here

Per unit (pu) value is introduced to simplify calculationand simulation The datum voltage 119881

119886V and datum capacity119878119886V respectively are set as 330V and 1MVar The net voltage119906119873119889

under voltage orientation is chosen as 220V and thus119870119902119899

with respect to (17) is calculated as minus1 Other parameters usedfor simulation are chosen as 119879sum = 05 119879

119865119902119899= 1 120573 = 1 and

1198700= 1 The PI control algorithm described in Ackermann

[15] for reactive power is rebuilt for comparison Three typesof working conditions are simulated here

41 Regulating under Steady Working Condition In realapplication DFIG can work as a compensator to provideconstant reactive power Based on this fact we set the desiredreactive power output as 1 puThe desired reactive power andactual one are together plotted in Figure 4

Compared with the adopted PI scheme adaptive methodeliminates overshoot and has a shorter regulate time anda longer rise time Both of the control schemes can obtainstabilization

42 Regulating under Modelling Uncertainty In this kind ofsimulation the influence of modelling uncertainty is investi-gated Assume that the time constant 119879sum has a deviation of5 then 119879sum = 0525 The result of the proposed adaptivemethod is illustrated in Figure 5 and the PI method is shownin Figure 6

The result shows that by utilizing the proposed schemethe tracking trajectory almost remains the same even undersuch modelling uncertainty and parameter deviation Fromthe point of detail the proposed adaptive scheme enjoys abetter tracking trajectory compared with the PI controllerunder such condition

43 Tracking under Dynamic Reactive Power CompensationIf the voltage begins to drop during an unsymmetrical gridfault the DFIG will work as a dynamic reactive compensatorto keep the voltage to a certain degree Once the fault ismoved the reactive power generating capacity of DFIG willbe resumed [24] Based on this fact we set the desired reactivepower output as illustrated in Figure 7 Specifically the DFIGis ordered to provide 1 pu before 7 s and provide an extra

0 2 4 6 8 100

1

14

12

08

06

04

02

Simulation time (s)

Reac

tive p

ower

(pu

)

AdaptivePIDesired

Figure 4 Reactive power regulation trajectories

0 2 4 6 8 10

Simulation time (s)

Reac

tive p

ower

(pu

)

Without modelling uncertaintyUnder modelling uncertainty

0

1

08

07

09

06

05

03

04

02

01

Figure 5 Adaptive controller tracking trajectories

02 pu reactive power in the time from 7 s to 8 s becausethe proposed scheme uses the derivative of the desired inputThe reference has been smoothed before putting into thecontroller

The result of PI controller is also compared to the pro-posed adaptive method Figure 7 illustrates that comparedwith PI controller the proposedmethod can track the desiredreactive power better This is because PI controller fails totrack the reference value at the time of 7 s The result showsthat the adopted PI controller fails to deal with such problemwhich however the adaptive scheme proposed in this papercan deal with



0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)

0 2 4 6 8 10Simulation time (s)

Figure 6 PI controller tracking trajectories

0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)


AdaptivePIDesired

Figure 7 Reactive power tracking trajectories

5 Conclusions

Reactive power control for mains-side inverter (MSI) inDFIG represents an important issue in wind power gener-ation systems A robust adaptive control scheme for MSI isdeveloped As confirmed by theoretical analysis the proposedmethod is able to maintain satisfactory performance undervarying operation conditions without the need for manuallyredesigning or reprogramming the control gains Numericalsimulations also validate the correctness and benefits of theproposed algorithm

Conflict of Interests

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

Acknowledgments

This work was supported in part by the National BasicResearch Programof China (973 Programno 2012CB215200)and in part by the National Natural Science Foundation ofChina (no 51205046)

References

[1] Y She X She and M E Baran ldquoUniversal tracking control ofwind conversion system for purpose of maximum power acqui-sition under hierarchical control structurerdquo IEEE Transactionson Energy Conversion vol 26 no 3 pp 766ndash775 2011

[2] W Leonhard Control of Electrical Drives Springer 2000[3] Rabelo and B Hofmann W ldquoPower flow optimisation and grid

integration of wind turbines with the Doubly-Fed inductiongeneratorrdquo in Proceedings of the 36th International Conferenceon Power Electronics Specialists Conference pp 2930ndash2936Recife Brazil June 2005

[4] R Pena J C Clare and G M Asher ldquoDoubly fed induc-tion generator using back-to-back PWM converters and itsapplication to variable-speed wind-energy generationrdquo IEEProceedings Electric Power Applications vol 143 no 3 pp 231ndash241 1996

[5] M El Moursi G Joos and C Abbey ldquoA secondary voltagecontrol strategy for transmission level interconnection of windgenerationrdquo IEEE Transactions on Power Electronics vol 23 no3 pp 1178ndash1190 2008

[6] E O N Netz ldquoGrid code high and extra high voltagerdquo 2006[7] T K A Brekken andNMohan ldquoControl of a doubly fed induc-

tion wind generator under unbalanced grid voltage conditionsrdquoIEEE Transactions on Energy Conversion vol 22 no 1 pp 129ndash135 2007

[8] M Machmoum F Poitiers C Darengosse and A QuericldquoDynamic performances of a doubly-fed inductionmachine fora variable-speed wind energy generationrdquo in Proceedings of theInternational Conference on Power System Technology pp 2431ndash2436 October 2002

[9] S Engelhardt I Erlich C Feltes J Kretschmann and FShewarega ldquoReactive power capability of wind turbines basedon doubly fed induction generatorsrdquo IEEE Transactions onEnergy Conversion vol 26 no 1 pp 364ndash372 2011

[10] J G Slootweg S W H de Haan H Polinder and W L KlingldquoVoltage control methods with grid connected wind turbines atutorial reviewrdquo Wind Engineering vol 25 no 6 pp 353ndash3662001

[11] A Tapia G Tapia J XabierOstolaza and J R Saenz ldquoModelingand control of a wind turbine driven doubly fed inductiongeneratorrdquo IEEE Transactions on Energy Conversion vol 18 no2 pp 194ndash204 2003

[12] M Kayikci and J V Milanovic ldquoReactive power controlstrategies for DFIG-based plantsrdquo IEEE Transactions on EnergyConversion vol 22 no 2 pp 389ndash396 2007

[13] B C Rabelo Jr W Hofmann J L da Silva R G de Oliveiraand S R Silva ldquoReactive power control design in doubly fed


induction generators for wind turbinesrdquo IEEE Transactions onIndustrial Electronics vol 56 no 10 pp 4154ndash4162 2009

[14] V Akhmatov Analysis of Dynamic Behaviour of Electric PowerSystems with Large Amount of Wind Power Electric PowerEngineering Oslashrsted-DTU Technical University of Denmark2003

[15] T Ackermann Ed Wind Power in Power Systems John Wileyand Sons New York NY USA 2005

[16] Y Song X Li and W Cai ldquoAdaptive and fault-tolerant reactivepower compensation in power systemsrdquo International Journalof Innovative Computing Information and Control vol 9 no 8pp 3403ndash3413 2013

[17] D Niu and Y Wei ldquoA novel social-environmental-economicdispatch model for thermalwind power generation and appli-cationrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 7 pp 3005ndash3014 2013

[18] Y Chen G Mei G Ma S Lin and J Gao ldquoRobust adaptiveinverse dynamics control for uncertain robot manipulatorrdquoInternational Journal of Innovative Computing Information andControl vol 10 no 2 pp 575ndash587 2014

[19] Y-D Song H-N Chen and D-Y Li ldquoVirtual-point-basedfault-tolerant lateral and longitudinal control of 4W-steeringvehiclesrdquo IEEE Transactions on Intelligent Transportation Sys-tems vol 12 no 4 pp 1343ndash1351 2011

[20] V Kaura and V Blasko ldquoOperation of a phase locked loopsystemunder distorted utility conditionsrdquo IEEE Transactions onIndustry Applications vol 33 no 1 pp 58ndash63 1997

[21] O Follinger Regelungstechnik Springer 8th edition 1999[22] J J Slotine andW LeiAppliedNonlinear Control Prentice-Hall

1991[23] W Cai X H Liao and Y D Song ldquoIndirect robust adaptive

fault-tolerant control for attitude tracking of spacecraftrdquo Journalof Guidance Control andDynamics vol 31 no 5 pp 1456ndash14632008

[24] S Seman J Niiranen and A Arkkio ldquoRide-through analysis ofdoubly fed induction wind-power generator under unsymmet-rical network disturbancerdquo IEEETransactions onPower Systemsvol 21 no 4 pp 1782ndash1789 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


DC-link

Control

Gearbox

LN

is

in

ir

iN

uN

CfRSI un

usPm

ur

Lrf

MSI

P120596

Figure 1 DFIG drive topology

β

120572120579N

d

q

120596N

uN

Figure 2 Voltage orientation

An interesting effort has been made on using both the RSIand the MSI to design reactive power regulator in DFIG[13] There are several other coordinated RSI and MSI basedvoltage-control methods suggested in the literature (see egAkhmatov [14] and Ackermann [15] and the references citedtherein)

While reactive power control of DFIG has been exten-sively studied during the past few years there are some openissues which have practical and theoretical importance inthis area For instance from a reliable operation and real-time implementation point of view currently there is nouniform framework for the design of a cost-effective andreliable method for reactive power adjustment As a matterof fact in most existing works either the control develop-ment and closed-loop system stability conditions are basedon largely oversimplified linear dynamics or the resultantcontrol algorithms are prohibitively too complex for real-timeimplementation One reason leading to such barrier is thefact that differential equations of DFIG are nonlinear andcomplex in nature To facilitate control design most existingmethods have carried on the tradition of using linear modelwithout fully recognizing modelling uncertainty externaldisturbance or implementation cost As such these controlmethods which heavily depend on linear model and precisesystem parameters seldom satisfactorily work in practice Itis interesting to notice that different applications of adaptivecontrol method have been studied in various fields such as inpower systems [16 17] and in robot controlling [18] Adaptivecontrol method can help to solve the above problems

This paper proposes a computationally inexpensive con-trol algorithm for controlling reactive power in DFIG forwind turbines The main interest in the mentioned methodis primarily motivated by some practical implementationsituations where algorithm cost-effectiveness appears to bethe prior concern Meanwhile there exist the possibilitiesthat the system parameters and dynamics are not alwaysfully available for the sake of some constraints A dynamicmodel which reflects the electrical connection effects of theMSI of DIFG wind turbine system is established in thispaper Inspired by the recent work on using core informationfor control design [19] a simple yet effective robust adap-tive control scheme is developed The superior features ofthe resultant control scheme consist in the significance indealing with unpredictable lumped disturbances and simplestructure In fact only little information of the parame-tersdynamics is necessary for the construction of the controlalgorithm Meanwhile complicated and painful trail-and-error process for control gains determination is no longerneeded These friendly advantages are favourable in practicalimplementations

2 System Topology and Dynamic Equations

Drive topologies of DFIG have been intensively studied in theliterature [13] Drive topology of DFIG containing the currentand power flow is shown in Figure 1 Wind power capturedfrom the wind turbine transfers into electric power throughthe gear box and the induction generator The inductiongenerator is quite special since it has a dual converter whichis made of electric devices such as IGBTs The size of theconverter is determined according to the desired speed rangeWith a proper control for the RSI and MSI the separatecontrol of reactive power from active power is achieved TheDFIG employs some inductors between the rotor terminalsand the RSI as filters In order to suppress harmonics outputfilters are also used in the DFIG

21 Voltage Orientation Considering the deep couplingnature between reactive and active power a well-chosen ori-entation can help to control the two variables independentlyIn this paper the coordinate system rotates synchronouslyalong with mains voltage vector By adopting the phase-locked loop (PLL) scheme [20] the mains voltage vector iswell tracked by 119889-axis in the frameThus we obtain 119906

119873= 119906119873119889

and 119906119873119902

= 0 Based on such voltage orientation currentcomponents on the 119902- and 119889-axis are considered separatelyas reactive component and active componentThus the 119902-axiscurrent component is responsible for controlling the reactivepower production which will be used in later discussionThisimplication is illustrated in Figure 2 and 120596

119873is the angular

speed of 119906119873

22 Generator Model In this work we follow the modellingmethods as described in Rabelo et al [13] The stator voltagefrequency is the same as the net frequency that is 120596

119873= 120596119878

and the slip frequency is determined by 120596119903= 119878120596119873 119906119904and




uNd

Control plant GPqn

Gci

middot

GFqn

Qnqninq




1199061015840

119903= 1198771015840

1199031198941015840

119903+119889120595119903

119889119905+ 119895119878119908

119873120595119903

119906119904= 119877119904119894119904+119889120595119904

119889119905+ 119895119908119873120595119904

(1)



120595119903= 1198711015840

1199031198941015840

119903+ 119871119898119894119904 120595

119904= 119871119904119894119904+ 1198711198981198941015840

119903 (2)


119879119890=3

2119899119871119898

119871119904

sdotI 1205951199041198941015840lowast

119903 (3)




119894119904119889= minus

119871119898

119871119904

119894119903119889

119894119904119902=120595119904119902

119871119904

minus119871119898

119871119904

119894119903119902

(4)




119906119903119902= 119877119903119894119903119902+ 120590119871119903

119889119894119903119902

119889119905minus 119878119908119873120590119871119903119894119903119902

119906119903119889= 119877119903119894119903119889+ 120590119871119903

119889119894119903119889

119889119905+ 119878119908119873120590119871119903119894119903119902+ 119878119908119873

119871119898

119871119904

120595119904119902

(5)




follows

119906119873119902

= 119877119891119894119899119902+ 119871119891

119889119894119899119902

119889119905minus 119908119873119871119891119894119899119889+ 119906119899119902= 0 (6)

119906119873119889

= 119877119891119894119899119889+ 119871119891

119889119894119899119889

119889119905minus 119908119873119871119891119894119899119902+ 119906119899119889 (7)




11990211989911987911198792+ 119902119899(1198791+ 1198792) + 119902119899+ ℎ (sdot) = 119870119902119899119894

lowast

119899119902 (8)




as

119866119894 (119904) =

119870119894

119904119879119894+ 1

(9)

where 119870119894= 1119877 and 119879





119866sum (119904) =1

119904119879sum + 1 (10)



119866119900119894 (119904) = 119866119877119894119866119894119866sum = 119870

119875119894

119904119879119868119894+ 1

119904119879119868119894

119870119894

119904119879119894+ 1

1

119904119879sum + 1 (11)

where 119870119875119894


grated time



119875119894and119879119868119894

as follows

119870119875119894=

119871

2119879sum 119879

119868119894= 119879119894 (12)


119866119888119894 (119904) =

119866119900119894

1 + 119866119900119894

=1

21198792sum1199042 + 2119879sum119904 + 1

(13)



119866119888119894 (119904) cong

1

2119879sum119904 + 1 (14)


119876119899=3

2I 119906119899119894lowast

119899 =

3

2(119906119873119902119894119899119889minus 119906119873119889119894119899119902) = minus

3

2119906119873119889119894119899119902 (15)


119866119865119902119899

=119902119899

119876119899

=1

(119904119879119865119902119899

+ 1)

(16)


filter and 119879119865119902119899



119866119875119902119899

= 119870119902119899119866119888119894119866119865119902119899

=119870119902119899

(1199042119879sum + 1) (119904119879119865119902119899

+ 1)

(17)

where

119870119902119899= minus(

3

2) 119906119873119889 (18)




119899asymptoti-


119902119899

119894lowast119899119902

=119870119899119902

(1199041198791+ 1) (119904119879

2+ 1)

(19)

where 1198791= 2119879sum 1198792 = 119879



119899119902 (20)








119890 = 119902119899minus 119902lowast

119899 (21)


120576 = 120573119890 + 119890 (22)




120576 = 120573 119890 + 119890 (23)


obtain

120576 =119870119902119899

11987911198792

119894lowast

119899119902+

1

11987911198792


lowast

119899

(24)

Or equivalently

120576 = 119861119906 + 119871 (sdot) (25)


119861 = minus119870119902119899

11987911198792

119871 (sdot) =1

11987911198792


lowast

119899

(26)



119871 (sdot) le

10038161003816100381610038161003816100381610038161003816

1

11987911198792

10038161003816100381610038161003816100381610038161003816[|ℎ (sdot)| +

10038161003816100381610038161199021198991003816100381610038161003816 +

1003816100381610038161003816(1198791 + 1198792)10038161003816100381610038161003816100381610038161003816 1199021198991003816100381610038161003816] +

1003816100381610038161003816120573 1198901003816100381610038161003816 +

1003816100381610038161003816 119902lowast

119899

1003816100381610038161003816

(27)







1

11987911198792

10038161003816100381610038161003816100381610038161003816le 1198862lt infin

1003816100381610038161003816(1198791 + 1198792)1003816100381610038161003816 le 1198863 lt infin (29)



|119871 (sdot)| le 119886120593 (119902119899) (30)

where

119886 = max (1198862sdot 1198861 1198862 1198862sdot 1198863 1) (31)

120593 (119902119899) = 1 +

10038161003816100381610038161199021198991003816100381610038161003816 +

1003816100381610038161003816 1199021198991003816100381610038161003816 + 120573 | 119890| +

1003816100381610038161003816 119902lowast

119899

1003816100381610038161003816 (32)


119906 = minus (1198700+ ) 120576 (33a)


=119886120593

|120576| (33b)


119886 = |120576| 120593 (33c)



119881 =1

21205762+

1

2119861min(119886 minus 119886119861min)

2 (34)



= 120576 120576 minus (119886 minus 119886119861min)119886 (35)


120576 = minus1198611198700120576 minus 119861120576 + 119871 (36)



119886 (37)





(38)

Because we have

|119871 (sdot)| le 119886120593 0 lt 119861min le 119861 (39)

we can get

le minus119861min11987001205762lt 0 (40)



119886 = minus1205901119886 + 1205902

12057621205932

|120576| 120593 + 120591 1205901gt 0 120590

2gt 0 (41a)



119906 = minus (1198700+ ) 120576 (41b)


=119886120593

|120576| + 120591 (41c)




0gt 0 and 120573 gt 0




1 1198792


1 1198862 1198863 Such fact can









119865119902119899= 1 120573 = 1 and








0 2 4 6 8 100

1

14

12

08

06

04

02

Simulation time (s)

Reac

tive p

ower

(pu

)

AdaptivePIDesired


0 2 4 6 8 10

Simulation time (s)

Reac

tive p

ower

(pu

)


0

1

08

07

09

06

05

03

04

02

01






0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)



0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)


AdaptivePIDesired


5 Conclusions




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












uNd

Control plant GPqn

Gci

middot

GFqn

Qnqninq




1199061015840

119903= 1198771015840

1199031198941015840

119903+119889120595119903

119889119905+ 119895119878119908

119873120595119903

119906119904= 119877119904119894119904+119889120595119904

119889119905+ 119895119908119873120595119904

(1)



120595119903= 1198711015840

1199031198941015840

119903+ 119871119898119894119904 120595

119904= 119871119904119894119904+ 1198711198981198941015840

119903 (2)


119879119890=3

2119899119871119898

119871119904

sdotI 1205951199041198941015840lowast

119903 (3)




119894119904119889= minus

119871119898

119871119904

119894119903119889

119894119904119902=120595119904119902

119871119904

minus119871119898

119871119904

119894119903119902

(4)




119906119903119902= 119877119903119894119903119902+ 120590119871119903

119889119894119903119902

119889119905minus 119878119908119873120590119871119903119894119903119902

119906119903119889= 119877119903119894119903119889+ 120590119871119903

119889119894119903119889

119889119905+ 119878119908119873120590119871119903119894119903119902+ 119878119908119873

119871119898

119871119904

120595119904119902

(5)




follows

119906119873119902

= 119877119891119894119899119902+ 119871119891

119889119894119899119902

119889119905minus 119908119873119871119891119894119899119889+ 119906119899119902= 0 (6)

119906119873119889

= 119877119891119894119899119889+ 119871119891

119889119894119899119889

119889119905minus 119908119873119871119891119894119899119902+ 119906119899119889 (7)




11990211989911987911198792+ 119902119899(1198791+ 1198792) + 119902119899+ ℎ (sdot) = 119870119902119899119894

lowast

119899119902 (8)




as

119866119894 (119904) =

119870119894

119904119879119894+ 1

(9)

where 119870119894= 1119877 and 119879





119866sum (119904) =1

119904119879sum + 1 (10)



119866119900119894 (119904) = 119866119877119894119866119894119866sum = 119870

119875119894

119904119879119868119894+ 1

119904119879119868119894

119870119894

119904119879119894+ 1

1

119904119879sum + 1 (11)

where 119870119875119894


grated time



119875119894and119879119868119894

as follows

119870119875119894=

119871

2119879sum 119879

119868119894= 119879119894 (12)


119866119888119894 (119904) =

119866119900119894

1 + 119866119900119894

=1

21198792sum1199042 + 2119879sum119904 + 1

(13)



119866119888119894 (119904) cong

1

2119879sum119904 + 1 (14)


119876119899=3

2I 119906119899119894lowast

119899 =

3

2(119906119873119902119894119899119889minus 119906119873119889119894119899119902) = minus

3

2119906119873119889119894119899119902 (15)


119866119865119902119899

=119902119899

119876119899

=1

(119904119879119865119902119899

+ 1)

(16)


filter and 119879119865119902119899



119866119875119902119899

= 119870119902119899119866119888119894119866119865119902119899

=119870119902119899

(1199042119879sum + 1) (119904119879119865119902119899

+ 1)

(17)

where

119870119902119899= minus(

3

2) 119906119873119889 (18)




119899asymptoti-


119902119899

119894lowast119899119902

=119870119899119902

(1199041198791+ 1) (119904119879

2+ 1)

(19)

where 1198791= 2119879sum 1198792 = 119879



119899119902 (20)








119890 = 119902119899minus 119902lowast

119899 (21)


120576 = 120573119890 + 119890 (22)




120576 = 120573 119890 + 119890 (23)


obtain

120576 =119870119902119899

11987911198792

119894lowast

119899119902+

1

11987911198792


lowast

119899

(24)

Or equivalently

120576 = 119861119906 + 119871 (sdot) (25)


119861 = minus119870119902119899

11987911198792

119871 (sdot) =1

11987911198792


lowast

119899

(26)



119871 (sdot) le

10038161003816100381610038161003816100381610038161003816

1

11987911198792

10038161003816100381610038161003816100381610038161003816[|ℎ (sdot)| +

10038161003816100381610038161199021198991003816100381610038161003816 +

1003816100381610038161003816(1198791 + 1198792)10038161003816100381610038161003816100381610038161003816 1199021198991003816100381610038161003816] +

1003816100381610038161003816120573 1198901003816100381610038161003816 +

1003816100381610038161003816 119902lowast

119899

1003816100381610038161003816

(27)







1

11987911198792

10038161003816100381610038161003816100381610038161003816le 1198862lt infin

1003816100381610038161003816(1198791 + 1198792)1003816100381610038161003816 le 1198863 lt infin (29)



|119871 (sdot)| le 119886120593 (119902119899) (30)

where

119886 = max (1198862sdot 1198861 1198862 1198862sdot 1198863 1) (31)

120593 (119902119899) = 1 +

10038161003816100381610038161199021198991003816100381610038161003816 +

1003816100381610038161003816 1199021198991003816100381610038161003816 + 120573 | 119890| +

1003816100381610038161003816 119902lowast

119899

1003816100381610038161003816 (32)


119906 = minus (1198700+ ) 120576 (33a)


=119886120593

|120576| (33b)


119886 = |120576| 120593 (33c)



119881 =1

21205762+

1

2119861min(119886 minus 119886119861min)

2 (34)



= 120576 120576 minus (119886 minus 119886119861min)119886 (35)


120576 = minus1198611198700120576 minus 119861120576 + 119871 (36)



119886 (37)





(38)

Because we have

|119871 (sdot)| le 119886120593 0 lt 119861min le 119861 (39)

we can get

le minus119861min11987001205762lt 0 (40)



119886 = minus1205901119886 + 1205902

12057621205932

|120576| 120593 + 120591 1205901gt 0 120590

2gt 0 (41a)



119906 = minus (1198700+ ) 120576 (41b)


=119886120593

|120576| + 120591 (41c)




0gt 0 and 120573 gt 0




1 1198792


1 1198862 1198863 Such fact can









119865119902119899= 1 120573 = 1 and








0 2 4 6 8 100

1

14

12

08

06

04

02

Simulation time (s)

Reac

tive p

ower

(pu

)

AdaptivePIDesired


0 2 4 6 8 10

Simulation time (s)

Reac

tive p

ower

(pu

)


0

1

08

07

09

06

05

03

04

02

01






0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)



0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)


AdaptivePIDesired


5 Conclusions




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875119894and119879119868119894

as follows

119870119875119894=

119871

2119879sum 119879

119868119894= 119879119894 (12)


119866119888119894 (119904) =

119866119900119894

1 + 119866119900119894

=1

21198792sum1199042 + 2119879sum119904 + 1

(13)



119866119888119894 (119904) cong

1

2119879sum119904 + 1 (14)


119876119899=3

2I 119906119899119894lowast

119899 =

3

2(119906119873119902119894119899119889minus 119906119873119889119894119899119902) = minus

3

2119906119873119889119894119899119902 (15)


119866119865119902119899

=119902119899

119876119899

=1

(119904119879119865119902119899

+ 1)

(16)


filter and 119879119865119902119899



119866119875119902119899

= 119870119902119899119866119888119894119866119865119902119899

=119870119902119899

(1199042119879sum + 1) (119904119879119865119902119899

+ 1)

(17)

where

119870119902119899= minus(

3

2) 119906119873119889 (18)




119899asymptoti-


119902119899

119894lowast119899119902

=119870119899119902

(1199041198791+ 1) (119904119879

2+ 1)

(19)

where 1198791= 2119879sum 1198792 = 119879



119899119902 (20)








119890 = 119902119899minus 119902lowast

119899 (21)


120576 = 120573119890 + 119890 (22)




120576 = 120573 119890 + 119890 (23)


obtain

120576 =119870119902119899

11987911198792

119894lowast

119899119902+

1

11987911198792


lowast

119899

(24)

Or equivalently

120576 = 119861119906 + 119871 (sdot) (25)


119861 = minus119870119902119899

11987911198792

119871 (sdot) =1

11987911198792


lowast

119899

(26)



119871 (sdot) le

10038161003816100381610038161003816100381610038161003816

1

11987911198792

10038161003816100381610038161003816100381610038161003816[|ℎ (sdot)| +

10038161003816100381610038161199021198991003816100381610038161003816 +

1003816100381610038161003816(1198791 + 1198792)10038161003816100381610038161003816100381610038161003816 1199021198991003816100381610038161003816] +

1003816100381610038161003816120573 1198901003816100381610038161003816 +

1003816100381610038161003816 119902lowast

119899

1003816100381610038161003816

(27)







1

11987911198792

10038161003816100381610038161003816100381610038161003816le 1198862lt infin

1003816100381610038161003816(1198791 + 1198792)1003816100381610038161003816 le 1198863 lt infin (29)



|119871 (sdot)| le 119886120593 (119902119899) (30)

where

119886 = max (1198862sdot 1198861 1198862 1198862sdot 1198863 1) (31)

120593 (119902119899) = 1 +

10038161003816100381610038161199021198991003816100381610038161003816 +

1003816100381610038161003816 1199021198991003816100381610038161003816 + 120573 | 119890| +

1003816100381610038161003816 119902lowast

119899

1003816100381610038161003816 (32)


119906 = minus (1198700+ ) 120576 (33a)


=119886120593

|120576| (33b)


119886 = |120576| 120593 (33c)



119881 =1

21205762+

1

2119861min(119886 minus 119886119861min)

2 (34)



= 120576 120576 minus (119886 minus 119886119861min)119886 (35)


120576 = minus1198611198700120576 minus 119861120576 + 119871 (36)



119886 (37)





(38)

Because we have

|119871 (sdot)| le 119886120593 0 lt 119861min le 119861 (39)

we can get

le minus119861min11987001205762lt 0 (40)



119886 = minus1205901119886 + 1205902

12057621205932

|120576| 120593 + 120591 1205901gt 0 120590

2gt 0 (41a)



119906 = minus (1198700+ ) 120576 (41b)


=119886120593

|120576| + 120591 (41c)




0gt 0 and 120573 gt 0




1 1198792


1 1198862 1198863 Such fact can









119865119902119899= 1 120573 = 1 and








0 2 4 6 8 100

1

14

12

08

06

04

02

Simulation time (s)

Reac

tive p

ower

(pu

)

AdaptivePIDesired


0 2 4 6 8 10

Simulation time (s)

Reac

tive p

ower

(pu

)


0

1

08

07

09

06

05

03

04

02

01






0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)



0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)


AdaptivePIDesired


5 Conclusions




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















1

11987911198792

10038161003816100381610038161003816100381610038161003816le 1198862lt infin

1003816100381610038161003816(1198791 + 1198792)1003816100381610038161003816 le 1198863 lt infin (29)



|119871 (sdot)| le 119886120593 (119902119899) (30)

where

119886 = max (1198862sdot 1198861 1198862 1198862sdot 1198863 1) (31)

120593 (119902119899) = 1 +

10038161003816100381610038161199021198991003816100381610038161003816 +

1003816100381610038161003816 1199021198991003816100381610038161003816 + 120573 | 119890| +

1003816100381610038161003816 119902lowast

119899

1003816100381610038161003816 (32)


119906 = minus (1198700+ ) 120576 (33a)


=119886120593

|120576| (33b)


119886 = |120576| 120593 (33c)



119881 =1

21205762+

1

2119861min(119886 minus 119886119861min)

2 (34)



= 120576 120576 minus (119886 minus 119886119861min)119886 (35)


120576 = minus1198611198700120576 minus 119861120576 + 119871 (36)



119886 (37)





(38)

Because we have

|119871 (sdot)| le 119886120593 0 lt 119861min le 119861 (39)

we can get

le minus119861min11987001205762lt 0 (40)



119886 = minus1205901119886 + 1205902

12057621205932

|120576| 120593 + 120591 1205901gt 0 120590

2gt 0 (41a)



119906 = minus (1198700+ ) 120576 (41b)


=119886120593

|120576| + 120591 (41c)




0gt 0 and 120573 gt 0




1 1198792


1 1198862 1198863 Such fact can









119865119902119899= 1 120573 = 1 and








0 2 4 6 8 100

1

14

12

08

06

04

02

Simulation time (s)

Reac

tive p

ower

(pu

)

AdaptivePIDesired


0 2 4 6 8 10

Simulation time (s)

Reac

tive p

ower

(pu

)


0

1

08

07

09

06

05

03

04

02

01






0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)



0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)


AdaptivePIDesired


5 Conclusions




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 1198792


1 1198862 1198863 Such fact can









119865119902119899= 1 120573 = 1 and








0 2 4 6 8 100

1

14

12

08

06

04

02

Simulation time (s)

Reac

tive p

ower

(pu

)

AdaptivePIDesired


0 2 4 6 8 10

Simulation time (s)

Reac

tive p

ower

(pu

)


0

1

08

07

09

06

05

03

04

02

01






0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)



0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)


AdaptivePIDesired


5 Conclusions




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)



0

1

14

12

08

06

04

02

Reac

tive p

ower

(pu

)


AdaptivePIDesired


5 Conclusions




Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article robust adaptive reactive power control...

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