Research Article Research on Improved Adaptive Control for...

Research ArticleResearch on Improved Adaptive Control for Static SynchronousCompensator in Power System

Chao Zhang1 Aimin Zhang2 Hang Zhang1 Yingsan Geng1 and Yunfei Bai1

1State Key Laboratory of Electrical Insulation and Power Equipment Xirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China2School of Electronic and Information Engineering Xirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China

Correspondence should be addressed to Aimin Zhang zhangammailxjtueducn

Received 5 March 2015 Revised 15 April 2015 Accepted 15 April 2015

Academic Editor Xiaosong Hu

Copyright copy 2015 Chao Zhang 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

This paper deals with the problems of ldquoexplosion of termrdquo uncertain parameter in static synchronous compensator (STATCOM)system with nonlinear time-delay An improved adaptive controller is proposed to enhance the transient stability of system statesand reduce computational complexity of STATCOM control system In contrast to backstepping control scheme in high ordersystems the problem of ldquoexplosion of termrdquo is avoided by designing dynamic surface controller The low pass filter is includedto allow a design where the model is not differentiated and thus has prevented the mathematical complexities effectively Inaddition unlike the traditional adaptive control schemes the certainty equivalence principle is not required for estimating theuncertain parameter by system immersion and manifold invariant (IampI) adaptive control A smooth function is added to ensurethat the estimation error converges to zero in finite timeThe effectiveness of the proposed controller is verified by the simulationsCompared with adaptive backstepping and proportion integration differentiation (PID) the oscillation amplitudes of transientresponse are reduced by nearly half and the time of reaching steady state is shortened by at least 11

1 Introduction

As amember of flexible alternate current transmission system(FACTS) static synchronous compensator (STATCOM) isbeing widely used to compensate the reactive power withfast response time by generating or absorbing reactive powercontinuously in a power system [1ndash3] Particularly in sus-tainable transportation systems the STATCOM can also beused such as energy storage [4] renewables integration [5]synergy between electric vehicles and power systems [6] Itis an effective way of reducing voltage flicker emissions atthe point of common coupling (PCC) removing the externalfluctuations and improving the transient stability of systemstates However as dealing with the real STATCOM controlproblem the designers are unavoidably to face the difficultiesinvolving uncertain parameter nonlinear time delay andcomplex mathematical models The nonlinear time-delay iscaused by smoothing the harmonic currents and harmonicvoltage by using the digital filtering calculatingmean value ofeach DC voltage and transmitting to protection control unit

The time-delay ranges from 001 s to 004 s Furthermorethe damping coefficient is difficult to measure accurately inpractice which can be seen as uncertain parameter Lastbut most important since the mathematical modeling ismore complicated the computing complexity is improvedsignificantly in designing STATCOM controller Therefore itis a hotspot to study an advanced reliable and low complexnonlinear control scheme for STATCOM

Recently adaptive backstepping control has been shownas an effective scheme to design STATCOM controllerswhile keeping better performance [7 8] This method isa vast of research which is adapted by some intelligentmethods for example least-squares estimation [9] and var-ious kalman filtering approaches [10ndash12] Some STATCOMcontrollers are designed by using backstepping techniqueinvolving the nonlinear robust control law and a new esti-mator to estimate the uncertain parameters [13ndash15] Howeverin the real STATCOM system nonlinear time-delay is animportant factor that these adaptive backstepping schemesdid not include Moreover the adaptive backstepping is

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 746903 9 pageshttpdxdoiorg1011552015746903

2 Mathematical Problems in Engineering

flawed with two shortcomings Frist the strong couplingbetween state variables and estimation errors existed inconstructing the control Lyapunov function (CLF) for esti-mating the uncertain parameter If the estimator is fixedthe estimation error will be accumulated and the errorof coupling term will in turn be accumulated as runningtime increases [16] Second backstepping technique involvesmodel differentiation in STATCOM control system andthus has suffered the ldquoexplosion of termrdquo So the com-putational processes of nonlinear controller leads to highcomplexity

The method named the system immersion and manifoldinvariant (IampI) adaptive control is developed in [17] Since theknowledge of the CLF is not required by this method theproblem of the oscillation of states caused by the couplingbetween state variables and estimation error can be avoidedIt has been proved that transient stability of the systemcan be guaranteed even if estimators reach the limit of itscapacity [14] Therefore for nonlinear STATCOM system inadaptive law design this IampI adaptive control can be adoptedto estimate uncertain parameter by designing a smoothfunction to offset the estimation error Moreover the stabilityand the convergence of the adaptive law are achieved byrequiring the estimation error to converge to zero in the finitetime

To overcome the ldquoexplosion of termrdquo caused by adaptivebackstepping control the method called dynamic surfacecontrol was developed to simplify the controller designwheremodel differentiation can be avoided in higher systems [18]The low pass filters are included to allow a design where themodel is not differentiated at the same time avoiding thehigh computing complexity caused by the ldquoexplosion of termrdquoFor a class of pure-feedback nonlinear systems the problemexplosion of complexity has been solved by using the dynamicsurface technique in [19] Moreover semiglobal uniformultimate boundedness of all signals is guaranteed in theclosed-loop system However the dynamic surface controlcannot be applied to nonlinear STATCOM systems withuncertain parameter it also did not consider the problem ofnonlinear time delay

In this paper an improved adaptive controller is proposedto improve stability of STATCOM system by simultane-ously addressing the problems involving ldquoexplosion of termrdquouncertain parameter and nonlinear time-delay The low passfilters are included to allow a design where the model isnot differentiated based on dynamic surface control at thesame time avoiding the ldquoexplosion of termrdquo caused by modeldifferentiation Furthermore based on the IampI adaptivecontrol the existing uncertain parameter is estimated inadaptive law design The designed adaptive law can ensurethat the estimation error converges to zero in finite timeMoreover the nonnegative time-delay function is introducedto overcome the effect of nonlinear time delay and achievethe best possible control performance It is proved that allthe state variables are globally bounded and converge tothe equilibrium points by using the proposed controllerThe simulation results show that transient stability and theconvergence speed of the system state variables are improvedeffectively by the proposed controller

G T

STATCOM

Inverter

X1X2

VS

Figure 1 STATCOM single machine infinite system model

2 System Model and Control Objective

Consider the single-machine infinite-bus system with STAT-COM shown in Figure 1 [20] where 119866 is the alternatingcurrent generator and 119879 is the transformer

It is clearly shown that the STATCOM is installed on thegrid which can instantly and continuously provide variablereactive power in response to voltage transients supportingthe stability of grid voltage

Its mathematical equivalent system dynamic model canbe expressed in (1) by the following nonlinear differentialequations

= 120596 minus 1205960

=1205960

119867

[[[

[

119875119898minus

1198641015840

119902119881119904sin 120575

1198831+ 1198832

(1

+11988311198832119868119902

radic(11988321198641015840119902)2

+ (1198831119881119904)2+ 2119883111988321198641015840119902119881119878cos 120575

)

minus119863

119867(120596 minus 120596

0)]]]

]

119902=

1

119879119902

(minus119868119902(119905 minus 119889) + 119868

1199020+ 119906119861)

(1)

where 119889 is delay time and the three state variables aregenerator rotor angle 120575 generator rotor angular speed 120596 andreactive current 119868

119902where transient responses will be tracked

It is noted that the vector [1205750 1205960 1198681199020]119879 is the steady-state

operating point This implies that the steady operation pointis the desired value or objective value [20 21]The parametersin (1) are expressed in Appendix A

To simplify model (1) three state variables are redefinedas 1199091= 120575 minus 120575

0 1199092= 120596 minus 120596

0 and 119909

3= 119868119902minus 1198681199020 The model (1)

can be rewritten as

1= 1199092

2= 1205791199092+ 1198961119875119898

Mathematical Problems in Engineering 3

minus 1198962sin (1205750+ 1199091) [1 + 119891 (119909

1) (1199093+ 1198681199020)]

3= 1198963(minus1199093(119905 minus 119889) + 119906

119861)

(2)

where

119891 (1199091)

=11988311198832

radic(11988321198641015840119902)2

+ (1198831119881119878)2+ 2119883111988321198641015840119902119881119878cos (119909

1+ 1205750)

1198961=1205960

119867

1198962=

12059601198641015840

119902119881119878

119867(1198831+ 1198832)

1198963=

1

119879119902

(3)

The damping coefficient 119863 cannot be measured accu-rately in STATCOM system and the inertia 119867 is a constantTherefore the expression 120579 = minus119863119867 is also the uncertainparameter

The objective of designing STATCOM controller is toguarantee that all the state variables are globally bounded andconverge to the desired points This implies that generatorrotor angle generator rotor angular speed and reactivecurrent of the STATCOM can be adjusted to the equilibriumsin the finite time

3 Design of STATCOM Controller

Three are three sections to introduce our proposed controllerin designing robust controller In Section 31 IampI adaptivecontrol is adopted for designing adaptive law In Section 32dynamic surface control is used for designing control lawIn Section 33 the stability of STATCOM control system isverified

31 Design of the Adaptive Law The method IampI adaptivecontrol can be adopted to the estimate uncertain parameterwith the adaptive law design By adopting this method theuncertain parameter is estimated in the following steps

Define a manifold as

119890120579= 120579 minus 120579 + 120573 (119909

1 1199092) (4)

where 120579 is the uncertain parameter 120579 is the estimation valueof 120579 and 120573(119909

1 1199092) is the smooth function to be designedThe

derivative of (4) is

119890120579=

120579 +

2

sum

119896=1

120597120573

120597119909119896

times 119896=

120579 +

120597120573

1205971199091

1199092+

120597120573

1205971199092

(1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)))

(5)

In order to cancel the parameter-independent terms 120579 is

designed as

120579 = minus

120597120573

1205971199091

1199092minus

120597120573

1205971199092

((120579 + 120573) 1199092+ 1198961119875119898

sdot minus1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)))

(6)

Substituting (6) into (5) (5) can be rewritten as

119890120579= minus

120597120573

1205971199092

1198901205791199092 (7)

Lemma 1 Define a candidate Lyapunov function (CLF)

119881 (119890120579) =

1

2119890120579

2 (8)

By selecting the smooth function 120573(1199091 1199092) we have

lim119905rarrinfin

119890120579(119905) = 0

Proof Theoretically we have large flexibility in selecting120573(1199091 1199092) For simplicity we let 120573(119909

1 1199092) = (12)120588119909

2

2 with120588 gt 0

(119890120579) = 119890120579119890120579= minus

120597120573

1205971199092

119890120579

21199092= minus119890120579

21199092

2le 0 (9)

Since the derivative of the CLF (8) is negative semidefinitethe manifold 119890

120579can converge to zero in finite time based on

Lyapunov theorem As a result we have lim119905rarrinfin

119890120579(119905) = 0

Lemma 1 holds

Remark 2 By using the designed smooth function themanifold 119890

120579can converge to zero in finite time that is

lim119905rarrinfin

119890120579(119905) = 0 based on Lyapunov stability theorem

Therefore based on the theory of immersion and manifoldinvariant (IampI) adaptive control the manifold 119890

120579(119905) = 120579 minus

120579 + 120573(1199091 1199092) = 0 is invariant when lim

119905rarrinfin119890120579(119905) = 0 and

thus the parametric form manifold 119868119890= (119909 120579) isin 119877

3times 1198771|

120579 minus 120579 + 120573(1199091 1199092) = 0 is invariant and attractive [9 10]

32 Design of the Control Law Based on dynamic surfacecontrol we can design control law (119906

119861) in three steps

Step 1 Error variables 119911119894(119894 = 1 2 3) can be defined as the

following

1199111= 1199091

1199112= 1199092minus 1199092

lowast

1199113= 1199093minus 1199093

lowast

(10)


where 1199091 1199092 and 119909

3are the virtual controls and 119909

2

lowast and 1199093

lowast

are the stabilizing functions The derivative of error variables119911119894(119894 = 1 2 3) with (2) is

1= 1199092

2= 2minus 2

lowast

= 1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)) minus

2

lowast

3= 3minus 3

lowast= 1198963(minus1199093(119905 minus 119889) + 119906

119861) minus 3

lowast

(11)

Choose the first CLF as

1198811=1

21199111

2 (12)

The derivative of 1198811along with (12) is

1= 11991111199112+ 11991111199092

lowast (13)

Take the stabilizing function 1199092

lowast as

1199092

lowast= minus11988811199091 (14)

where 1198881is a nonnegative constant It can be seen clearly that

1le 0 if 119911

2= 0

Step 2 The second energy storage function with respect toLyapunov function is

1198812=1

21199111

2+1

21199112

2 (15)

The derivative of 1198812is

2= 11991111+ 11991122 (16)

Substituting (11) and (14) into (16) the equation above ismanipulated as

2= 11991111+ 11991122= 11991111199112minus 11988811199111

2+ 1199112(2minus 2

lowast)

= 11991111199112minus 11988811199111

2+ 1199112(2+ 11988811199112minus 1198881

21199111) = (1 minus 119888

1

2)

sdot 11991111199112minus 11988811199111

2+ 1199112[1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)) + 11988811199112]

(17)

To guarantee this second-order subsystem satisfying Lya-punov stability the stabilizing function 119909

3

lowast must make (17)satisfy the inequality that 119881

2le 0 And then 119909

3

lowast is

1199093

lowast

= (

(120579 + 120573) 1199092+ 1198961119875119898+ (1 minus 119888

1

2) 1199111+ 11988811199112+ 11988821199112

1198962119891 (1199091) sin (120575

0+ 1199091)

)

minus1

119891 (1199091)minus 1198681199020

(18)

Remark 3 It would be a tremendous expansion of terms ifderivative of (18) is calculatedThe problem of high computa-tional complexity can be caused in the following control lawdesign by using backstepping control The dynamic surfacecontrol can be introduced to design the control law and solvethe problem of ldquoexplosion of termrdquo

The low pass filter 1(120591119904 + 1) is included to design con-trol law without model differentiation which can avoid theproblem of ldquoexplosion of termrdquo that has made other methodsdifficult to implement in practice

The stabilizing function 1199093

lowast is the output of low-passfilter and the119909

3is the input of low-pass filterThe relationship

between 1199093

lowast and 1199093is

120591 (1199093

lowast)1015840+ 1199093

lowast= 1199093

1199093

lowast(0) = 119909

3 (0)

(19)

From (19) we can obtain (1199093

lowast)1015840= (1199093minus 1199093

lowast)120591 The

filtering error can be defined as

119910 = 1199093

lowastminus 1199093 (20)

A CLF involving time-delay nonlinearity error variablesand filtering error is designed as

1198813=1

21199111

2+1

21199112

2+1

21199113

2+1

21199102+ int

119905

119905minus119889

119902 (119909 (120572)) 119889120572 (21)

where 119902(119909(119905)) is a nonnegative function We have the deriva-tive of (21) being

3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))

minus 119902 (119909 (119905 minus 119889))

(22)

Substituting 3= 1198963(minus1199093(119905)+119906

119861)minus3

lowast = (1199093minus119909lowast

3)120591minus

3= minus119910120591 + 119861

3into (22) we can obtain

3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))

minus 119902 (119909 (119905 minus 119889)) = (1 minus 1198881

2) 11991111199112minus 11988811199111

2+ 1199112[1205791199092

+ 1198961119875119898minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020))

+ 11988811199112] + 1199113(1198963(minus1199093 (119905 minus 119889) + 119906

119861) minus 3

lowast) + 119910 (

minus119910

120591

+ 1198613) + 119902 (119909 (119905)) minus 119902 (119909 (119905 minus 119889))

(23)

where 1198613= minus3 Define ℎ(119909

3(119905 minus 119889)) = |119896

31199093(119905 minus 119889)| where

ℎ(1199093(119905minus119889)) is a nonnegative time-delay functionwhich can be

compensated in the adaptive nonlinear controller designThenonnegative function can be defined as 119902(119909(119905)) = |119896

311991131199093(119905)|

which is a reduced form of satisfying Lyapunov stability butnot the only form

Based on Cauchy-Schwartz inequality theorem a rela-tional expression can be obtained

minus1199113ℎ (1199093(119905 minus 119889)) le

1003816100381610038161003816119896311991131199093 (119905 minus 119889)1003816100381610038161003816

le10038161003816100381610038161199113

1003816100381610038161003816100381610038161003816100381611989631199093 (119905 minus 119889)

1003816100381610038161003816

(24)


By 119902(119909(119905)) = |119896311991131199093(119905)| we can get 119902(119909(119905 minus 119889)) =

|119896311991131199093(119905 minus 119889)| Substituting 119902(119909(119905)) 119902(119909(119905 minus 119889)) and (24) into

(23) we can obtain

3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))

minus 119902 (119909 (119905 minus 119889)) le (1 minus 1198881

2) 11991111199112minus 11988811199111

2+ 1199112[1205791199092

+ 1198961119875119898minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020))

+ 11988811199112] + 1199113(1198963119906119861minus 3

lowast) + 119910 (

minus119910

120591+ 1198613)

+1003816100381610038161003816119896311991131199093 (119905)

1003816100381610038161003816

(25)

The control law is designed as

119906119861=

1

1198963

3

lowastminus

1198883

1198963

1199113+ 120582

10038161003816100381610038161199093 (119905)1003816100381610038161003816 (26)

where 1198883gt 0 and 120582 is a sign function which is defined as

120582 = minus1 when 1199113gt 0 and 120582 = 1 when 119911

3lt 0

33 Proof of System Stability

Lemma 4 All of state variables of the closed-loop system arebounded and converge to the equilibrium point if 119881(0) le 119901119901 gt 0

Proof Let 119881 = (12)1199111

2+ (12)119911

2

2+ (12)119911

3

2+ (12)119910

2= 119901

1198613is bounded which is denoted as 119872

3 and then we have

1198613

21198723

2minus 1 le 0 Substituting (18) and (26) to (25) we can

obtain

= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988831199113

2+ 119910(

minus119910

120591+ 1198613)

(27)

Based on Cauchy-Schwartz inequality theorem (27) canbe rewritten as follows

= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1003816100381610038161003816119910100381610038161003816100381610038161003816100381610038161198613

1003816100381610038161003816

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1

21198613

21199102

+1

2

= minus11988811199111

2minus 11988821199112

2minus 11988831199113

2minus 11991121199092119890120579

+ (1

21198613

2minus1

120591) 1199102+1

2

(28)

By designing 1198881ge 119903 1198882ge 119903 1198883ge 119903 and 1120591 ge (12)119872

3

2+119903

119903 ge 0 we can obtain

le minus1199031199111

2minus 1199031199112

2minus 1199031199113

2+ (

1

21198613

2minus1198723

2

2minus 119903)119910

2

+1

2= minus2119903119881 + (

1198723

21198723

1198613

2minus1198723

2

2)1199102+1

2

= minus2119903119881 + (1198613

2

1198723

minus 1)1198723

21199102

2+1

2

(29)

Substituting 119903 ge 14119901 into (29) (29) can be rewritten as

le minus21

4119901119901 +

1

2= 0 (30)

From (30) we have 119881(119905) le 119901 if 119881(0) le 119901 where 119905 ge 0Lemma 4 holds

In addition the convergence analysis is also given From(29) we can get

le minus2119903119881 +1

2 (31)

Solve this differential equation as

119881 le1

4119903+ (119881 (0) minus

1

4119903) 119890minus2119903119905

(32)

If 119905 rarr infin 119881 rarr 14119903 and then we have 119881 rarr 14119903when 119903 rarr infin Furthermore due to 1120591 ge (12)119872

3

2+12+119903

we can obtain 119903 rarr infin when 120591 rarr 0 It is an important basisfor design of low pass filter 1(120591119904 + 1)

4 Simulation Results and Discussion

In this section the simulation model of adaptive nonlinearcontroller has been established under theMATLABSimulinkenvironment for nonlinear STATCOM with nonlinear time-delay The parameters in (1) are given as follows

Consider119867 = 8 s 1198641015840119902= 1108 pu 119875

119898= 10 kw 119881

119904= 1 pu

1198831= 084 pu119883

2= 052 pu 119879

119902= 003 s 119901 = 2 119888

1= 1 1198882= 1

1198883= 1 and 119889 = 002 s 004 s The steady operation points are

given as 1205750= 571

∘ 1205960= 314159 rads 119868

0= 0 and 119910(0) = 0

The transient responses of the nonlinear STATCOMsystem with time-delay are then discussed A comparisonanalysis with the conventional nonlinear controller is alsoprovided under the same conditions

(1) Different Control Approaches The comparison betweenthe proposed dynamic surface method for nonlinear STAT-COM with time-delay based on system immersion andmanifold invariant methodology (DSMII) approach andtwo approaches involving adaptive backstepping (AB) [21]and proportion integration differentiation (PID) [22] wereinvestigated when 119889 = 002 s

Figures 2(a)ndash2(c) show the comparison between theproposed controller and the two controllers when 119889 = 002 s


5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

DSMII

120575(∘

)

ABPID

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

DSMIIABPID

(b)

0 02 04 06 08 1 12 14 16 1804

06

08

1

12

14

16

18

2

Time (s)

I q(a

)

DSMIIABPID

(c)

Figure 2 (a) Transient responses of rotor angle when 119889 = 002 s (b) Transient responses of rotor angular speed when 119889 = 002 s (c) Transientresponses of reactive current when 119889 = 002 s

For the proposed controller it can be clearly seen that theconvergences of transient responses trajectories are achievedand the system tend to be stable state more rapidly after avery short time Taking Figure 2(c) for example the transientresponses fluctuate fast and tend to be stable after 16 s ormoreunder AB and PID Instead by using the proposed controllertransient responses fluctuate more smoothly and converge tostable state after 06 s suggesting that the proposed controllerresults in better system performance

(2) Different Time Delay Simulations of our proposed con-troller are performed at 119889

1= 002 and 119889

1= 004 respectively

In Figures 3(a)ndash3(c) we simulated the model in twodifferent delay times to investigate its influences The tran-sient trajectories depart from the initial state and fluctuatestrongly without an appropriate control From the compar-ison between Figures 2(a)ndash2(c) and Figures 3(a)ndash3(c) alltransient trajectories fluctuate faster and system reaches thestable state more quickly when 119889 = 002 s Moreover more

time is spent for the transient responses to converge to thestable state when 119889 = 004 s It is noted that the transienttrajectories fluctuate powerfully and cannot reach steady statein finite time under AB Consequently the delay time 119889 is acrucial nonlinear factor impacting the transient and steadyperformance of the STATCOM system A larger 119889 can resultin a poorer robustness and worse convergence This result isconsistent with the theoretical analysis

5 Conclusions

This paper presents an improved adaptive controller toaddress the problems of ldquoexplosion of termrdquo and uncertainparameter in static synchronous compensator (STATCOM)with nonlinear time delay Improvements are achieved inthree aspects as follows

(1) The uncertain parameter is estimated by IampI adaptivecontrol in designing adaptive law which can ensure


5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

120575(∘

)

d1 = 002 (s)d2 = 004 (s)

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

d1 = 002 (s)d2 = 004 (s)

(b)

00

02

02

04 06 08 1 12 14 16 18

04

06

08

1

12

14

2

Time (s)

I q(a

)

d1 = 002 (s)d2 = 004 (s)

(c)

Figure 3 (a) Transient responses of rotor angle in different time delay (b) Transient responses of rotor angular speed in different time delay(c) Transient responses of reactive current in different time delay

that the estimation error converges to zero in finitetime

(2) With regards to ldquoexplosion of termrdquo caused by back-stepping technology a low pass filter is included toallow a design where the model is not differentiatedby using dynamic surface control

(3) Furthermore the proposed method can add a non-negative time-delay function to compensate the time-delay term which can avoid the influence of time-delay term and achieve the best possible controlperformance

By comparing with some conventional controller theproposed controller has advantages in terms of enhancingtransient stability and reducing computational complexitySimulations results show that the proposed controller notonly is insensitive to time-delay term but also reduces theconvergence time and oscillation amplitude

Appendices

A Nomenclature

120575 Generator rotor angle120596 Generator rotor angular speed119868119902 Reactive current

119889 Time delay119888 Adjustable parameter119910 Output119867 Inertia constant1198641015840

119902 Transient electromotive force

119875119898 Mechanical power

119863 Damping coefficient11988312 Equivalent impedance

119906119861 Equivalence input


119879119902 Time constant

119881119904 Infinite bus voltage

B AB and PID Controller

The AB controller with the control law is

119906119861= 1199093+ 119879

[1

1198991

11989811199092+ (1198982+ 120579) (120579119909

2+ 1198961119875119898

minus 1198991(1 + 119891 (119909

1) (1199093+ 1198681199020))) +

1205791199092]

minus1

11989912[11990921198992(11989811199091+ 11989821199092+ 1205791199092+ 1198961119875119898)]

sdot1

119891 (1199091)+11989911198963119891 (1199091) 1199092

1198961

[1

1198991

(11989811199091+ 11989821199092

+ 1205791199092+ 1198961119875119898) minus 1] minus [

(1198982+ 120579)

1198991119891 (1199091) 120574

]

2

1198903minus1198903

1205742

minus 120583100381610038161003816100381611989631199093 (119905)

1003816100381610038161003816

(B1)

The parameters in simulations are given as follows119867 = 8 s 1198641015840

119902= 1108 pu 119879

119902= 003 s 119888

1= 2 120574 = 02 120588 = 2

1199021= 04 119902

2= 06 120590 = 1 1198641015840

119902= 1108 pu119881

119904= 1119883

1= 084 pu

1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and 119868

0= 0

The PID controller with the control law is

119906119861= 119877 (119883) +

119867119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575

sdot (minus119896119897intΔ1205961015840119889119905 minus 119896

119901Δ1205961015840minus

119896119889119889 (Δ120596

1015840)

119889119905)

(B2)

where

119877 (119883) =119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575

119898minus

119883Σ

2119879119902119863

119864101584011990211988111990411988311198832sin 120575

Δ

minus119883Σ119879119902con120575

11988311198832sin 120575

Δ120596 minus 119868119902+ 1198681199020

(B3)

The PID parameter is set as

[119896119889 119896119901 119896119897]119879

= [13 315 42]119879 (B4)

The parameters in simulations are given as follows 119867 =

8 s 1198641015840119902= 1108 pu 119879

119902= 003 s 119875

119898= 10 kw 119881

119904= 1 pu 119883

1=

084 pu 1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and

1198680= 0

Conflict of Interests

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

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (51177126 61105126) Major Technologi-cal Innovation Project Special Fund of Shaanxi Province(2008ZKC01-09) andApplied Fundamental Research Project(SYG201201)

References

[1] P Rao M L Crow and Z Yang ldquoSTATCOM control forpower system voltage control applicationsrdquo IEEE Transactionson Power Delivery vol 15 no 4 pp 1311ndash1317 2000

[2] A Astolfi D Karagiannis and R OrtegaNonlinear and Adapt-ive Control with Applications Springer Science amp BusinessMedia 2007

[3] W Song and A Q Huang ldquoFault-tolerant design and con-trol strategy for cascaded H-bridge multilevel converter-basedSTATCOMrdquo IEEE Transactions on Industrial Electronics vol 57no 8 pp 2700ndash2708 2010

[4] X Hu S Li H Peng and F Sun ldquoCharging time and lossoptimization for LiNMC and LiFePO

4batteries based on

equivalent circuit modelsrdquo Journal of Power Sources vol 239pp 449ndash457 2013

[5] X S Hu N Murgovski L M Johannesson and B EgardtldquoComparison of three electrochemical energy buffers applied toa hybrid bus powertrain with simultaneous optimal sizing andenergy managementrdquo IEEE Intelligent Transportation SystemsMagazine vol 15 no 3 pp 1193ndash1205 2014

[6] L Zhang ZWang X Hu F Sun and D G Dorrell ldquoA compar-ative study of equivalent circuit models of ultracapacitors forelectric vehiclesrdquo Journal of Power Sources vol 274 pp 899ndash906 2015

[7] M A Mohd Basri A R Husain and K A DanapalasingamldquoIntelligent adaptive backstepping control for MIMO uncertainnon-linear quadrotor helicopter systemsrdquo Transactions of theInstitute of Measurement and Control vol 37 no 3 pp 345ndash3612015

[8] J Zhou C Wen and Y Zhang ldquoAdaptive backstepping con-trol of a class of uncertain nonlinear systems with unknownbacklash-like hysteresisrdquo IEEE Transactions on Automatic Con-trol vol 49 no 10 pp 1751ndash1757 2004

[9] X-S Hu F-C Sun and Y Zou ldquoOnline model identificationof lithium-ion battery for electric vehiclesrdquo Journal of CentralSouth University of Technology vol 18 no 5 pp 1525ndash1531 2011

[10] X S Hu F Sun and Y Zou ldquoComparison between two model-based algorithms for Li-ion battery SOC estimation in electricvehiclesrdquo Simulation Modelling Practice andTheory vol 34 pp1ndash11 2013

[11] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash3540 2011

[12] L Zhang ZWang F Sun andDGDorrell ldquoOnline parameteridentification of ultracapacitor models using the extendedKalman filterrdquo Energies vol 7 no 5 pp 3204ndash3217 2014

[13] Y Zhang CWen andY C Soh ldquoAdaptive backstepping controldesign for systems with unknown high-frequency gainrdquo IEEETransactions on Automatic Control vol 45 no 12 pp 2350ndash2354 2000

[14] H Farokhi Moghadam and N Vasegh ldquoRobust PID stabiliza-tion of linear neutral time-delay systemsrdquo International Journal


of Computers Communications amp Control vol 9 no 2 pp 201ndash208 2014

[15] L Zhang A Zhang Z Ren G Li C Zhang and J HanldquoHybrid adaptive robust control of static var compensator inpower systemsrdquo International Journal of Robust and NonlinearControl vol 24 no 12 pp 1707ndash1723 2014

[16] D Karagiannis and A Astolfi ldquoNonlinear adaptive control ofsystems in feedback form an alternative to adaptive backstep-pingrdquo SystemsampControl Letters vol 57 no 9 pp 733ndash739 2008

[17] A Astolfi and R Ortega ldquoImmersion and invariance a newtool for stabilization and adaptive control of nonlinear systemsrdquoIEEE Transactions on Automatic Control vol 48 no 4 pp 590ndash606 2003

[18] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000

[19] S J Yoo J B Park and Y H Choi ldquoAdaptive dynamic surfacecontrol for stabilization of parametric strict-feedback nonlinearsystems with unknown time delaysrdquo IEEE Transactions onAutomatic Control vol 52 no 12 pp 2360ndash2364 2007

[20] N Jiang S Li T Liu andXDong ldquoNonlinear large disturbanceattenuation controller design for the power systems with STAT-COMrdquo Applied Mathematics and Computation vol 219 no 20pp 10378ndash10386 2013

[21] W L Li Y W Jing X P Liu and B Wang ldquoNonlinear robustcontrol based on adaptive backstepping design for STATCOMrdquoJournal of Northeastern University (Natural Science) vol 24 pp221ndash224 2003

[22] C Zhang Z Aimin Z Hang et al ldquoAn advanced adaptivebackstepping control method for STATCOMrdquo in Proceedings ofthe 26th Chinese Control And Decision Conference (CCDC rsquo14)pp 1822ndash1827 IEEE Changsha China May 2014

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


flawed with two shortcomings Frist the strong couplingbetween state variables and estimation errors existed inconstructing the control Lyapunov function (CLF) for esti-mating the uncertain parameter If the estimator is fixedthe estimation error will be accumulated and the errorof coupling term will in turn be accumulated as runningtime increases [16] Second backstepping technique involvesmodel differentiation in STATCOM control system andthus has suffered the ldquoexplosion of termrdquo So the com-putational processes of nonlinear controller leads to highcomplexity

The method named the system immersion and manifoldinvariant (IampI) adaptive control is developed in [17] Since theknowledge of the CLF is not required by this method theproblem of the oscillation of states caused by the couplingbetween state variables and estimation error can be avoidedIt has been proved that transient stability of the systemcan be guaranteed even if estimators reach the limit of itscapacity [14] Therefore for nonlinear STATCOM system inadaptive law design this IampI adaptive control can be adoptedto estimate uncertain parameter by designing a smoothfunction to offset the estimation error Moreover the stabilityand the convergence of the adaptive law are achieved byrequiring the estimation error to converge to zero in the finitetime

To overcome the ldquoexplosion of termrdquo caused by adaptivebackstepping control the method called dynamic surfacecontrol was developed to simplify the controller designwheremodel differentiation can be avoided in higher systems [18]The low pass filters are included to allow a design where themodel is not differentiated at the same time avoiding thehigh computing complexity caused by the ldquoexplosion of termrdquoFor a class of pure-feedback nonlinear systems the problemexplosion of complexity has been solved by using the dynamicsurface technique in [19] Moreover semiglobal uniformultimate boundedness of all signals is guaranteed in theclosed-loop system However the dynamic surface controlcannot be applied to nonlinear STATCOM systems withuncertain parameter it also did not consider the problem ofnonlinear time delay

In this paper an improved adaptive controller is proposedto improve stability of STATCOM system by simultane-ously addressing the problems involving ldquoexplosion of termrdquouncertain parameter and nonlinear time-delay The low passfilters are included to allow a design where the model isnot differentiated based on dynamic surface control at thesame time avoiding the ldquoexplosion of termrdquo caused by modeldifferentiation Furthermore based on the IampI adaptivecontrol the existing uncertain parameter is estimated inadaptive law design The designed adaptive law can ensurethat the estimation error converges to zero in finite timeMoreover the nonnegative time-delay function is introducedto overcome the effect of nonlinear time delay and achievethe best possible control performance It is proved that allthe state variables are globally bounded and converge tothe equilibrium points by using the proposed controllerThe simulation results show that transient stability and theconvergence speed of the system state variables are improvedeffectively by the proposed controller

G T

STATCOM

Inverter

X1X2

VS

Figure 1 STATCOM single machine infinite system model

2 System Model and Control Objective

Consider the single-machine infinite-bus system with STAT-COM shown in Figure 1 [20] where 119866 is the alternatingcurrent generator and 119879 is the transformer

It is clearly shown that the STATCOM is installed on thegrid which can instantly and continuously provide variablereactive power in response to voltage transients supportingthe stability of grid voltage

Its mathematical equivalent system dynamic model canbe expressed in (1) by the following nonlinear differentialequations

= 120596 minus 1205960

=1205960

119867

[[[

[

119875119898minus

1198641015840

119902119881119904sin 120575

1198831+ 1198832

(1

+11988311198832119868119902

radic(11988321198641015840119902)2

+ (1198831119881119904)2+ 2119883111988321198641015840119902119881119878cos 120575

)

minus119863

119867(120596 minus 120596

0)]]]

]

119902=

1

119879119902

(minus119868119902(119905 minus 119889) + 119868

1199020+ 119906119861)

(1)

where 119889 is delay time and the three state variables aregenerator rotor angle 120575 generator rotor angular speed 120596 andreactive current 119868

119902where transient responses will be tracked

It is noted that the vector [1205750 1205960 1198681199020]119879 is the steady-state

operating point This implies that the steady operation pointis the desired value or objective value [20 21]The parametersin (1) are expressed in Appendix A

To simplify model (1) three state variables are redefinedas 1199091= 120575 minus 120575

0 1199092= 120596 minus 120596

0 and 119909

3= 119868119902minus 1198681199020 The model (1)

can be rewritten as

1= 1199092

2= 1205791199092+ 1198961119875119898


minus 1198962sin (1205750+ 1199091) [1 + 119891 (119909

1) (1199093+ 1198681199020)]

3= 1198963(minus1199093(119905 minus 119889) + 119906

119861)

(2)

where

119891 (1199091)

=11988311198832

radic(11988321198641015840119902)2

+ (1198831119881119878)2+ 2119883111988321198641015840119902119881119878cos (119909

1+ 1205750)

1198961=1205960

119867

1198962=

12059601198641015840

119902119881119878

119867(1198831+ 1198832)

1198963=

1

119879119902

(3)







119890120579= 120579 minus 120579 + 120573 (119909

1 1199092) (4)




119890120579=

120579 +

2

sum

119896=1

120597120573

120597119909119896

times 119896=

120579 +

120597120573

1205971199091

1199092+

120597120573

1205971199092

(1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)))

(5)


designed as

120579 = minus

120597120573

1205971199091

1199092minus

120597120573

1205971199092

((120579 + 120573) 1199092+ 1198961119875119898

sdot minus1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)))

(6)


119890120579= minus

120597120573

1205971199092

1198901205791199092 (7)


119881 (119890120579) =

1

2119890120579

2 (8)


lim119905rarrinfin

119890120579(119905) = 0


1 1199092) = (12)120588119909

2

2 with120588 gt 0

(119890120579) = 119890120579119890120579= minus

120597120573

1205971199092

119890120579

21199092= minus119890120579

21199092

2le 0 (9)




119890120579(119905) = 0

Lemma 1 holds



lim119905rarrinfin



120579(119905) = 120579 minus


119905rarrinfin119890120579(119905) = 0 and


3times 1198771|





following

1199111= 1199091

1199112= 1199092minus 1199092

lowast

1199113= 1199093minus 1199093

lowast

(10)


where 1199091 1199092 and 119909


2

lowast and 1199093

lowast


1= 1199092

2= 2minus 2

lowast

= 1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)) minus

2

lowast

3= 3minus 3


119861) minus 3

lowast

(11)


1198811=1

21199111

2 (12)


1= 11991111199112+ 11991111199092

lowast (13)


lowast as

1199092

lowast= minus11988811199091 (14)


1le 0 if 119911

2= 0


1198812=1

21199111

2+1

21199112

2 (15)


2= 11991111+ 11991122 (16)


2= 11991111+ 11991122= 11991111199112minus 11988811199111

2+ 1199112(2minus 2

lowast)

= 11991111199112minus 11988811199111

2+ 1199112(2+ 11988811199112minus 1198881

21199111) = (1 minus 119888

1

2)

sdot 11991111199112minus 11988811199111

2+ 1199112[1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)) + 11988811199112]

(17)


3



3

lowast is

1199093

lowast

= (

(120579 + 120573) 1199092+ 1198961119875119898+ (1 minus 119888

1

2) 1199111+ 11988811199112+ 11988821199112

1198962119891 (1199091) sin (120575

0+ 1199091)

)

minus1

119891 (1199091)minus 1198681199020

(18)






between 1199093


120591 (1199093

lowast)1015840+ 1199093

lowast= 1199093

1199093

lowast(0) = 119909

3 (0)

(19)


lowast)1015840= (1199093minus 1199093

lowast)120591 The


119910 = 1199093



1198813=1

21199111

2+1

21199112

2+1

21199113

2+1

21199102+ int

119905

119905minus119889

119902 (119909 (120572)) 119889120572 (21)


3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))

minus 119902 (119909 (119905 minus 119889))

(22)


119861)minus3


3)120591minus

3= minus119910120591 + 119861


3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))


2) 11991111199112minus 11988811199111

2+ 1199112[1205791199092

+ 1198961119875119898minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020))

+ 11988811199112] + 1199113(1198963(minus1199093 (119905 minus 119889) + 119906

119861) minus 3

lowast) + 119910 (

minus119910

120591

+ 1198613) + 119902 (119909 (119905)) minus 119902 (119909 (119905 minus 119889))

(23)


3(119905 minus 119889)) = |119896

31199093(119905 minus 119889)| where



311991131199093(119905)|



minus1199113ℎ (1199093(119905 minus 119889)) le

1003816100381610038161003816119896311991131199093 (119905 minus 119889)1003816100381610038161003816

le10038161003816100381610038161199113

1003816100381610038161003816100381610038161003816100381611989631199093 (119905 minus 119889)

1003816100381610038161003816

(24)




(23) we can obtain

3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))


2) 11991111199112minus 11988811199111

2+ 1199112[1205791199092

+ 1198961119875119898minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020))

+ 11988811199112] + 1199113(1198963119906119861minus 3

lowast) + 119910 (

minus119910

120591+ 1198613)

+1003816100381610038161003816119896311991131199093 (119905)

1003816100381610038161003816

(25)


119906119861=

1

1198963

3

lowastminus

1198883

1198963

1199113+ 120582

10038161003816100381610038161199093 (119905)1003816100381610038161003816 (26)



3lt 0



Proof Let 119881 = (12)1199111

2+ (12)119911

2

2+ (12)119911

3

2+ (12)119910

2= 119901


3 and then we have

1198613

21198723


obtain

= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988831199113

2+ 119910(

minus119910

120591+ 1198613)

(27)


= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1003816100381610038161003816119910100381610038161003816100381610038161003816100381610038161198613

1003816100381610038161003816

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1

21198613

21199102

+1

2

= minus11988811199111

2minus 11988821199112

2minus 11988831199113

2minus 11991121199092119890120579

+ (1

21198613

2minus1

120591) 1199102+1

2

(28)


3

2+119903


le minus1199031199111

2minus 1199031199112

2minus 1199031199113

2+ (

1

21198613

2minus1198723

2

2minus 119903)119910

2

+1

2= minus2119903119881 + (

1198723

21198723

1198613

2minus1198723

2

2)1199102+1

2

= minus2119903119881 + (1198613

2

1198723

minus 1)1198723

21199102

2+1

2

(29)


le minus21

4119901119901 +

1

2= 0 (30)



le minus2119903119881 +1

2 (31)


119881 le1

4119903+ (119881 (0) minus

1

4119903) 119890minus2119903119905

(32)


3

2+12+119903




Consider119867 = 8 s 1198641015840119902= 1108 pu 119875

119898= 10 kw 119881

119904= 1 pu

1198831= 084 pu119883

2= 052 pu 119879

119902= 003 s 119901 = 2 119888

1= 1 1198882= 1


given as 1205750= 571

∘ 1205960= 314159 rads 119868

0= 0 and 119910(0) = 0





5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

DSMII

120575(∘

)

ABPID

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

DSMIIABPID

(b)

0 02 04 06 08 1 12 14 16 1804

06

08

1

12

14

16

18

2

Time (s)

I q(a

)

DSMIIABPID

(c)




1= 002 and 119889

1= 004 respectively



5 Conclusions




5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

120575(∘

)

d1 = 002 (s)d2 = 004 (s)

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

d1 = 002 (s)d2 = 004 (s)

(b)

00

02

02

04 06 08 1 12 14 16 18

04

06

08

1

12

14

2

Time (s)

I q(a

)

d1 = 002 (s)d2 = 004 (s)

(c)






Appendices

A Nomenclature












119906119861= 1199093+ 119879

[1

1198991

11989811199092+ (1198982+ 120579) (120579119909

2+ 1198961119875119898

minus 1198991(1 + 119891 (119909

1) (1199093+ 1198681199020))) +

1205791199092]

minus1

11989912[11990921198992(11989811199091+ 11989821199092+ 1205791199092+ 1198961119875119898)]

sdot1

119891 (1199091)+11989911198963119891 (1199091) 1199092

1198961

[1

1198991

(11989811199091+ 11989821199092

+ 1205791199092+ 1198961119875119898) minus 1] minus [

(1198982+ 120579)

1198991119891 (1199091) 120574

]

2

1198903minus1198903

1205742

minus 120583100381610038161003816100381611989631199093 (119905)

1003816100381610038161003816

(B1)


119902= 1108 pu 119879

119902= 003 s 119888

1= 2 120574 = 02 120588 = 2

1199021= 04 119902

2= 06 120590 = 1 1198641015840

119902= 1108 pu119881

119904= 1119883

1= 084 pu

1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and 119868

0= 0


119906119861= 119877 (119883) +

119867119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575


119901Δ1205961015840minus

119896119889119889 (Δ120596

1015840)

119889119905)

(B2)

where

119877 (119883) =119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575

119898minus

119883Σ

2119879119902119863

119864101584011990211988111990411988311198832sin 120575

Δ

minus119883Σ119879119902con120575

11988311198832sin 120575

Δ120596 minus 119868119902+ 1198681199020

(B3)


[119896119889 119896119901 119896119897]119879

= [13 315 42]119879 (B4)


8 s 1198641015840119902= 1108 pu 119879

119902= 003 s 119875

119898= 10 kw 119881

119904= 1 pu 119883

1=

084 pu 1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and

1198680= 0



Acknowledgments


References





4batteries based on





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus 1198962sin (1205750+ 1199091) [1 + 119891 (119909

1) (1199093+ 1198681199020)]

3= 1198963(minus1199093(119905 minus 119889) + 119906

119861)

(2)

where

119891 (1199091)

=11988311198832

radic(11988321198641015840119902)2

+ (1198831119881119878)2+ 2119883111988321198641015840119902119881119878cos (119909

1+ 1205750)

1198961=1205960

119867

1198962=

12059601198641015840

119902119881119878

119867(1198831+ 1198832)

1198963=

1

119879119902

(3)







119890120579= 120579 minus 120579 + 120573 (119909

1 1199092) (4)




119890120579=

120579 +

2

sum

119896=1

120597120573

120597119909119896

times 119896=

120579 +

120597120573

1205971199091

1199092+

120597120573

1205971199092

(1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)))

(5)


designed as

120579 = minus

120597120573

1205971199091

1199092minus

120597120573

1205971199092

((120579 + 120573) 1199092+ 1198961119875119898

sdot minus1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)))

(6)


119890120579= minus

120597120573

1205971199092

1198901205791199092 (7)


119881 (119890120579) =

1

2119890120579

2 (8)


lim119905rarrinfin

119890120579(119905) = 0


1 1199092) = (12)120588119909

2

2 with120588 gt 0

(119890120579) = 119890120579119890120579= minus

120597120573

1205971199092

119890120579

21199092= minus119890120579

21199092

2le 0 (9)




119890120579(119905) = 0

Lemma 1 holds



lim119905rarrinfin



120579(119905) = 120579 minus


119905rarrinfin119890120579(119905) = 0 and


3times 1198771|





following

1199111= 1199091

1199112= 1199092minus 1199092

lowast

1199113= 1199093minus 1199093

lowast

(10)


where 1199091 1199092 and 119909


2

lowast and 1199093

lowast


1= 1199092

2= 2minus 2

lowast

= 1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)) minus

2

lowast

3= 3minus 3


119861) minus 3

lowast

(11)


1198811=1

21199111

2 (12)


1= 11991111199112+ 11991111199092

lowast (13)


lowast as

1199092

lowast= minus11988811199091 (14)


1le 0 if 119911

2= 0


1198812=1

21199111

2+1

21199112

2 (15)


2= 11991111+ 11991122 (16)


2= 11991111+ 11991122= 11991111199112minus 11988811199111

2+ 1199112(2minus 2

lowast)

= 11991111199112minus 11988811199111

2+ 1199112(2+ 11988811199112minus 1198881

21199111) = (1 minus 119888

1

2)

sdot 11991111199112minus 11988811199111

2+ 1199112[1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)) + 11988811199112]

(17)


3



3

lowast is

1199093

lowast

= (

(120579 + 120573) 1199092+ 1198961119875119898+ (1 minus 119888

1

2) 1199111+ 11988811199112+ 11988821199112

1198962119891 (1199091) sin (120575

0+ 1199091)

)

minus1

119891 (1199091)minus 1198681199020

(18)






between 1199093


120591 (1199093

lowast)1015840+ 1199093

lowast= 1199093

1199093

lowast(0) = 119909

3 (0)

(19)


lowast)1015840= (1199093minus 1199093

lowast)120591 The


119910 = 1199093



1198813=1

21199111

2+1

21199112

2+1

21199113

2+1

21199102+ int

119905

119905minus119889

119902 (119909 (120572)) 119889120572 (21)


3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))

minus 119902 (119909 (119905 minus 119889))

(22)


119861)minus3


3)120591minus

3= minus119910120591 + 119861


3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))


2) 11991111199112minus 11988811199111

2+ 1199112[1205791199092

+ 1198961119875119898minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020))

+ 11988811199112] + 1199113(1198963(minus1199093 (119905 minus 119889) + 119906

119861) minus 3

lowast) + 119910 (

minus119910

120591

+ 1198613) + 119902 (119909 (119905)) minus 119902 (119909 (119905 minus 119889))

(23)


3(119905 minus 119889)) = |119896

31199093(119905 minus 119889)| where



311991131199093(119905)|



minus1199113ℎ (1199093(119905 minus 119889)) le

1003816100381610038161003816119896311991131199093 (119905 minus 119889)1003816100381610038161003816

le10038161003816100381610038161199113

1003816100381610038161003816100381610038161003816100381611989631199093 (119905 minus 119889)

1003816100381610038161003816

(24)




(23) we can obtain

3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))


2) 11991111199112minus 11988811199111

2+ 1199112[1205791199092

+ 1198961119875119898minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020))

+ 11988811199112] + 1199113(1198963119906119861minus 3

lowast) + 119910 (

minus119910

120591+ 1198613)

+1003816100381610038161003816119896311991131199093 (119905)

1003816100381610038161003816

(25)


119906119861=

1

1198963

3

lowastminus

1198883

1198963

1199113+ 120582

10038161003816100381610038161199093 (119905)1003816100381610038161003816 (26)



3lt 0



Proof Let 119881 = (12)1199111

2+ (12)119911

2

2+ (12)119911

3

2+ (12)119910

2= 119901


3 and then we have

1198613

21198723


obtain

= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988831199113

2+ 119910(

minus119910

120591+ 1198613)

(27)


= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1003816100381610038161003816119910100381610038161003816100381610038161003816100381610038161198613

1003816100381610038161003816

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1

21198613

21199102

+1

2

= minus11988811199111

2minus 11988821199112

2minus 11988831199113

2minus 11991121199092119890120579

+ (1

21198613

2minus1

120591) 1199102+1

2

(28)


3

2+119903


le minus1199031199111

2minus 1199031199112

2minus 1199031199113

2+ (

1

21198613

2minus1198723

2

2minus 119903)119910

2

+1

2= minus2119903119881 + (

1198723

21198723

1198613

2minus1198723

2

2)1199102+1

2

= minus2119903119881 + (1198613

2

1198723

minus 1)1198723

21199102

2+1

2

(29)


le minus21

4119901119901 +

1

2= 0 (30)



le minus2119903119881 +1

2 (31)


119881 le1

4119903+ (119881 (0) minus

1

4119903) 119890minus2119903119905

(32)


3

2+12+119903




Consider119867 = 8 s 1198641015840119902= 1108 pu 119875

119898= 10 kw 119881

119904= 1 pu

1198831= 084 pu119883

2= 052 pu 119879

119902= 003 s 119901 = 2 119888

1= 1 1198882= 1


given as 1205750= 571

∘ 1205960= 314159 rads 119868

0= 0 and 119910(0) = 0





5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

DSMII

120575(∘

)

ABPID

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

DSMIIABPID

(b)

0 02 04 06 08 1 12 14 16 1804

06

08

1

12

14

16

18

2

Time (s)

I q(a

)

DSMIIABPID

(c)




1= 002 and 119889

1= 004 respectively



5 Conclusions




5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

120575(∘

)

d1 = 002 (s)d2 = 004 (s)

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

d1 = 002 (s)d2 = 004 (s)

(b)

00

02

02

04 06 08 1 12 14 16 18

04

06

08

1

12

14

2

Time (s)

I q(a

)

d1 = 002 (s)d2 = 004 (s)

(c)






Appendices

A Nomenclature












119906119861= 1199093+ 119879

[1

1198991

11989811199092+ (1198982+ 120579) (120579119909

2+ 1198961119875119898

minus 1198991(1 + 119891 (119909

1) (1199093+ 1198681199020))) +

1205791199092]

minus1

11989912[11990921198992(11989811199091+ 11989821199092+ 1205791199092+ 1198961119875119898)]

sdot1

119891 (1199091)+11989911198963119891 (1199091) 1199092

1198961

[1

1198991

(11989811199091+ 11989821199092

+ 1205791199092+ 1198961119875119898) minus 1] minus [

(1198982+ 120579)

1198991119891 (1199091) 120574

]

2

1198903minus1198903

1205742

minus 120583100381610038161003816100381611989631199093 (119905)

1003816100381610038161003816

(B1)


119902= 1108 pu 119879

119902= 003 s 119888

1= 2 120574 = 02 120588 = 2

1199021= 04 119902

2= 06 120590 = 1 1198641015840

119902= 1108 pu119881

119904= 1119883

1= 084 pu

1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and 119868

0= 0


119906119861= 119877 (119883) +

119867119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575


119901Δ1205961015840minus

119896119889119889 (Δ120596

1015840)

119889119905)

(B2)

where

119877 (119883) =119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575

119898minus

119883Σ

2119879119902119863

119864101584011990211988111990411988311198832sin 120575

Δ

minus119883Σ119879119902con120575

11988311198832sin 120575

Δ120596 minus 119868119902+ 1198681199020

(B3)


[119896119889 119896119901 119896119897]119879

= [13 315 42]119879 (B4)


8 s 1198641015840119902= 1108 pu 119879

119902= 003 s 119875

119898= 10 kw 119881

119904= 1 pu 119883

1=

084 pu 1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and

1198680= 0



Acknowledgments


References





4batteries based on





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 1199091 1199092 and 119909


2

lowast and 1199093

lowast


1= 1199092

2= 2minus 2

lowast

= 1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)) minus

2

lowast

3= 3minus 3


119861) minus 3

lowast

(11)


1198811=1

21199111

2 (12)


1= 11991111199112+ 11991111199092

lowast (13)


lowast as

1199092

lowast= minus11988811199091 (14)


1le 0 if 119911

2= 0


1198812=1

21199111

2+1

21199112

2 (15)


2= 11991111+ 11991122 (16)


2= 11991111+ 11991122= 11991111199112minus 11988811199111

2+ 1199112(2minus 2

lowast)

= 11991111199112minus 11988811199111

2+ 1199112(2+ 11988811199112minus 1198881

21199111) = (1 minus 119888

1

2)

sdot 11991111199112minus 11988811199111

2+ 1199112[1205791199092+ 1198961119875119898

minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020)) + 11988811199112]

(17)


3



3

lowast is

1199093

lowast

= (

(120579 + 120573) 1199092+ 1198961119875119898+ (1 minus 119888

1

2) 1199111+ 11988811199112+ 11988821199112

1198962119891 (1199091) sin (120575

0+ 1199091)

)

minus1

119891 (1199091)minus 1198681199020

(18)






between 1199093


120591 (1199093

lowast)1015840+ 1199093

lowast= 1199093

1199093

lowast(0) = 119909

3 (0)

(19)


lowast)1015840= (1199093minus 1199093

lowast)120591 The


119910 = 1199093



1198813=1

21199111

2+1

21199112

2+1

21199113

2+1

21199102+ int

119905

119905minus119889

119902 (119909 (120572)) 119889120572 (21)


3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))

minus 119902 (119909 (119905 minus 119889))

(22)


119861)minus3


3)120591minus

3= minus119910120591 + 119861


3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))


2) 11991111199112minus 11988811199111

2+ 1199112[1205791199092

+ 1198961119875119898minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020))

+ 11988811199112] + 1199113(1198963(minus1199093 (119905 minus 119889) + 119906

119861) minus 3

lowast) + 119910 (

minus119910

120591

+ 1198613) + 119902 (119909 (119905)) minus 119902 (119909 (119905 minus 119889))

(23)


3(119905 minus 119889)) = |119896

31199093(119905 minus 119889)| where



311991131199093(119905)|



minus1199113ℎ (1199093(119905 minus 119889)) le

1003816100381610038161003816119896311991131199093 (119905 minus 119889)1003816100381610038161003816

le10038161003816100381610038161199113

1003816100381610038161003816100381610038161003816100381611989631199093 (119905 minus 119889)

1003816100381610038161003816

(24)




(23) we can obtain

3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))


2) 11991111199112minus 11988811199111

2+ 1199112[1205791199092

+ 1198961119875119898minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020))

+ 11988811199112] + 1199113(1198963119906119861minus 3

lowast) + 119910 (

minus119910

120591+ 1198613)

+1003816100381610038161003816119896311991131199093 (119905)

1003816100381610038161003816

(25)


119906119861=

1

1198963

3

lowastminus

1198883

1198963

1199113+ 120582

10038161003816100381610038161199093 (119905)1003816100381610038161003816 (26)



3lt 0



Proof Let 119881 = (12)1199111

2+ (12)119911

2

2+ (12)119911

3

2+ (12)119910

2= 119901


3 and then we have

1198613

21198723


obtain

= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988831199113

2+ 119910(

minus119910

120591+ 1198613)

(27)


= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1003816100381610038161003816119910100381610038161003816100381610038161003816100381610038161198613

1003816100381610038161003816

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1

21198613

21199102

+1

2

= minus11988811199111

2minus 11988821199112

2minus 11988831199113

2minus 11991121199092119890120579

+ (1

21198613

2minus1

120591) 1199102+1

2

(28)


3

2+119903


le minus1199031199111

2minus 1199031199112

2minus 1199031199113

2+ (

1

21198613

2minus1198723

2

2minus 119903)119910

2

+1

2= minus2119903119881 + (

1198723

21198723

1198613

2minus1198723

2

2)1199102+1

2

= minus2119903119881 + (1198613

2

1198723

minus 1)1198723

21199102

2+1

2

(29)


le minus21

4119901119901 +

1

2= 0 (30)



le minus2119903119881 +1

2 (31)


119881 le1

4119903+ (119881 (0) minus

1

4119903) 119890minus2119903119905

(32)


3

2+12+119903




Consider119867 = 8 s 1198641015840119902= 1108 pu 119875

119898= 10 kw 119881

119904= 1 pu

1198831= 084 pu119883

2= 052 pu 119879

119902= 003 s 119901 = 2 119888

1= 1 1198882= 1


given as 1205750= 571

∘ 1205960= 314159 rads 119868

0= 0 and 119910(0) = 0





5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

DSMII

120575(∘

)

ABPID

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

DSMIIABPID

(b)

0 02 04 06 08 1 12 14 16 1804

06

08

1

12

14

16

18

2

Time (s)

I q(a

)

DSMIIABPID

(c)




1= 002 and 119889

1= 004 respectively



5 Conclusions




5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

120575(∘

)

d1 = 002 (s)d2 = 004 (s)

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

d1 = 002 (s)d2 = 004 (s)

(b)

00

02

02

04 06 08 1 12 14 16 18

04

06

08

1

12

14

2

Time (s)

I q(a

)

d1 = 002 (s)d2 = 004 (s)

(c)






Appendices

A Nomenclature












119906119861= 1199093+ 119879

[1

1198991

11989811199092+ (1198982+ 120579) (120579119909

2+ 1198961119875119898

minus 1198991(1 + 119891 (119909

1) (1199093+ 1198681199020))) +

1205791199092]

minus1

11989912[11990921198992(11989811199091+ 11989821199092+ 1205791199092+ 1198961119875119898)]

sdot1

119891 (1199091)+11989911198963119891 (1199091) 1199092

1198961

[1

1198991

(11989811199091+ 11989821199092

+ 1205791199092+ 1198961119875119898) minus 1] minus [

(1198982+ 120579)

1198991119891 (1199091) 120574

]

2

1198903minus1198903

1205742

minus 120583100381610038161003816100381611989631199093 (119905)

1003816100381610038161003816

(B1)


119902= 1108 pu 119879

119902= 003 s 119888

1= 2 120574 = 02 120588 = 2

1199021= 04 119902

2= 06 120590 = 1 1198641015840

119902= 1108 pu119881

119904= 1119883

1= 084 pu

1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and 119868

0= 0


119906119861= 119877 (119883) +

119867119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575


119901Δ1205961015840minus

119896119889119889 (Δ120596

1015840)

119889119905)

(B2)

where

119877 (119883) =119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575

119898minus

119883Σ

2119879119902119863

119864101584011990211988111990411988311198832sin 120575

Δ

minus119883Σ119879119902con120575

11988311198832sin 120575

Δ120596 minus 119868119902+ 1198681199020

(B3)


[119896119889 119896119901 119896119897]119879

= [13 315 42]119879 (B4)


8 s 1198641015840119902= 1108 pu 119879

119902= 003 s 119875

119898= 10 kw 119881

119904= 1 pu 119883

1=

084 pu 1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and

1198680= 0



Acknowledgments


References





4batteries based on





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(23) we can obtain

3= 11991111+ 11991122+ 11991133+ 119910 + 119902 (119909 (119905))


2) 11991111199112minus 11988811199111

2+ 1199112[1205791199092

+ 1198961119875119898minus 1198962sin (1205750+ 1199091) (1 + 119891 (119909

1) (1199093+ 1198681199020))

+ 11988811199112] + 1199113(1198963119906119861minus 3

lowast) + 119910 (

minus119910

120591+ 1198613)

+1003816100381610038161003816119896311991131199093 (119905)

1003816100381610038161003816

(25)


119906119861=

1

1198963

3

lowastminus

1198883

1198963

1199113+ 120582

10038161003816100381610038161199093 (119905)1003816100381610038161003816 (26)



3lt 0



Proof Let 119881 = (12)1199111

2+ (12)119911

2

2+ (12)119911

3

2+ (12)119910

2= 119901


3 and then we have

1198613

21198723


obtain

= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988831199113

2+ 119910(

minus119910

120591+ 1198613)

(27)


= 11991111+ 11991122+ 11991133+ 119910

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1003816100381610038161003816119910100381610038161003816100381610038161003816100381610038161198613

1003816100381610038161003816

le minus11988811199111

2minus 11988821199112

2minus 11991121199092119890120579minus 11988821199113

2minus1199102

120591+1

21198613

21199102

+1

2

= minus11988811199111

2minus 11988821199112

2minus 11988831199113

2minus 11991121199092119890120579

+ (1

21198613

2minus1

120591) 1199102+1

2

(28)


3

2+119903


le minus1199031199111

2minus 1199031199112

2minus 1199031199113

2+ (

1

21198613

2minus1198723

2

2minus 119903)119910

2

+1

2= minus2119903119881 + (

1198723

21198723

1198613

2minus1198723

2

2)1199102+1

2

= minus2119903119881 + (1198613

2

1198723

minus 1)1198723

21199102

2+1

2

(29)


le minus21

4119901119901 +

1

2= 0 (30)



le minus2119903119881 +1

2 (31)


119881 le1

4119903+ (119881 (0) minus

1

4119903) 119890minus2119903119905

(32)


3

2+12+119903




Consider119867 = 8 s 1198641015840119902= 1108 pu 119875

119898= 10 kw 119881

119904= 1 pu

1198831= 084 pu119883

2= 052 pu 119879

119902= 003 s 119901 = 2 119888

1= 1 1198882= 1


given as 1205750= 571

∘ 1205960= 314159 rads 119868

0= 0 and 119910(0) = 0





5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

DSMII

120575(∘

)

ABPID

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

DSMIIABPID

(b)

0 02 04 06 08 1 12 14 16 1804

06

08

1

12

14

16

18

2

Time (s)

I q(a

)

DSMIIABPID

(c)




1= 002 and 119889

1= 004 respectively



5 Conclusions




5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

120575(∘

)

d1 = 002 (s)d2 = 004 (s)

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

d1 = 002 (s)d2 = 004 (s)

(b)

00

02

02

04 06 08 1 12 14 16 18

04

06

08

1

12

14

2

Time (s)

I q(a

)

d1 = 002 (s)d2 = 004 (s)

(c)






Appendices

A Nomenclature












119906119861= 1199093+ 119879

[1

1198991

11989811199092+ (1198982+ 120579) (120579119909

2+ 1198961119875119898

minus 1198991(1 + 119891 (119909

1) (1199093+ 1198681199020))) +

1205791199092]

minus1

11989912[11990921198992(11989811199091+ 11989821199092+ 1205791199092+ 1198961119875119898)]

sdot1

119891 (1199091)+11989911198963119891 (1199091) 1199092

1198961

[1

1198991

(11989811199091+ 11989821199092

+ 1205791199092+ 1198961119875119898) minus 1] minus [

(1198982+ 120579)

1198991119891 (1199091) 120574

]

2

1198903minus1198903

1205742

minus 120583100381610038161003816100381611989631199093 (119905)

1003816100381610038161003816

(B1)


119902= 1108 pu 119879

119902= 003 s 119888

1= 2 120574 = 02 120588 = 2

1199021= 04 119902

2= 06 120590 = 1 1198641015840

119902= 1108 pu119881

119904= 1119883

1= 084 pu

1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and 119868

0= 0


119906119861= 119877 (119883) +

119867119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575


119901Δ1205961015840minus

119896119889119889 (Δ120596

1015840)

119889119905)

(B2)

where

119877 (119883) =119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575

119898minus

119883Σ

2119879119902119863

119864101584011990211988111990411988311198832sin 120575

Δ

minus119883Σ119879119902con120575

11988311198832sin 120575

Δ120596 minus 119868119902+ 1198681199020

(B3)


[119896119889 119896119901 119896119897]119879

= [13 315 42]119879 (B4)


8 s 1198641015840119902= 1108 pu 119879

119902= 003 s 119875

119898= 10 kw 119881

119904= 1 pu 119883

1=

084 pu 1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and

1198680= 0



Acknowledgments


References





4batteries based on





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

DSMII

120575(∘

)

ABPID

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

DSMIIABPID

(b)

0 02 04 06 08 1 12 14 16 1804

06

08

1

12

14

16

18

2

Time (s)

I q(a

)

DSMIIABPID

(c)




1= 002 and 119889

1= 004 respectively



5 Conclusions




5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

120575(∘

)

d1 = 002 (s)d2 = 004 (s)

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

d1 = 002 (s)d2 = 004 (s)

(b)

00

02

02

04 06 08 1 12 14 16 18

04

06

08

1

12

14

2

Time (s)

I q(a

)

d1 = 002 (s)d2 = 004 (s)

(c)






Appendices

A Nomenclature












119906119861= 1199093+ 119879

[1

1198991

11989811199092+ (1198982+ 120579) (120579119909

2+ 1198961119875119898

minus 1198991(1 + 119891 (119909

1) (1199093+ 1198681199020))) +

1205791199092]

minus1

11989912[11990921198992(11989811199091+ 11989821199092+ 1205791199092+ 1198961119875119898)]

sdot1

119891 (1199091)+11989911198963119891 (1199091) 1199092

1198961

[1

1198991

(11989811199091+ 11989821199092

+ 1205791199092+ 1198961119875119898) minus 1] minus [

(1198982+ 120579)

1198991119891 (1199091) 120574

]

2

1198903minus1198903

1205742

minus 120583100381610038161003816100381611989631199093 (119905)

1003816100381610038161003816

(B1)


119902= 1108 pu 119879

119902= 003 s 119888

1= 2 120574 = 02 120588 = 2

1199021= 04 119902

2= 06 120590 = 1 1198641015840

119902= 1108 pu119881

119904= 1119883

1= 084 pu

1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and 119868

0= 0


119906119861= 119877 (119883) +

119867119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575


119901Δ1205961015840minus

119896119889119889 (Δ120596

1015840)

119889119905)

(B2)

where

119877 (119883) =119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575

119898minus

119883Σ

2119879119902119863

119864101584011990211988111990411988311198832sin 120575

Δ

minus119883Σ119879119902con120575

11988311198832sin 120575

Δ120596 minus 119868119902+ 1198681199020

(B3)


[119896119889 119896119901 119896119897]119879

= [13 315 42]119879 (B4)


8 s 1198641015840119902= 1108 pu 119879

119902= 003 s 119875

119898= 10 kw 119881

119904= 1 pu 119883

1=

084 pu 1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and

1198680= 0



Acknowledgments


References





4batteries based on





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5755

575

5745

574

5735

573

5725

572

5715

571

57050 02 04 06 08 1 12 14 16 18 2

Time (s)

120575(∘

)

d1 = 002 (s)d2 = 004 (s)

(a)

0 02 04 06 08 1 12 14 16 18 2

Time (s)

120596(r

ads

)

315

3145

314

3135

313

3125

312

d1 = 002 (s)d2 = 004 (s)

(b)

00

02

02

04 06 08 1 12 14 16 18

04

06

08

1

12

14

2

Time (s)

I q(a

)

d1 = 002 (s)d2 = 004 (s)

(c)






Appendices

A Nomenclature












119906119861= 1199093+ 119879

[1

1198991

11989811199092+ (1198982+ 120579) (120579119909

2+ 1198961119875119898

minus 1198991(1 + 119891 (119909

1) (1199093+ 1198681199020))) +

1205791199092]

minus1

11989912[11990921198992(11989811199091+ 11989821199092+ 1205791199092+ 1198961119875119898)]

sdot1

119891 (1199091)+11989911198963119891 (1199091) 1199092

1198961

[1

1198991

(11989811199091+ 11989821199092

+ 1205791199092+ 1198961119875119898) minus 1] minus [

(1198982+ 120579)

1198991119891 (1199091) 120574

]

2

1198903minus1198903

1205742

minus 120583100381610038161003816100381611989631199093 (119905)

1003816100381610038161003816

(B1)


119902= 1108 pu 119879

119902= 003 s 119888

1= 2 120574 = 02 120588 = 2

1199021= 04 119902

2= 06 120590 = 1 1198641015840

119902= 1108 pu119881

119904= 1119883

1= 084 pu

1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and 119868

0= 0


119906119861= 119877 (119883) +

119867119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575


119901Δ1205961015840minus

119896119889119889 (Δ120596

1015840)

119889119905)

(B2)

where

119877 (119883) =119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575

119898minus

119883Σ

2119879119902119863

119864101584011990211988111990411988311198832sin 120575

Δ

minus119883Σ119879119902con120575

11988311198832sin 120575

Δ120596 minus 119868119902+ 1198681199020

(B3)


[119896119889 119896119901 119896119897]119879

= [13 315 42]119879 (B4)


8 s 1198641015840119902= 1108 pu 119879

119902= 003 s 119875

119898= 10 kw 119881

119904= 1 pu 119883

1=

084 pu 1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and

1198680= 0



Acknowledgments


References





4batteries based on





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119906119861= 1199093+ 119879

[1

1198991

11989811199092+ (1198982+ 120579) (120579119909

2+ 1198961119875119898

minus 1198991(1 + 119891 (119909

1) (1199093+ 1198681199020))) +

1205791199092]

minus1

11989912[11990921198992(11989811199091+ 11989821199092+ 1205791199092+ 1198961119875119898)]

sdot1

119891 (1199091)+11989911198963119891 (1199091) 1199092

1198961

[1

1198991

(11989811199091+ 11989821199092

+ 1205791199092+ 1198961119875119898) minus 1] minus [

(1198982+ 120579)

1198991119891 (1199091) 120574

]

2

1198903minus1198903

1205742

minus 120583100381610038161003816100381611989631199093 (119905)

1003816100381610038161003816

(B1)


119902= 1108 pu 119879

119902= 003 s 119888

1= 2 120574 = 02 120588 = 2

1199021= 04 119902

2= 06 120590 = 1 1198641015840

119902= 1108 pu119881

119904= 1119883

1= 084 pu

1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and 119868

0= 0


119906119861= 119877 (119883) +

119867119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575


119901Δ1205961015840minus

119896119889119889 (Δ120596

1015840)

119889119905)

(B2)

where

119877 (119883) =119883Σ

2119879119902

119864101584011990211988111990411988311198832sin 120575

119898minus

119883Σ

2119879119902119863

119864101584011990211988111990411988311198832sin 120575

Δ

minus119883Σ119879119902con120575

11988311198832sin 120575

Δ120596 minus 119868119902+ 1198681199020

(B3)


[119896119889 119896119901 119896119897]119879

= [13 315 42]119879 (B4)


8 s 1198641015840119902= 1108 pu 119879

119902= 003 s 119875

119898= 10 kw 119881

119904= 1 pu 119883

1=

084 pu 1198832= 052 pu 120575

0= 571

∘ 1205960= 314159 rads and

1198680= 0



Acknowledgments


References





4batteries based on





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Research on Improved Adaptive Control for...

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