research article discrete-time nonlinear control of vsc...

Research ArticleDiscrete-Time Nonlinear Control of VSC-HVDC System

TianTian Qian and ShiHong Miao

State Key Laboratory of Advanced Electromagnetic Engineering and TechnologyHuazhong University of Science and Technology Wuhan 430074 China

Correspondence should be addressed to ShiHong Miao shmiaohusteducn

Received 17 March 2015 Revised 19 May 2015 Accepted 28 May 2015

Academic Editor Miguel A F Sanjuan

Copyright copy 2015 T Qian and S MiaoThis 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

Because VSC-HVDC is a kind of strong nonlinear coupling and multi-input multioutput (MIMO) system its control problem isalways attractingmuch attention from scholars And a lot of papers have done research on its control strategy in the continuous-timedomain But the control system is implemented through the computer discrete sampling in practical engineering It is necessary tostudy themathematicalmodel and control algorithm in the discrete-time domainThediscretemathematicalmodel based onoutputfeedback linearization and discrete sliding mode control algorithm is proposed in this paper And to ensure the effectiveness of thecontrol system in the quasi sliding mode state the fast output sampling method is used in the output feedback The results fromsimulation experiment in MATLABSIMULINK prove that the proposed discrete control algorithm can make the VSC-HVDCsystem have good static dynamic and robust characteristics in discrete-time domain

1 Introduction

Voltage source converter based high voltage direct current(VSC-HVDC) technology which uses advanced (insulatedgate bipolar transistor) IGBT device and pulse width modu-lation (PWM) method overcomes the disadvantage of tradi-tional HVDC It can not only adjust the active and reactivepower independently but also supply power to passivenetwork system Therefore the VSC-HVDC technology hasa broad application prospect in the fields of distributed gen-eration asynchronous AC network interconnection remote-area power supply and so forth [1ndash5]

Because VSC-HVDC is a multiple input multiple out-put strong coupling nonlinear system its control problemis always attracting much attention from the scholars Inchronological order its research process can be classified intothe following two phases

(1) The steady state mathematical model and basic con-trol strategy of VSC-HVDC system [6] developedthe steady state model and proposed control strategywhich combined an inverse model controller witha PI controller Reference [7] presented an equiv-alent continuous-time state space model of VSC-HVDC in the synchronous 119889-119902 reference frame and

proposed a decoupled PI control strategy using thefeed-forward compensation method Reference [8]presented the elements of VSC-HVDC and proposeda feed-forward decoupled current control strategyIn this phase the study is mainly based on thetraditional PI controllers under the linear decouplingcontrol strategy The design method of this kind ofcontrol strategy is easy And it shows good adjust-ment ability But the parameters of PI controllers arefixed and their adjustment ability is limited Whensystem suffers from large disturbance they showweakrobustness and dynamic characteristics Some severecases can lead to sustained oscillation of the systemTherefore some scholars started the second phasestudy

(2) Optimizing and nonlinear control strategy of theVSC-HVDC system [9] proposed an adaptive controldesign to improve dynamic performances of VSC-HVDC systems The adaptive controllers designedfor nonlinear characteristics of VSC-HVDC systemswhich were based on back stepping method con-sidered parameters uncertainties Reference [10] pre-sented a robust nonlinear controller for VSC-HVDCtransmission link using input-output linearization

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 929467 11 pageshttpdxdoiorg1011552015929467

2 Mathematical Problems in Engineering

and sliding mode control strategy The feedbacklinearization was used to cancel nonlinearity and thesliding mode control offered invariant stability tomodeling uncertainties Reference [11] proposed thecontroller design based on 119867infin control methodolo-gies to deal with the nonlinearities introduced byrequirements to power flow and line voltage Refer-ence [12] addressed two different robust nonlinearcontrol methodologies based on slidingmode controlfor the VSC-HVDC transmission systemrsquos perfor-mance enhancement and stability improvement Inthis phase the studies are based on the nonlinearcontrol robust control and optimizing control of theVSC-HVDC system The experimental results fromresearch papers show these control strategies canimprove static dynamic and robust performance ofVSC-HVDC system

However the referred literatures are based on the continuous-time state In practice nowadays most controllers are imple-mented in discrete time It is known that the realization of acontroller using digital elements and complex programmablelogic devices can achieve maximum reproducibility at min-imum cost So it is necessary to do research on the discretemodel and control strategy In [13] the discrete PI controllersof VSC-HVDC system were established Reference [14] stud-ied the discrete PI control algorithm of VSC-HVDC systemwhich supplied power to the passive network

But so far the discrete-time mathematic model and con-trol strategy based on nonlinear control methods are seldomstudied by scholars Through proper feedback linearizationcontrol strategy complex nonlinear system synthesis prob-lems can be transformed to linear system synthesis problemsAs a kind of robust control method sliding mode controlis versatile to linear and nonlinear system It is easy to bedesigned and carried out Because of its complete robustnessto the parameters change and external disturbances whichmeet the match condition it gets extensive attention fromscholars and engineers [15]

Therefore the discrete mathematical model of the VSC-HVDC system based on nonlinear feedback linearizationand discrete sliding mode control algorithm are proposedin this paper And the fast output sampling (FOS) tech-nique is adopted in the output feedback which ensuresthe stability of the closed loop system in the condition ofquasi slidingmode controlThe simulation results performedin MATLABSIMULINK show that the proposed discretecontrol strategy can make the VSC-HVDC system have goodoperation performance

2 The Mathematical Model of VSC-HVDC

21 Continuous-Time State Space Model The structure dia-gram of VSC-HVDC is shown in Figure 1 The more widelyused continuous-time mathematical models based on thesynchronous reference coordinates are employed in thispaper which are shown as (1) and (7)

211 Rectifier Side Choose state variables x12 = (119894119903119889 119894119903119902)119879

controlled variablesu12 = (119880119888119903119889 119880119888119903119902)119879 and output variables

y12 = (119894119903119889 119894119903119902)119879Themathematical model of the rectifier side

is shown as

1 =minus119877119903

119871119903

1199091 +1205961199031199092 +119880119904119903119889

119871119903

minus1119871119903

1199061

2 =minus119877119903

119871119903

1199092 minus1205961199031199091 +119880119904119903119902

119871119903

minus1119871119903

1199062

1199101 = 1199091

1199102 = 1199092

(1)

where 119894119903119889

119894119903119902

and 119880119904119903119889

119880119904119903119902

are the 119889-119902 axis currents andvoltages on the rectifier side respectively 119880

119888119903119889

and 119880119888119903119902

arerespectively the 119889-119902 axis control inputs on the rectifier side119877119903

and 119871119903

are the corresponding equivalent resistance and theinductance on the rectifier side120596

119903

is theAC system frequencyon the rectifier side

Here the output variables on the rectifier side shouldbe (119875119903

119876119903)119879 119875119903

and 119876119903

are respectively the output valuesof active and reactive power on the rectifier side Forconvenience define 119880

119904119903

as the line voltage effective valueof the rectifier power supply Define the 119889 axis in thesynchronization reference frame coincidence with 119886 axis inthe three-phase reference coordinate so119880

119904119903119889

= 119880119904119903

119880119904119903119902

= 0[119875119903

119876119903

] = [119880119904119903119889119880119904119903119902

minus119880119904119903119902119880119904119903119889

] [119894119903119889

119894119903119902

] = [119880119904119903119894119903119889

119880119904119903119894119903119902

] Therefore the output

variables can be chosen as y12 = (119894119903119889 119894119903119902)119879

Define

1198911 =minus119877119903

119871119903

1199091 +1205961199031199092 +119880119904119903119889

119871119903

1198912 =minus119877119903

119871119903

1199092 minus1205961199031199091 +119880119904119903119902

119871119903

f119903

= (1198911 1198912)119879

g1 = (minus1119871119903

0)119879

g2 = (0 minus1119871119903

)

119879

(2)

Therefore (1) can be reorganized by

x12 = f119903

+ g11199061 + g21199062

y12 = (ℎ1 (119909)

ℎ2 (119909)) = (

1199091

1199092)

(3)

Then do exact feedback linearization on system equation(3) By calculation the relative degree of 1199101 is denoted by1205741 = 1 The relative degree of 1199102 is denoted by 1205742 = 1 Basedon the feedback linearization theory [16] the exact feedback

Mathematical Problems in Engineering 3

AC1

Rectifier

C

C

C

C

stationInverterstation

Usr

T1

AC2T2

RrRiLr

Li

Ucr Uci Usi

Rdc

Rdc

Ldc

Ldc

Figure 1 Structure diagram of VSC-HVDC system

linearization state equations of the rectifier side are obtainedas

(1199101

1199102) = (

1

2) = (

1205921

1205922)

= (

1198711205741119891119903

ℎ1

1198711205742119891119903

ℎ2)

+(

11987111989211198711205741minus1119891119903

ℎ1 11987111989221198711205741minus1119891119903

ℎ1

11987111989211198711205742minus1119891119903

ℎ2 11987111989221198711205742minus1119891119903

ℎ2

)(1199061

1199062)

(1

2) = (

1205921

1205922) = (

1198911

1198912)+(

minus1119871119903

0

0 minus1119871119903

)(1199061

1199062)

(4)

Because 10038161003816100381610038161003816minus1119871119903

00 minus1119871

119903

10038161003816100381610038161003816= 11198712

119903

= 0 ( minus1119871119903 00 minus1119871

119903

) is nonsingu-lar

The controlled variables 1199061 and 1199062 can be denoted byvirtual control variables 1205921 and 1205922

(1199061

1199062) = (

119871119903

(1198911 minus 1205921)

119871119894

(1198912 minus 1205922)) (5)

To make it convenient for using FOS (fast output sampling)technique in Section 31 [17] substitute (5) into (3) then thesystem equation of rectifier side can be denoted by

1 = 1205921

2 = 1205922

1199101 = 1199091

1199102 = 1199092

(6)

212 Inverter Side Choose state variables x345 =

(119894119894119889

119894119894119902

1198801198891198882)119879 controlled variables u34 = (119880119888119894119889 119880119888119894119902)

119879 andoutput variables y34 = (1198801198891198882 119894119894119902)

119879 The mathematical modelof inverter side is shown as

3 =minus119877119894

119871119894

1199093 +1205961198941199094 +119880119904119894119889

119871119894

minus1119871119894

1199063

4 =minus119877119894

119871119894

1199094 minus1205961198941199093 +119880119904119894119902

119871119894

minus1119871119894

1199064

5 =119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990932+ 1199094

2)

1198621199095+1198801198891198881 minus 1199095119862119877119889119888

1199103 = 1199095

1199104 = 1199094

(7)

where 119894119894119889

119894119894119902

and 119880119904119894119889

119880119904119894119902

are the 119889-119902 axis currents andvoltages on the inverter side respectively 119880

119888119894119889

and 119880119888119894119902

arerespectively the control inputs on the inverter side 119877

119894

and 119871119894

are the corresponding equivalent resistance and the induc-tance on the inverter side 119880

1198891198881 and1198801198891198882 are respectively theDC voltages on the rectifier side and inverter side 119862 and 119877

119889119888

are respectively the converter station capacitance and DCline resistance 120596

119894

is the AC system frequency on the inverterside

Here the output variables on the inverter side should be(1198801198891198882 119876119894) 119876119894 is the output value of reactive power on the

inverter side For convenience define 119880119904119894

as the line voltageeffective value of the inverter side power source Define the119889 axis in the synchronization reference frame that coincideswith 119886 axis in the three-phase reference coordinate so 119880

119904119894119889

=

119880119904119894

119880119904119894119902

= 0 and 119876119894

= minus119880119904119894119902

119894119894119889

+ 119880119904119894119889

119894119894119902

= 119880119904119894

119894119894119902

Thereforethe output variables can be chosen as y34 = (1198801198891198882 119894119894119902)

119879Define

1198913 =minus119877119894

119871119894

1199093 +1205961198941199094 +119880119904119894119889

119871119894

1198914 =minus119877119894

119871119894

1199094 minus1205961198941199093 +119880119904119894119902

119871119894

1198915 =119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990923 + 119909

24)

1198621199095+1198801198891198881 minus 1199095119862119877119889119888

f119894

= (1198913 1198914 1198915)119879

g3 = (minus1119871119894

0 0)119879

g4 = (0 minus1119871119894

0)119879

(8)



x345 = f119894

+ g31199063 + g41199064

y34 = (ℎ3 (119909)

ℎ4 (119909)) = (

1199095

1199094)

(9)

Then do exact feedback linearization on system (9) Bycalculation the relative degree of 1199103 is denoted by 1205743 = 2The relative degree of 1199104 is denoted by 1205744 = 1 Based uponthe feedback linearization theory [16] the exact feedbacklinearization state equations of inverter side are obtained as

(1199103

1199104) = (

5

4) = (

1205923

1205924)

= (

1198711205743119891119894

ℎ3

1198711205744119891119894

ℎ4)

+(

11987111989231198711205743minus1119891119894

ℎ3 11987111989241198711205743minus1119891119894

ℎ3

11987111989231198711205744minus1119891119894

ℎ4 11987111989241198711205744minus1119891119894

ℎ4)(

1199063

1199064)

(5

4) = (

1205923

1205924)

= (11988611198913 + 11988621198914 + 11988631198915

1198914)

+(

minus1198861119871119894

minus1198862119871119894

0 minus1119871119894

)(1199063

1199064)

(10)

where

1198861 =12059711989151205971199093

=119880119904119894119889

minus 2119877119894

11990931198621199095

1198862 =12059711989151205971199094

=119880119904119894119902

minus 2119877119894

1199094

1198621199095

1198863 =12059711989151205971199095

= minus119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990923 + 119909

24)

11986211990925minus

1119862119877119889119888

(11)

Because 10038161003816100381610038161003816minus1198861119871 119894 minus1198862119871 119894

0 minus1119871119894

10038161003816100381610038161003816= 1198861119871

2119894

= 0 ( minus1198861119871 119894 minus1198862119871 1198940 minus1119871119894

) isnonsingular


(1199063

1199064) = (

(11988611198911 + 11988631198913 minus 1205923 + 11988621205924) sdot119871119894

1198861119871119894

(1198914 minus 1205924)) (12)

To make it convenient for using FOS (fast output sampling)technology in Section 31 [17] substitute 1198861 1198862 1198863 and (12)

into (7) then the system equation of inverter side can bedenoted by

3 = (119886231198861+

11988631198861119862119877119889119888

)1199095 +111988611205923 minus

119886211988611205924

4 = 1205924

5 = (minus1198863 minus1

119862119877119889119888

)1199095

1199103 = 1199095

1199104 = 1199094

(13)

Here 5 should be 5 = (minus1198863minus1119862119877119889119888)1199095+(1198801198891198881minus1199095)119862119877119889119888Since during normal operation 119880

1198891198881 is approximately equalto 1199095 for convenient calculation 5 is simplified into 5 =(minus1198863 minus 1119862119877

119889119888

)1199095

22 Discrete-Time State Space Model Discretize the virtualcontrol variables 1205921 1205922 1205923 and 1205924 with sampling time 120591

(1205921 (119896)

1205922 (119896)) = (

1199101 (119896 + 1) minus 1199101 (119896)120591

1199102 (119896 + 1) minus 1199102 (119896)120591

)

=(

1199091 (119896 + 1) minus 1199091 (119896)120591

1199092 (119896 + 1) minus 1199092 (119896)120591

)

= (1198911 (119896)

1198912 (119896))+(

minus1119871119903

0

0 minus1119871119903

)(1199061 (119896)

1199062 (119896))

(14)

(1205923 (119896)

1205924 (119896)) = (

1199103 (119896 + 2) minus 21199103 (119896 + 1) + 1199103 (119896)1205912

1199104 (119896 + 1) minus 1199104 (119896)120591

)

=(

1199095 (119896 + 2) minus 21199095 (119896 + 1) + 1199095 (119896)1205912

1199094 (119896 + 1) minus 1199094 (119896)120591

)

= (11988611198913 (119896) + 11988621198914 (119896) + 11988631198915 (119896)

1198914 (119896))

+(

minus1198861119871119894

minus1198862119871119894

0 minus1119871119894

)(1199063 (119896)

1199064 (119896))

(15)


Discretize system equations (6) (13) with sampling time 120591shown as

1199091 (119896 + 1) = 1199091 (119896) + 1205911205921 (119896)

1199092 (119896 + 1) = 1199092 (119896) + 1205911205922 (119896)

1199101 (119896) = 1199091 (119896)

1199102 (119896) = 1199092 (119896)

1199093 (119896 + 1) = 1199093 (119896) +(120591119886

231198861+

12059111988631198861119862119877119889119888

)1199095 (119896)

+120591

11988611205923 (119896) minus

120591119886211988611205924 (119896)

1199094 (119896 + 1) = 1199094 (119896) + 1205911205924 (119896)

1199095 (119896 + 1) = (1minus 1205911198863 minus120591

119862119877119889119888

)1199095 (119896)

1199103 (119896) = 1199095 (119896)

1199104 (119896) = 1199094 (119896)

(16)

Rearrange (16) and then get

x12 (k + 1) = Ar120591x12 (k) +Br12059112059212 (k)

y12 (k) = Crx12 (k)

x345 (k + 1) = Ai120591x345 (k) +Bi12059112059234 (k)

y34 (k) = Cix345 (k)

(17)

where

Ar120591 = (1 00 1

)

Br120591 = (120591 00 120591

)

Cr = (1 00 1

)

Ai120591 =(

1 01205911198863

2

1198861+

12059111988631198861119862119877119889119888

0 1 0

0 0 1 minus 1205911198863 minus120591

119862119877119889119888

)

Bi120591 = (

120591

1198861minus12059111988621198861

0 120591

0 0

)

Ci = (0 0 10 1 0

)

(18)

Assume that the pairs (Ar120591Br120591) (Ai120591Bi120591) are controllableand the pairs (Ar120591Cr) (Ai120591Ci) are observable throughproperly sampling output variables

3 Discrete Sliding Mode Control ofVSC-HVDC System

In ideal continuous-time case the SMC (sliding mode con-trol) switches at infinite frequency and forces the states toslide on the so-called switching hyperplane In practicalapplications direct implementation of continuous-time SMCschemes using digital elements which are considered asthe device for imperfect switching will inevitably inducechattering phenomenon and deteriorate performance or eveninduce instability Chattering will cause serious harmonicswhich is undesirable in VSC-HVDC systems Hence thecontroller design using the discrete-time SMC (DSMC)algorithm is desirable for a successful implementation ofthe VSC-HVDC control systems And due to the finitesampling frequency the controller inputs are calculated onceper sampling period and held constant during that intervalUnder such a circumstance the trajectories of the systemstates of interest are unable to preciselymove along the slidingsurface which will lead to a quasi sliding mode motion only[18] Therefore only using static output feedback technologyhas not effectively ensured the control effect of discrete slidingmode controlThe fast output sampling technology should beused [17 19]

31 Fast Output Sampling (FOS) Technology Compared withthe static output feedback technology FOS not only keeps itsadvantage but also can randomly configure the system polesand always make the closed loop system stable And thenFOS can ensure the effectiveness of the discrete sliding modecontrol In the FOS every sampling period 120591 is divided into119873subintervals Here Δ = 120591119873 and119873 is equal to or greater thanthe observable index of system (AB) The output variablesare measured at time instants 119905 = 119897Δ 119897 = 0 1 119873 minus 1Consider the discrete-time system having be at time 119905 = 119896120591the fast output samples are obtained as [17 19]

yk =[[[[[[

[

119910 ((119896 minus 1) 120591)119910 ((119896 minus 1) 120591 + Δ)

119910 ((119896 minus 1) 120591 + (119873 minus 1) Δ)

]]]]]]

]

(19)

Then the rectifier side can be expressed as

x12 (k) = Ar120591x12 (k minus 1) +Br12059112059212 (k minus 1)

y12k = Cr0x12 (k minus 1) +Dr012059212 (k minus 1) (20)


where

Cr0 =

[[[[[[[

[

Cr

CrAr

CrArN1minus1

]]]]]]]

]

Dr0 =

[[[[[[[[[[

[

0CrBr

Cr

N1minus2sum

j=0Ar

jBr

]]]]]]]]]]

]

(21)

And assume thatCr0 andDr0 are invertible through appropri-ate choice (Ar Br Cr) is the system parameter matrix withsampling rate 1Δ 1 Δ 1 = 1205911198731

Define y12k = (1199101((119896minus1)120591)

1199102((119896minus1)120591)

)This means 1199101 is sampled onceand 1199102 is sampled once in each sampling period 120591

The inverter side can be expressed as

x345 (k) = Ai120591x345 (k minus 1) +Bi12059112059234 (k minus 1)

y34k = Ci0x345 (k minus 1) +Di012059234 (k minus 1) (22)

where

Ci0 =

[[[[[[[

[

Ci

CiAi

CiAiN2minus1

]]]]]]]

]

Di0 =

[[[[[[[[[[

[

0CiBi

Ci

N2minus2sum

j=0Ai

jBi

]]]]]]]]]]

]

(23)

And assume thatCi0 andDi0 are invertible through appropri-ate choice (Ai Bi Ci) is the system parameter matrix withsampling rate 1Δ 2 Δ 2 = 1205911198732

Define y34k = (1199103((119896minus1)120591)

1199103((119896minus1)120591+Δ

2)

119910

4

((119896minus1)120591)

) This means 1199103 is sampled

twice and 1199104 is sampled once in each sampling period 120591

Usr

+ +

+

minusPr

ND

kp

kis

Prrefirdref

(a) The reference value of 119889 axis current

Usr

+ +

+

minus

ND

Qr

kp

kis

Qrrefirqref

(b) The reference value of 119902 axis current

Figure 2 Control block diagrams of output references value at therectifier station

According to (20) and (22) state vectors 11990912(119896) and119909345(119896) can be deduced as

x12 (k) = Ar120591Cr0minus1y12k

+ (Br120591 minusAr120591Cr0minus1Dr0) 12059212 (k minus 1)

(24)

x345 (k) = Ai120591Ci0minus1y34k

+ (Bi120591 minusAi120591Ci0minus1Di0) 12059234 (k minus 1)

(25)

32 The Rectifier Side Control The differences between out-put and reference variables are denoted by (26) Because therelative degrees of 1199101 and 1199102 are respectively equal to 1 1 thesliding mode surfaces are defined as (27) Consider

e12 = (1198901

1198902) = (

1199101

(119896) minus 119894119903119889ref (119896)

1199102

(119896) minus 119894119903119902ref (119896)

) (26)

where 119894119903119889ref(119896) and 119894119903119902ref(119896) are the reference values Their

calculation processes are shown in Figure 2 119875119903

119875119903ref 119876119903 and

119876119903ref are respectively the output value and reference value of

active and reactive power Consider

s12 = (1199041

1199042) = (

1198901

(119896)

1198902

(119896)) (27)


Usi

+ +

+

minus

ND

Qi

kp

kis

Qiref iiqref

Figure 3 Control block diagram of output reference value at theinverter station

Desired state trajectories of the discrete variable structuresystem shown as in (28) can be obtained by control law basedon reaching law method

1199041

(119896 + 1) minus 1199041

(119896) = minus 1205881

1205911199041

(119896) minus 1205761

120591 sgn (1199041

(119896))

1199042

(119896 + 1) minus 1199042

(119896) = minus 1205882

1205911199042

(119896) minus 1205762

120591 sgn (1199042

(119896))

(28)

where 1205881 1205882 1205761 and 1205762 are all greater than zero And 1minus1205881120591 gt0 1 minus 1205882120591 gt 0

Now replacing (26) with (27) and then with (28) thevirtual control variables can be denoted by

1205921 (119896) = minus 12058811199041 (119896) minus 1205761 sgn (1199041 (119896))

1205922 (119896) = minus 12058821199042 (119896) minus 1205762 sgn (1199042 (119896)) (29)

The control variables shown as in (30) can be deduced by (14)(24) and (29)

1199061

(119896) = minus119877119903

1198761199031

+120596119871119903

1198761199032

minus119871119903

1205921

+119880119904119903119889

1199062

(119896) = minus120596119871119903

1198761199031

minus119877119903

1198761199032

minus119871119903

1205922

+119880119904119903119902

(30)

where 1198761199031 and 1198761199032 are the first and second row of x12(k) in

Section 31 1199061(119896) 1199062(119896) can be expressed by y12(k) throughFOS

33The Inverter SideControl Thedifferences between outputand reference variables are denoted by (31) Because therelative degrees of 1199103 and 1199104 are respectively equal to 2 1the sliding mode surfaces are defined as (32) Consider

e34 = (1198903

1198904) = (

1199103 (119896) minus 119880119889119888ref (119896)

1199104 (119896) minus 119894119894119902ref (119896)) (31)

where 119880119889119888ref(119896) and 119894119894119902ref(119896) are the reference values Usually

119880119889119888ref(119896) is knownThe calculation process of 119894

119894119902ref(119896) is shownin Figure 3 119876

119894

and 119876119894ref are respectively the output and

reference value of reactive power

s34 = (1199043

1199044

) = (1198883

1198903

(119896) +1198903

(119896 + 1) minus 1198903

(119896)

120591

1198904

(119896)

) (32)

Similar to the rectifier side desired state trajectories of thediscrete variable structure system shown as in (33) can beobtained by control law based on reaching law method

1199043

(119896 + 1) minus 1199043

(119896) = minus 1205883

1205911199043

(119896) minus 1205763

120591 sgn (1199043

(119896))

1199044

(119896 + 1) minus 1199044

(119896) = minus 1205884

1205911199044

(119896) minus 1205764

120591 sgn (1199044

(119896))

(33)



1205923 (119896) = minus 12058831199043 (119896) minus 1205763 sgn (1199043 (119896))

minus1198883 [1199103 (119896 + 1) minus 1199103 (119896)]

120591

1205924 (119896) = minus 12058841199044 (119896) minus 1205764 sgn (1199044 (119896))

(34)


1199063

(119896) = minus119877119894

1198761198943

+120596119894

119871119894

1198761198944

minus119871119894

1198863

1198861

(1198863

+2

119862119877119889119888

)1198761198945

minus119871119894

1198861

1205923

+119871119894

1198862

1198861

1205924

+119871119894

1198863

1198801198891198881

1198861

119862119877119889119888

+119880119904119894119889

1199064

(119896) = minus120596119894

119871119894

1198761198943

minus119877119894

1198761198944

minus119871119894

1205924

+119880119904119894119902

(35)

where 1198761198943 1198761198944 and 1198761198945 are the first second and third row of

x345(k) in Section 31 1199063(119896) and 1199064(119896) can be expressed byy34(k) through FOS

4 Simulation Results

The typical VSC-HVDC system composed of two converterstations is taken as example and the detailed parameters areshown in Table 1 The simulation experiment is performedin MATLABSIMULINK In the per-unit value system thebased power is 200MW the based voltage at AC side is8165 KV and the based voltage at DC side is 100KV Thesampling period 120591 = 74 120583s Also 1198731 = 1 and 119873

2

= 2 Thediscrete SMC controllers parameters are 1205881 = 1205882 = 11001205761 = 1205762 = 3 1205883 = 1205884 = 1000 1205763 = 1205764 = 3 and 1198883 = 1200

41 The Steady State Operation In steady state operation119875119903ref 119880119889119888ref 119876119894ref and 119876119894ref are respectively 1 pu 0 pu 1 pu

and 0 pu As shown in Figure 4 the active reactive powerand DC voltage can track their reference values effectivelyThe proposed mathematical model and control strategy canbe proved to make the system operate well in steady statecondition


00

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

05

10

PrQr

t (s)

PrQ

r(p

u)

(a) The active and reactive power at rectifier side

05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

PiQi

t (s)

PiQi

(pu)

(b) The active and reactive power at inverter side

00

0

1

02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

Udc1

(pu)

Udc1

(c) The DC voltage

Figure 4 Responses of the rectifier and inverter in steady state operation

Table 1 The system parameters of three-level VSC-HVDC

Parameters ValueAC voltages 119880

119904119903

119880119904119894

230 kv120596119903

120596119894

2 lowast pi lowast 50 radsTransformer ratio 1198791 1198792 230 kv100 kv119877119903

119877119894

0325Ω119871119903

119871119894

0072HRated voltage in DC side plusmn100 kv119862 70120583FUnit resistance of DC line 139119890 minus 002ΩkmDC line length 75 lowast 2 kmSwitching frequency 1350HZ

42 Reversion and Step Changes As shown in Figure 5 thechange process of 119875

119903ref is as follows the 119875119903ref keeps 1 pu from

0 s to 15 s At 15 s it steps to minus1 pu and then keeps thisvalue until 30 s At last it steps to 1 pu at 30 s From theexperimental results the referred changes least affect thereactive power on the rectifier side and reactive power onthe inverter side As shown in Figure 6 the change processof119876119903ref is as follows the119876119903ref keeps 0 pu from 0 s to 15 s And

then steps to 01 pu at 15 s The change process of 119876119894ref is as

follows the 119876119894ref keeps 0 pu from 0 s to 25 s And then steps

to minus01 pu at 25 s From the experimental results the referredchanges least affect the active power on the rectifier side andthe active power on the inverter side These can prove thatthe proposed discrete SMC strategy can make the active andreactive power decoupled and independent

43 Robustness Test The equivalent resistance and induc-tance at the converter stations both reduce 20 The exper-imental results are shown in Figure 7 The reference values


0

1

minus1

00 05 10 15 20 25 30 35 40 5045t (s)

PrQr

PrQ

r(p

u)


10

05

00

PiQi

minus10

minus05

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

(pu)


Figure 5 Responses of the rectifier and inverter when active power reverses

00

000102030405060708091011

05 10 15 20 25 30 35 40 5045t (s)

minus01minus02minus03minus04

PrQr

PrQ

r(p

u)


minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01

0102030405

00

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

PiQi

(pu)


Figure 6 Responses of the rectifier and inverter when the reactive power step changes

are the same as those in Section 41The active reactive powerandDC voltage can track their reference values smoothly andquicklyThis can prove that the proposed control strategy hasgood robustness

5 Conclusions

This paper deduces the discrete mathematical model of VSC-HVDC system by nonlinear input-output feedback lineariza-tion method Based on this the discrete sliding mode robustcontrollers are designed And to ensure the effectiveness ofthe controllers in quasi sliding mode condition the FOS

technology is used in output feedback The results fromsimulation experiment in MATLABSIMULINK prove theeffectiveness of proposed discrete mathematical model andcontrol strategy Since the actual computer control system isdiscrete sampling and SMC has a promising prospect theproposed discrete mathematical model and control strategyhave certain practical application prospect

Conflict of Interests

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


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage

Figure 7 Responses of the rectifier and inverter when the inner parameters change at both converter stations

Acknowledgment

This work is supported by the Natural Science Foundation ofChina (51377068)

References

[1] N Flourentzou V G Agelidis and G D Demetriades ldquoVSC-based HVDC power transmission systems an overviewrdquo IEEETransactions on Power Electronics vol 24 no 3 pp 592ndash6022009

[2] Y Wei Q He Y Sun and C Ji ldquoImproved power flowalgorithm forVSC-HVDCsystembased onhigh-order newton-type methodrdquoMathematical Problems in Engineering vol 2013Article ID 235316 10 pages 2013

[3] L Zhang L Harnefors and H-P Nee ldquoInterconnectionof two very weak AC systems by VSC-HVDC links using

power-synchronization controlrdquo IEEE Transactions on PowerSystems vol 26 no 1 pp 344ndash355 2011

[4] L Zhang L Harnefors and H-P Nee ldquoModeling and controlof VSC-HVDC links connected to island systemsrdquo IEEE Trans-actions on Power Systems vol 26 no 2 pp 783ndash793 2011

[5] F Xu and Z Xu ldquoA modular multilevel power flow controllerfor meshed HVDC gridsrdquo Science China Technological Sciencesvol 57 no 9 pp 1773ndash1784 2014

[6] G Zhang and Z Xu ldquoSteady-state model for VSC basedHVDC and its controller designrdquo in Proceedings of the IEEEPower Engineering SocietyWinterMeeting vol 3 pp 1085ndash1090February 2001

[7] M Yin G-Y Li T-Y Niu G-K Li H-F Liang and M ZhouldquoContinuous-time state-space model of VSC-HVDC and itscontrol strategyrdquo Proceedings of the Chinese Society of ElectricalEngineering vol 25 no 18 pp 34ndash39 2005

[8] R Song C Zheng R Li and Z Xiaoxin ldquoVSCs basedHVDC and its control strategyrdquo in Proceedings of the IEEEPES


Transmission and Distribution Conference and Exhibition Asiaand Pacific pp 1ndash6 August 2005

[9] S-Y Ruan G-J Li X-H Jiao Y-Z Sun and T T Lie ldquoAdaptivecontrol design for VSC-HVDC systems based on backsteppingmethodrdquo Electric Power Systems Research vol 77 no 5-6 pp559ndash565 2007

[10] A Moharana and P K Dash ldquoInput-output linearization androbust sliding-mode controller for the VSC-HVDC transmis-sion linkrdquo IEEE Transactions on Power Delivery vol 25 no 3pp 1952ndash1961 2010

[11] N Nayak S K Routray and P K Rout ldquoState feedback robust119867infin

controller for transient stability enhancement of Vsc-Hvdctransmission systemsrdquo Procedia Technology vol 4 pp 652ndash6602012 Proceedings of the 2nd International Conference onCom-puter Communication Control and Information Technology(C3IT 12) on February 25-26 2012

[12] H S Ramadan H Siguerdidjane M Petit and R Kacz-marek ldquoPerformance enhancement and robustness assessmentof VSCndashHVDC transmission systems controllers under uncer-taintiesrdquo International Journal of Electrical Power amp EnergySystems vol 35 no 1 pp 34ndash46 2012

[13] X-G Wei G-F Tang and J-C Zheng ldquoStudy of VSC-HVDCdiscrete model and its control strategiesrdquo Proceedings of theChinese Society of Electrical Engineering vol 27 no 28 pp 6ndash112007

[14] H Yang N Zhang and M J Ye ldquoStudy of VSC-HVDCconnected to passive network discrete model and its controlstrategiesrdquo Power System Protection and Control vol 40 no 4pp 37ndash42 2012

[15] V I Utkin and H-C Chang ldquoSliding mode control on electro-mechanical systemsrdquo Mathematical Problems in Engineeringvol 8 no 4-5 pp 451ndash473 2002

[16] H K Khalil and J W Grizzle Nonlinear Systems Prentice hallUpper Saddle River NJ USA 1996

[17] H Werner ldquoMultimodel robust control by fast outputsamplingmdashan lmi approachrdquo Automatica vol 34 no 12 pp1625ndash1630 1998

[18] T-L Tai and J-S Chen ldquoUPS inverter design using discrete-time sliding-mode control schemerdquo IEEE Transactions onIndustrial Electronics vol 49 no 1 pp 67ndash75 2002

[19] M C Saaj B Bandyopadhyay and H Unbehauen ldquoA newalgorithm for discrete-time sliding-mode control using fastoutput sampling feedbackrdquo IEEE Transactions on IndustrialElectronics vol 49 no 3 pp 518ndash523 2002

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


and sliding mode control strategy The feedbacklinearization was used to cancel nonlinearity and thesliding mode control offered invariant stability tomodeling uncertainties Reference [11] proposed thecontroller design based on 119867infin control methodolo-gies to deal with the nonlinearities introduced byrequirements to power flow and line voltage Refer-ence [12] addressed two different robust nonlinearcontrol methodologies based on slidingmode controlfor the VSC-HVDC transmission systemrsquos perfor-mance enhancement and stability improvement Inthis phase the studies are based on the nonlinearcontrol robust control and optimizing control of theVSC-HVDC system The experimental results fromresearch papers show these control strategies canimprove static dynamic and robust performance ofVSC-HVDC system

However the referred literatures are based on the continuous-time state In practice nowadays most controllers are imple-mented in discrete time It is known that the realization of acontroller using digital elements and complex programmablelogic devices can achieve maximum reproducibility at min-imum cost So it is necessary to do research on the discretemodel and control strategy In [13] the discrete PI controllersof VSC-HVDC system were established Reference [14] stud-ied the discrete PI control algorithm of VSC-HVDC systemwhich supplied power to the passive network

But so far the discrete-time mathematic model and con-trol strategy based on nonlinear control methods are seldomstudied by scholars Through proper feedback linearizationcontrol strategy complex nonlinear system synthesis prob-lems can be transformed to linear system synthesis problemsAs a kind of robust control method sliding mode controlis versatile to linear and nonlinear system It is easy to bedesigned and carried out Because of its complete robustnessto the parameters change and external disturbances whichmeet the match condition it gets extensive attention fromscholars and engineers [15]

Therefore the discrete mathematical model of the VSC-HVDC system based on nonlinear feedback linearizationand discrete sliding mode control algorithm are proposedin this paper And the fast output sampling (FOS) tech-nique is adopted in the output feedback which ensuresthe stability of the closed loop system in the condition ofquasi slidingmode controlThe simulation results performedin MATLABSIMULINK show that the proposed discretecontrol strategy can make the VSC-HVDC system have goodoperation performance

2 The Mathematical Model of VSC-HVDC

21 Continuous-Time State Space Model The structure dia-gram of VSC-HVDC is shown in Figure 1 The more widelyused continuous-time mathematical models based on thesynchronous reference coordinates are employed in thispaper which are shown as (1) and (7)

211 Rectifier Side Choose state variables x12 = (119894119903119889 119894119903119902)119879

controlled variablesu12 = (119880119888119903119889 119880119888119903119902)119879 and output variables

y12 = (119894119903119889 119894119903119902)119879Themathematical model of the rectifier side

is shown as

1 =minus119877119903

119871119903

1199091 +1205961199031199092 +119880119904119903119889

119871119903

minus1119871119903

1199061

2 =minus119877119903

119871119903

1199092 minus1205961199031199091 +119880119904119903119902

119871119903

minus1119871119903

1199062

1199101 = 1199091

1199102 = 1199092

(1)

where 119894119903119889

119894119903119902

and 119880119904119903119889

119880119904119903119902

are the 119889-119902 axis currents andvoltages on the rectifier side respectively 119880

119888119903119889

and 119880119888119903119902

arerespectively the 119889-119902 axis control inputs on the rectifier side119877119903

and 119871119903

are the corresponding equivalent resistance and theinductance on the rectifier side120596

119903

is theAC system frequencyon the rectifier side

Here the output variables on the rectifier side shouldbe (119875119903

119876119903)119879 119875119903

and 119876119903

are respectively the output valuesof active and reactive power on the rectifier side Forconvenience define 119880

119904119903

as the line voltage effective valueof the rectifier power supply Define the 119889 axis in thesynchronization reference frame coincidence with 119886 axis inthe three-phase reference coordinate so119880

119904119903119889

= 119880119904119903

119880119904119903119902

= 0[119875119903

119876119903

] = [119880119904119903119889119880119904119903119902

minus119880119904119903119902119880119904119903119889

] [119894119903119889

119894119903119902

] = [119880119904119903119894119903119889

119880119904119903119894119903119902

] Therefore the output

variables can be chosen as y12 = (119894119903119889 119894119903119902)119879

Define

1198911 =minus119877119903

119871119903

1199091 +1205961199031199092 +119880119904119903119889

119871119903

1198912 =minus119877119903

119871119903

1199092 minus1205961199031199091 +119880119904119903119902

119871119903

f119903

= (1198911 1198912)119879

g1 = (minus1119871119903

0)119879

g2 = (0 minus1119871119903

)

119879

(2)


x12 = f119903

+ g11199061 + g21199062

y12 = (ℎ1 (119909)

ℎ2 (119909)) = (

1199091

1199092)

(3)

Then do exact feedback linearization on system equation(3) By calculation the relative degree of 1199101 is denoted by1205741 = 1 The relative degree of 1199102 is denoted by 1205742 = 1 Basedon the feedback linearization theory [16] the exact feedback


AC1

Rectifier

C

C

C

C


Usr

T1

AC2T2

RrRiLr

Li

Ucr Uci Usi

Rdc

Rdc

Ldc

Ldc



(1199101

1199102) = (

1

2) = (

1205921

1205922)

= (

1198711205741119891119903

ℎ1

1198711205742119891119903

ℎ2)

+(

11987111989211198711205741minus1119891119903

ℎ1 11987111989221198711205741minus1119891119903

ℎ1

11987111989211198711205742minus1119891119903

ℎ2 11987111989221198711205742minus1119891119903

ℎ2

)(1199061

1199062)

(1

2) = (

1205921

1205922) = (

1198911

1198912)+(

minus1119871119903

0

0 minus1119871119903

)(1199061

1199062)

(4)

Because 10038161003816100381610038161003816minus1119871119903

00 minus1119871

119903

10038161003816100381610038161003816= 11198712

119903

= 0 ( minus1119871119903 00 minus1119871

119903

) is nonsingu-lar


(1199061

1199062) = (

119871119903

(1198911 minus 1205921)

119871119894

(1198912 minus 1205922)) (5)


1 = 1205921

2 = 1205922

1199101 = 1199091

1199102 = 1199092

(6)


(119894119894119889

119894119894119902




3 =minus119877119894

119871119894

1199093 +1205961198941199094 +119880119904119894119889

119871119894

minus1119871119894

1199063

4 =minus119877119894

119871119894

1199094 minus1205961198941199093 +119880119904119894119902

119871119894

minus1119871119894

1199064

5 =119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990932+ 1199094

2)

1198621199095+1198801198891198881 minus 1199095119862119877119889119888

1199103 = 1199095

1199104 = 1199094

(7)

where 119894119894119889

119894119894119902

and 119880119904119894119889

119880119904119894119902


119888119894119889

and 119880119888119894119902


119894

and 119871119894



119889119888


119894





119904119894119889

=

119880119904119894

119880119904119894119902

= 0 and 119876119894

= minus119880119904119894119902

119894119894119889

+ 119880119904119894119889

119894119894119902

= 119880119904119894

119894119894119902


119879Define

1198913 =minus119877119894

119871119894

1199093 +1205961198941199094 +119880119904119894119889

119871119894

1198914 =minus119877119894

119871119894

1199094 minus1205961198941199093 +119880119904119894119902

119871119894

1198915 =119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990923 + 119909

24)

1198621199095+1198801198891198881 minus 1199095119862119877119889119888

f119894

= (1198913 1198914 1198915)119879

g3 = (minus1119871119894

0 0)119879

g4 = (0 minus1119871119894

0)119879

(8)



x345 = f119894

+ g31199063 + g41199064

y34 = (ℎ3 (119909)

ℎ4 (119909)) = (

1199095

1199094)

(9)


(1199103

1199104) = (

5

4) = (

1205923

1205924)

= (

1198711205743119891119894

ℎ3

1198711205744119891119894

ℎ4)

+(

11987111989231198711205743minus1119891119894

ℎ3 11987111989241198711205743minus1119891119894

ℎ3

11987111989231198711205744minus1119891119894

ℎ4 11987111989241198711205744minus1119891119894

ℎ4)(

1199063

1199064)

(5

4) = (

1205923

1205924)

= (11988611198913 + 11988621198914 + 11988631198915

1198914)

+(

minus1198861119871119894

minus1198862119871119894

0 minus1119871119894

)(1199063

1199064)

(10)

where

1198861 =12059711989151205971199093

=119880119904119894119889

minus 2119877119894

11990931198621199095

1198862 =12059711989151205971199094

=119880119904119894119902

minus 2119877119894

1199094

1198621199095

1198863 =12059711989151205971199095

= minus119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990923 + 119909

24)

11986211990925minus

1119862119877119889119888

(11)


0 minus1119871119894

10038161003816100381610038161003816= 1198861119871

2119894


) isnonsingular


(1199063

1199064) = (

(11988611198911 + 11988631198913 minus 1205923 + 11988621205924) sdot119871119894

1198861119871119894

(1198914 minus 1205924)) (12)



3 = (119886231198861+

11988631198861119862119877119889119888

)1199095 +111988611205923 minus

119886211988611205924

4 = 1205924


119862119877119889119888

)1199095

1199103 = 1199095

1199104 = 1199094

(13)



119889119888

)1199095


(1205921 (119896)

1205922 (119896)) = (

1199101 (119896 + 1) minus 1199101 (119896)120591

1199102 (119896 + 1) minus 1199102 (119896)120591

)

=(

1199091 (119896 + 1) minus 1199091 (119896)120591

1199092 (119896 + 1) minus 1199092 (119896)120591

)

= (1198911 (119896)

1198912 (119896))+(

minus1119871119903

0

0 minus1119871119903

)(1199061 (119896)

1199062 (119896))

(14)

(1205923 (119896)

1205924 (119896)) = (

1199103 (119896 + 2) minus 21199103 (119896 + 1) + 1199103 (119896)1205912

1199104 (119896 + 1) minus 1199104 (119896)120591

)

=(

1199095 (119896 + 2) minus 21199095 (119896 + 1) + 1199095 (119896)1205912

1199094 (119896 + 1) minus 1199094 (119896)120591

)

= (11988611198913 (119896) + 11988621198914 (119896) + 11988631198915 (119896)

1198914 (119896))

+(

minus1198861119871119894

minus1198862119871119894

0 minus1119871119894

)(1199063 (119896)

1199064 (119896))

(15)



1199091 (119896 + 1) = 1199091 (119896) + 1205911205921 (119896)

1199092 (119896 + 1) = 1199092 (119896) + 1205911205922 (119896)

1199101 (119896) = 1199091 (119896)

1199102 (119896) = 1199092 (119896)

1199093 (119896 + 1) = 1199093 (119896) +(120591119886

231198861+

12059111988631198861119862119877119889119888

)1199095 (119896)

+120591

11988611205923 (119896) minus

120591119886211988611205924 (119896)

1199094 (119896 + 1) = 1199094 (119896) + 1205911205924 (119896)

1199095 (119896 + 1) = (1minus 1205911198863 minus120591

119862119877119889119888

)1199095 (119896)

1199103 (119896) = 1199095 (119896)

1199104 (119896) = 1199094 (119896)

(16)


x12 (k + 1) = Ar120591x12 (k) +Br12059112059212 (k)

y12 (k) = Crx12 (k)

x345 (k + 1) = Ai120591x345 (k) +Bi12059112059234 (k)

y34 (k) = Cix345 (k)

(17)

where

Ar120591 = (1 00 1

)

Br120591 = (120591 00 120591

)

Cr = (1 00 1

)

Ai120591 =(

1 01205911198863

2

1198861+

12059111988631198861119862119877119889119888

0 1 0

0 0 1 minus 1205911198863 minus120591

119862119877119889119888

)

Bi120591 = (

120591

1198861minus12059111988621198861

0 120591

0 0

)

Ci = (0 0 10 1 0

)

(18)





yk =[[[[[[

[

119910 ((119896 minus 1) 120591)119910 ((119896 minus 1) 120591 + Δ)

119910 ((119896 minus 1) 120591 + (119873 minus 1) Δ)

]]]]]]

]

(19)





where

Cr0 =

[[[[[[[

[

Cr

CrAr

CrArN1minus1

]]]]]]]

]

Dr0 =

[[[[[[[[[[

[

0CrBr

Cr

N1minus2sum

j=0Ar

jBr

]]]]]]]]]]

]

(21)


Define y12k = (1199101((119896minus1)120591)

1199102((119896minus1)120591)





where

Ci0 =

[[[[[[[

[

Ci

CiAi

CiAiN2minus1

]]]]]]]

]

Di0 =

[[[[[[[[[[

[

0CiBi

Ci

N2minus2sum

j=0Ai

jBi

]]]]]]]]]]

]

(23)


Define y34k = (1199103((119896minus1)120591)

1199103((119896minus1)120591+Δ

2)

119910

4

((119896minus1)120591)



Usr

+ +

+

minusPr

ND

kp

kis

Prrefirdref


Usr

+ +

+

minus

ND

Qr

kp

kis

Qrrefirqref






(24)



(25)


e12 = (1198901

1198902) = (

1199101

(119896) minus 119894119903119889ref (119896)

1199102

(119896) minus 119894119903119902ref (119896)

) (26)



119875119903ref 119876119903 and



s12 = (1199041

1199042) = (

1198901

(119896)

1198902

(119896)) (27)


Usi

+ +

+

minus

ND

Qi

kp

kis

Qiref iiqref



1199041

(119896 + 1) minus 1199041

(119896) = minus 1205881

1205911199041

(119896) minus 1205761

120591 sgn (1199041

(119896))

1199042

(119896 + 1) minus 1199042

(119896) = minus 1205882

1205911199042

(119896) minus 1205762

120591 sgn (1199042

(119896))

(28)



1205921 (119896) = minus 12058811199041 (119896) minus 1205761 sgn (1199041 (119896))

1205922 (119896) = minus 12058821199042 (119896) minus 1205762 sgn (1199042 (119896)) (29)


1199061

(119896) = minus119877119903

1198761199031

+120596119871119903

1198761199032

minus119871119903

1205921

+119880119904119903119889

1199062

(119896) = minus120596119871119903

1198761199031

minus119877119903

1198761199032

minus119871119903

1205922

+119880119904119903119902

(30)




e34 = (1198903

1198904) = (

1199103 (119896) minus 119880119889119888ref (119896)

1199104 (119896) minus 119894119894119902ref (119896)) (31)




119894



s34 = (1199043

1199044

) = (1198883

1198903

(119896) +1198903

(119896 + 1) minus 1198903

(119896)

120591

1198904

(119896)

) (32)


1199043

(119896 + 1) minus 1199043

(119896) = minus 1205883

1205911199043

(119896) minus 1205763

120591 sgn (1199043

(119896))

1199044

(119896 + 1) minus 1199044

(119896) = minus 1205884

1205911199044

(119896) minus 1205764

120591 sgn (1199044

(119896))

(33)



1205923 (119896) = minus 12058831199043 (119896) minus 1205763 sgn (1199043 (119896))

minus1198883 [1199103 (119896 + 1) minus 1199103 (119896)]

120591

1205924 (119896) = minus 12058841199044 (119896) minus 1205764 sgn (1199044 (119896))

(34)


1199063

(119896) = minus119877119894

1198761198943

+120596119894

119871119894

1198761198944

minus119871119894

1198863

1198861

(1198863

+2

119862119877119889119888

)1198761198945

minus119871119894

1198861

1205923

+119871119894

1198862

1198861

1205924

+119871119894

1198863

1198801198891198881

1198861

119862119877119889119888

+119880119904119894119889

1199064

(119896) = minus120596119894

119871119894

1198761198943

minus119877119894

1198761198944

minus119871119894

1205924

+119880119904119894119902

(35)





2





00

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

05

10

PrQr

t (s)

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

PiQi

t (s)

PiQi

(pu)


00

0

1

02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

Udc1

(pu)

Udc1

(c) The DC voltage




119904119903

119880119904119894

230 kv120596119903

120596119894


119877119894

0325Ω119871119903

119871119894










0

1

minus1

00 05 10 15 20 25 30 35 40 5045t (s)

PrQr

PrQ

r(p

u)


10

05

00

PiQi

minus10

minus05

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

(pu)



00

000102030405060708091011

05 10 15 20 25 30 35 40 5045t (s)


PrQr

PrQ

r(p

u)



0102030405

00

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

PiQi

(pu)




5 Conclusions






00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










AC1

Rectifier

C

C

C

C


Usr

T1

AC2T2

RrRiLr

Li

Ucr Uci Usi

Rdc

Rdc

Ldc

Ldc



(1199101

1199102) = (

1

2) = (

1205921

1205922)

= (

1198711205741119891119903

ℎ1

1198711205742119891119903

ℎ2)

+(

11987111989211198711205741minus1119891119903

ℎ1 11987111989221198711205741minus1119891119903

ℎ1

11987111989211198711205742minus1119891119903

ℎ2 11987111989221198711205742minus1119891119903

ℎ2

)(1199061

1199062)

(1

2) = (

1205921

1205922) = (

1198911

1198912)+(

minus1119871119903

0

0 minus1119871119903

)(1199061

1199062)

(4)

Because 10038161003816100381610038161003816minus1119871119903

00 minus1119871

119903

10038161003816100381610038161003816= 11198712

119903

= 0 ( minus1119871119903 00 minus1119871

119903

) is nonsingu-lar


(1199061

1199062) = (

119871119903

(1198911 minus 1205921)

119871119894

(1198912 minus 1205922)) (5)


1 = 1205921

2 = 1205922

1199101 = 1199091

1199102 = 1199092

(6)


(119894119894119889

119894119894119902




3 =minus119877119894

119871119894

1199093 +1205961198941199094 +119880119904119894119889

119871119894

minus1119871119894

1199063

4 =minus119877119894

119871119894

1199094 minus1205961198941199093 +119880119904119894119902

119871119894

minus1119871119894

1199064

5 =119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990932+ 1199094

2)

1198621199095+1198801198891198881 minus 1199095119862119877119889119888

1199103 = 1199095

1199104 = 1199094

(7)

where 119894119894119889

119894119894119902

and 119880119904119894119889

119880119904119894119902


119888119894119889

and 119880119888119894119902


119894

and 119871119894



119889119888


119894





119904119894119889

=

119880119904119894

119880119904119894119902

= 0 and 119876119894

= minus119880119904119894119902

119894119894119889

+ 119880119904119894119889

119894119894119902

= 119880119904119894

119894119894119902


119879Define

1198913 =minus119877119894

119871119894

1199093 +1205961198941199094 +119880119904119894119889

119871119894

1198914 =minus119877119894

119871119894

1199094 minus1205961198941199093 +119880119904119894119902

119871119894

1198915 =119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990923 + 119909

24)

1198621199095+1198801198891198881 minus 1199095119862119877119889119888

f119894

= (1198913 1198914 1198915)119879

g3 = (minus1119871119894

0 0)119879

g4 = (0 minus1119871119894

0)119879

(8)



x345 = f119894

+ g31199063 + g41199064

y34 = (ℎ3 (119909)

ℎ4 (119909)) = (

1199095

1199094)

(9)


(1199103

1199104) = (

5

4) = (

1205923

1205924)

= (

1198711205743119891119894

ℎ3

1198711205744119891119894

ℎ4)

+(

11987111989231198711205743minus1119891119894

ℎ3 11987111989241198711205743minus1119891119894

ℎ3

11987111989231198711205744minus1119891119894

ℎ4 11987111989241198711205744minus1119891119894

ℎ4)(

1199063

1199064)

(5

4) = (

1205923

1205924)

= (11988611198913 + 11988621198914 + 11988631198915

1198914)

+(

minus1198861119871119894

minus1198862119871119894

0 minus1119871119894

)(1199063

1199064)

(10)

where

1198861 =12059711989151205971199093

=119880119904119894119889

minus 2119877119894

11990931198621199095

1198862 =12059711989151205971199094

=119880119904119894119902

minus 2119877119894

1199094

1198621199095

1198863 =12059711989151205971199095

= minus119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990923 + 119909

24)

11986211990925minus

1119862119877119889119888

(11)


0 minus1119871119894

10038161003816100381610038161003816= 1198861119871

2119894


) isnonsingular


(1199063

1199064) = (

(11988611198911 + 11988631198913 minus 1205923 + 11988621205924) sdot119871119894

1198861119871119894

(1198914 minus 1205924)) (12)



3 = (119886231198861+

11988631198861119862119877119889119888

)1199095 +111988611205923 minus

119886211988611205924

4 = 1205924


119862119877119889119888

)1199095

1199103 = 1199095

1199104 = 1199094

(13)



119889119888

)1199095


(1205921 (119896)

1205922 (119896)) = (

1199101 (119896 + 1) minus 1199101 (119896)120591

1199102 (119896 + 1) minus 1199102 (119896)120591

)

=(

1199091 (119896 + 1) minus 1199091 (119896)120591

1199092 (119896 + 1) minus 1199092 (119896)120591

)

= (1198911 (119896)

1198912 (119896))+(

minus1119871119903

0

0 minus1119871119903

)(1199061 (119896)

1199062 (119896))

(14)

(1205923 (119896)

1205924 (119896)) = (

1199103 (119896 + 2) minus 21199103 (119896 + 1) + 1199103 (119896)1205912

1199104 (119896 + 1) minus 1199104 (119896)120591

)

=(

1199095 (119896 + 2) minus 21199095 (119896 + 1) + 1199095 (119896)1205912

1199094 (119896 + 1) minus 1199094 (119896)120591

)

= (11988611198913 (119896) + 11988621198914 (119896) + 11988631198915 (119896)

1198914 (119896))

+(

minus1198861119871119894

minus1198862119871119894

0 minus1119871119894

)(1199063 (119896)

1199064 (119896))

(15)



1199091 (119896 + 1) = 1199091 (119896) + 1205911205921 (119896)

1199092 (119896 + 1) = 1199092 (119896) + 1205911205922 (119896)

1199101 (119896) = 1199091 (119896)

1199102 (119896) = 1199092 (119896)

1199093 (119896 + 1) = 1199093 (119896) +(120591119886

231198861+

12059111988631198861119862119877119889119888

)1199095 (119896)

+120591

11988611205923 (119896) minus

120591119886211988611205924 (119896)

1199094 (119896 + 1) = 1199094 (119896) + 1205911205924 (119896)

1199095 (119896 + 1) = (1minus 1205911198863 minus120591

119862119877119889119888

)1199095 (119896)

1199103 (119896) = 1199095 (119896)

1199104 (119896) = 1199094 (119896)

(16)


x12 (k + 1) = Ar120591x12 (k) +Br12059112059212 (k)

y12 (k) = Crx12 (k)

x345 (k + 1) = Ai120591x345 (k) +Bi12059112059234 (k)

y34 (k) = Cix345 (k)

(17)

where

Ar120591 = (1 00 1

)

Br120591 = (120591 00 120591

)

Cr = (1 00 1

)

Ai120591 =(

1 01205911198863

2

1198861+

12059111988631198861119862119877119889119888

0 1 0

0 0 1 minus 1205911198863 minus120591

119862119877119889119888

)

Bi120591 = (

120591

1198861minus12059111988621198861

0 120591

0 0

)

Ci = (0 0 10 1 0

)

(18)





yk =[[[[[[

[

119910 ((119896 minus 1) 120591)119910 ((119896 minus 1) 120591 + Δ)

119910 ((119896 minus 1) 120591 + (119873 minus 1) Δ)

]]]]]]

]

(19)





where

Cr0 =

[[[[[[[

[

Cr

CrAr

CrArN1minus1

]]]]]]]

]

Dr0 =

[[[[[[[[[[

[

0CrBr

Cr

N1minus2sum

j=0Ar

jBr

]]]]]]]]]]

]

(21)


Define y12k = (1199101((119896minus1)120591)

1199102((119896minus1)120591)





where

Ci0 =

[[[[[[[

[

Ci

CiAi

CiAiN2minus1

]]]]]]]

]

Di0 =

[[[[[[[[[[

[

0CiBi

Ci

N2minus2sum

j=0Ai

jBi

]]]]]]]]]]

]

(23)


Define y34k = (1199103((119896minus1)120591)

1199103((119896minus1)120591+Δ

2)

119910

4

((119896minus1)120591)



Usr

+ +

+

minusPr

ND

kp

kis

Prrefirdref


Usr

+ +

+

minus

ND

Qr

kp

kis

Qrrefirqref






(24)



(25)


e12 = (1198901

1198902) = (

1199101

(119896) minus 119894119903119889ref (119896)

1199102

(119896) minus 119894119903119902ref (119896)

) (26)



119875119903ref 119876119903 and



s12 = (1199041

1199042) = (

1198901

(119896)

1198902

(119896)) (27)


Usi

+ +

+

minus

ND

Qi

kp

kis

Qiref iiqref



1199041

(119896 + 1) minus 1199041

(119896) = minus 1205881

1205911199041

(119896) minus 1205761

120591 sgn (1199041

(119896))

1199042

(119896 + 1) minus 1199042

(119896) = minus 1205882

1205911199042

(119896) minus 1205762

120591 sgn (1199042

(119896))

(28)



1205921 (119896) = minus 12058811199041 (119896) minus 1205761 sgn (1199041 (119896))

1205922 (119896) = minus 12058821199042 (119896) minus 1205762 sgn (1199042 (119896)) (29)


1199061

(119896) = minus119877119903

1198761199031

+120596119871119903

1198761199032

minus119871119903

1205921

+119880119904119903119889

1199062

(119896) = minus120596119871119903

1198761199031

minus119877119903

1198761199032

minus119871119903

1205922

+119880119904119903119902

(30)




e34 = (1198903

1198904) = (

1199103 (119896) minus 119880119889119888ref (119896)

1199104 (119896) minus 119894119894119902ref (119896)) (31)




119894



s34 = (1199043

1199044

) = (1198883

1198903

(119896) +1198903

(119896 + 1) minus 1198903

(119896)

120591

1198904

(119896)

) (32)


1199043

(119896 + 1) minus 1199043

(119896) = minus 1205883

1205911199043

(119896) minus 1205763

120591 sgn (1199043

(119896))

1199044

(119896 + 1) minus 1199044

(119896) = minus 1205884

1205911199044

(119896) minus 1205764

120591 sgn (1199044

(119896))

(33)



1205923 (119896) = minus 12058831199043 (119896) minus 1205763 sgn (1199043 (119896))

minus1198883 [1199103 (119896 + 1) minus 1199103 (119896)]

120591

1205924 (119896) = minus 12058841199044 (119896) minus 1205764 sgn (1199044 (119896))

(34)


1199063

(119896) = minus119877119894

1198761198943

+120596119894

119871119894

1198761198944

minus119871119894

1198863

1198861

(1198863

+2

119862119877119889119888

)1198761198945

minus119871119894

1198861

1205923

+119871119894

1198862

1198861

1205924

+119871119894

1198863

1198801198891198881

1198861

119862119877119889119888

+119880119904119894119889

1199064

(119896) = minus120596119894

119871119894

1198761198943

minus119877119894

1198761198944

minus119871119894

1205924

+119880119904119894119902

(35)





2





00

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

05

10

PrQr

t (s)

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

PiQi

t (s)

PiQi

(pu)


00

0

1

02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

Udc1

(pu)

Udc1

(c) The DC voltage




119904119903

119880119904119894

230 kv120596119903

120596119894


119877119894

0325Ω119871119903

119871119894










0

1

minus1

00 05 10 15 20 25 30 35 40 5045t (s)

PrQr

PrQ

r(p

u)


10

05

00

PiQi

minus10

minus05

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

(pu)



00

000102030405060708091011

05 10 15 20 25 30 35 40 5045t (s)


PrQr

PrQ

r(p

u)



0102030405

00

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

PiQi

(pu)




5 Conclusions






00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











x345 = f119894

+ g31199063 + g41199064

y34 = (ℎ3 (119909)

ℎ4 (119909)) = (

1199095

1199094)

(9)


(1199103

1199104) = (

5

4) = (

1205923

1205924)

= (

1198711205743119891119894

ℎ3

1198711205744119891119894

ℎ4)

+(

11987111989231198711205743minus1119891119894

ℎ3 11987111989241198711205743minus1119891119894

ℎ3

11987111989231198711205744minus1119891119894

ℎ4 11987111989241198711205744minus1119891119894

ℎ4)(

1199063

1199064)

(5

4) = (

1205923

1205924)

= (11988611198913 + 11988621198914 + 11988631198915

1198914)

+(

minus1198861119871119894

minus1198862119871119894

0 minus1119871119894

)(1199063

1199064)

(10)

where

1198861 =12059711989151205971199093

=119880119904119894119889

minus 2119877119894

11990931198621199095

1198862 =12059711989151205971199094

=119880119904119894119902

minus 2119877119894

1199094

1198621199095

1198863 =12059711989151205971199095

= minus119880119904119894119889

1199093 + 1198801199041198941199021199094 minus 119877119894 (11990923 + 119909

24)

11986211990925minus

1119862119877119889119888

(11)


0 minus1119871119894

10038161003816100381610038161003816= 1198861119871

2119894


) isnonsingular


(1199063

1199064) = (

(11988611198911 + 11988631198913 minus 1205923 + 11988621205924) sdot119871119894

1198861119871119894

(1198914 minus 1205924)) (12)



3 = (119886231198861+

11988631198861119862119877119889119888

)1199095 +111988611205923 minus

119886211988611205924

4 = 1205924


119862119877119889119888

)1199095

1199103 = 1199095

1199104 = 1199094

(13)



119889119888

)1199095


(1205921 (119896)

1205922 (119896)) = (

1199101 (119896 + 1) minus 1199101 (119896)120591

1199102 (119896 + 1) minus 1199102 (119896)120591

)

=(

1199091 (119896 + 1) minus 1199091 (119896)120591

1199092 (119896 + 1) minus 1199092 (119896)120591

)

= (1198911 (119896)

1198912 (119896))+(

minus1119871119903

0

0 minus1119871119903

)(1199061 (119896)

1199062 (119896))

(14)

(1205923 (119896)

1205924 (119896)) = (

1199103 (119896 + 2) minus 21199103 (119896 + 1) + 1199103 (119896)1205912

1199104 (119896 + 1) minus 1199104 (119896)120591

)

=(

1199095 (119896 + 2) minus 21199095 (119896 + 1) + 1199095 (119896)1205912

1199094 (119896 + 1) minus 1199094 (119896)120591

)

= (11988611198913 (119896) + 11988621198914 (119896) + 11988631198915 (119896)

1198914 (119896))

+(

minus1198861119871119894

minus1198862119871119894

0 minus1119871119894

)(1199063 (119896)

1199064 (119896))

(15)



1199091 (119896 + 1) = 1199091 (119896) + 1205911205921 (119896)

1199092 (119896 + 1) = 1199092 (119896) + 1205911205922 (119896)

1199101 (119896) = 1199091 (119896)

1199102 (119896) = 1199092 (119896)

1199093 (119896 + 1) = 1199093 (119896) +(120591119886

231198861+

12059111988631198861119862119877119889119888

)1199095 (119896)

+120591

11988611205923 (119896) minus

120591119886211988611205924 (119896)

1199094 (119896 + 1) = 1199094 (119896) + 1205911205924 (119896)

1199095 (119896 + 1) = (1minus 1205911198863 minus120591

119862119877119889119888

)1199095 (119896)

1199103 (119896) = 1199095 (119896)

1199104 (119896) = 1199094 (119896)

(16)


x12 (k + 1) = Ar120591x12 (k) +Br12059112059212 (k)

y12 (k) = Crx12 (k)

x345 (k + 1) = Ai120591x345 (k) +Bi12059112059234 (k)

y34 (k) = Cix345 (k)

(17)

where

Ar120591 = (1 00 1

)

Br120591 = (120591 00 120591

)

Cr = (1 00 1

)

Ai120591 =(

1 01205911198863

2

1198861+

12059111988631198861119862119877119889119888

0 1 0

0 0 1 minus 1205911198863 minus120591

119862119877119889119888

)

Bi120591 = (

120591

1198861minus12059111988621198861

0 120591

0 0

)

Ci = (0 0 10 1 0

)

(18)





yk =[[[[[[

[

119910 ((119896 minus 1) 120591)119910 ((119896 minus 1) 120591 + Δ)

119910 ((119896 minus 1) 120591 + (119873 minus 1) Δ)

]]]]]]

]

(19)





where

Cr0 =

[[[[[[[

[

Cr

CrAr

CrArN1minus1

]]]]]]]

]

Dr0 =

[[[[[[[[[[

[

0CrBr

Cr

N1minus2sum

j=0Ar

jBr

]]]]]]]]]]

]

(21)


Define y12k = (1199101((119896minus1)120591)

1199102((119896minus1)120591)





where

Ci0 =

[[[[[[[

[

Ci

CiAi

CiAiN2minus1

]]]]]]]

]

Di0 =

[[[[[[[[[[

[

0CiBi

Ci

N2minus2sum

j=0Ai

jBi

]]]]]]]]]]

]

(23)


Define y34k = (1199103((119896minus1)120591)

1199103((119896minus1)120591+Δ

2)

119910

4

((119896minus1)120591)



Usr

+ +

+

minusPr

ND

kp

kis

Prrefirdref


Usr

+ +

+

minus

ND

Qr

kp

kis

Qrrefirqref






(24)



(25)


e12 = (1198901

1198902) = (

1199101

(119896) minus 119894119903119889ref (119896)

1199102

(119896) minus 119894119903119902ref (119896)

) (26)



119875119903ref 119876119903 and



s12 = (1199041

1199042) = (

1198901

(119896)

1198902

(119896)) (27)


Usi

+ +

+

minus

ND

Qi

kp

kis

Qiref iiqref



1199041

(119896 + 1) minus 1199041

(119896) = minus 1205881

1205911199041

(119896) minus 1205761

120591 sgn (1199041

(119896))

1199042

(119896 + 1) minus 1199042

(119896) = minus 1205882

1205911199042

(119896) minus 1205762

120591 sgn (1199042

(119896))

(28)



1205921 (119896) = minus 12058811199041 (119896) minus 1205761 sgn (1199041 (119896))

1205922 (119896) = minus 12058821199042 (119896) minus 1205762 sgn (1199042 (119896)) (29)


1199061

(119896) = minus119877119903

1198761199031

+120596119871119903

1198761199032

minus119871119903

1205921

+119880119904119903119889

1199062

(119896) = minus120596119871119903

1198761199031

minus119877119903

1198761199032

minus119871119903

1205922

+119880119904119903119902

(30)




e34 = (1198903

1198904) = (

1199103 (119896) minus 119880119889119888ref (119896)

1199104 (119896) minus 119894119894119902ref (119896)) (31)




119894



s34 = (1199043

1199044

) = (1198883

1198903

(119896) +1198903

(119896 + 1) minus 1198903

(119896)

120591

1198904

(119896)

) (32)


1199043

(119896 + 1) minus 1199043

(119896) = minus 1205883

1205911199043

(119896) minus 1205763

120591 sgn (1199043

(119896))

1199044

(119896 + 1) minus 1199044

(119896) = minus 1205884

1205911199044

(119896) minus 1205764

120591 sgn (1199044

(119896))

(33)



1205923 (119896) = minus 12058831199043 (119896) minus 1205763 sgn (1199043 (119896))

minus1198883 [1199103 (119896 + 1) minus 1199103 (119896)]

120591

1205924 (119896) = minus 12058841199044 (119896) minus 1205764 sgn (1199044 (119896))

(34)


1199063

(119896) = minus119877119894

1198761198943

+120596119894

119871119894

1198761198944

minus119871119894

1198863

1198861

(1198863

+2

119862119877119889119888

)1198761198945

minus119871119894

1198861

1205923

+119871119894

1198862

1198861

1205924

+119871119894

1198863

1198801198891198881

1198861

119862119877119889119888

+119880119904119894119889

1199064

(119896) = minus120596119894

119871119894

1198761198943

minus119877119894

1198761198944

minus119871119894

1205924

+119880119904119894119902

(35)





2





00

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

05

10

PrQr

t (s)

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

PiQi

t (s)

PiQi

(pu)


00

0

1

02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

Udc1

(pu)

Udc1

(c) The DC voltage




119904119903

119880119904119894

230 kv120596119903

120596119894


119877119894

0325Ω119871119903

119871119894










0

1

minus1

00 05 10 15 20 25 30 35 40 5045t (s)

PrQr

PrQ

r(p

u)


10

05

00

PiQi

minus10

minus05

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

(pu)



00

000102030405060708091011

05 10 15 20 25 30 35 40 5045t (s)


PrQr

PrQ

r(p

u)



0102030405

00

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

PiQi

(pu)




5 Conclusions






00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199091 (119896 + 1) = 1199091 (119896) + 1205911205921 (119896)

1199092 (119896 + 1) = 1199092 (119896) + 1205911205922 (119896)

1199101 (119896) = 1199091 (119896)

1199102 (119896) = 1199092 (119896)

1199093 (119896 + 1) = 1199093 (119896) +(120591119886

231198861+

12059111988631198861119862119877119889119888

)1199095 (119896)

+120591

11988611205923 (119896) minus

120591119886211988611205924 (119896)

1199094 (119896 + 1) = 1199094 (119896) + 1205911205924 (119896)

1199095 (119896 + 1) = (1minus 1205911198863 minus120591

119862119877119889119888

)1199095 (119896)

1199103 (119896) = 1199095 (119896)

1199104 (119896) = 1199094 (119896)

(16)


x12 (k + 1) = Ar120591x12 (k) +Br12059112059212 (k)

y12 (k) = Crx12 (k)

x345 (k + 1) = Ai120591x345 (k) +Bi12059112059234 (k)

y34 (k) = Cix345 (k)

(17)

where

Ar120591 = (1 00 1

)

Br120591 = (120591 00 120591

)

Cr = (1 00 1

)

Ai120591 =(

1 01205911198863

2

1198861+

12059111988631198861119862119877119889119888

0 1 0

0 0 1 minus 1205911198863 minus120591

119862119877119889119888

)

Bi120591 = (

120591

1198861minus12059111988621198861

0 120591

0 0

)

Ci = (0 0 10 1 0

)

(18)





yk =[[[[[[

[

119910 ((119896 minus 1) 120591)119910 ((119896 minus 1) 120591 + Δ)

119910 ((119896 minus 1) 120591 + (119873 minus 1) Δ)

]]]]]]

]

(19)





where

Cr0 =

[[[[[[[

[

Cr

CrAr

CrArN1minus1

]]]]]]]

]

Dr0 =

[[[[[[[[[[

[

0CrBr

Cr

N1minus2sum

j=0Ar

jBr

]]]]]]]]]]

]

(21)


Define y12k = (1199101((119896minus1)120591)

1199102((119896minus1)120591)





where

Ci0 =

[[[[[[[

[

Ci

CiAi

CiAiN2minus1

]]]]]]]

]

Di0 =

[[[[[[[[[[

[

0CiBi

Ci

N2minus2sum

j=0Ai

jBi

]]]]]]]]]]

]

(23)


Define y34k = (1199103((119896minus1)120591)

1199103((119896minus1)120591+Δ

2)

119910

4

((119896minus1)120591)



Usr

+ +

+

minusPr

ND

kp

kis

Prrefirdref


Usr

+ +

+

minus

ND

Qr

kp

kis

Qrrefirqref






(24)



(25)


e12 = (1198901

1198902) = (

1199101

(119896) minus 119894119903119889ref (119896)

1199102

(119896) minus 119894119903119902ref (119896)

) (26)



119875119903ref 119876119903 and



s12 = (1199041

1199042) = (

1198901

(119896)

1198902

(119896)) (27)


Usi

+ +

+

minus

ND

Qi

kp

kis

Qiref iiqref



1199041

(119896 + 1) minus 1199041

(119896) = minus 1205881

1205911199041

(119896) minus 1205761

120591 sgn (1199041

(119896))

1199042

(119896 + 1) minus 1199042

(119896) = minus 1205882

1205911199042

(119896) minus 1205762

120591 sgn (1199042

(119896))

(28)



1205921 (119896) = minus 12058811199041 (119896) minus 1205761 sgn (1199041 (119896))

1205922 (119896) = minus 12058821199042 (119896) minus 1205762 sgn (1199042 (119896)) (29)


1199061

(119896) = minus119877119903

1198761199031

+120596119871119903

1198761199032

minus119871119903

1205921

+119880119904119903119889

1199062

(119896) = minus120596119871119903

1198761199031

minus119877119903

1198761199032

minus119871119903

1205922

+119880119904119903119902

(30)




e34 = (1198903

1198904) = (

1199103 (119896) minus 119880119889119888ref (119896)

1199104 (119896) minus 119894119894119902ref (119896)) (31)




119894



s34 = (1199043

1199044

) = (1198883

1198903

(119896) +1198903

(119896 + 1) minus 1198903

(119896)

120591

1198904

(119896)

) (32)


1199043

(119896 + 1) minus 1199043

(119896) = minus 1205883

1205911199043

(119896) minus 1205763

120591 sgn (1199043

(119896))

1199044

(119896 + 1) minus 1199044

(119896) = minus 1205884

1205911199044

(119896) minus 1205764

120591 sgn (1199044

(119896))

(33)



1205923 (119896) = minus 12058831199043 (119896) minus 1205763 sgn (1199043 (119896))

minus1198883 [1199103 (119896 + 1) minus 1199103 (119896)]

120591

1205924 (119896) = minus 12058841199044 (119896) minus 1205764 sgn (1199044 (119896))

(34)


1199063

(119896) = minus119877119894

1198761198943

+120596119894

119871119894

1198761198944

minus119871119894

1198863

1198861

(1198863

+2

119862119877119889119888

)1198761198945

minus119871119894

1198861

1205923

+119871119894

1198862

1198861

1205924

+119871119894

1198863

1198801198891198881

1198861

119862119877119889119888

+119880119904119894119889

1199064

(119896) = minus120596119894

119871119894

1198761198943

minus119877119894

1198761198944

minus119871119894

1205924

+119880119904119894119902

(35)





2





00

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

05

10

PrQr

t (s)

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

PiQi

t (s)

PiQi

(pu)


00

0

1

02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

Udc1

(pu)

Udc1

(c) The DC voltage




119904119903

119880119904119894

230 kv120596119903

120596119894


119877119894

0325Ω119871119903

119871119894










0

1

minus1

00 05 10 15 20 25 30 35 40 5045t (s)

PrQr

PrQ

r(p

u)


10

05

00

PiQi

minus10

minus05

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

(pu)



00

000102030405060708091011

05 10 15 20 25 30 35 40 5045t (s)


PrQr

PrQ

r(p

u)



0102030405

00

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

PiQi

(pu)




5 Conclusions






00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

Cr0 =

[[[[[[[

[

Cr

CrAr

CrArN1minus1

]]]]]]]

]

Dr0 =

[[[[[[[[[[

[

0CrBr

Cr

N1minus2sum

j=0Ar

jBr

]]]]]]]]]]

]

(21)


Define y12k = (1199101((119896minus1)120591)

1199102((119896minus1)120591)





where

Ci0 =

[[[[[[[

[

Ci

CiAi

CiAiN2minus1

]]]]]]]

]

Di0 =

[[[[[[[[[[

[

0CiBi

Ci

N2minus2sum

j=0Ai

jBi

]]]]]]]]]]

]

(23)


Define y34k = (1199103((119896minus1)120591)

1199103((119896minus1)120591+Δ

2)

119910

4

((119896minus1)120591)



Usr

+ +

+

minusPr

ND

kp

kis

Prrefirdref


Usr

+ +

+

minus

ND

Qr

kp

kis

Qrrefirqref






(24)



(25)


e12 = (1198901

1198902) = (

1199101

(119896) minus 119894119903119889ref (119896)

1199102

(119896) minus 119894119903119902ref (119896)

) (26)



119875119903ref 119876119903 and



s12 = (1199041

1199042) = (

1198901

(119896)

1198902

(119896)) (27)


Usi

+ +

+

minus

ND

Qi

kp

kis

Qiref iiqref



1199041

(119896 + 1) minus 1199041

(119896) = minus 1205881

1205911199041

(119896) minus 1205761

120591 sgn (1199041

(119896))

1199042

(119896 + 1) minus 1199042

(119896) = minus 1205882

1205911199042

(119896) minus 1205762

120591 sgn (1199042

(119896))

(28)



1205921 (119896) = minus 12058811199041 (119896) minus 1205761 sgn (1199041 (119896))

1205922 (119896) = minus 12058821199042 (119896) minus 1205762 sgn (1199042 (119896)) (29)


1199061

(119896) = minus119877119903

1198761199031

+120596119871119903

1198761199032

minus119871119903

1205921

+119880119904119903119889

1199062

(119896) = minus120596119871119903

1198761199031

minus119877119903

1198761199032

minus119871119903

1205922

+119880119904119903119902

(30)




e34 = (1198903

1198904) = (

1199103 (119896) minus 119880119889119888ref (119896)

1199104 (119896) minus 119894119894119902ref (119896)) (31)




119894



s34 = (1199043

1199044

) = (1198883

1198903

(119896) +1198903

(119896 + 1) minus 1198903

(119896)

120591

1198904

(119896)

) (32)


1199043

(119896 + 1) minus 1199043

(119896) = minus 1205883

1205911199043

(119896) minus 1205763

120591 sgn (1199043

(119896))

1199044

(119896 + 1) minus 1199044

(119896) = minus 1205884

1205911199044

(119896) minus 1205764

120591 sgn (1199044

(119896))

(33)



1205923 (119896) = minus 12058831199043 (119896) minus 1205763 sgn (1199043 (119896))

minus1198883 [1199103 (119896 + 1) minus 1199103 (119896)]

120591

1205924 (119896) = minus 12058841199044 (119896) minus 1205764 sgn (1199044 (119896))

(34)


1199063

(119896) = minus119877119894

1198761198943

+120596119894

119871119894

1198761198944

minus119871119894

1198863

1198861

(1198863

+2

119862119877119889119888

)1198761198945

minus119871119894

1198861

1205923

+119871119894

1198862

1198861

1205924

+119871119894

1198863

1198801198891198881

1198861

119862119877119889119888

+119880119904119894119889

1199064

(119896) = minus120596119894

119871119894

1198761198943

minus119877119894

1198761198944

minus119871119894

1205924

+119880119904119894119902

(35)





2





00

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

05

10

PrQr

t (s)

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

PiQi

t (s)

PiQi

(pu)


00

0

1

02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

Udc1

(pu)

Udc1

(c) The DC voltage




119904119903

119880119904119894

230 kv120596119903

120596119894


119877119894

0325Ω119871119903

119871119894










0

1

minus1

00 05 10 15 20 25 30 35 40 5045t (s)

PrQr

PrQ

r(p

u)


10

05

00

PiQi

minus10

minus05

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

(pu)



00

000102030405060708091011

05 10 15 20 25 30 35 40 5045t (s)


PrQr

PrQ

r(p

u)



0102030405

00

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

PiQi

(pu)




5 Conclusions






00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Usi

+ +

+

minus

ND

Qi

kp

kis

Qiref iiqref



1199041

(119896 + 1) minus 1199041

(119896) = minus 1205881

1205911199041

(119896) minus 1205761

120591 sgn (1199041

(119896))

1199042

(119896 + 1) minus 1199042

(119896) = minus 1205882

1205911199042

(119896) minus 1205762

120591 sgn (1199042

(119896))

(28)



1205921 (119896) = minus 12058811199041 (119896) minus 1205761 sgn (1199041 (119896))

1205922 (119896) = minus 12058821199042 (119896) minus 1205762 sgn (1199042 (119896)) (29)


1199061

(119896) = minus119877119903

1198761199031

+120596119871119903

1198761199032

minus119871119903

1205921

+119880119904119903119889

1199062

(119896) = minus120596119871119903

1198761199031

minus119877119903

1198761199032

minus119871119903

1205922

+119880119904119903119902

(30)




e34 = (1198903

1198904) = (

1199103 (119896) minus 119880119889119888ref (119896)

1199104 (119896) minus 119894119894119902ref (119896)) (31)




119894



s34 = (1199043

1199044

) = (1198883

1198903

(119896) +1198903

(119896 + 1) minus 1198903

(119896)

120591

1198904

(119896)

) (32)


1199043

(119896 + 1) minus 1199043

(119896) = minus 1205883

1205911199043

(119896) minus 1205763

120591 sgn (1199043

(119896))

1199044

(119896 + 1) minus 1199044

(119896) = minus 1205884

1205911199044

(119896) minus 1205764

120591 sgn (1199044

(119896))

(33)



1205923 (119896) = minus 12058831199043 (119896) minus 1205763 sgn (1199043 (119896))

minus1198883 [1199103 (119896 + 1) minus 1199103 (119896)]

120591

1205924 (119896) = minus 12058841199044 (119896) minus 1205764 sgn (1199044 (119896))

(34)


1199063

(119896) = minus119877119894

1198761198943

+120596119894

119871119894

1198761198944

minus119871119894

1198863

1198861

(1198863

+2

119862119877119889119888

)1198761198945

minus119871119894

1198861

1205923

+119871119894

1198862

1198861

1205924

+119871119894

1198863

1198801198891198881

1198861

119862119877119889119888

+119880119904119894119889

1199064

(119896) = minus120596119894

119871119894

1198761198943

minus119877119894

1198761198944

minus119871119894

1205924

+119880119904119894119902

(35)





2





00

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

05

10

PrQr

t (s)

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

PiQi

t (s)

PiQi

(pu)


00

0

1

02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

Udc1

(pu)

Udc1

(c) The DC voltage




119904119903

119880119904119894

230 kv120596119903

120596119894


119877119894

0325Ω119871119903

119871119894










0

1

minus1

00 05 10 15 20 25 30 35 40 5045t (s)

PrQr

PrQ

r(p

u)


10

05

00

PiQi

minus10

minus05

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

(pu)



00

000102030405060708091011

05 10 15 20 25 30 35 40 5045t (s)


PrQr

PrQ

r(p

u)



0102030405

00

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

PiQi

(pu)




5 Conclusions






00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

05

10

PrQr

t (s)

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

PiQi

t (s)

PiQi

(pu)


00

0

1

02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

Udc1

(pu)

Udc1

(c) The DC voltage




119904119903

119880119904119894

230 kv120596119903

120596119894


119877119894

0325Ω119871119903

119871119894










0

1

minus1

00 05 10 15 20 25 30 35 40 5045t (s)

PrQr

PrQ

r(p

u)


10

05

00

PiQi

minus10

minus05

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

(pu)



00

000102030405060708091011

05 10 15 20 25 30 35 40 5045t (s)


PrQr

PrQ

r(p

u)



0102030405

00

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

PiQi

(pu)




5 Conclusions






00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

minus1

00 05 10 15 20 25 30 35 40 5045t (s)

PrQr

PrQ

r(p

u)


10

05

00

PiQi

minus10

minus05

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

(pu)



00

000102030405060708091011

05 10 15 20 25 30 35 40 5045t (s)


PrQr

PrQ

r(p

u)



0102030405

00

00 05 10 15 20 25 30 35 40 5045t (s)

PiQi

PiQi

(pu)




5 Conclusions






00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30

00

02

04

06

08

10

12

t (s)

minus02

minus04

PrQr

PrQ

r(p

u)


05

00

minus05

minus10

00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30t (s)

PiQi

PiQi

(pu)


00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 3000

05

10

15

t (s)

Udc1

(pu)

Udc1

(c) The DC voltage


Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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