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)
4 Mathematical Problems in Engineering
Therefore (7) can be reorganized by
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
The controlled variables 1199063 and 1199064 can be denoted byvirtual control variables 1205923 and 1205924
(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)
Mathematical Problems in Engineering 5
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)
6 Mathematical Problems in Engineering
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)
Mathematical Problems in Engineering 7
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)
where 1205883 1205884 1205763 and 1205764 are all greater than zero And 1minus1205883120591 gt0 1 minus 1205884120591 gt 0
Now replacing (31) with (32) and then with (33) thevirtual control variables can be denoted by
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)
The control variables shown as in (35) can be deduced by (15)(25) and (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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
0
1
minus1
00 05 10 15 20 25 30 35 40 5045t (s)
PrQr
PrQ
r(p
u)
(a) The active and reactive power at rectifier side
10
05
00
PiQi
minus10
minus05
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
(pu)
(b) The active and reactive power at inverter side
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)
(a) The active and reactive power at rectifier side
minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01
0102030405
00
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Mathematical Problems in Engineering
Therefore (7) can be reorganized by
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
The controlled variables 1199063 and 1199064 can be denoted byvirtual control variables 1205923 and 1205924
(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)
Mathematical Problems in Engineering 5
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)
6 Mathematical Problems in Engineering
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)
Mathematical Problems in Engineering 7
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)
where 1205883 1205884 1205763 and 1205764 are all greater than zero And 1minus1205883120591 gt0 1 minus 1205884120591 gt 0
Now replacing (31) with (32) and then with (33) thevirtual control variables can be denoted by
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)
The control variables shown as in (35) can be deduced by (15)(25) and (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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
0
1
minus1
00 05 10 15 20 25 30 35 40 5045t (s)
PrQr
PrQ
r(p
u)
(a) The active and reactive power at rectifier side
10
05
00
PiQi
minus10
minus05
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
(pu)
(b) The active and reactive power at inverter side
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)
(a) The active and reactive power at rectifier side
minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01
0102030405
00
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Mathematical Problems in Engineering
Therefore (7) can be reorganized by
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
The controlled variables 1199063 and 1199064 can be denoted byvirtual control variables 1205923 and 1205924
(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)
Mathematical Problems in Engineering 5
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)
6 Mathematical Problems in Engineering
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)
Mathematical Problems in Engineering 7
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)
where 1205883 1205884 1205763 and 1205764 are all greater than zero And 1minus1205883120591 gt0 1 minus 1205884120591 gt 0
Now replacing (31) with (32) and then with (33) thevirtual control variables can be denoted by
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)
The control variables shown as in (35) can be deduced by (15)(25) and (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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
0
1
minus1
00 05 10 15 20 25 30 35 40 5045t (s)
PrQr
PrQ
r(p
u)
(a) The active and reactive power at rectifier side
10
05
00
PiQi
minus10
minus05
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
(pu)
(b) The active and reactive power at inverter side
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)
(a) The active and reactive power at rectifier side
minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01
0102030405
00
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Therefore (7) can be reorganized by
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
The controlled variables 1199063 and 1199064 can be denoted byvirtual control variables 1205923 and 1205924
(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)
Mathematical Problems in Engineering 5
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)
6 Mathematical Problems in Engineering
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)
Mathematical Problems in Engineering 7
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)
where 1205883 1205884 1205763 and 1205764 are all greater than zero And 1minus1205883120591 gt0 1 minus 1205884120591 gt 0
Now replacing (31) with (32) and then with (33) thevirtual control variables can be denoted by
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)
The control variables shown as in (35) can be deduced by (15)(25) and (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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
0
1
minus1
00 05 10 15 20 25 30 35 40 5045t (s)
PrQr
PrQ
r(p
u)
(a) The active and reactive power at rectifier side
10
05
00
PiQi
minus10
minus05
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
(pu)
(b) The active and reactive power at inverter side
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)
(a) The active and reactive power at rectifier side
minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01
0102030405
00
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
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)
6 Mathematical Problems in Engineering
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)
Mathematical Problems in Engineering 7
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)
where 1205883 1205884 1205763 and 1205764 are all greater than zero And 1minus1205883120591 gt0 1 minus 1205884120591 gt 0
Now replacing (31) with (32) and then with (33) thevirtual control variables can be denoted by
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)
The control variables shown as in (35) can be deduced by (15)(25) and (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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
0
1
minus1
00 05 10 15 20 25 30 35 40 5045t (s)
PrQr
PrQ
r(p
u)
(a) The active and reactive power at rectifier side
10
05
00
PiQi
minus10
minus05
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
(pu)
(b) The active and reactive power at inverter side
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)
(a) The active and reactive power at rectifier side
minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01
0102030405
00
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
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)
Mathematical Problems in Engineering 7
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)
where 1205883 1205884 1205763 and 1205764 are all greater than zero And 1minus1205883120591 gt0 1 minus 1205884120591 gt 0
Now replacing (31) with (32) and then with (33) thevirtual control variables can be denoted by
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)
The control variables shown as in (35) can be deduced by (15)(25) and (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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
0
1
minus1
00 05 10 15 20 25 30 35 40 5045t (s)
PrQr
PrQ
r(p
u)
(a) The active and reactive power at rectifier side
10
05
00
PiQi
minus10
minus05
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
(pu)
(b) The active and reactive power at inverter side
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)
(a) The active and reactive power at rectifier side
minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01
0102030405
00
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
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)
where 1205883 1205884 1205763 and 1205764 are all greater than zero And 1minus1205883120591 gt0 1 minus 1205884120591 gt 0
Now replacing (31) with (32) and then with (33) thevirtual control variables can be denoted by
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)
The control variables shown as in (35) can be deduced by (15)(25) and (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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
0
1
minus1
00 05 10 15 20 25 30 35 40 5045t (s)
PrQr
PrQ
r(p
u)
(a) The active and reactive power at rectifier side
10
05
00
PiQi
minus10
minus05
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
(pu)
(b) The active and reactive power at inverter side
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)
(a) The active and reactive power at rectifier side
minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01
0102030405
00
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
0
1
minus1
00 05 10 15 20 25 30 35 40 5045t (s)
PrQr
PrQ
r(p
u)
(a) The active and reactive power at rectifier side
10
05
00
PiQi
minus10
minus05
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
(pu)
(b) The active and reactive power at inverter side
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)
(a) The active and reactive power at rectifier side
minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01
0102030405
00
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
0
1
minus1
00 05 10 15 20 25 30 35 40 5045t (s)
PrQr
PrQ
r(p
u)
(a) The active and reactive power at rectifier side
10
05
00
PiQi
minus10
minus05
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
(pu)
(b) The active and reactive power at inverter side
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)
(a) The active and reactive power at rectifier side
minus11minus10minus09minus08minus07minus06minus05minus04minus03minus02minus01
0102030405
00
00 05 10 15 20 25 30 35 40 5045t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
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)
(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 30t (s)
PiQi
PiQi
(pu)
(b) The active and reactive power at inverter side
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
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of