research article improved power flow algorithm for vsc

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 235316 10 pageshttpdxdoiorg1011552013235316

Research ArticleImproved Power Flow Algorithm for VSC-HVDC SystemBased on High-Order Newton-Type Method

Yanfang Wei1 Qiang He2 Yonghui Sun3 Yanzhou Sun1 and Cong Ji4

1 School of Electrical Engineering and Automation Henan Polytechnic University Jiaozuo 454000 China2 Economics and Business College Qingdao Technological University Qingdao 266520 China3 College of Energy and Electrical Engineering Hohai University Nanjing 210098 China4 Jiangsu Frontier Electric Technology Co Ltd Nanjing 211102 China

Correspondence should be addressed to Yanfang Wei weiyanfanghpueducn

Received 4 April 2013 Accepted 12 May 2013

Academic Editor Guanghui Wen

Copyright copy 2013 Yanfang Wei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Voltage source converter (VSC) based high-voltage direct-current (HVDC) system is a new transmission technique which hasthe most promising applications in the fields of power systems and power electronics Considering the importance of powerflow analysis of the VSC-HVDC system for its utilization and exploitation the improved power flow algorithms for VSC-HVDCsystem based on third-order and sixth-order Newton-typemethod are presentedThe steady powermodel of VSC-HVDC system isintroduced firstly Then the derivation solving formats of multivariable matrix for third-order and sixth-order Newton-type powerflow method of VSC-HVDC system are given The formats have the feature of third-order and sixth-order convergence based onNewton method Further based on the automatic differentiation technology and third-order Newton method a new improvedalgorithm is given which will help in improving the program development computation efficiency maintainability and flexibilityof the power flow Simulations of ACDC power systems in two-terminal multi-terminal and multi-infeed DC with VSC-HVDCare carried out for the modified IEEE bus systems which show the effectiveness and practicality of the presented algorithms forVSC-HVDC system

1 Introduction

Voltage source converter (VSC) based high-voltage direct-current (HVDC) is a new technology of HVDC transmissionsystem Based on pulse width modulation and VSC theVSC-HVDC system has many merits and attracted widepublicity worldwide [1ndash3] Since the first pilot project appli-cation in 1997 the VSC-HVDC system is widely applied ininterconnected power system the connection of distributedgeneration to power grid the supply of electric power toislands or offshore drilling platform the distribution of powerto urban power network and so forth

The main advantages of VSC-HVDC system are as fol-lows no synchronization problem of AC system the featureof supplying power to passive network the simultaneous andindependent control for active power and reactive power theeasy achievement of inversion for power flow the more flexi-ble control modes the suitable application formulti-terminal

andmulti-infeed system and so on In the near future a seriesof newVSC-HVDC transmission systemwill be built and putinto operation worldwide [1 4] As a fundamental analyticalmethod for operation and analyzing of power system thereliable power flow calculation algorithm is an indispensabletool forACDC interconnected power systems [5ndash10] A greatdeal of research has been conducted in this field Now thereare two criticalmethods for the power flowanalysis ofACDCsystems the unified iteration technique and the alternativeiteration technique [10ndash14] The former has the precision ofquadratic convergence and has the better convergence for avariety of controlmodes ofHVDC system But the realizationof the control modes switching for HVDC system is difficultThe latter has the trait of easy programming especially for thevarying process of HVDC control modes But the alternativeiteration technique is sensitive to the operation ways andcontrol modes of the HVDC system and is inclined to theconvergence problem

2 Mathematical Problems in Engineering

The Newton method is a fundamental and importanttechnology to solve the power flow of power system [6ndash9 15ndash17] In [6] a Newton-Raphson power flow algorithmis proposed for the VSC-HVDC system In [7] the steadypower flow of VSC-HVDC is presented based on Newtonmethod and alternative iteration technique In [8] an optimalpower flow (OPF) model suitable for VSC-HVDC systemis presented based on Newton-Raphson algorithm In [9]based on Newton method a new model that considers theoperational constraints associated with the MVA ratings ofthe converters for OPF analysis of VSC-HVDC system isintroduced In these papers the modeling approach basedon Newton algorithm only has the first-order or quadraticconvergence and the convergence precision needs to befurther improved

In recent years the solution of nonlinear equation hasmade great progress especially themodifiedNewtonmethodwith high-order convergence performance [18ndash23] In [23]the power flow algorithm with cubic convergence is analyzedfor power systemDespite the fact that a greatmany improvedalgorithms for power flow analysis of ACDC interconnectednetworks with VSC-HVDC have been presented in manyaspects few papers use the Newton method of high-orderconvergence to analyze the power flow of VSC-HVDCsystem Moreover with the VSC-HVDC adding to ACDCsystem the kind and number of the system variables are mul-tiplied causing the modeling of ACDC system to becomemore complex And with the increase of DC line of VSC-HVDC the dimension and scale of Jacobian matrix andequations of ACDC system are increased obviously causingthe efficiency of hand codes to decrease The automaticdifferentiation (AD) technology overcomes the shortcomingsof hand codes Compared with other differential methodssuch as numerical differentiation and symbolic differentia-tion methods the AD has the advantages of no truncationerror the exact solution of Jacobian matrix the less workof hand codes and so on [24ndash26] Motivated by the abovediscussions we will investigate the problem of power flowfor ACDC systems with VSC-HVDC Novel third-order andsixth-order convergence of power flow technique based onNewton method will be derived and the AD technology willbe introduced in this paper based on third-order Newtonmethod to raise the efficiency of programming The effec-tiveness of the presented arithmetics for VSC-HVDC systemis also discussed These results and observations will helppromote the practical applications of high-order Newton-type method in ACDC systems with VSC-HVDC

The remaining of this paper is arranged as follows InSection 2 a mathematical model for the VSC-HVDC systemis presented in which all the AC system equations the VSCequations and the control modes of VSC are analyzed basingon steady model In Section 3 the power flow and converterequations of VSC-HVDC the mathematical description ofNewton method third-order and sixth-order convergence ofNewton-type methods and the improved third-order New-tonmethod based on automatic differentiation are presentedIn Section 4 the methods are applied to the modified IEEEbus test systems with VSC-HVDC This paper ends with aconclusion finally

Ii

Psi Qsi

Pci QciRi

Xli

Udi

2Cdi

2CdiXfi

Converter stationsDC

systemAC

system

Idi

Usiang120579i Uciang120579ci

Figure 1 Schematic diagram of steady state physical model formulti-terminal VSC-HVDC

2 Mathematical Steady Model ofVSC-HVDC System

21 Per-Unit Value System of VSC-HVDC For the simulationand calculation of ACDC hybrid power systems the unifiedper-unit value system should be adopted both for AC systemand DC system In this paper the per-unit value system isintroduced as follows [7]

119875dB = 119878B

119880dB = 119880B

119868dB = radic3119868B

119877dB = 119885B

(1)

where 119878B 119875dB are the reference power of AC system and DCsystem respectively 119880B 119868B and 119885B are the reference voltagereference current and reference impedance of AC side of theconverter respectively 119880dB 119868dB and 119877dB are the referencevoltage reference current and reference impedance of DCside of the converter respectively

22 Mathematical Steady Model of VSC-HVDC System TheVSC-HVDC system consists of at least two VSC stationsone operating as a rectifier station and the other as aninverter station The VSC stations can be connected as two-terminal multi-terminal or multi-infeed DC system withVSC-HVDC depending on the various different applicationsfields [1ndash3] The steady state physical model for a multi-terminal DC system with VSC-HVDC is shown schemati-cally in Figure 1 The steady models of VSC-HVDC are givenin the per-unit system (pu) as follows

I119894=Us119894 minus Uc119894119877119894+ 119895119883119897119894

(2)

119878s119894 = 119875s119894 + 119895119876s119894 = Us119894Ilowast

119894 (3)

119875s119894 = minus1003816100381610038161003816119884119894

1003816100381610038161003816 119880s119894119880c119894 cos (120575119894 + 120572119894) +

10038161003816100381610038161198841198941003816100381610038161003816 1198802

s119894 cos120572119894 (4)

Mathematical Problems in Engineering 3

119876s119894 = minus1003816100381610038161003816119884119894

1003816100381610038161003816 119880s119894119880c119894 sin (120575119894+ 120572119894) +

10038161003816100381610038161198841198941003816100381610038161003816 1198802

s119894 sin120572119894+

1198802

s119894119883f119894

(5)

119875c119894 =1003816100381610038161003816119884119894

1003816100381610038161003816 119880s119894119880c119894 cos (120575119894 minus 120572119894) minus

10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894 (6)

119876c119894 = minus1003816100381610038161003816119884119894

1003816100381610038161003816 119880s119894119880c119894 sin (120575119894minus 120572119894) minus

10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 sin120572119894 (7)

119875d119894 = 119880d119894119868d119894 =1003816100381610038161003816119884119894


10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894 (8)

119880c119894 =radic6119872119894119880d119894

4 (9)

The variables in the equations of (2)ndash(9) are referenced to theliterature [14]

23 Steady-State Control Modes of VSC-HVDC Owning tohaving full controllable power electronic switch semiconduc-tors such as insulated gate bipolar transistor and gate turn-offthyristor VSC-HVDC has the ability to independent controlactive and reactive power at its terminal So for each VSC acouple of regular used control goals can be set [27]

(1) AC active power control determines the active powerexchanged with the AC system

(2) DC voltage control is used to keep the DC voltagecontrol constant

(3) AC reactive power control determines the reactivepower exchanged with the AC system

(4) AC voltage control instead of controlling reactivepower AC voltage can be directly controlled deter-mining the voltage of the system bus

The general used control means of VSC include thefollowing four categories

A constant DC voltage control constant AC reactivepower controlB constant DC voltage control constant AC voltagecontrolC constant AC active power control constant ACreactive power controlD constant AC active power control constant ACvoltage control

3 The Improved Power Flow Algorithms ofACDC Systems with VSC-HVDC Based onHigh-Order Newton-Type Method

31 Steady Mathematical Model of Power Flow Calculationof VSC-HVDC System For the ACDC systems with VSC-HVDC the power flow equations are given as follows [14]

Pure AC bus equation

Δ119875a119894 = 119875a119894 minus 119880a119894sum119895isin119894

119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

(10)

DC bus equation

Δ119875t119894 = 119875t119894 minus 119880t119894sum119895isin119894

119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

(11)

VSC converter equation

Δ1198891198961

= 119875t119896 +radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 + 120572

119896)

minus 1198802

t119896 |119884| cos120572119896

Δ1198891198962

= 119876t119896 +radic6

4119872119896119880t119896119880d119896 |119884| sin (120575

119896+ 120572119896)

minus 1198802

t119896 |119884| sin120572119896minus

1198802

t119896119883f119896

Δ1198891198963

= 119880t119896119868d119896 minusradic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 minus 120572

119896)

+3

8(119872119896119880d119896)2|119884| cos120572119896

Id = GdUd

(12)

DC network equation

Δ1198891198964

= plusmn119868d119896 minus

119899AC

sum

119904=1

119892d119896119904119880d119904 = 0 (13)

The variables in the equations of (10)ndash(13) are referenced tothe literature [14]

32 The Mathematical Description of Newton Method Themathematical description of multivariable iterative form forNewton method is given by

Δ119909(119896)

= minus [1198651015840(119909(119896)

)minus1

] 1198651015840(119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(14)

The formula (14) has the second-order convergence [23 28]The equivalence form of linear equation solution for (14)

is given by

[1198651015840(119909(119896)

)] Δ119909(119896)

= minus1198651015840(119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(15)

where 1198651015840(119909(119896)

) is the matrix variable of first-order partialderivative of 119865(119909) and 119896 is the number of iterations

33 The Newton-Type Method of Third-Order Convergence(Algorithm 1) The single variable iterative algorithm formatbased on modified Newton-type method is given by

119909119896+1

= 119909119896minus

119891 (119909119896+ 119891 (119909

119896) 1198911015840(119909119896)) minus 119891 (119909

119896)

1198911015840 (119909119896)

(16)


The iterative format of (16) has the trait of third-orderconvergence [29]

The multivariable matrix equivalent form of (16) is givenby

1198651015840(119909(119896)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

minus Δ119909(119896)

1198651015840(119909(119896)

) Δ119909(119896)

= minus1198651015840(119909(119896+1)

) + 119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(17)

The gotten Jacobian matrix and its triangular factoriza-tion are being utilized fully in the algorithm iterative processof (17)

34 The Newton-Type Method of Third-Order Convergence(Algorithm 2) Another single variable iterative algorithmformat with third-order convergence based on Newton-typemethod is given by

119909119896+1

= 119909119896minus 05119891 (119909

119896)

times [1

1198911015840 (119909119896minus 119891 (119909

119896) 1198911015840 (119909

119896))

+1

1198911015840 (119909119896)]

(18)



1198651015840(119909(119896)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

1198651015840(119909(119896+1)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ 05 (Δ119909(119896)

+ Δ119909(119896)

)

(19)

35 The Newton-Type Method of Sixth-Order ConvergenceFor the above presented Algorithm 1 and Algorithm 2 thetwo iterative formats have the advantages for fast convergencespeed of Newtonmethod and less computations of simplifiedNewton method The application of Algorithm 1 and Algo-rithm 2 is a two-step process

Step 1 The prediction based on the Newton method [23]

Δ119909(119896)

= minus[1198651015840(119909(119896)

)]minus1

119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(20)

Step 2 The correction for the obtained predicted value of119909(119896+1)

Δ119909(119896)

= minus[1198651015840(119909(119896)

)]minus1

[119865 (119909(119896)

) + 119865 (119909(119896+1)

)]

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(21)

The simplified realization of the iterative procedure for(20) and (21) is given by

119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119911119896= 119909119896minus

2119891 (119909119896)

1198911015840 (119909119896) + 1198911015840 (119910

119896)

(22)

where 119911119896is the iterative result of 119909

119896+1at the 119896 iterative cycle

An effective implement of the iterative process of (22) isgiven by [31]

119909119896+1

= 119911119896minus

119891 (119911119896)

1198911015840 (119911119896) (23)

The approximate value of 1198911015840(119909) at 119911119896is given by

1198911015840(119911119896) asymp

1198911015840(119909119896) (31198911015840(119910119896) minus 1198911015840(119909119896))

1198911015840 (119909119896) + 1198911015840 (119910

119896)

(24)

The (16) or (18) (23) and (24) comprise the new sixth-order convergence method [31]

119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119911119896= 119909119896minus

2119891 (119909119896)

1198911015840 (119909119896) + 1198911015840 (119910

119896)

119909119896+1

= 119911119896minus

119891 (119911119896)

1198911015840 (119909119896)

1198911015840(119909119896) + 1198911015840(119910119896)

31198911015840 (119910119896) minus 1198911015840 (119909

119896)

(25)

36 The Automatic Differentiation Algorithm Based on Third-Order Newton-TypeMethod TheAD technique could alwaysbe decomposed to complex computations of basic func-tions and basic mathematical operations such as the fourarithmetic operations of add subtract multiply and dividethe basic functions of trigonometric function exponentialfunction and logarithmic function Here an instance is givento illustrate the application of ADThe function expression ofa certain model is given by

119891 (119909) = sin1199091+ 11989011990911199092 (26)

The independent variables and intermediate variablesof (26) are given in Table 1 For the use of independentvariables and intermediate variables the function of (26) isdecomposed to a series of basic functions If the value ofindependent variables is given the exact value of 119910 is gottenby top-down solution order in Table 1 Given the value of

1

and 2 the differentiation of (26) can be obtained mechan-

ically through the chain rule of differentiation calculationAt present there are two main modes for the application ofAD the forward mode and the backward mode as shown inTable 2 And in Table 2 119909

119894119895= 120597119909119894120597119909119895 119901119894= 120597119910120597119909

119894

Now there are two kinds of implementation methodfor AD the source code transform method and operator


Table 1 Independent variables and intermediate variables of (26)

Independent variables Intermediate variables1199091

1199093= sin119909

1

1199092

1199094= 11990911199092

1199095= 1198901199094

1199096= 1199093+ 1199095

119910 = 1199096

Table 2 Forward and backward mode

Forward mode Backward mode1199093= con (119909

1) sdot 1199091

1199011= 1199011+ 119909311199013

1199094= 11990911199092+ 1199091

1199092

1199011= 1199011+ 119909411199014 1199012= 1199012+ 119909421199014

1199095= 1199095

1199094

1199014= 1199014+ 119909541199015

1199096= 1199093+ 1199095

1199013= 1199013+ 119909631199016 1199015= 1199015+ 119909651199016

119910 = 1199096

1199017= 1

overloading method The typical representative softwares forthe former is ADIFOR and ADIC The typical representativesoftware for the latter is ADOL-C and ADC The method ofADOL-C realizes the differentiation of C++ program auto-matically by using operator overloading and can calculateany order derivative by forward and backward mode In thispaper the ADOL-C method is used to realize the differentialoperation [32]

The steps of the improved AD algorithm based on third-order Newton method are listed below

Step 1 Read network parameter including bus num-ber active and reactive power of load compensatecapacitance branch number of line resistor andreactance in series and ratio and impedance oftransformerStep 2 (initialization) Form the admittance matrix ofthe DC and AC systemsStep 3 Distribute space for AD and state active vari-ables including independent variables and dependentvariablesStep 4 Transmit the value of system variable to activevariableStep 5 Form the expression of dependent variable byusing independent variableStep 6 Judge the maximum of imbalance equationwhether to meet the error precision or not If yes exitthe loop If not the loop goes onStep 7 Call the function of Jacobian and Hessian ofADStep 8 Solve the equation of (17) or (19)

Return to Step 3

4 Case Studies

In this part in order to validate the correctness and suitabilityof the proposed algorithms three sections are presented

828 27

30 2926

25

2224

21

20

10

6

11 17

16

19

18

23

7

5

2

1 3

4

1314

15

12

VSC

VSC

G

G

G

G

G

G

VSC VSC

9

Figure 2 The modified IEEE-30 bus ACDC system with VSC-HVDC

(1) For the modified high-order Newton methods themodified IEEE 30-bus system with two-terminal andmulti-infeed VSC-HVDC is analyzed in detail firstly

(2) Then the simulation results of performance compar-isons for the improved high-order Newton methodsare presented among the modified IEEE 5-bus IEEE9-bus IEEE 14-bus IEEE 57-bus and IEEE 118- bustext systems

(3) At last the AD based on third-order Newton methodis evaluated for themodified IEEE 30-bus systemwithtwo-terminal of VSC-HVDC

41 The Modified IEEE 30-Bus System with Two-Terminal andMulti-Infeed VSC-HVDC The proposed method has beenapplied to the modified IEEE 30-bus system [33] The wiringdiagram is shown in Figure 2 In this section two casesare considered and compared In Figure 2 the dotted linesrepresent the possible positions of VSC stations and thespecific positions are located as follows

(1) In the system with two-terminal VSC-HVDC theVSC1 and VSC2 are connected to AC line of bus 29and bus 30 respectively

(2) In the systemwith two-infeedVSC-HVDC theVSC1VSC2 VSC3 and VSC4 are connected to AC line ofbus 12 bus 14 bus 29 and bus 30 respectively


Table 3 Results of the power flow calculation of AC system

Control mode MethodBus Number

1 2 3 4119881 120579 119881 120579 119881 120579 119881 120579

A +C

Newton 10600 0 10450 minus55025 10333 minus81365 10271 minus98132

Algorithm 1 10600 0 10450 minus55025 10333 minus81365 10271 minus98132


Sixth-order Newton 10600 0 10450 minus55025 10333 minus81365 10271 minus98132

A +D





B +C





B +D





Table 4 Results of the power flow calculation of DC system

DC variable Converter number Control modeA +C A +D B +C B +D

119880119889

Newton VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 1 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 2 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Sixth-order Newton VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

120575

Newton VSC1 03556 03402 03381 03322

VSC2 minus03701 minus02960 minus03495 minus03047

Algorithm 1 VSC1 00062 00059 00059 00058


Algorithm 2 VSC1 00062 00059 00059 00058




119872

Newton VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 1 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 2 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161


VSC2 07574 08188 07794 08161


Table 5 Comparisons of iteration times and computing time

Control mode Iteration times Computing time (ms)Newton Algorithm 1 Algorithm 2 Sixth-order Newton Newton Algorithm 1 Algorithm 2 Sixth-order Newton

A +C 4 3 3 2 86885 88367 01040 126763A +D 4 3 3 2 98464 56506 01023 95408B +C 4 3 3 15lowast 93356 56121 01045 84520B +D 4 3 3 15lowast 94767 56403 00990 82271

Table 6 Comparison of iteration times and computing time

Control mode Iteration times Computing time (ms)

VSC1 + VSC2 VSC3 + VSC4 Newton Algorithm 1 Algorithm 2 Sixth-orderNewton Newton Algorithm 1 Algorithm 2 Sixth-order

NewtonA +C A +C 4 3 3 15 100649 86786 01186 04648

A +C A +D 4 3 3 15 94347 68694 01110 04870

A +C B +C 4 3 3 15 94438 60041 01086 04399

A +C B +D 4 3 3 15 98407 67326 01136 04384

A +D A +D 4 3 3 15 94073 63733 01164 04885

A +D B +C 4 3 3 15 101531 59505 01113 04499

A +D B +D 4 3 3 15 58884 60551 01121 04273

B +C B +C 4 3 3 15 97429 58435 01330 04283

B +C B +D 4 3 3 15 98400 61026 01216 03860

B +D B +D 4 3 3 15 97773 62020 01168 04181

411 The Modified IEEE 30-Bus System with Two-TerminalVSC-HVDC The results of the power flow calculation of theAC system and DC system under different control modes forNewton third-order and sixth-order Newton methods areshown in Tables 3 and 4 In Table 3 the simulation results ofbus number of 1 2 3 and 4 are presented only and the otherbuses of the modified IEEE 30-bus system are not includedfor the sake of brevity There are two methods of third-orderNewton in Tables 3 and 4 Algorithm 1 and Algorithm 2In Table 3 the ldquoNewtonrdquo method is the algorithm in thereferences of [5ndash10] the unit of 119881 is pu and the unit of 120579is ∘

It can be seen from Table 3 the voltage amplitudes of bus1 and bus 2 are all the same for the proposed four methodsThis is due to the node type of bus 1 and bus 2 for the IEEE30-bus system Bus 1 is the equilibrium node And Bus 2 is thePV node So in the iterative process of the proposed differentmethods if the generators of bus 1 and bus 2 have not reachedthe limit of reactive power the voltage amplitudes of bus 1 andbus 2 remain the same

In Table 4 the DC variables of Ud are the same fordifferent control modes and Newton methods the reasonis that for the general used four control modes of VSC allthe mode combinations of A + C A + D B + C andB +D contain the constant DC voltage control category Asa result in the operation process of VSC-HVDC system theDC voltage of VSC remains constant

Both in Tables 3 and 4 for the proposed algorithms ofNewton Algorithm 1 Algorithm 2 and sixth-order Newtonmethod the results of the power flow calculation for the fourdifferent control modes remain the same fundamentally And

the operational parameters of DC system are all in the normalrange The simulation results also illustrate the flexible appli-cation of the third-order and sixth-orderNewtonmethods forthe ACDC system with VSC-HVDC

The comparisons of iteration times and computing timefor the four proposed Newton methods are shown in Table 5In Table 5 the lowast indicates that the criterion for convergenceis met only at the front half iteration procedure As seen inTable 5 for the modified IEEE 30-bus text system with two-terminalVSC-HVDC the iteration times of sixth-orderNew-ton method are evidently less than other Newton methodsAnd the CPU computing time of the third-order Newton ofAlgorithm 2 is smaller than other Newton methods underfour different control modes The reason is that for themodified IEEE 30-bus text system the computation task ofJacobian matrix formation and triangular factorization forhigh-order Newton method is less than Newton method

412 The Modified IEEE 30-Bus System with Multi-InfeedVSC-HVDC For the flexible control performance and par-ticular technical advantages the VSC-HVDC is suitable forapplication in multi-infeed system [5 34] In this sectionthe comparison of iteration times and computing time underdifferent control modes for improved Newton methods isshown in Table 6 for the modified IEEE 30-bus system withtwo-infeed VSC-HVDC For two-infeed VSC-HVDC thecombinations ways of VSC have ten different types as shownin Table 6

It can be seen from Table 6 both for the iteration timesand the computing time the high-order Newton-type of


Table 7 The topology and parameter settings for different IEEE text systems

Topology Two-terminal Two-infeed Three-terminalModifiedtest system IEEE-5 IEEE-9 IEEE-14 IEEE-57 IEEE-118 IEEE-14 IEEE-14 IEEE-57

Bus number VSC1 VSC2 VSC1 VSC2 VSC1 VSC2 VSC1 VSC2 VSC1 VSC2 VSC1 VSC2 VSC3 VSC4 VSC1 VSC2 VSC3 VSC1 VSC2 VSC34 5 8 9 13 14 56 57 75 118 12 14 29 30 12 13 14 55 56 41

Table 8 Performance comparison for different text systems

Topology Modified test system Iteration times Computing time (ms)Newton Algorithm 1 Sixth-order Newton Newton Algorithm 1 Sixth-order Newton

Two-terminal IEEE-5 5 4 15 48287 44651 53692Two-terminal IEEE-9 4 3 15 31646 22094 27866Two-terminal IEEE-14 4 3 15 41375 26509 41890Two-infeed IEEE-14 4 3 15 43529 31544 39907Three-terminal IEEE-14 4 3 15 43546 29592 41260Two-terminal IEEE-57 4 4 2 220698 142854 236236Three-terminal IEEE-57 4 4 2 289858 247143 290658Two-terminal IEEE-118 4 3 15 1208938 708048 1151420


DC variable Converternumber

Control modesofA +C

119880119889

Algorithm 1 + AD VSC1 20000

VSC2 19994


VSC2 19994

120575


VSC2 minus00064


VSC2 minus00064

119872


VSC2 07589


VSC2 07589

third-order and sixth-order Newton methods is less than theNewton method And the advantage of computing time forAlgorithm2 is obvious Table 6 also shows the proposed high-order methods suitable for the ACDC systems with multi-infeed VSC-HVDC

42 The Simulations of Modified IEEE 5- 9- 14- 57- and118-Bus Systems The modified IEEE 5- 9- 14- 57- and 118-bus systems are analyzed in this section [33] The topologyand parameter settings for those different IEEE text systemsare shown in Table 7 The simulation results of performancecomparisons for those IEEE text systems among improvedNewtonmethods are shown in Table 8The system topologiesof two-terminal two-infeed and three-terminal are analyzedin this section

It can be seen in Table 8 as the size and scale of the IEEEtext systems grow the iteration times keep mostly unchange-able for the third-order Newton and sixth-order Newtonmethods And the computing time of the third-order orsixth-order Newton method is less than the Newton methodThe validity and usability of the proposed improved Newtonmethod suitable for VSC-HVDC system are certified

43 The Simulation Results of AD Based on Third-OrderNewton Method In this part the simulation results forAD algorithm based on Algorithm 1 and Algorithm 2 ofthird-order Newton method are presented The IEEE-30 bustext system with two-terminal VSC-HVDC is employed todemonstrate the validity of the proposed AD algorithm Theresults of the power flow calculation of DC system of controlmode A + C are shown in Table 9 The comparison ofiteration times and computing time for the proposed ADalgorithm is shown in Table 10

From the results of Tables 9 and 10 the following can beseen

(1) Compared with the results of Algorithm 1 and Algo-rithm 2 of third-order Newton method as shownin Table 4 the improved AD algorithm satisfies theoperation requirements of VSC parameter

(2) Compared with the results of Table 5 for the iterationtimes and computing time the improved AD algo-rithm has certain advantages

(3) The result shows that AD technology is suitablefor use in the third-order Newton method of VSC-HVDC systemAnd the application ofAD technologyreduces the work of hand code greatly The efficiencyof code programming is improved



Control mode Iteration times Computing time (ms) VSC1 + VSC2 Algorithm 1 + AD Algorithm 2 + AD Algorithm 1 + AD Algorithm 2 + ADA +C 2 2 015 031A +D 2 2 031 031B +C 2 2 031 032B +D 2 2 031 032

5 Conclusions

In this paper based on the steady mathematical model ofVSC-HVDC the modified third-order Newton and sixth-order Newton methods have been presented to calculate thepower flow of ACDC systems with VSC-HVDC The mul-tivariate iteration matrix forms of the presented algorithmssuitable for VSC-HVDC system are given The proposedhigh-order Newton method has the third-order and sixth-order convergence without solving the Hessian matrix Thetask of the calculation is greatly reduced and the efficiencyis improved Based on the third-order Newton methodthe automatic differentiation technology is used to increasethe efficiency of hand code Some numerical examples onthe modified IEEE bus systems with two-terminal multi-terminal and multi-infeed VSC-HVDC have demonstratedthe computational performance of the power flow algorithmswith incorporation of VSC-HVDC models

Acknowledgments

This work is supported in part by the National NaturalScience Foundation of China (Grant no 61104045 andU1204506) in part by the Youth Project of National SocialScience Fund (Grant no 09CJY007) and in part by theFundamental Research Funds for the Central Universities ofChina (Grant no 2012B03514)

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] P Haugland Itrsquos Time to ConnectmdashTechnical Description ofHVDC Light Technology ABB 2008

[3] A L P de Oliveira C E Tiburcio M N Lemes and DRetzmann ldquoProspects of Voltage-Sourced Converters (VSC)applications in DC transmission systemsrdquo in Proceedings ofthe IEEEPES Transmission and Distribution Conference andExposition Latin America (T and D-LA rsquo10) pp 491ndash495 SaoPaulo Brazil November 2010

[4] J M Maza-Ortega A Gomez-Exposito M Barragan-VillarejoE Romero-Ramos and A Marano-Marcolini ldquoVoltage sourceconverter-based topologies to further integrate renewableenergy sources in distribution systemsrdquo IET Renewable PowerGeneration vol 6 no 6 pp 435ndash445 2012

[5] X P Zhang ldquoMultiterminal voltage-sourced converter-basedHVDC models for power flow analysisrdquo IEEE Transactions onPower Systems vol 19 no 4 pp 1877ndash1884 2004

[6] C Angeles-Camacho O L Tortelli E Acha and C R Fuerte-Esquivel ldquoInclusion of a high voltage DC-voltage source con-verter model in a Newton-Raphson power flow algorithmrdquo IEEProceedings Generation Transmission andDistribution vol 150no 6 pp 691ndash696 2003

[7] C Zheng X Zhou R Li and C Sheng ldquoStudy on the steadycharacteristic and algorithm of power flow for VSC-HVDCrdquoProceedings of the CSEE vol 25 no 6 pp 1ndash5 2005

[8] A Pizano-Martinez C R Fuerte-Esquivel H Ambriz-Perezand E Acha ldquoModeling of VSC-based HVDC systems for aNewton-Raphson OPF algorithmrdquo IEEE Transactions on PowerSystems vol 22 no 4 pp 1794ndash1803 2007

[9] A Pizano-Martinez C R Fuerte-Esquivel and C Angeles-Camacho ldquoVoltage source converter based high-voltage DCsystemmodeling for optimal power flow studiesrdquo Electric PowerComponents and Systems vol 40 no 3 pp 312ndash320 2012

[10] C Liu B Zhang Y Hou F F Wu and Y Liu ldquoAn improvedapproach for AC-DC power flow calculation with multi-infeedDC systemsrdquo IEEE Transactions on Power Systems vol 26 no2 pp 862ndash869 2011

[11] X Wang Y Song and M Irving Modern Power SystemsAnalysis Springer Berlin Germany 2008

[12] J Beerten S Cole and R Belmans ldquoGeneralized steady-statevsc mtdc model for sequential acdc power flow algorithmsrdquoIEEE Transactions on Power Systems vol 27 no 2 pp 821ndash8292012

[13] T Smed G Andersson G B Sheble and L L Grigsby ldquoA newapproach to ACDC power flowrdquo IEEE Transactions on PowerSystems vol 6 no 3 pp 1238ndash1244 1991

[14] Y Wei Z Zheng Y Sun Z Wei and G Sun ldquoVoltage stabilitybifurcation analysis for ACDC systems with VSC-HVDCrdquoAbstract and Applied Analysis vol 2013 Article ID 387167 9pages 2013

[15] C R Fuerte-Esquivel E Acha and H Ambriz-Perez ldquoA com-prehensive Newton-Raphson UPFC model for the quadraticpower flow solution of practical power networksrdquo IEEE Trans-actions on Power Systems vol 15 no 1 pp 102ndash109 2000

[16] H Ambriz-Perez E Acha and C R Fuerte-EsquivelldquoAdvanced SVC models for Newton-Raphson load flow andNewton optimal power flow studiesrdquo IEEE Transactions onPower Systems vol 15 no 1 pp 129ndash136 2000

[17] C R Fuerte-Esquivel and E Acha ldquoA Newton-type algorithmfor the control of power flow in electrical power networksrdquo IEEETransactions onPower Systems vol 12 no 4 pp 1474ndash1480 1997

[18] M A Noor K I Noor E Al-Said and M Waseem ldquoSomenew iterative methods for nonlinear equationsrdquo MathematicalProblems in Engineering vol 2010 Article ID 198943 12 pages2010

[19] P Tang and X Wang ldquoAn iteration method with generally con-vergent property for cubic polynomialsrdquo International Journalof Bifurcation and Chaos vol 19 no 1 pp 395ndash401 2009


[20] A Cordero J L Hueso E Martınez and J R TorregrosaldquoIncreasing the convergence order of an iterative method fornonlinear systemsrdquo Applied Mathematics Letters vol 25 no 12pp 2369ndash2374 2012

[21] D Herceg and D Herceg ldquoMeans based modifications ofNewtonrsquos method for solving nonlinear equationsrdquo AppliedMathematics and Computation vol 219 no 11 pp 6126ndash61332013

[22] F Soleymani S Karimi Vanani and A Afghani ldquoA generalthree-step class of optimal iterations for nonlinear equationsrdquoMathematical Problems in Engineering vol 2011 Article ID469512 10 pages 2011

[23] Z Y Sun Y Sun and W H Ning ldquoPower flow algorithm withcubic convergencerdquo Power System Protection and Control vol37 no 4 pp 5ndash8+28 2009

[24] T Orfanogianni and R Bacher ldquoUsing automatic code differen-tiation in power flow algorithmsrdquo IEEE Transactions on PowerSystems vol 14 no 1 pp 138ndash144 1999

[25] M Bartholomew-Biggs S Brown B Christianson and LDixon ldquoAutomatic differentiation of algorithmsrdquo Journal ofComputational and Applied Mathematics vol 124 no 1-2 pp171ndash190 2000

[26] Q Jiang GGeng CGuo andYCao ldquoAn efficient implementa-tion of automatic differentiation in interior point optimal powerflowrdquo IEEETransactions on Power Systems vol 25 no 1 pp 147ndash155 2010

[27] S Cole Steady-State and Dynamic Modeling of VSC-HVDCSystems for Power System Simulation K U Leuven LeuvenBelgium 2010

[28] AM Ostrowski Solution of Equations in Euclidean and BanachSpaces Academic Press New York NY USA 1973

[29] J Kou Y Li and X Wang ldquoA modification of Newton methodwith third-order convergencerdquo Applied Mathematics and Com-putation vol 181 no 2 pp 1106ndash1111 2006

[30] H H H Homeier ldquoOn Newton-type methods with cubic con-vergencerdquo Journal of Computational and Applied Mathematicsvol 176 no 2 pp 425ndash432 2005

[31] S K Parhi andD KGupta ldquoA sixth ordermethod for nonlinearequationsrdquoAppliedMathematics and Computation vol 203 no1 pp 50ndash55 2008

[32] A Walther and A Griewank A Package for the Auto-matic Differentiation of Algorithms Written in CC++ 2009httpwwwcoin-ororgprojectsADOL-Cxml

[33] Power Systems Test Case Archive University of Washingtonhttpwwweewashingtoneduresearchpstca

[34] Y Wei Z Wei and G Sun ldquoA commentary on voltagestability of multi-infeed ACDC system with VSC-HDVCrdquo inProceedings of the Asia-Pacific Power and Energy EngineeringConference (APPEEC rsquo11) pp 1ndash4 Wuhan China March 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


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International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The Newton method is a fundamental and importanttechnology to solve the power flow of power system [6ndash9 15ndash17] In [6] a Newton-Raphson power flow algorithmis proposed for the VSC-HVDC system In [7] the steadypower flow of VSC-HVDC is presented based on Newtonmethod and alternative iteration technique In [8] an optimalpower flow (OPF) model suitable for VSC-HVDC systemis presented based on Newton-Raphson algorithm In [9]based on Newton method a new model that considers theoperational constraints associated with the MVA ratings ofthe converters for OPF analysis of VSC-HVDC system isintroduced In these papers the modeling approach basedon Newton algorithm only has the first-order or quadraticconvergence and the convergence precision needs to befurther improved

In recent years the solution of nonlinear equation hasmade great progress especially themodifiedNewtonmethodwith high-order convergence performance [18ndash23] In [23]the power flow algorithm with cubic convergence is analyzedfor power systemDespite the fact that a greatmany improvedalgorithms for power flow analysis of ACDC interconnectednetworks with VSC-HVDC have been presented in manyaspects few papers use the Newton method of high-orderconvergence to analyze the power flow of VSC-HVDCsystem Moreover with the VSC-HVDC adding to ACDCsystem the kind and number of the system variables are mul-tiplied causing the modeling of ACDC system to becomemore complex And with the increase of DC line of VSC-HVDC the dimension and scale of Jacobian matrix andequations of ACDC system are increased obviously causingthe efficiency of hand codes to decrease The automaticdifferentiation (AD) technology overcomes the shortcomingsof hand codes Compared with other differential methodssuch as numerical differentiation and symbolic differentia-tion methods the AD has the advantages of no truncationerror the exact solution of Jacobian matrix the less workof hand codes and so on [24ndash26] Motivated by the abovediscussions we will investigate the problem of power flowfor ACDC systems with VSC-HVDC Novel third-order andsixth-order convergence of power flow technique based onNewton method will be derived and the AD technology willbe introduced in this paper based on third-order Newtonmethod to raise the efficiency of programming The effec-tiveness of the presented arithmetics for VSC-HVDC systemis also discussed These results and observations will helppromote the practical applications of high-order Newton-type method in ACDC systems with VSC-HVDC

The remaining of this paper is arranged as follows InSection 2 a mathematical model for the VSC-HVDC systemis presented in which all the AC system equations the VSCequations and the control modes of VSC are analyzed basingon steady model In Section 3 the power flow and converterequations of VSC-HVDC the mathematical description ofNewton method third-order and sixth-order convergence ofNewton-type methods and the improved third-order New-tonmethod based on automatic differentiation are presentedIn Section 4 the methods are applied to the modified IEEEbus test systems with VSC-HVDC This paper ends with aconclusion finally

Ii

Psi Qsi

Pci QciRi

Xli

Udi

2Cdi

2CdiXfi

Converter stationsDC

systemAC

system

Idi

Usiang120579i Uciang120579ci

Figure 1 Schematic diagram of steady state physical model formulti-terminal VSC-HVDC

2 Mathematical Steady Model ofVSC-HVDC System

21 Per-Unit Value System of VSC-HVDC For the simulationand calculation of ACDC hybrid power systems the unifiedper-unit value system should be adopted both for AC systemand DC system In this paper the per-unit value system isintroduced as follows [7]

119875dB = 119878B

119880dB = 119880B

119868dB = radic3119868B

119877dB = 119885B

(1)

where 119878B 119875dB are the reference power of AC system and DCsystem respectively 119880B 119868B and 119885B are the reference voltagereference current and reference impedance of AC side of theconverter respectively 119880dB 119868dB and 119877dB are the referencevoltage reference current and reference impedance of DCside of the converter respectively

22 Mathematical Steady Model of VSC-HVDC System TheVSC-HVDC system consists of at least two VSC stationsone operating as a rectifier station and the other as aninverter station The VSC stations can be connected as two-terminal multi-terminal or multi-infeed DC system withVSC-HVDC depending on the various different applicationsfields [1ndash3] The steady state physical model for a multi-terminal DC system with VSC-HVDC is shown schemati-cally in Figure 1 The steady models of VSC-HVDC are givenin the per-unit system (pu) as follows

I119894=Us119894 minus Uc119894119877119894+ 119895119883119897119894

(2)

119878s119894 = 119875s119894 + 119895119876s119894 = Us119894Ilowast

119894 (3)

119875s119894 = minus1003816100381610038161003816119884119894

1003816100381610038161003816 119880s119894119880c119894 cos (120575119894 + 120572119894) +

10038161003816100381610038161198841198941003816100381610038161003816 1198802

s119894 cos120572119894 (4)


119876s119894 = minus1003816100381610038161003816119884119894

1003816100381610038161003816 119880s119894119880c119894 sin (120575119894+ 120572119894) +

10038161003816100381610038161198841198941003816100381610038161003816 1198802

s119894 sin120572119894+

1198802

s119894119883f119894

(5)

119875c119894 =1003816100381610038161003816119884119894


10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894 (6)

119876c119894 = minus1003816100381610038161003816119884119894


10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 sin120572119894 (7)

119875d119894 = 119880d119894119868d119894 =1003816100381610038161003816119884119894


10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894 (8)

119880c119894 =radic6119872119894119880d119894

4 (9)













119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

(10)

DC bus equation


119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

(11)


Δ1198891198961

= 119875t119896 +radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 + 120572

119896)

minus 1198802

t119896 |119884| cos120572119896

Δ1198891198962

= 119876t119896 +radic6

4119872119896119880t119896119880d119896 |119884| sin (120575

119896+ 120572119896)

minus 1198802

t119896 |119884| sin120572119896minus

1198802

t119896119883f119896

Δ1198891198963

= 119880t119896119868d119896 minusradic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 minus 120572

119896)

+3

8(119872119896119880d119896)2|119884| cos120572119896

Id = GdUd

(12)

DC network equation

Δ1198891198964


119899AC

sum

119904=1

119892d119896119904119880d119904 = 0 (13)



Δ119909(119896)

= minus [1198651015840(119909(119896)

)minus1

] 1198651015840(119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(14)


is given by

[1198651015840(119909(119896)

)] Δ119909(119896)

= minus1198651015840(119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(15)

where 1198651015840(119909(119896)



119909119896+1

= 119909119896minus

119891 (119909119896+ 119891 (119909

119896) 1198911015840(119909119896)) minus 119891 (119909

119896)

1198911015840 (119909119896)

(16)




1198651015840(119909(119896)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

minus Δ119909(119896)

1198651015840(119909(119896)

) Δ119909(119896)

= minus1198651015840(119909(119896+1)

) + 119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(17)



119909119896+1

= 119909119896minus 05119891 (119909

119896)

times [1

1198911015840 (119909119896minus 119891 (119909

119896) 1198911015840 (119909

119896))

+1

1198911015840 (119909119896)]

(18)



1198651015840(119909(119896)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

1198651015840(119909(119896+1)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ 05 (Δ119909(119896)

+ Δ119909(119896)

)

(19)



Δ119909(119896)

= minus[1198651015840(119909(119896)

)]minus1

119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(20)


Δ119909(119896)

= minus[1198651015840(119909(119896)

)]minus1

[119865 (119909(119896)

) + 119865 (119909(119896+1)

)]

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(21)


119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119911119896= 119909119896minus

2119891 (119909119896)

1198911015840 (119909119896) + 1198911015840 (119910

119896)

(22)




119909119896+1

= 119911119896minus

119891 (119911119896)

1198911015840 (119911119896) (23)


1198911015840(119911119896) asymp

1198911015840(119909119896) (31198911015840(119910119896) minus 1198911015840(119909119896))

1198911015840 (119909119896) + 1198911015840 (119910

119896)

(24)


119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119911119896= 119909119896minus

2119891 (119909119896)

1198911015840 (119909119896) + 1198911015840 (119910

119896)

119909119896+1

= 119911119896minus

119891 (119911119896)

1198911015840 (119909119896)

1198911015840(119909119896) + 1198911015840(119910119896)

31198911015840 (119910119896) minus 1198911015840 (119909

119896)

(25)


119891 (119909) = sin1199091+ 11989011990911199092 (26)


1



119894119895= 120597119909119894120597119909119895 119901119894= 120597119910120597119909

119894





1199093= sin119909

1

1199092

1199094= 11990911199092

1199095= 1198901199094

1199096= 1199093+ 1199095

119910 = 1199096



1) sdot 1199091

1199011= 1199011+ 119909311199013

1199094= 11990911199092+ 1199091

1199092

1199011= 1199011+ 119909411199014 1199012= 1199012+ 119909421199014

1199095= 1199095

1199094

1199014= 1199014+ 119909541199015

1199096= 1199093+ 1199095

1199013= 1199013+ 119909631199016 1199015= 1199015+ 119909651199016

119910 = 1199096

1199017= 1




Return to Step 3

4 Case Studies


828 27

30 2926

25

2224

21

20

10

6

11 17

16

19

18

23

7

5

2

1 3

4

1314

15

12

VSC

VSC

G

G

G

G

G

G

VSC VSC

9











1 2 3 4119881 120579 119881 120579 119881 120579 119881 120579

A +C





A +D





B +C





B +D







119880119889

Newton VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 1 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 2 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994


VSC2 19994 19994 19994 19994

120575

Newton VSC1 03556 03402 03381 03322


Algorithm 1 VSC1 00062 00059 00059 00058


Algorithm 2 VSC1 00062 00059 00059 00058




119872

Newton VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 1 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 2 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161


VSC2 07574 08188 07794 08161








NewtonA +C A +C 4 3 3 15 100649 86786 01186 04648

A +C A +D 4 3 3 15 94347 68694 01110 04870

A +C B +C 4 3 3 15 94438 60041 01086 04399

A +C B +D 4 3 3 15 98407 67326 01136 04384

A +D A +D 4 3 3 15 94073 63733 01164 04885

A +D B +C 4 3 3 15 101531 59505 01113 04499

A +D B +D 4 3 3 15 58884 60551 01121 04273

B +C B +C 4 3 3 15 97429 58435 01330 04283

B +C B +D 4 3 3 15 98400 61026 01216 03860

B +D B +D 4 3 3 15 97773 62020 01168 04181


















Control modesofA +C

119880119889


VSC2 19994


VSC2 19994

120575


VSC2 minus00064


VSC2 minus00064

119872


VSC2 07589


VSC2 07589












5 Conclusions


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119876s119894 = minus1003816100381610038161003816119884119894

1003816100381610038161003816 119880s119894119880c119894 sin (120575119894+ 120572119894) +

10038161003816100381610038161198841198941003816100381610038161003816 1198802

s119894 sin120572119894+

1198802

s119894119883f119894

(5)

119875c119894 =1003816100381610038161003816119884119894


10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894 (6)

119876c119894 = minus1003816100381610038161003816119884119894


10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 sin120572119894 (7)

119875d119894 = 119880d119894119868d119894 =1003816100381610038161003816119884119894


10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894 (8)

119880c119894 =radic6119872119894119880d119894

4 (9)













119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

(10)

DC bus equation


119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

(11)


Δ1198891198961

= 119875t119896 +radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 + 120572

119896)

minus 1198802

t119896 |119884| cos120572119896

Δ1198891198962

= 119876t119896 +radic6

4119872119896119880t119896119880d119896 |119884| sin (120575

119896+ 120572119896)

minus 1198802

t119896 |119884| sin120572119896minus

1198802

t119896119883f119896

Δ1198891198963

= 119880t119896119868d119896 minusradic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 minus 120572

119896)

+3

8(119872119896119880d119896)2|119884| cos120572119896

Id = GdUd

(12)

DC network equation

Δ1198891198964


119899AC

sum

119904=1

119892d119896119904119880d119904 = 0 (13)



Δ119909(119896)

= minus [1198651015840(119909(119896)

)minus1

] 1198651015840(119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(14)


is given by

[1198651015840(119909(119896)

)] Δ119909(119896)

= minus1198651015840(119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(15)

where 1198651015840(119909(119896)



119909119896+1

= 119909119896minus

119891 (119909119896+ 119891 (119909

119896) 1198911015840(119909119896)) minus 119891 (119909

119896)

1198911015840 (119909119896)

(16)




1198651015840(119909(119896)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

minus Δ119909(119896)

1198651015840(119909(119896)

) Δ119909(119896)

= minus1198651015840(119909(119896+1)

) + 119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(17)



119909119896+1

= 119909119896minus 05119891 (119909

119896)

times [1

1198911015840 (119909119896minus 119891 (119909

119896) 1198911015840 (119909

119896))

+1

1198911015840 (119909119896)]

(18)



1198651015840(119909(119896)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

1198651015840(119909(119896+1)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ 05 (Δ119909(119896)

+ Δ119909(119896)

)

(19)



Δ119909(119896)

= minus[1198651015840(119909(119896)

)]minus1

119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(20)


Δ119909(119896)

= minus[1198651015840(119909(119896)

)]minus1

[119865 (119909(119896)

) + 119865 (119909(119896+1)

)]

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(21)


119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119911119896= 119909119896minus

2119891 (119909119896)

1198911015840 (119909119896) + 1198911015840 (119910

119896)

(22)




119909119896+1

= 119911119896minus

119891 (119911119896)

1198911015840 (119911119896) (23)


1198911015840(119911119896) asymp

1198911015840(119909119896) (31198911015840(119910119896) minus 1198911015840(119909119896))

1198911015840 (119909119896) + 1198911015840 (119910

119896)

(24)


119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119911119896= 119909119896minus

2119891 (119909119896)

1198911015840 (119909119896) + 1198911015840 (119910

119896)

119909119896+1

= 119911119896minus

119891 (119911119896)

1198911015840 (119909119896)

1198911015840(119909119896) + 1198911015840(119910119896)

31198911015840 (119910119896) minus 1198911015840 (119909

119896)

(25)


119891 (119909) = sin1199091+ 11989011990911199092 (26)


1



119894119895= 120597119909119894120597119909119895 119901119894= 120597119910120597119909

119894





1199093= sin119909

1

1199092

1199094= 11990911199092

1199095= 1198901199094

1199096= 1199093+ 1199095

119910 = 1199096



1) sdot 1199091

1199011= 1199011+ 119909311199013

1199094= 11990911199092+ 1199091

1199092

1199011= 1199011+ 119909411199014 1199012= 1199012+ 119909421199014

1199095= 1199095

1199094

1199014= 1199014+ 119909541199015

1199096= 1199093+ 1199095

1199013= 1199013+ 119909631199016 1199015= 1199015+ 119909651199016

119910 = 1199096

1199017= 1




Return to Step 3

4 Case Studies


828 27

30 2926

25

2224

21

20

10

6

11 17

16

19

18

23

7

5

2

1 3

4

1314

15

12

VSC

VSC

G

G

G

G

G

G

VSC VSC

9











1 2 3 4119881 120579 119881 120579 119881 120579 119881 120579

A +C





A +D





B +C





B +D







119880119889

Newton VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 1 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 2 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994


VSC2 19994 19994 19994 19994

120575

Newton VSC1 03556 03402 03381 03322


Algorithm 1 VSC1 00062 00059 00059 00058


Algorithm 2 VSC1 00062 00059 00059 00058




119872

Newton VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 1 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 2 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161


VSC2 07574 08188 07794 08161








NewtonA +C A +C 4 3 3 15 100649 86786 01186 04648

A +C A +D 4 3 3 15 94347 68694 01110 04870

A +C B +C 4 3 3 15 94438 60041 01086 04399

A +C B +D 4 3 3 15 98407 67326 01136 04384

A +D A +D 4 3 3 15 94073 63733 01164 04885

A +D B +C 4 3 3 15 101531 59505 01113 04499

A +D B +D 4 3 3 15 58884 60551 01121 04273

B +C B +C 4 3 3 15 97429 58435 01330 04283

B +C B +D 4 3 3 15 98400 61026 01216 03860

B +D B +D 4 3 3 15 97773 62020 01168 04181


















Control modesofA +C

119880119889


VSC2 19994


VSC2 19994

120575


VSC2 minus00064


VSC2 minus00064

119872


VSC2 07589


VSC2 07589












5 Conclusions


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198651015840(119909(119896)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

minus Δ119909(119896)

1198651015840(119909(119896)

) Δ119909(119896)

= minus1198651015840(119909(119896+1)

) + 119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(17)



119909119896+1

= 119909119896minus 05119891 (119909

119896)

times [1

1198911015840 (119909119896minus 119891 (119909

119896) 1198911015840 (119909

119896))

+1

1198911015840 (119909119896)]

(18)



1198651015840(119909(119896)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

1198651015840(119909(119896+1)

) Δ119909(119896)

= minus119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ 05 (Δ119909(119896)

+ Δ119909(119896)

)

(19)



Δ119909(119896)

= minus[1198651015840(119909(119896)

)]minus1

119865 (119909(119896)

)

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(20)


Δ119909(119896)

= minus[1198651015840(119909(119896)

)]minus1

[119865 (119909(119896)

) + 119865 (119909(119896+1)

)]

119909(119896+1)

= 119909(119896)

+ Δ119909(119896)

(21)


119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119911119896= 119909119896minus

2119891 (119909119896)

1198911015840 (119909119896) + 1198911015840 (119910

119896)

(22)




119909119896+1

= 119911119896minus

119891 (119911119896)

1198911015840 (119911119896) (23)


1198911015840(119911119896) asymp

1198911015840(119909119896) (31198911015840(119910119896) minus 1198911015840(119909119896))

1198911015840 (119909119896) + 1198911015840 (119910

119896)

(24)


119910119896= 119909119896minus

119891 (119909119896)

1198911015840 (119909119896)

119911119896= 119909119896minus

2119891 (119909119896)

1198911015840 (119909119896) + 1198911015840 (119910

119896)

119909119896+1

= 119911119896minus

119891 (119911119896)

1198911015840 (119909119896)

1198911015840(119909119896) + 1198911015840(119910119896)

31198911015840 (119910119896) minus 1198911015840 (119909

119896)

(25)


119891 (119909) = sin1199091+ 11989011990911199092 (26)


1



119894119895= 120597119909119894120597119909119895 119901119894= 120597119910120597119909

119894





1199093= sin119909

1

1199092

1199094= 11990911199092

1199095= 1198901199094

1199096= 1199093+ 1199095

119910 = 1199096



1) sdot 1199091

1199011= 1199011+ 119909311199013

1199094= 11990911199092+ 1199091

1199092

1199011= 1199011+ 119909411199014 1199012= 1199012+ 119909421199014

1199095= 1199095

1199094

1199014= 1199014+ 119909541199015

1199096= 1199093+ 1199095

1199013= 1199013+ 119909631199016 1199015= 1199015+ 119909651199016

119910 = 1199096

1199017= 1




Return to Step 3

4 Case Studies


828 27

30 2926

25

2224

21

20

10

6

11 17

16

19

18

23

7

5

2

1 3

4

1314

15

12

VSC

VSC

G

G

G

G

G

G

VSC VSC

9











1 2 3 4119881 120579 119881 120579 119881 120579 119881 120579

A +C





A +D





B +C





B +D







119880119889

Newton VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 1 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 2 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994


VSC2 19994 19994 19994 19994

120575

Newton VSC1 03556 03402 03381 03322


Algorithm 1 VSC1 00062 00059 00059 00058


Algorithm 2 VSC1 00062 00059 00059 00058




119872

Newton VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 1 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 2 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161


VSC2 07574 08188 07794 08161








NewtonA +C A +C 4 3 3 15 100649 86786 01186 04648

A +C A +D 4 3 3 15 94347 68694 01110 04870

A +C B +C 4 3 3 15 94438 60041 01086 04399

A +C B +D 4 3 3 15 98407 67326 01136 04384

A +D A +D 4 3 3 15 94073 63733 01164 04885

A +D B +C 4 3 3 15 101531 59505 01113 04499

A +D B +D 4 3 3 15 58884 60551 01121 04273

B +C B +C 4 3 3 15 97429 58435 01330 04283

B +C B +D 4 3 3 15 98400 61026 01216 03860

B +D B +D 4 3 3 15 97773 62020 01168 04181


















Control modesofA +C

119880119889


VSC2 19994


VSC2 19994

120575


VSC2 minus00064


VSC2 minus00064

119872


VSC2 07589


VSC2 07589












5 Conclusions


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1199093= sin119909

1

1199092

1199094= 11990911199092

1199095= 1198901199094

1199096= 1199093+ 1199095

119910 = 1199096



1) sdot 1199091

1199011= 1199011+ 119909311199013

1199094= 11990911199092+ 1199091

1199092

1199011= 1199011+ 119909411199014 1199012= 1199012+ 119909421199014

1199095= 1199095

1199094

1199014= 1199014+ 119909541199015

1199096= 1199093+ 1199095

1199013= 1199013+ 119909631199016 1199015= 1199015+ 119909651199016

119910 = 1199096

1199017= 1




Return to Step 3

4 Case Studies


828 27

30 2926

25

2224

21

20

10

6

11 17

16

19

18

23

7

5

2

1 3

4

1314

15

12

VSC

VSC

G

G

G

G

G

G

VSC VSC

9











1 2 3 4119881 120579 119881 120579 119881 120579 119881 120579

A +C





A +D





B +C





B +D







119880119889

Newton VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 1 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 2 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994


VSC2 19994 19994 19994 19994

120575

Newton VSC1 03556 03402 03381 03322


Algorithm 1 VSC1 00062 00059 00059 00058


Algorithm 2 VSC1 00062 00059 00059 00058




119872

Newton VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 1 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 2 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161


VSC2 07574 08188 07794 08161








NewtonA +C A +C 4 3 3 15 100649 86786 01186 04648

A +C A +D 4 3 3 15 94347 68694 01110 04870

A +C B +C 4 3 3 15 94438 60041 01086 04399

A +C B +D 4 3 3 15 98407 67326 01136 04384

A +D A +D 4 3 3 15 94073 63733 01164 04885

A +D B +C 4 3 3 15 101531 59505 01113 04499

A +D B +D 4 3 3 15 58884 60551 01121 04273

B +C B +C 4 3 3 15 97429 58435 01330 04283

B +C B +D 4 3 3 15 98400 61026 01216 03860

B +D B +D 4 3 3 15 97773 62020 01168 04181


















Control modesofA +C

119880119889


VSC2 19994


VSC2 19994

120575


VSC2 minus00064


VSC2 minus00064

119872


VSC2 07589


VSC2 07589












5 Conclusions


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 2 3 4119881 120579 119881 120579 119881 120579 119881 120579

A +C





A +D





B +C





B +D







119880119889

Newton VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 1 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994

Algorithm 2 VSC1 20000 20000 20000 20000

VSC2 19994 19994 19994 19994


VSC2 19994 19994 19994 19994

120575

Newton VSC1 03556 03402 03381 03322


Algorithm 1 VSC1 00062 00059 00059 00058


Algorithm 2 VSC1 00062 00059 00059 00058




119872

Newton VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 1 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161

Algorithm 2 VSC1 07663 07838 08282 08257

VSC2 07574 08188 07794 08161


VSC2 07574 08188 07794 08161








NewtonA +C A +C 4 3 3 15 100649 86786 01186 04648

A +C A +D 4 3 3 15 94347 68694 01110 04870

A +C B +C 4 3 3 15 94438 60041 01086 04399

A +C B +D 4 3 3 15 98407 67326 01136 04384

A +D A +D 4 3 3 15 94073 63733 01164 04885

A +D B +C 4 3 3 15 101531 59505 01113 04499

A +D B +D 4 3 3 15 58884 60551 01121 04273

B +C B +C 4 3 3 15 97429 58435 01330 04283

B +C B +D 4 3 3 15 98400 61026 01216 03860

B +D B +D 4 3 3 15 97773 62020 01168 04181


















Control modesofA +C

119880119889


VSC2 19994


VSC2 19994

120575


VSC2 minus00064


VSC2 minus00064

119872


VSC2 07589


VSC2 07589












5 Conclusions


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















NewtonA +C A +C 4 3 3 15 100649 86786 01186 04648

A +C A +D 4 3 3 15 94347 68694 01110 04870

A +C B +C 4 3 3 15 94438 60041 01086 04399

A +C B +D 4 3 3 15 98407 67326 01136 04384

A +D A +D 4 3 3 15 94073 63733 01164 04885

A +D B +C 4 3 3 15 101531 59505 01113 04499

A +D B +D 4 3 3 15 58884 60551 01121 04273

B +C B +C 4 3 3 15 97429 58435 01330 04283

B +C B +D 4 3 3 15 98400 61026 01216 03860

B +D B +D 4 3 3 15 97773 62020 01168 04181


















Control modesofA +C

119880119889


VSC2 19994


VSC2 19994

120575


VSC2 minus00064


VSC2 minus00064

119872


VSC2 07589


VSC2 07589












5 Conclusions


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Control modesofA +C

119880119889


VSC2 19994


VSC2 19994

120575


VSC2 minus00064


VSC2 minus00064

119872


VSC2 07589


VSC2 07589












5 Conclusions


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












5 Conclusions


Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article improved power flow algorithm for vsc

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