improving power system stability with gramian matrix-based

Research ArticleImproving Power System Stability with GramianMatrix-Based Optimal Setting of a Single Series FACTS DeviceFeasibility Study in Vietnamese Power System

Le Van Dai12 Doan Duc Tung3 Tran Le Thang Dong1 and Le Cao Quyen45

1Duy Tan University Da Nang 550000 Vietnam2Faculty of Electrical Technology Inductrial University of Ho Chi Minh City Ho Chi Minh City 700000 Vietnam3Faculty of Engineering and Technology Quy Nhon University Binh Dinh 820000 Vietnam4Department forManagement of Science andTechnologyDevelopment TonDucThangUniversityHoChiMinhCity 700000 Vietnam5Faculty of Electrical amp Electronics Engineering Ton Duc Thang University Ho Chi Minh City 700000 Vietnam

Correspondence should be addressed to Le Cao Quyen lecaoquyentdteduvn

Received 17 October 2016 Revised 11 November 2016 Accepted 23 November 2016 Published 29 January 2017

Academic Editor Dimitri Volchenkov

Copyright copy 2017 Le Van Dai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The Vietnamese power system has experienced instabilities due to the effect of increase in peak load demand or contingency gridfaults hence using flexible alternating-current transmission systems (FACTS) devices is a best choice for improving the stabilitymargins Among the FACTS devices the thyristor-controlled series capacitor (TCSC) is a series connected FACTS device widelyused in power systemsHowever in practice its influence and ability depend on setting For solving the problem this paper proposesa relevant method for optimal setting of a single TCSC for the purpose of damping the power system oscillations This proposedmethod is developed from the combination between the energy method and Hankel-norm approximation approach based on thecontrollability Gramian matrix considering the Lyapunov equation to search for a number of feasible locations on the small-signalstability analysis The transient stability analysis is used to compare and determine appropriate settings through various simulationcases The effectiveness of the proposed method is confirmed by the simulation results based on the power system simulationengineering (PSSE) andMATLAB programsThe obtained results show that the proposed method can apply to immediately solvethe difficulties encountering in the Vietnamese power system

1 Introduction

Motivation When the social political and technologicalaspects develop the demand of electric power grows rapidlyresulting in the increase in the scale and complexity ofpower systems Some characteristics such as long transmis-sion distances over weak grids highly variable generationpatterns and heavy load tend to increase the wide-area elec-tromechanical oscillations These oscillations threaten thesecure operation of the power systems and if not controlledefficiently can lead to generator outages line tripping andeven large-scale blackouts

During over several decades a number of electrical powersystems in the world have been faced with the serious power

system blackouts such as Indian on July 30 and 31 2012 [1]Dubai on June 9 2005 Malaysia on Jan 13 2005 Kuwaiton Nov 01 2004 Australia on Aug 14 2004 Shanghai onAug 27 2003 SwedenDenmark on Sept 23 2003 Finlandon Aug 1992 Western France on Jan 12 1987 Florida onJan 12 1987 Florida USA on May 17 1985 Belgium onAug 4 1982 [2ndash5] Typically the event is on August 10 1996western blackout of Western Systems Coordinating Council(WSCC) interconnection caused by the negative damping ofthe 025Hz western interareamode [6] As a result leading tothe split of the system into four large islands over 75 millioncustomers experienced outages ranging from a few minutesto nine hours and total load loss is 30500MW because ofthe poor oscillations damping Figures 1(a) and 1(b) show

HindawiComplexityVolume 2017 Article ID 3014510 21 pageshttpsdoiorg10115520173014510

2 Complexity

4000

4200

4400

4600

Activ

e pow

er (M

W)

10 20 30 40 50 60 70 80 900Time (sec)

(a)

4000

4200

4400

4600

Activ

e pow

er (M

W)

0 10 20 30Time (sec) 40 50 60 70 80 90

(b)

Figure 1 The event of the small-signal instability during the blackout WSCC system on Aug 10 1996 (a) the observed California OregonInterconnections power (Dittmer control center) (b) the simulated California Oregon Interconnections power (initial base case)

the measured and simulated power swings of this event Themeasured data show that the standard planningmodels couldbe unreliable predictors of oscillatory behavior

In recent years the economic tempo in the Vietnamhas been developed rapidly the total load has increasedcontinuously Accordingly the Vietnamese power system hasbeen faced with the serious power system blackouts such ason Dec 27 2006 July 20 2007 Apr 09 2007 and the latestevent on May 22 2013 All of the technical problems that theVietnamese power system identified after these events hadalready been reported [7 8]

The power system oscillations occur in the power systemsbecause of the contingencies such as the grid faults andsudden load changes the dampening of these oscillationsis necessary for a secure system operation If the controlledsystems react quickly against faults the power system stabilitywill enhance significantly The advanced power electronicshas led to a new design called flexible alternating-currenttransmission systems (FACTS) by Electrical Power ResearchInstitute (EPRI) The FACTS devices make more use of theexiting capacities in the power system and enhance thepower system stability For example the parameters in thepower system are controlled and the load flow is modifiedto preclude the overload of transmission lines after the gridfaults The FACTS devices are widely used to improve theefficiency of power system operation However the benefitsderived from FACTS controllers such as the small-signalstability and transient stability that depend greatly on theiroptimal placement in the power systems [9 10] Thereforelooking for the optimal placement of FACTS devices inthe large-scale power systems is an interesting researchtopic

State of the Science The damping of electromechanical oscil-lations in the power systems is a matter to belong with powerengineers and become one of the most interesting researchtopics [11 12] The PSSs are a profound influence solutionon improving the power system stability This solution isbased on the excitation system and the generator speeddeviation signals in which the speed deviation is used asthe supplementary control signals in order to provide to the

automatic voltage regulators The fast progress in the fieldof power electronics had contributed to the developmentof the electrical power industry based on the controllableutilization of FACTS [13] This device can dampen both thelocal- and interarea oscillations [10] whereas PSSs could onlydampen the local-area oscillations [14] Thus FACTS is thebest choice to enhance the stability margin of the existingpower systems [9 15] and especially to TCSC This deviceis considered as one of the most effective FACTS and iswidely used to regulate the power flow in transmission linesdampen the power oscillations mitigate subsynchronousresonance (SSR) enhance power system stability and so forthby changing the reactance of transmission line [16] There ishard sledding for the researchers regarding how to determineoptimal location of FACTS

The methods for solving location problems can be clas-sified into two categories (i) analytical techniques and (ii)heuristic optimization approaches [17ndash19] Among heuristicsParticle Swarm Optimization (PSO) has some advantages asless control parameters no require preconditions (continuityor differentiability of objective functions) and slow com-putational burden which make it popular to solve optimallocation and design problems of FACTS [20ndash24] HoweverPSO had drawback as slow convergence in search stage andlimit of local search ability furthermore the algorithm willnot work out the problems of scattering and optimization[15] Genetic algorithm (GA) has been proposed to solveoptimal location and design problems of FACTS [9 15 19 25]This method has been applied to obtain promising resultsHowever downside of GA is that the requested run timeis very long when studying the large-scale systems [26] In[24] optimal location of TCSC is determined via PSO whoseobjective is to maximize small-signal stability In [9 23] amathematical objective function is used to derive objectivefunction for GA and PSO by considering the load-ability inthe system security margins respectively In [27] the authorsused a method based on hybrid between PSO and GA foroptimal allocation of TCSC to enhance voltage stability andreduce power system losses The brainstorm optimizationalgorithm (BSOA) is new heuristic optimization algorithmproposed by the authors in [28] to seek the optimal location of

Complexity 3

TCSC for enhancing the voltage profile The obtained resultsare better than PSO and GA

For the analytical approaches the modal controllabilityindex has been developed by the authors in [29] to findsuitable location of TCSC for dampening interarea modelof oscillations However the authors consider the simulationwith or without FACTS placed in the power system to calcu-late the maximum controllability index values correspondingto critical mode Particularly their main interest is in theinput signals that do not know what is occurring at outputof FACTS through observability index values In [30] Vaidyaand Rajderkar used the sensitivity-based method and theauthors in [31] applied real power flow performance indexsensitivity to determine the optimal location of TCSC toenhance the power system stability In [32] the eigenvaluesensitivity method is utilized to find optimal location ofcontrollable series capacitors for dampening power systemlocation The eigenvalue sensitivity values are calculatedbased on the modal controllability and observability indicesof series reactance modulation In [33] the suitable locationof series compensators is determined based on trajectorysensitivity analysis The objective is to maximize the benefitof series compensators in order to improve the generatorrotor angle stability The energy method based on Gramianmatrices is another technique developed by the authors in[34 35] to determine the optimal setting of TCSC and staticVAR compensator (SVC) obtained with promising resultHowever the difficult of this method is when calculating thetoo large-scale power systems

It is observed that most of existing methods in theprevious literatures have been proposed recently for thelocation of FACTS These methods have several drawbacksfirstly the computation of critical modesmay be questionablein case of large-scale power system since they may not beunique Moreover the computation of them also depends onthe local or interarea modes Secondly the computation ofparticipation factors is only based on the state variables andneglects the input-output behavior Thirdly it just focuses onanalyzing the small-scale power systems Therefore in orderto overcome these drawbacks this paper is a continuation of[34] and combines the Hankel-norm approximation methodto determine the best location for installing TCSCwith objec-tive for damping power system oscillation of the practicalpower system to wit Vietnamese power system It indicatesthat using the Gramian matrix-based method to calculatethe complexity and large-scale power systems takes a lot oftime since the system state matrix is very large Thereforethe Hankel-norm approximation method [36] is proposedto solve such problem The selection of the input signal forTCSC controller is an important consideration for seekingthe optimal location to dampen the interarea oscillationsin which the line reactive and active power line currentand bus voltage are all good selections In this study theactive power in transmission line is considered as an effectiveinput signal for TCSC controller The contingency cases areconsidered based on the active power perturbation signalsin the transmission lines that were selected on the basis ofthe real power line flow performance index (PI) introducedin [37] In addition in order to determine the parameters

of the TCSC controller the optimal method in [38] is alsoconsidered in this paper

Contribution In this paper a relevant method for determin-ing the optimal placement of TCSC controller is proposedto enhance the stability of large-scale power systems Thisproposed method is developed from the energy approachbased on the controllability Gramian matrix of the linearizedsystem The multimachine power system with TCSC con-troller is expressed in the form of a differential algebraicequation (DAE) model The controllability Gramian matrixis obtained from the unique positive definite solutions ofthe Lyapunov equation and it depends on the control input-output and state matrixes of system The optimal locationis selected based on the maximum total Gramian energycalculated from the contingency outage cases (disturbancein lines) on the small-signal stability analysis that meansthe control input must insure the smallest control energyIn line with this purpose the Hankel-norm approximationmethod is applied to reduce the number of state variableswhen dealing with large-scale power systems

Themain new contributions of this paper are summarizedas follows

(i) To develop a relevant method to determine the opti-mal location of FACTS on the small-signal stabilityanalysis

(ii) To propose an association between proposed methodand the Hankel-norm approximationmethod to limitthe time calculation so that the proposedmethod canbe easily implemented for complexity and large-scalepower systems

The remainder of this paper is organized as followsSection 2 addresses the principle of characteristics of theVietnamese power system and the differential algebraic equa-tion (DAE) model of the power system with and withoutTCSC controller The proposed method for optimal locationof TCSC controller based on the Gramian matrices andthe Hankel-norm approximation method are introduced inSection 3 The case studies and conclusions are given inSections 4 and 5 respectively Finally the algorithm for thetransformation matrix can be found in the Appendix

2 Theoretical Analysis

21 Characteristics of the Vietnamese Power System TheViet-namese 500 kV power system operated in 1994 The lengthof transmission line of about 1500 km was connected fromthe HoaBinh to PhuLam power stations with total rating of2700MVA It reached the length of about 5690 km and totalcapacity of all the 500 kV substations of 22800MVA in 2015Several areas of 500 kV transmission lines are compensatedby shunt reactors of about 70 and series capacitors of about60 The total generation capacity of the Vietnamese powersystem could reach 60000MW by the end of 2020 [39] Theresult of load flow calculation on the 500 kV power system isshown in Figure 2

4 Complexity

LongPhu

0 + j0

2942 + j1279

2942 + j1278

minus606 minus j938

2303 + j168

minus5385 + j798

359 + j1297

1295

minus j2

973

8299 + j2641

5642 + j125

227 + j305 5534 minus j62

1200 + j56

2400 + j9876

4955 minus j687

900 + j3765

600 + j1594

1000 + j132

1860 + j1080600 + j1836

600 + j2988

764 minus j1175

720 + j3788

335 + j169

4812 + j4836924 minus j556

5657 + j03

720 + j888

5367 minus j1454

3352 minus j142

1492 minus j314 2446 + j82

2402 + j1358

2276 minus j725

2822 minus j123182 + j2722

13816 + j6534

319 + j5338

754 minus j2314

11446 minus j3809

7096 minus j3679

13711 minus j7312

281 + j1917

10167 minus j2432

1079

4 minus

j520

4574 minus j2212

13056 + j1896

583 + j124

10594 + j353

1546 minus j3265371 minus j5563

5493 + j3645

6528 + j3692

8812 + j1088

8416 + j303

646

9 + j1

266

4042 + j1491

8728 + j2066

855 + j2012

8444 + j4532

15564 + j2107561 + j623

730 + j516 minus932 + j16

6264 + j1372

831 + j1152

14766 + j2381

999

6 + j4

52

DakNong

ThanhMy

DocSoi

DucHoa

PhuLamMyTho

Pleiku

CauBong NhaBe

OMonDuyenHai

PhuMy

SongMay

DiLinh

DaNang

TanDinh

ThuongTin

PhoNoi

VungAng

HoaBinh

NhoQuanPitoong

VietTriHiepHoa

ThangLongQuangNinh

MongDuongSonLa

2729 + j1124

1168 minus j47

8255 minus j5352

1079

4 minus

j520

204 + j2716HaTinh

7219 minus j1846

7219 minus j1846

598 minus j923

598 minus j923

1028

2 minus

j97

8

minus8 minus j1227

1200 minus j1458

1022 + j345

1215

minus j4

28

10204 + j3898TanUyen

VinhTan

519 + j741

855 minus j423

5083 kV

5022 kV

4924 kV

5114 kV

5104 kV

5021 kV

5008 kV

5129 kV

5132 kV4993 kV

4923 kV

5094 kV4888 kV

491 kV

5088 kV

503 kV

5059 kV

5167 kV

504 kV

5019 kV

518 kV

5018 kV

5033 kV522 kV

5046 kV

5071 kV

5052 kV 5027 kV

5138 kV

5122 kV

5019 kV

5009 kV

Figure 2 The result of load flow calculation on the Vietnamese 500220 kV power system 2020

22 DAE Model of Power System Themethodological meth-od for dynamic modeling of general 119898-machine and 119899-buspower system has been described in [40] and is appliedfor this work In this model each synchronous generatoris represented by two-axis flux decay dynamic model alongwith IEEE type I The differential algebraic equation (DAE)model of the power system without TCSC controller can beexpressed as follows

119909sys = 119891 (119909sys 119910sys 119906sys) 119909sys (0) = 11990900 = 119892 (119909sys 119910sys 119906sys) 119910sys (0) = 1199100 (1)

in which 119909sys 119910sys and 119906sys are respectively the statealgebraic and input vectors and are defined as

119909sys = [120575119894 120596119894 1198641015840119902119894 1198641015840119889119894 119864119891119889119894 119881119877119894 119877119865119894]T 119910sys = [119881119895 120579119895 119868119889119894 119868119902119894]T 119906sys = [119879119872119894 119881REF119894]T

119894 = 1 119898 119895 = 1 119899

(2)

where

T is the transpose operator120575 is the rotor angle of generator

120596 is the speed of generator

119881 is the voltage magnitude of bus

120579 is the power angle of bus119868119889 are the 119889-axis components of the current ofgenerator

119868119902 are the 119902-axis components of the current of gener-ator

119881119877 is the input amplifier voltage of the excitation ofgenerator

119877119865 is the stabilizer feedback variable of the excitationof generator

119879119872 is the electrical power of generator

1198641015840119902 is the 119902-axis component of the internal voltage ofgenerator


119864119891119889 is the 119889-axis component of the field voltages ofgenerator

119881REF is the reference voltage of generator

Complexity 5

Thyristors

TCSC

Controller

Vk ang 120579k Vt ang 120579t

Ikt Itk

L

C

Zline

(a)

Ikt ItkminusjXtcsc

Xnew_line

Rline + jXline

(b)

Inductive region

Capacitive region

1284

90 110 120 180100 150 160 170130 140

120572

minus10

minus5

0

5

10

15

Xtcsc

(c)

Figure 3 The TCSC controller (a) structure (b) equivalent (c) reactance versus firing angle characteristic cure

Next the linearized model is given as [40]

[[[[

Δsys00]]]]= [[[[

1198601015840sys 1198611015840sys1 1198611015840sys21198621015840sys1 1198631015840sys11 1198631015840sys121198621015840sys2 1198631015840sys21 1198631015840sys22]]]][[[[

Δ119883sysΔ119910sys1Δ119910sys2]]]]

+ [[[119864sys00]]]Δ119880sys

(3)

It can be identified as119860 sys = 1198601015840sys119861sys = [1198611015840sys1 1198611015840sys2] 119862sys = [119862

1015840sys11198621015840sys2]

119863sys = [1198631015840sys11 1198631015840sys121198631015840sys21 1198631015840sys22]

(4)

119864sys = [[[119864sys100]]] (5)

Equation (3) can be changed to another form as follows[40]

Δsys = 119860 sysΔ119909sys + 119861sysΔ119910sys + 119864sysΔ119906sys0 = 119862sysΔ119909sys + 119863sysΔ119910sys (6)

where

Δ119909sys = [Δ120575119894 Δ120596119894 Δ1198641015840119902119894 Δ1198641015840119889119894 Δ119864119891119889119894 Δ119881119877119894 Δ119877119865119894]T Δ119910sys = [Δ119910sys1 Δ119910sys1]

= [ 1205791 1198811 sdot sdot sdot 119881119898 | 1205792 sdot sdot sdot 120579119899 119881119898+1 sdot sdot sdot 119881119899 ]T Δ119906sys = [Δ119879119872119894 Δ119881REF119894]T

(7)

23 DAE Model of Power System with TCSC Controller

231 TCSC Controller The main role of TCSC is to controlfast the active power flow increase the power transfer ontransmission line and enhance the stability of the powersystem The basic structure TCSC consists of a fixed seriescapacitor bank C in parallel with a thyristor-controlledreactor (TCR) as shown in Figure 3(a) It can controlthe continuous power flow on the alternating-current (AC)line with a variable series capacitive reactance This seriesreactance is adjusted through variation of firing angle 120572the effective reactance 119883tcsc depends on three regions (i)inductive region which starts increasing from 119883119871119883119862 valueto infinity (119883119871(120572) = 119883119862) (ii) capacitive region which startsincreasing from infinity to capacitive reactance 119883119862 and (iii)resonance region which occurs between these two regionsIn order to avoid the resonance region the steady state limitsof the firing angle are chosen to be 90∘ le 120572 le 180∘ withthe resonant point 120572119903 = 1284∘ In this paper for investigatingpower system stability after a fault TCSC should operate incapacitive region and the limit of the steady state to the firingangle is chosen to be 120572119903 le 120572 le 180∘ as shown in Figure 3(c)

6 Complexity

1205900

PlineK

1

1 + sT1

sTw

1 + sTw

1 + sT2

1 + sT3

++

Δ120590Xmin

Xmax

1

1 + sTtcsc

OutputXtcsc

Figure 4 The transfer function mode of TCSC controller

The variable 119883tcsc can be obtained by using some of thecontrol strategies and the feedback signal of the TCSC con-troller This feedback signal can be the signal of active powerreactive power current of transmission line or transmissionangle [41] In this paper the main objective is to dampen thepower oscillations The active power signal in transmissionline is chosen [42] and the control strategy for TCSC is shownin Figure 4 [34]

The new equivalent impedance of the line when placedTCSC as shown in Figure 3(b) can be obtained as [43]

119883new line = 119883line + 119883tcsc (8)

in which the relationship between the firing angle 120572 and theimpedance of the TCSC at fundamental frequency can bederived as follows [44]

119883tcsc = minus119883119862 + 1198701 (2120590 + sin 2120590)minus 1198702cos2120590 (120603 tan (120603120590) minus tan120590)

120590 = 120587 minus 120572 120603 = radic119883119862119883119871 119883119871119862 =119883119862119883119871119883119862 minus 119883119871 1198701

= 119883119862 + 119883119871119862120587 1198702 = 41198832119871119862120587119883119871

(9)

where120590 is the conduction angle120603 is theTCSC ratio119883119871 = 120596119871is the reactance of the inductor and 119883119871 = 1120596119862 is the fixedcapacitive impedance

232 The Unification of TCSC Controller in Power SystemTheTCSChas been instated on the transmission line betweenbases 119896 and 119905 of an 119899-bus power system The injected powerflow into buses 119896 and 119905 is given ib the following equations [45]

At bus 119896119875119896 = Δ119866119896119905 100381610038161003816100381611988111989610038161003816100381610038162 minus 10038161003816100381610038161198811198961003816100381610038161003816 10038161003816100381610038161198811199051003816100381610038161003816sdot [Δ119866119896119905 cos (120579119896 minus 120579119905) + Δ119861119896119905 sin (120579119896 minus 120579119905)]

119876119896 = minusΔ119861119896119905 100381610038161003816100381611988111989610038161003816100381610038162 minus 10038161003816100381610038161198811198961003816100381610038161003816 10038161003816100381610038161198811199051003816100381610038161003816sdot [Δ119866119896119905 sin (120579119896 minus 120579119905) minus Δ119861119896119905 cos (120579119896 minus 120579119905)]

(10)

Similarly at bus 119905

119875119905 = 100381610038161003816100381611988111990510038161003816100381610038162 Δ119866119896119905 minus 10038161003816100381610038161198811198961003816100381610038161003816 10038161003816100381610038161198811199051003816100381610038161003816sdot [Δ119866119905119896 cos (120579119896 minus 120579119905) + Δ119861119905119896 sin (120579119896 minus 120579119905)]

119876119905 = minus 100381610038161003816100381611988111990510038161003816100381610038162 Δ119861119896119905 + 10038161003816100381610038161198811198961003816100381610038161003816 10038161003816100381610038161198811199051003816100381610038161003816sdot [Δ119866119896119905 sin (120579119896 minus 120579119905) minus Δ119861119905119896 cos (120579119896 minus 120579119905)]

(11)

where 119875119896 119876119896 119875119905 and 119876119905 are the active and reactive powersinjected at buses 119896 and 119905 respectively Also 119881119896 120579119896 119881119905 and120579119905 are voltage magnitudes and phase angles of buses 119896 and 119905respectively Δ119866 and Δ119861 depend on TCSC reactance and aregiven as

Δ119866119905119896 = 119883tcsc119877line (119883tcsc minus 2119883line)(1198772line + 1198832line) [1198772line + (119883tcsc minus 119883line)2]

Δ119861119896119905 = 119883tcsc [1198772line minus 1198832line + 119883tcsc119883line](1198772line + 1198832line) [1198772line + (119883tcsc minus 119883line)2]

(12)

where119877line and119883line represent the resistance and reactance ofthe line respectively and119883tcsc is the optimal value of TCSC

The linearized model of TCSC is given as follows

Δtcsc = 119860 tcscΔ119883tcsc + 119861tcsc [Δ120579119896 ΔV119896 Δ120579119905 Δ119881119905]T [Δ119875119896 Δ119876119896 Δ119875119905 Δ119876119905]T= 119862tcscΔ119883tcsc + 119863tcsc [Δ120579119896 Δ119881119896 Δ120579119905 Δ119881119905]T

(13)

Complexity 7

Incorporating (3) and (13) the DAE model of themultimachine power system with TCSC controller can bedescribed as follows

[[[[[[

Δsys

Δtcsc00

]]]]]]

=[[[[[[[

1198601015840sys 1198601tcsc 1198611015840sys1 1198611015840sys21198602tcsc 119860 tcsc 11986110158401tcsc 11986110158402tcsc1198621015840sys1 1198621tcsc 1198631015840sys11 1198631015840sys121198621015840sys2 1198622tcsc 11986310158401tcsc 11986310158402tcsc

]]]]]]]

[[[[[[

Δ119883sysΔ119883tcscΔ119910sys1Δ119910sys2

]]]]]]

+ [[[[[[

119864sys-tcsc000

]]]]]]Δ119880sys-tcsc

(14)

It can be identified as

119860new = [ 1198601015840sys 1198601tcsc1198602tcsc 119860 tcsc

]

119861new = [1198611015840sys1 1198611015840sys21198611015840tcsc 1198611015840tcsc]

119862new = [1198621015840sys1 1198621tcsc1198621015840sys2 1198622tcsc]


119864new = [119864sys-tcsc 0 0 0]T

(15)

Therefore (14) can be changed to another form as follows

Δnew = 119860newΔ119909 + 119861newΔ119910 + 119864newΔ1199060 = 119862newΔ119909 + 119863newΔ119910 (16)

3 The Proposed Approach

31 Gramian-Based Controllability and Observability Thesystem often has two properties controllability and observ-ability which play an important role regarding the determi-nation of the optimal location of TCSC in the power systemFrom that the input and output variables need to be used inorder to observe and control the system Therefore (16) canbe redescribed in a state-space form as follows

(119905) = 119860119909 (119905) + 119861119906 (119905) 119910 (119905) = 119862119909 (119905) + 119863119906 (119905) (17)

where

119909(119905) is the state vector119910(119905) is the output vector119906(119905) is the input vector119860 is the state matrix119861 is the control matrix119862 is the output matrix119863 is the feed-forward matrix

311 Controllability

Definition 1 System (17) is controllable over the interval[1199050 1199051] if any states (1199090 1199091 isin R) there exists the input 119906(119905)[1199050 1199051] rarrR that drives the system from 1199090(1199050) to 1199091(1199051) thisis true the system is completely controllable despite the initialtime and state

Every actuator in the power systems is the energy thatcan be limited such that controllability matrix has been usedfor the purpose of dealing with the amount of input energyThis input energy is required to reach a given state fromthe origin The property of controllability of the system canbe described in a quantitative manner by the controllabilityfunction Typically it is defined for dynamic system as theminimum input energy that necessary drives the system fromstate 119909(minus119879) = 0 to state 119909(0) = 1199090 and can be given as [46]

119871119888 (1199090) = minimize119906isin1198712(minus1198790)

int0minus119879119906 (119905)2 119889119905

subject to (119905) = 119860119909 (119905) + 119861119906 (119905) 119909 (minus119879) = 0119909 (0) = 1199090

(18)

It can be proven that the above system (17) has thetransient controllability function which is given as follows

119871119888 (1199090) = 12119909T0119866minus1119888 (119879) 1199090 (19)

Remark 2 For the real dynamic system 119871119888(1199090) is largeand the state 1199090 is difficult to reach Thus the system isuncontrollable

Remark 3 Matrices 119860 and 119861 of the above system (17) arecontrollable in the time range (1199050 1199051) if and only if thecontrollability matrix Δ defined as (20) has full rank 119899 where119899 is the number of states Notice that the controllabilitymatrixhas dimension 119899times119899119898 where119898 is the dimension of the inputvector 119906

Δ = [119861 119860119861 1198602119861 sdot sdot sdot 119860119899minus1119861] (20)

Remark 4 Matrices 119860 and 119861 of the above system (17) arecontrollable if and only if the controllability Gramian matrix

8 Complexity

119866119888(119879) on horizon T defined as (21) has full rank 119899 and ispositive definite

119866119888 (119879) = int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905 (21)

312 Observability The system states are the internal vari-ables which are hard to directly measure but the outputscan be measured easily at the same time The observabilityproperty plays an important role in the analysis of optimallocation

Definition 5 System (17) is observable over the interval[1199050 1199051] if any states 1199090 1199091 isin R there exists the output 119910(119905)1199050 le 119905 le 1199051 lt infin assuming that the input 119906(119905) is known thisis true the system is observable despite the initial time andstate

The observability property of the system can be character-ized in a quantitative manner by the observability function Itis defined for dynamic system as output energy generated bythe state 119909(0) = 1199090 (when 119906 = 0) and is given as [46]

119871119900 (1199090) = 12 int0

minus119879

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905subject to 119909 (0) = 1199090

119906 equiv 0(22)

It can be proven that system (17) has the transientobservability function which is given as follows

119871119900 (1199090) = 12119909T0119866minus1119900 1199090 (23)

Remark 6 For the real dynamic system 119871119900(1199090)measures theeffect of the initial condition on the output if 119871119900(1199090) is smallthe effect of1199090 in the output is confined such that it is difficultto reach state 1199090 Thus the system is unobservable

Remark 7 Matrices 119860 and 119862 of the above system (17)are observable in the interval [1199050 1199051] if and only if theobservability matrix nabla defined as (24) has full rank 119899 where119899 is the number of states Notice that the observability matrixhas dimension 119899119901times119899 where 119901 is the dimension of the outputvector 119910

nabla =[[[[[[[[[[[

1198621198621198601198621198602

119862119860119899minus1

]]]]]]]]]]]

(24)

Remark 8 Matrices 119860 and 119862 of the above system (17) areobservable if and only if the observability Gramian matrix119866119900(119879) on horizon T defined as (25) has full rank 1198991015840 and ispositive definite

119866119900 (119879) = int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905 (25)

If system (17) is asymptotically stability around the originthe controllability and observability functions can be given by

119871119888 = minimize119906isin1198712(minusinfin0)119909(0)=1199090119909(minusinfin)=0

12 int0

minusinfin119906 (119905)2 119889119905

119871119900 = 12 intinfin

0

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905 119909 (0) = 1199090 119906 equiv 0(26)

Formal remarks for controllability and observability arewhite and black In reality some states are very calamitousto control or have little effect on the outputs The degree ofobservability and controllability can be evaluated by the sizesof 119866119888 and 119866119900 with infinite time horizon

119866119888 (0infin) = lim119879rarrinfin

int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905


int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905

(27)

Notice that 119866119888 and 119866119900 satisfy the following differentialexpression

119889119866119888 (0 119905)119889119905 = 119860119866119888 (0 119905) + 119866119888 (0 119905) 119860T + 119861119861T119889119866119900 (0 119905)119889119905 = 119860T119866119900 (0 119905) + 119866119900 (0 119905) 119860 + 119862T119862

(28)

It is hard to directly calculate the Gramian matrices fromexpression (27) because they consist of an exponential matrixand an integral If all the eigenvalues of 119860 have strictlynegative real parts as 119879 rarr infin there exists an easier way tocalculate these matrices by solving the Lyapunov expression(29) below [47] which can be solved using MATLAB viagramThe obtained solutions are the unique positive definite

119860119866119888 + 119866119888119860T + 119861119861T = 0119860119879119866119900 + 119866119900119860 + 119862T119862 = 0 (29)

Obviously expression (29) shows that (i) the property ofobservability Gramian matrix depends on the output matrix119860 and state matrix 119862 accordingly the output energy can beaffected by properly choosing this matrix (ii) The propertyof controllability Gramian matrix depends on the controlmatrix 119860 and state matrix 119861 accordingly the control energycan be affected by properly choosing this matrix (iii) Whenthe system is only detectable the Gramian matrices will beonly some nonnegative definite matrices

In particular we can pose the problem seeking theminimum input energy (ie the control energy) that mustderive system (17) from the initial state 1199090 to a final state 119909119905119891at time 119905 = 119905119891 and this energy is defined by [34]

119869 = (exp (119860119905119891) 1199090 minus 119909119905119891)Tsdot (119866119888 minus exp (119860119905119891)119866119888 exp (119860T119905119891))Tsdot (exp (119860119905119891) 1199090 minus 119909119905119891)

(30)

Complexity 9

Clearly expression (30) shows that minimizing 119869 is thesame as maximizing the Gramian matrix 119866119888 Furthermoremaximizing the transmitted total energy (potential andkinetic energy) from the actuators to the structure for a giveninput energy it can be obtained as follows

119864 = intinfin0(119864119901 (119905) + 119864119896 (119905)) 119889119905 = intinfin

0119909 (119905) 119909T (119905) 119889119905 (31)

where 119864119896 is the kinetic energy and 119864119901 is the potential energyIn this paper we suppose that the input is aDirac impulse thatexcites all frequencies and 119909(119905) = exp(119860119905)119861 is the impulsematrix Hence the obtained energy is the weighted trace ofcontrollability Gramian [34 48]

119864 = trace (119866119888) = 119899sum119894=1

120590119894 (32)

where 120590119894 is the 119894th Hankel singular value The decompositionof this value denotes spatial energy decomposition containedin the impulse response The sum of the singular valuesdenotes the energy total in which each of the singular valuerepresents the energy in a particular direction The optimallocation is determined based on themaximization of the totalenergy of the system at the fault clearing timeTherefore (32)is a basis for selecting optimal control input

32 Hankel-Norm Method In order to compute easilywe focus on analyzing the infinite horizon Gramian TheGramian matrices are computed by using expression (29)rather than solving expressions (21) and (25) However it maybe more difficult for analyzing the Vietnamese power systembecause that is a large-scale networkWith such networks thenumber of state variables could be big leading to lost timefor computing In reality determining the optimal locationwe just need the number of major state variables that playan important role in order to analyze the linear system onthe small-signal stability Therefore in order to solve thisproblem the order reduction method is applied based onthe computation of controllability and observability Gramianmatrices introduced in [36]

321 Balanced Realization Considering system (17) in orderto make effect for this study the so-called similarity trans-formation 119872 must need to modify the state variables andmatrices of the system but the output and input behaviorsremain unchanged such that the controllability and observ-ability Gramian matrices satisfy the following

119866119888 = 119866119900 =[[[[[[[

1205901 0 0 sdot sdot sdot 00 1205902 0 sdot sdot sdot 0 d

0 0 0 sdot sdot sdot 120590119899

]]]]]]]

with 1205901 ge 1205902 ge sdot sdot sdot ge 120590119899 ge 0

(33)

The so-called similarity transformation transforms thecontrollability and observability Gramian matrices in thefollowing way

119866119888 = 119872119866119888119872T119866119900 = (119872minus1)T 119866119888119872minus1 (34)

However the result turns out to be the invariant outputand input behaviors that is119872119866119888119872T = (119872minus1)T119866119888119872minus1 (ieit relates to the Gramian of the original system by119872)

Expression (34) constructs a state-space transformationthat is considered with the form 119891(119905) = 119872119909(119905) substitutingthis state-space transformation into (17) the new system canbe obtained

(119905) = 119860119891 (119905) + 119861119906 (119905) 119910 (119905) = 119862119891 (119905) + 119863119906 (119905) (35)

where 119860 = 119872119860119872minus1 119861 = 119872119861 119862 = 119862119872minus1 119863 = 119863Therefore system (35) is a balanced form according to thegiven nonsingular transformation matrix 119872 to change thesystem in such a way that the controllability and observabilityGramian matrices satisfy the following condition

119866119888 = 119866119900 = diag 1205901 1205901 120590119899 1205901 ge 1205901 ge sdot sdot sdot ge 120590119899 (36)

in which 1205901 1205902 120590119899 are real and positive numberscalled the Hankel singular values and the transformationmatrix is determined by using the algorithm [47] as shownin the Appendix

322 Order Reduction Realization After obtaining the newsystem in a balanced form as shown in (35) the orderreduction can be performed The first step is to choose theorder of 119896 based on the purposes of reduction The author in[36] has considered general situation that theHankel singularvalues satisfy the condition that is

1205901 ge 1205902 ge sdot sdot sdot ge 120590119896 ge 120590119896+1 = 120590119896+2 = sdot sdot sdot = 120590119896+119903gt 120590119896+119903+1 ge sdot sdot sdot ge 120590119899 gt 0 (37)

In this study the system reduction is performed byeliminating all state variables corresponding to the Hankelsingular values that are smaller than 10minus3 Therefore thesystem can be converted to the following order reductionform

1 (119905) = 119860111198911 (119905) + 119860121198912 (119905) + 1198611119906 (119905)2 (119905) = 119860211198911 (119905) + 119860221198912 (119905) + 1198612119906 (119905) 119910 (119905) = 11986211198911 (119905) + 11986221198912 (119905) + 119863119906 (119905)

119866119888 = 119866119900 = Σ(38)

where Σ = [(diag1205901 120590119896 120590119896+119903+1 120590119899 )2 minus 1205902119896+1Ψ119903] =(Σ21 minus 1205902119896+1Ψ119903) is the nonsingular and diagonal matrix The

10 Complexity

vectors 1198911(119905) and 1198912(119905) are obtained by decomposing the statevariable 119891(119905) (ie 119891(119905) = [1198911(119905)1198912(119905)]T) and have dimensions119899 minus 119903 and 119903 respectively Ψ119903 is the identity matrix and hasdimension 119903Thematrices (11986011 11986012 11986021 11986022)2times2 (1198611 1198612)1times2and (1198621 1198622)2times1 are respectively partitioned from matrices119860 119861 and 119862 in balanced system (35) corresponding to thepartitioned Gramian

If the number of inputs is more than that of outputs (theso-called all-pass dilation of the system) then 119860 119861 119862 and119863 are replaced by 119860T 119861T 119862T and 119863T respectively [49]Thus the components for the so-called all-pass dilation of thesystem are defined as follows

= Σminus1 (1205902119896+1119860T11 + Σ111986011Σ1 minus 120590119896+1119862T

1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)

where 119880 is a unitary matrix and is solution of the followingexpression

1198612 + 119862T2119880 = 0 (40)

According to the conclusion in [49] the all-pass dilationof the system might be unstable The final step is to computethe stable part of the all-pass dilation of the systems 10158401015840 1015840 and 1015840 based on the Hankel-norm method As weconsider a system that the number of inputs is more thanthe number of outputs matrices 1015840 1015840 1015840 and 1015840 can bereplaced by 1015840T 1015840T 1015840T and 1015840T based on the followingerror of approximation

1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)

where119879(119904) and119879119896(119904) are the transfer functions of the originaland the 119896th-order reduced systems

Clearly expression (41) shows that the largest error for allfrequencies is equal to theHankel singular value 119896+1 It allowsus to select a suitable value of 119896Therefore the matrices of thereduced system are obtained as follows

119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)

33 Algorithm The main objective is to dampen the activepower and generator angle oscillations in the power system

when the active power perturbation occurs in the transmis-sion line The proposed algorithm is used to determine theoptimal location of TCSC based on the Gramian criticalenergy and the flowchart of the algorithm is shown inFigure 5 The principles and details have been presented inSections 31 and 32 and below is the interpretation

First Start

Step 1 Place the TCSC controller in the line 119894th (119894 = 1 23 119899)Step 2 Run steady state

Step 3 Generate the active power perturbation signal inthe transmission line 119895th (119895 = 1 2 3 119898) in which 119898is large enough For each contingency of the active powerperturbation 119895 the observed output 120592119895 and control input 120585119895were analyzed

Step 4 Compute the matrices 119860 119861 and 119862 corresponding tothe placement 119894 and the active power perturbation signal 119895Step 5 Perform the order reduction for system based on theHankel-norm method

Step 6 Estimate the stable condition based on the statematrix (119860119896)119894119895 to exclude the unstable cases If the conditionis satisfied Step 6 will be performed and vice versa Step 1will be iterated

Step 7 Compute the controllability Gramian matrix of thenew system (after performing the order reduction) corre-sponding to the active power perturbation signal 119895thStep 8 Iterate the steps from 2 to 7 for computing a set of theactive power perturbations (119895 = 1 2 3 119898)Step 9 Compute the energy for each placement corre-sponding to a set of the active power perturbations (119895 =1 2 3 119898)Step 10 Iterate the steps from 1 to 7 to calculate all thelocations (119894 = 1 2 3 119899)Step 11 Compare the maximum total energy to evaluate theoptimal placement for TCSC controller

Finally End

4 Case Study

The Vietnamese 500220 kV transmission system is usedin the study to illustrate the effectiveness of the proposedmethod This system consists of 29 substations of 500 kV 162substations of 220 kV 16 double-circuit lines and 20 single-circuit lines of 500 kV 205 double-circuit lines and 67 single-circuit lines of 220 kV and 179 generator unitsThe generatedtotal power is about 42179MW the peak load demand is

Complexity 11

Start

Run steady state

Place the TCSC in the transmission line

Generate the active power perturbation signal in

Calculate the matrices

Evaluate the stable condition

Yes

No

End

Calculate the energy

Eqs (41) and (42)

Assess the optimal location for TCSC

Calculate the controllability Gramian

Eq (17)(a) Find M(b) Perform similarity transformation f(t) = Mx(t)

(a) Select k calculate r(b) Partition balanced system states

(b) If the number of inputs is more than the number of outputs


Hankel-norm approximation algorithm

line ith

the transmission line jth

Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m

(Ak)ij (Gc)ij + (Gc)ij (ATk)ij + (Bk)ij (B

Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2

1 minus 1205902k+1Ψr

then A B C and D are replaced by AT B

T C

T and DT

(a) Compute state part of all-pass dilation and A998400 B

998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D

998400 are replaced by A998400T B

998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)

Figure 5 Flowchart of the proposed algorithm

12 Complexity

Table 1 The existing compensation ratio on the 500 kV system

Casenumber

The line betweenbuses

Compensationratio () On circuit

(1) NhoQuan-Hatinh(NQ-HT) 577 1

(2) NQ-HT 544 2

(3) VungAng-Danang(VA-DN) 64 1

(4) VA-DN 64 2

(5) Danang-Docsoi(DN-DS) 63 mdash

(6) Docsoi-Pleiku(DS-Plei) 58 mdash

(7) Thanhmy-Pleiku(TM-Plei) 52 mdash

(8) Pleiku-Daknong(Plei-DakN) 76 mdash

(9) Daknong-Caubong(DakN-CB) 61 mdash

(10) Pleiku-Dilinh(Plei-DL) 70 mdash

(11) Dilinh-Tandinh(DL-TD) 59 mdash

(12) Pleiku-Caubong(Plei-CB) 70 mdash

about 40703MWThe power system simulation engineering(PSSE) program is used to analyze the transient and small-signal stability and MATLAB is used to calculate Gramianmatrices and perform order reduction All dynamic modelssuch as generators excitation systems transmission linesand loads are modeled by using PSSE (from dynamic modellibrary) [50] and all dynamic parameters are taken from [2]and the Vietnamese national load dispatch center (NLDC)The single-line diagram and the result of load flow calculationof the 500 kV voltage level are given in Figure 2

The optimal placement of TCSC in the Vietnamesenetwork is determined based on (i) a combination of the con-trollability Gramian critical energy with the order reductionmethod on the small-signal stability analysis applied to searchfor several feasible locations (ii) the transient stability anal-ysis performed to compare and determine optimal locationsthrough various simulation cases

In addition the existing series compensation ratio on theVietnamese 500 kV power system is considered to calculateand listed out in Table 1

41 Determine Feasible Locations The correctness of the pro-posed method for the optimal location of TCSC is verifiedon the small-signal analysis the dynamic model of TCSCis chosen as shown in Figure 4 [34] We suppose thecompensation capacity of TCSC is set to the range [075 12]of the uncompensated line The active power perturbationsignal in the line is considered as the input signal for theTCSC controller [51]The optimalmethod in [38] was used todetermine the parameters for the TCSC controller and theseparameters are listed in Table 2 [34]

Table 2 Parameters of the TCSC controller

Parameter Value Parameter Value1198791 01 sec 119870 0751198792 01 sec 119883max 12119883line pu1198793 04 sec 119883min 025119883line pu119879119908 10 sec 119879tcsc 0015 sec

Mag

nitu

de

Eigen value of AEigen value of

minus20minus15minus10minus5

05

101520

0minus100 minus80 minus60 minus40 minus20minus120minus140Eigen value of A and A11

A11

Figure 6 Eigenvalue distribution before and after using balancedreduction

The studied contingency cases of active power perturba-tion were selected from single-line outage cases on the basisof the real power flow performance index (PI) introduced in[36] as follows

PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)

where 119875119897119894 and 119875max119897119894 are the real power and rated power flow

of the line 119894 respectively 120599 is the exponent and 120592119894 is a the realnonnegative weighting factor for line 119894 In this study the valueof the exponent was selected as 20 and 10 is the value of 120592119894

Observing from (43) that all the power flows in the linesare within their limits PI is small and vice versa it reachesout a high value when there are overloads The studiedcontingency cases are listed in Table 3 by using (43) thiscorresponds to the active power perturbation signal in theline of the Vietnamese network

The Vietnamese 500220 kV power system created thestate matrix (119860) having dimension [2371 times 2371] that corre-sponds with 179 generator units By performing the equiv-alence of 179 generator units into 45 equivalent generatorunits the state matrix now has dimension [515 times 515] ATCSC controller has three states as the transducer washoutand leadlag added in the system Therefore the state matrixdimension of the equivalent power system model is [518 times518]

Once again the state matrix of the system is reduced thedimension by applying the balanced order reduction methodto eliminate the singular values that are smaller than 10minus3Figure 6 shows the eigenvalue of the system state matrixbefore and after using balanced reductionTherefore the statematrix (11986011) of the new system has the dimension [45 times 45]

Complexity 13

Table 3 Studied contingency cases

Case number Active power perturbation signal in the linebetween buses

(1) NhoQuan-HoaBinh (NQ-HB)(2) VietTri-Pitoong (VT-PT)(3) ThangLong-PhoNoi (TL-PN)(4) NhoQuan-SonLa (NQ-SL)(5) Pitoong-HoaBinh (PT-HB)(6) QuangNinh-HiepHoa (QN-HH)(7) HiepHoa-VietTri (HH-VT)(8) HiepHoa-Pitoong (HH-PT)(9) Pitoong-SonLa (PT-SL)(10) QuangNinh-ThangLong (QN-TL)(11) QuangNinh-PhoNoi (QN-PN)(12) QuangNinh-MongDuong (QN-MD)(13) ThuongTin-Nho Quan (TT-NQ)(14) ThuongTin-PhoNoi (TT-PN)(15) VungAng-HaTinh (VA-HT)(16) DaNang-ThanhMy (DN-TM)(17) SongMay-TanUyen (SM-TY)(18) SongMay-TanDinh (SM-TD)(19) NhaBe-PhuLam (NB-PL)(20) TanDinh-CauBong (TD-CB)(21) PhuLam-MyTho (PL-MT)(22) PhuLam-DucHoa (PL-DH)(23) DucHoa-MyTho (DH-MT)(24) MyTho-DuyenHai (MT-DH)(25) OMon-MyTho (OM-MT)(26) VinhTan-SongMay (VT-SM)(27) PhuMy-NhaBe (PM-NB)(28) CauBong-PhuLam (CB-PL)(29) NhaBe-MyTho (NB-MT)(30) CauBong-DucHoa (CB-DH)(31) SongMay-PhuMy (SM-PM)

Figure 7 shows the frequency response of the systembefore and after using balanced reduction Obviously theinput-output behaviors of the system in both cases at thebandwidth range from 10minusinfin to 105 are the same with thesame control signal The bandwidth has a range from 105 to10infin the frequency response of the reduced system is flat(a straight line) since its order is smaller than the originalsystem Therefore we can conclude that two systems areequivalent in terms of the input-output behaviors

Figure 8 shows the Hankel singular values after perform-ing the balanced realization reduction technique It can beobserved from this figure that the 45 Hankel singular valueshave been finally retained in comparison with the 2374 statevariables of the original system (including state variables ofTCSC controller) This acquired singular values are smallerthan the chosen threshold 10minus3 (choice of this study) The

Sing

ular

val

ues (

dB)

The original systemThe new system

105104100 101 102 103 106 10710minus2 10minus110minus3

Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

Figure 7 Frequency response of the original and reduced system

0

05

1

15

2

25

Mag

nitu

de

times104

5 10 15 20 25 30 35 40 45 500Hankel singular values of state system

Figure 8 Distribution of Hankel singular values

eigenvalues distribution of the system before and after usingbalanced reduction is plotted in Figure 8 Therefore thereduced system has 45 state variables which can fully meeteffects of the network to calculate the Gramian matrices

The controllability Gramian energy indices were deter-mined based on the proposed algorithm as shown in Sec-tion 33 and the active power perturbation signals in the lineas listed in Table 3 when the transient horizon is equal toinfinity (119879 = infin) are given in Table 4 in which we onlygive several feasible locations having the high total energyvalue From this table it can be seen that the line PLei-CBis the suitable location for TCSC controller because it hasthe maximum total energy value it means that the energyneeded to drive the controllable state variables is smaller thanother cases Therefore the line between buses PLei and CB isconsidered as the best location to install the TCSC controller

42 Retest Transient Stability To verify the effectiveness ofthe proposed method several dynamic cases are analyzedbased on the transient stability to compare the suitablelocation (the line between buses PLei and CB) and otherfeasible ones (as shown in Table 4) The optimal locationof the TCSC controller is determined based on the rotorangle oscillation damping of the generator units having thelarge output power and the active power damping of thetransmission line In addition the power distribution on

14 Complexity

Table 4 Energy values according to several feasible locations of TCSC

Casenumber

Contingencycases (fromTable 3)

(Trace of 119866119862) times 105TCSC is placed on the line between buses

None NQ-HT DN-VA PLei-TM

PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD

The Northern region (1)(1) NQ-HB 00530 01354 01189 00780 00516 00614 00693 00780 00626 00290 00381(2) VT-PT 00020 00142 00132 00020 00067 00080 00103 00118 00089 00055 00064(3) TL-PN 00012 00947 00873 00374 00368 00483 00506 00526 00450 00215 00254(4) NQ-SL 00242 00923 00840 00520 00354 00429 00458 00502 00418 00185 00242(5) PT-HB 00034 00606 00575 00120 00240 00294 00307 00326 00282 00122 00156(6) QN-HH 00010 00108 00099 00011 00044 00043 00048 00059 00047 00029 00032(7) HH-VT 00012 00095 00097 00025 00044 00056 00061 00059 00054 00030 00031(8) HH-PT 00032 00129 00122 00040 00061 00074 00092 00101 00079 00060 00055(9) PT-SL 00010 00137 00117 00064 00059 00031 00074 00109 00071 00043 00057(10) QN-TL 00029 00613 00564 00135 00242 00307 00337 00364 00301 00156 00176(11) QN-PN 00032 00957 00887 00129 00372 00489 00514 00530 00458 00223 00253(12) QN-MD 00216 00361 00343 00160 00140 00197 00194 00188 00171 00082 00090(13) TT-NQ 00054 01523 01368 00850 00600 00766 00864 00941 00763 00382 00450(14) TT-PN 00152 01173 01072 00630 00460 00598 00646 00685 00574 00279 00330Total energy of (1) 01385 09067 08278 03858 03569 04461 04897 05288 04383 02151 02571

The Southern region (2)(15) VA-HT 00182 02777 02887 01660 01227 01616 01759 01900 01546 00780 00930(16) DN-TM 00287 02015 04708 04982 05948 01894 03267 03003 02733 01642 01462(17) SM-TY 00019 00037 00035 00002 00015 00022 00656 00125 00132 00296 00603(18) SM-TD 00085 00453 00646 00399 00412 00416 03961 01184 00281 01382 03472(19) NP-PL 00043 00683 00829 00492 00502 00460 01330 03648 01638 03823 00413(20) TD-CB 00039 00268 00380 00359 00248 00232 01742 01055 05285 01123 01103(21) PL-MT 00029 00158 00253 00229 00167 00188 00607 01077 00451 01217 00498(22) PL-DH 00013 00079 00062 00013 00030 00021 00381 00175 00092 00255 00262(23) DH-MT 00048 00180 00273 00095 00177 00189 00477 01083 00492 01166 00413(24) MT-DHa 00018 00077 00174 00136 00117 00133 00425 00612 00269 00659 00312(25) OM-MT 00023 00176 00256 00271 00166 00177 00672 00930 00419 01001 00498(26) VT-SM 00082 00120 00289 00124 00195 00198 00969 00826 00343 00803 00911(27) PM-NB 00011 00370 00341 00340 00184 00111 01370 01166 00620 00931 00798(28) CB-PL 00043 00815 01142 00150 00734 00774 00584 05821 02359 06319 01125(29) NB-MT 00034 00078 00114 00100 00077 00089 00774 00357 00123 00391 00502(30) CB-DH 00150 00346 00499 00380 00330 00357 00347 02696 01048 03000 00600(31) SM-PM 00112 00413 00424 00121 00245 00202 02338 00395 00303 00146 01432Total energy of (2) 01218 09043 13312 09853 10773 07079 21659 26053 18134 24934 15332Total energy of (1) + (2) 02604 18111 21590 13711 14342 11540 26556 31341 22517 27085 17903

500220 kV system at maximum andor minimum load isconsidered For maximum load the transferred power onthe 500 kV system from the North and Central to the Southis very high Thus when the three-phase fault occurs onthe lines such as Vungang (VA)-Danang (DN) and PLeiku(PLei)-DiLinh (DL) the system lost serious stability The

three-phase fault scenarios are considered in this study aslisted in Table 5 and can be summarized briefly as follows

Scenario Number 1 A solid three-phase fault occurs at 1 secon the line 500 kVDN-VA of circuit 1 close to the bus 500 kVDN and is cleared after 015 sec by tripping the faulted line

Complexity 15

TCSC placed in the line PLei-CBTCSC placed in the line PLei-TMWithout TCSC

minus60minus55minus50minus45minus40minus35minus30minus25minus20minus15minus10

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)

minus50minus45minus40minus35minus30minus25minus20minus15minus10

minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus30minus25minus20minus15minus10

minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)

TCSC placed in the line PLei-CBTCSC placed in the line NQ-HTTCSC placed in the line DN-VA

22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)

Figure 9 The relative angle oscillations of generators (a) Yaly (b) PhuMy-3 (c) VinhTan (d) QuangNinh

Table 5 Scenarios on three-phase fault for test of transient stability

Case number Fault is nearly bus Line outage(1) 500 kV DN 500 kV DN-VA(2) 500 kV Plei 500 kV PLei-DL(3) 500 kV CB 500 kV TD-CB(4) 500 kV NQ 500 kV NQ-TT

Scenario Number 2 A solid three-phase fault occurs at 1 secon the line 500 kV PLei-DL close to the bus 500 kV Plei andis cleared after 015 sec by tripping the faulted line

Scenario Number 3 A solid three-phase fault occurs at 1 secon the line 500 kV TD-CB close to the bus 500 kV TD andis cleared after 015 sec by tripping the faulted line

Scenario Number 4 A solid three-phase fault occurs at 1 secon the line 500 kV TT-NQ close to the bus 500 kV NQ andis cleared after 015 sec by tripping the faulted line

Case 1 The simulation was done on scenario number 1 basedon the relative angle oscillations of generator supposingthat the system has been operating at maximum load inorder to compare the difference locations of TCSC (thatare on the lines PLei-CB and PLei-TM) and without TCSCFigure 9 plots the relative angle oscillations of generatorsof Yaly PhuMy-3 VinhTan and QuangNinh Figures 9(a)ndash9(c) show the TCSC placed in the line PLei-CB the angleoscillations of generators Yaly PhuMy-3 and VinhTan aredamped out in about 6 sec compared to that in the line PLei-TM and without TCSC controller In particular it can beobserved from Figure 9(d) that the oscillations of generatorQuangNinh are damped out faster than the case when theTCSC is placed in the line PLei-CB

Case 2 The simulation of this case was done on scenarionumber 2 based on the active power of the line and the relativeangle oscillations of generators to compare the suitablelocation of TCSC and other feasible locations as NQ-HTand DN-VA In this case the system has been operating atmaximum load Observing from Figure 10 TCSC is placedin the line PLei-CB the transient response is significantly

16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)

Figure 10 The transient response (a) the relative angle oscillations of generator VinhTan (b) the active power oscillations in the lineCauBong-DucHoa (c) the active power oscillations in the line Pleilu-ThanhMy

improved compared with that in the two another locationsThis could easily be construed from Table 4 the total energyvalue when TCSC is placed in the line PLei-CB is thelarger

Case 3 From Table 4 it can be seen that the other feasiblelocations (DakN-CB and PLei-DL) have the second and thirdhighest total energy values respectivelyThe third simulationwas carried out for scenario number 3 to compare thesefeasible locations with the suitable location (PLei-CB) Inthis case the system is considered as operating at maximumload Figure 11 plots the system response as the relative angleoscillations of generators (VinhTan and DuyenHai) and theactive power oscillations in the lines (HaTinh-VungAng andPLeilu-ThanhMy) It can be observed from Figures 11(a)ndash11(d) that with TCSC placed in the line between PLei andCB the oscillations are damped faster compared with twoother instances Evidently the effect is obtained in terms ofoscillation damping from other two feasible locations to bethe same because they have the same value of total energy

value Figure 11(e) shows the response of TCSC controllerit can be observed in this figure that the TCSC has injectedthe reactance to the grid during the period of fault and thatthe best influence is obtained in terms of reactive power flowoutput of TCSC when it is placed in the line PLei-CB

Case 4 Observing from Table 4 the highest total energyvalue was obtained on the electric power supply system inthe Northern region corresponding to the TCSC controllerplaced in the line NQ-HT Contrarily in this controllerplaced in the line Plei-CB the highest total energy valuewas obtained on the electric power supply system in theSouthern region This simulation case was done on scenarionumber 4 to compare two locations having the highest totalenergy value at two regions In this case we suppose that thesystem has been operating at maximum load Figure 12 plotsthe relative angle oscillations of generators QuangNinh andVinhTan Evidently it can be observed from this figure thatTCSC is placed in line PLei-CB oscillations are damped fastercompared to placement in the line NQ-HT despite the far

Complexity 17

TCSC placed in the line PLei-CBTCSC placed in the line Plei-DLTCSC placed in the line DakN-CB

minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)

Figure 11 The transient response (a) the relative angle oscillations of generator VinhTan (b) the relative angle oscillations of generatorDuyenHai (c) the active power oscillations in the line HaTinh-VungAng (d) the active power oscillations in the line Pleiku-ThanhMy (e)the response of TCSC controller

18 Complexity

TCSC placed in the line NQ-HTTCSC placed in the line PLei-CB

18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)

Figure 12 The relative angle oscillations of generator (a) QuangNinh (b) VinhTan

TCSC placed in the line Plei-CBTCSC placed in the line DN-VA

2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)

Figure 13The transient response (a) the relative angle oscillations of generator SonLa (b) the active power oscillations in the line NhoQuan-SonLa (c) the active power oscillations in the line HaTinh-VungAng

fault location (in the line NQ-TT) compared with the TCSCplaced at the Southern region (in the line PLei-CB)

Case 5 Also for comparison of the other locations thesimulation for this case was done on scenario number 2

supposing that the system has been operating at minimumload Figure 13 plots the transient response of the systemFor maximum load the transferred power on the 500 kVsystem from theNorth region to the South region is very highConsequently the amplitude deviation of the generator rotor

Complexity 19

angle and active power oscillations is very large Howeverfor minimum load this deviation is negligible but it is stillaccordant with the proposed method It could be seen fromFigure 13 that the best effect is obtained in terms of oscillationdamping of the generator rotor angle of SonLa and theactive power in the lines of NhoQuan-SonLa and HaTinh-VungAng

5 Conclusions

In this paper a relevant stochastic method for the optimalplacement of TCSC controller has been presented to enhancethe rotor angle stability and dampen the power systemoscillations in the multimachine systems This proposedmethod is developed from the energy approach based onthe controllability and observability Gramian matrices ofthe linearized multimachine systemsThe optimal placementdepends on the trace indices of the Gramian matrices thathave been calculated on the active power perturbation in theline of the network (for this study the applied network is theVietnamese 500220 kV power system)

Theoptimal placement for TCSC controller is determinedbased on the Gramian critical energy values that have beencalculated on the small-signal stability analysis However theacquired results showed that the power system could operateperfectly under the influence of the transient conditions

The time-domain simulation results on the transientstability analysis show that the rotor angle oscillations of gen-erator and the power oscillations in the line are significantlydampened when the TCSC is placed in the line betweenPLeiku and CauBong which has the maximum total energyvalue

The Gramian-based reduction method has been alsointroduced for the purpose of reducing the calculation timeof the Gramian critical energy when dealing with the large-scale power systems

Appendix

Algorithm for the Transformation Matrix 119872First Step Calculate the decomposition of the controllabilityGramian

119866119888 = CTC (A1)

where C is an upper triangular matrix which has only zerosbelow the main diagonal and such decomposition is calledthe Cholesky decomposition and can be implemented inMATLAB as function C = chol(119866119888)Then Perform singular value decomposition for the productC119866119888CT

C119866119888CT = 119880Σ119880T (A2)

where119880 is orthogonal matrix and Σ = diag1205901 1205902 1205903 120590119899is the diagonal matrix

In the End The transformation matrix can be calculated asfollows

119872 = CT119880TΣminus14 (A3)

Disclosure

This paper is an applied and fully constituted version of thepaper published in International Transactions on ElectricalEnergy Systems 2016 Volume 26 Issue 7 pp 1493ndash1510

Competing Interests

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

Acknowledgments

The authors sincerely acknowledge the financial supportprovided by Duy Tan University Da Nang Vietnam TonDuc Thang University Ho Chi Minh Vietnam IndustrialUniversity of Ho Chi Minh City Ho Chi Minh Vietnam andQuy Nhon University Binh Dinh Vietnam for carrying outthis work

References

[1] L L LaiH T Zhang SMishraD Ramasubramanian C S Laiand F Y Xu ldquoLessons learned from July 2012 Indian blackoutrdquoin Proceedings of the 9th IET International Conference onAdvances in Power System Control Operation and Management(APSCOM rsquo12) pp 1ndash6 IEEE Hong Kong November 2012

[2] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1994

[3] G Andersson P Donalek R Farmer et al ldquoCauses of the 2003major grid blackouts in North America Europe and recom-mendedmeans to improve system dynamic performancerdquo IEEETransactions on Power Systems vol 20 no 4 pp 1922ndash19282005

[4] A Gheorghe M Masera M Weijnen and D L Vries CriticalInfrastructures at Risk vol 9 Kluwer Academic PublishersDordrecht Netherlands 2006

[5] S Larsson and E Ek ldquoThe black-out in southern Sweden andeastern Denmark September 23 2003rdquo in Proceedings of theIEEE Power Engineering Society General Meeting 2004

[6] D N Kosterev C W Taylor and W A Mittelstadt ldquoModelvalidation for the august 10 1996 wscc system outagerdquo IEEETransactions on Power Systems vol 14 no 3 pp 967ndash979 1999

[7] HNguyen-DucHCao-Duc CNguyen-Dinh andVNguyen-Xuan-Hoang ldquoSimulation of a power grid blackout event inVietnamrdquo in Proceedings of the IEEE Power and Energy SocietyGeneral Meeting (PESGM rsquo15) p 15 Denver Colo USA July2015

[8] V T Dinh and H H Le ldquoVietnamese 500kV power system andrecent blackoutsrdquo in Proceedings of the IEEE Power and EnergySociety General MeetingmdashConversion and Delivery of ElectricalEnergy in the 21st Century pp 1ndash5 IEEE Pittsburgh Pa USAJuly 2008

20 Complexity

[9] L J Cai I Erlich and G C Stamtsis ldquoOptimal choice andallocation of FACTS devices in deregulated electricity marketusing genetic algorithmsrdquo in Proceedings of the IEEE PES PowerSystems Conference and Exposition pp 201ndash207 New York NYUSA October 2004

[10] H Okamoto A Yokoyama and Y Sekine ldquoStabilizing controlof variable impedance power systems applications to variableseries capacitor systemsrdquo Electrical Engineering in Japan vol113 no 4 pp 89ndash100 1993

[11] J Ma T Wang S Wang et al ldquoApplication of dual youlaparameterization based adaptive wide-area damping control forpower systemoscillationsrdquo IEEETransactions on Power Systemsvol 29 no 4 pp 1602ndash1610 2014

[12] U P Mhaskar and A M Kulkarni ldquoPower oscillation dampingusing FACTS devices modal controllability observability inlocal signals and location of transfer function zerosrdquo IEEETransactions on Power Systems vol 21 no 1 pp 285ndash294 2006

[13] N Hingorani and L Gyugyi Understanding FACTS Conceptsand Technology of Flexible AC Transmission Systems JohnWileyamp Sons Hoboken NJ USA 2000

[14] K Tang and G K Venayagamoorthy ldquoDamping inter-areaoscillations using virtual generator based power system stabi-lizerrdquoElectric Power Systems Research vol 129 pp 126ndash141 2015

[15] S Gerbex R Cherkaoui and A J Germond ldquoOptimal locationof multi-type FACTS devices in a power system by means ofgenetic algorithmsrdquo IEEE Transactions on Power Systems vol16 no 3 pp 537ndash544 2001

[16] B Chaudhuri and B C Pal ldquoRobust damping of multiple swingmodes employing global stabilizing signals with a TCSCrdquo IEEETransactions on Power Systems vol 19 no 1 pp 499ndash506 2004

[17] A Rezaee Jordehi ldquoParticle swarm optimisation for dynamicoptimisation problems a reviewrdquoNeural Computing and Appli-cations vol 25 no 7-8 pp 1507ndash1516 2014

[18] A Rezaee Jordehi and J Jasni ldquoParameter selection in particleswarm optimisation a surveyrdquo Journal of Experimental ampTheoretical Artificial Intelligence vol 25 no 4 pp 527ndash542 2013

[19] E Ghahremani and I Kamwa ldquoOptimal placement of multiple-type FACTS devices to maximize power system loadabilityusing a generic graphical user interfacerdquo IEEE Transactions onPower Systems vol 28 no 2 pp 764ndash778 2013

[20] L Rocha R Castro and J M F de Jesus ldquoAn improved particleswarm optimization algorithm for optimal placement andsizing of STATCOMrdquo International Transactions on ElectricalEnergy Systems vol 26 no 4 pp 825ndash840 2015

[21] R Benabid M Boudour and M A Abido ldquoOptimal locationand setting of SVC and TCSC devices using non-dominatedsorting particle swarm optimizationrdquo Electric Power SystemsResearch vol 79 no 12 pp 1668ndash1677 2009

[22] K Ravi and M Rajaram ldquoOptimal location of FACTS devicesusing improved particle swarm optimizationrdquo InternationalJournal of Electrical Power amp Energy Systems vol 49 no 1 pp333ndash338 2013

[23] M Saravanan S M R Slochanal P Venkatesh and J P SAbraham ldquoApplication of particle swarm optimization tech-nique for optimal location of FACTS devices considering costof installation and system loadabilityrdquo Electric Power SystemsResearch vol 77 no 3-4 pp 276ndash283 2007

[24] D P Rini SM Shamsuddin and S S Yuhaniz ldquoParticle swarmoptimization technique system and challengesrdquo InternationalJournal of Computer Applications vol 14 no 1 pp 19ndash26 2011

[25] L Ippolito and P Siano ldquoSelection of optimal number andlocation of thyristor-controlled phase shifters using geneticbased algorithmsrdquo IEE Proceedings-Generation Transmissionand Distribution vol 151 no 5 pp 630ndash637 2004

[26] E S Ali and S M Abd-Elazim ldquoStability improvement ofmultimachine power systemvia new coordinated design of PSSsand SVCrdquo Complexity vol 21 no 2 pp 256ndash266 2015

[27] A E Dahej S Esmaeili and A Goroohi ldquoOptimal allocationof SVC and TCSC for improving voltage stability and reducingpower system losses using hybrid binary genetic algorithm andparticle swarm optimizationrdquo Canadian Journal on Electricaland Electronics Engineering vol 3 no 3 pp 100ndash107 2012

[28] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[29] B K Kumar S N Singh and S C Srivastava ldquoPlacement ofFACTS controllers using modal controllability indices to dampout power system oscillationsrdquo IET Generation Transmission ampDistribution vol 1 no 2 article 209 2007

[30] P S Vaidya and V P Rajderkar ldquoOptimal location of seriesFACTS devices for enhancing power system securityrdquo in Pro-ceedings of the 4th International Conference on Emerging Trendsin Engineering and Technology (ICETET rsquo11) pp 185ndash190 PortLouis Mauritius November 2011

[31] K S Verma S N Singh andHO Gupta ldquoFACTS devices loca-tion for enhancement of total transfer capabilityrdquo in Proceedingsof the IEEE Power Engineering SocietyWinterMeeting vol 3 pp522ndash527 Columbus Ohio USA February 2001

[32] L Rouco and F L Pagola ldquoAn eigenvalue sensitivity approach tolocation and controller design of controllable series capacitorsfor damping power system oscillationsrdquo IEEE Transactions onPower Systems vol 12 no 4 pp 1660ndash1666 1997

[33] A Nasri R Eriksson and M Ghandhari ldquoUsing trajectorysensitivity analysis to find suitable locations of series compen-sators for improving rotor angle stabilityrdquoElectric Power SystemsResearch vol 111 pp 1ndash8 2014

[34] V Le X Li Y Li Y Cao and C Le ldquoOptimal placement ofTCSC using controllability Gramian to damp power systemoscillationsrdquo International Transactions on Electrical EnergySystems vol 26 no 7 pp 1493ndash1510 2015

[35] L Van Dai X Li P Li and L C Quyen ldquoAn optimal locationof static VAr compensator based on Gramian critical energy fordamping oscillations in power systemsrdquo IEEJ Transactions onElectrical and Electronic Engineering vol 11 no 5 pp 577ndash5852016

[36] K Glover ldquoAll optimal Hankel-norm approximations of linearmultivariable systems and their Linfin-error boundsrdquo InternationalJournal of Control vol 39 no 6 pp 1115ndash1193 1984

[37] S N Singh and A K David ldquoA new approach for placement ofFACTS devices in open power marketsrdquo IEEE Power Engineer-ing Review vol 21 no 9 pp 58ndash60 2001

[38] M A Abido ldquoPole placement technique for PSS and TCSC-based stabilizer design using simulated annealingrdquo Interna-tional Journal of Electrical Power amp Energy System vol 22 no8 pp 543ndash554 2000

[39] Prime Minister of Vietnam ldquoApproval of the National MasterPlan for power development for the 2011ndash2020 period withvisions extended 2030rdquo Decision 428QETH-TTG 2016

[40] PW Sauer andM A Pai Power SystemDynamics and StabilityPrentice-Hall Upper Saddle River NJ USA 1998

Complexity 21

[41] D Chatterjee and A Ghosh ldquoTransient stability assessment ofpower systems containing series and shunt compensatorsrdquo IEEETransactions on Power Systems vol 22 no 3 pp 1210ndash12202007

[42] A B Leirbukt ldquoDamping control design based on time-domainidentified modelsrdquo IEEE Transactions on Power Systems vol 14no 1 pp 172ndash178 1999

[43] S Nagalakshmi and N Kamaraj ldquoSecured loadability enhance-ment with TCSC in transmission system using computationalintelligence techniques for pool and hybridmodelrdquoApplied SoftComputing Journal vol 11 no 8 pp 4748ndash4756 2011

[44] C R Fuerte-Esquivel E Acha and H Ambriz-Perez ldquoAthyristor controlled series compensator model for the powerflow solution of practical power networksrdquo IEEE Transactionson Power Systems vol 15 no 1 pp 58ndash64 2000

[45] Y Xiao Y H Song and Y Z Sun ldquoPower flow controlapproach to power systems with embedded FACTS devicesrdquoIEEE Transactions on Power Systems vol 17 no 4 pp 943ndash9502002

[46] D Georges ldquoUse of observability and controllability gramiansor functions for optimal sensor and actuator location in finite-dimensional systemsrdquo in Proceedings of the 34th IEEE Confer-ence on Decision and Control pp 3319ndash3324 December 1995

[47] G Obinata and B D O AndersonModel Reduction for ControlSystem Design Springer London UK 2001

[48] M A Wicks and R A DeCarlo ldquoAn energy approach tocontrollabilityrdquo in Proceedings of the 27th IEEE Conference onDecision and Control pp 2072ndash2077 December 1988

[49] A Ishchenko J M A Myrzik and W L Kling ldquoDynamicequivalencing of distribution networks with dispersed gen-eration using Hankel norm approximationrdquo IET GenerationTransmission amp Distribution vol 1 no 5 pp 818ndash825 2007

[50] Siemens PTI PSSE 302 Program Operational Manual vol 22005

[51] H Shayeghi H A Shayanfar S Jalilzadeh and A SafarildquoTCSC robust damping controller design based on particleswarm optimization for amulti-machine power systemrdquo EnergyConversion andManagement vol 51 no 10 pp 1873ndash1882 2010

Submit your manuscripts athttpswwwhindawicom

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

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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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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

2 Complexity

4000

4200

4400

4600

Activ

e pow

er (M

W)

10 20 30 40 50 60 70 80 900Time (sec)

(a)

4000

4200

4400

4600

Activ

e pow

er (M

W)

0 10 20 30Time (sec) 40 50 60 70 80 90

(b)

Figure 1 The event of the small-signal instability during the blackout WSCC system on Aug 10 1996 (a) the observed California OregonInterconnections power (Dittmer control center) (b) the simulated California Oregon Interconnections power (initial base case)

the measured and simulated power swings of this event Themeasured data show that the standard planningmodels couldbe unreliable predictors of oscillatory behavior

In recent years the economic tempo in the Vietnamhas been developed rapidly the total load has increasedcontinuously Accordingly the Vietnamese power system hasbeen faced with the serious power system blackouts such ason Dec 27 2006 July 20 2007 Apr 09 2007 and the latestevent on May 22 2013 All of the technical problems that theVietnamese power system identified after these events hadalready been reported [7 8]

The power system oscillations occur in the power systemsbecause of the contingencies such as the grid faults andsudden load changes the dampening of these oscillationsis necessary for a secure system operation If the controlledsystems react quickly against faults the power system stabilitywill enhance significantly The advanced power electronicshas led to a new design called flexible alternating-currenttransmission systems (FACTS) by Electrical Power ResearchInstitute (EPRI) The FACTS devices make more use of theexiting capacities in the power system and enhance thepower system stability For example the parameters in thepower system are controlled and the load flow is modifiedto preclude the overload of transmission lines after the gridfaults The FACTS devices are widely used to improve theefficiency of power system operation However the benefitsderived from FACTS controllers such as the small-signalstability and transient stability that depend greatly on theiroptimal placement in the power systems [9 10] Thereforelooking for the optimal placement of FACTS devices inthe large-scale power systems is an interesting researchtopic

State of the Science The damping of electromechanical oscil-lations in the power systems is a matter to belong with powerengineers and become one of the most interesting researchtopics [11 12] The PSSs are a profound influence solutionon improving the power system stability This solution isbased on the excitation system and the generator speeddeviation signals in which the speed deviation is used asthe supplementary control signals in order to provide to the

automatic voltage regulators The fast progress in the fieldof power electronics had contributed to the developmentof the electrical power industry based on the controllableutilization of FACTS [13] This device can dampen both thelocal- and interarea oscillations [10] whereas PSSs could onlydampen the local-area oscillations [14] Thus FACTS is thebest choice to enhance the stability margin of the existingpower systems [9 15] and especially to TCSC This deviceis considered as one of the most effective FACTS and iswidely used to regulate the power flow in transmission linesdampen the power oscillations mitigate subsynchronousresonance (SSR) enhance power system stability and so forthby changing the reactance of transmission line [16] There ishard sledding for the researchers regarding how to determineoptimal location of FACTS

The methods for solving location problems can be clas-sified into two categories (i) analytical techniques and (ii)heuristic optimization approaches [17ndash19] Among heuristicsParticle Swarm Optimization (PSO) has some advantages asless control parameters no require preconditions (continuityor differentiability of objective functions) and slow com-putational burden which make it popular to solve optimallocation and design problems of FACTS [20ndash24] HoweverPSO had drawback as slow convergence in search stage andlimit of local search ability furthermore the algorithm willnot work out the problems of scattering and optimization[15] Genetic algorithm (GA) has been proposed to solveoptimal location and design problems of FACTS [9 15 19 25]This method has been applied to obtain promising resultsHowever downside of GA is that the requested run timeis very long when studying the large-scale systems [26] In[24] optimal location of TCSC is determined via PSO whoseobjective is to maximize small-signal stability In [9 23] amathematical objective function is used to derive objectivefunction for GA and PSO by considering the load-ability inthe system security margins respectively In [27] the authorsused a method based on hybrid between PSO and GA foroptimal allocation of TCSC to enhance voltage stability andreduce power system losses The brainstorm optimizationalgorithm (BSOA) is new heuristic optimization algorithmproposed by the authors in [28] to seek the optimal location of

Complexity 3












4 Complexity

LongPhu

0 + j0

2942 + j1279

2942 + j1278

minus606 minus j938

2303 + j168

minus5385 + j798

359 + j1297

1295

minus j2

973

8299 + j2641

5642 + j125

227 + j305 5534 minus j62

1200 + j56

2400 + j9876

4955 minus j687

900 + j3765

600 + j1594

1000 + j132

1860 + j1080600 + j1836

600 + j2988

764 minus j1175

720 + j3788

335 + j169

4812 + j4836924 minus j556

5657 + j03

720 + j888

5367 minus j1454

3352 minus j142

1492 minus j314 2446 + j82

2402 + j1358

2276 minus j725

2822 minus j123182 + j2722

13816 + j6534

319 + j5338

754 minus j2314

11446 minus j3809

7096 minus j3679

13711 minus j7312

281 + j1917

10167 minus j2432

1079

4 minus

j520

4574 minus j2212

13056 + j1896

583 + j124

10594 + j353

1546 minus j3265371 minus j5563

5493 + j3645

6528 + j3692

8812 + j1088

8416 + j303

646

9 + j1

266

4042 + j1491

8728 + j2066

855 + j2012

8444 + j4532

15564 + j2107561 + j623

730 + j516 minus932 + j16

6264 + j1372

831 + j1152

14766 + j2381

999

6 + j4

52

DakNong

ThanhMy

DocSoi

DucHoa

PhuLamMyTho

Pleiku

CauBong NhaBe

OMonDuyenHai

PhuMy

SongMay

DiLinh

DaNang

TanDinh

ThuongTin

PhoNoi

VungAng

HoaBinh

NhoQuanPitoong

VietTriHiepHoa

ThangLongQuangNinh

MongDuongSonLa

2729 + j1124

1168 minus j47

8255 minus j5352

1079

4 minus

j520

204 + j2716HaTinh

7219 minus j1846

7219 minus j1846

598 minus j923

598 minus j923

1028

2 minus

j97

8

minus8 minus j1227

1200 minus j1458

1022 + j345

1215

minus j4

28


VinhTan

519 + j741

855 minus j423

5083 kV

5022 kV

4924 kV

5114 kV

5104 kV

5021 kV

5008 kV

5129 kV

5132 kV4993 kV

4923 kV

5094 kV4888 kV

491 kV

5088 kV

503 kV

5059 kV

5167 kV

504 kV

5019 kV

518 kV

5018 kV

5033 kV522 kV

5046 kV

5071 kV

5052 kV 5027 kV

5138 kV

5122 kV

5019 kV

5009 kV






119894 = 1 119898 119895 = 1 119899

(2)

where













Complexity 5

Thyristors

TCSC

Controller


Ikt Itk

L

C

Zline

(a)

Ikt ItkminusjXtcsc

Xnew_line

Rline + jXline

(b)

Inductive region

Capacitive region

1284

90 110 120 180100 150 160 170130 140

120572

minus10

minus5

0

5

10

15

Xtcsc

(c)



[[[[

Δsys00]]]]= [[[[


Δ119883sysΔ119910sys1Δ119910sys2]]]]

+ [[[119864sys00]]]Δ119880sys

(3)


1015840sys11198621015840sys2]


(4)

119864sys = [[[119864sys100]]] (5)



where



(7)



6 Complexity

1205900

PlineK

1

1 + sT1

sTw

1 + sTw

1 + sT2

1 + sT3

++

Δ120590Xmin

Xmax

1

1 + sTtcsc

OutputXtcsc








= 119883119862 + 119883119871119862120587 1198702 = 41198832119871119862120587119883119871

(9)





(10)




(11)




(12)




(13)

Complexity 7


[[[[[[

Δsys

Δtcsc00

]]]]]]

=[[[[[[[


]]]]]]]

[[[[[[


]]]]]]

+ [[[[[[

119864sys-tcsc000

]]]]]]Δ119880sys-tcsc

(14)



]




119864new = [119864sys-tcsc 0 0 0]T

(15)





(119905) = 119860119909 (119905) + 119861119906 (119905) 119910 (119905) = 119862119909 (119905) + 119863119906 (119905) (17)

where


311 Controllability




int0minus119879119906 (119905)2 119889119905

subject to (119905) = 119860119909 (119905) + 119861119906 (119905) 119909 (minus119879) = 0119909 (0) = 1199090

(18)


119871119888 (1199090) = 12119909T0119866minus1119888 (119879) 1199090 (19)





8 Complexity


119866119888 (119879) = int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905 (21)




119871119900 (1199090) = 12 int0

minus119879

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905subject to 119909 (0) = 1199090

119906 equiv 0(22)


119871119900 (1199090) = 12119909T0119866minus1119900 1199090 (23)



nabla =[[[[[[[[[[[

1198621198621198601198621198602

119862119860119899minus1

]]]]]]]]]]]

(24)


119866119900 (119879) = int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905 (25)



12 int0

minusinfin119906 (119905)2 119889119905

119871119900 = 12 intinfin

0

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905 119909 (0) = 1199090 119906 equiv 0(26)



int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905


int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905

(27)


119889119866119888 (0 119905)119889119905 = 119860119866119888 (0 119905) + 119866119888 (0 119905) 119860T + 119861119861T119889119866119900 (0 119905)119889119905 = 119860T119866119900 (0 119905) + 119866119900 (0 119905) 119860 + 119862T119862

(28)


119860119866119888 + 119866119888119860T + 119861119861T = 0119860119879119866119900 + 119866119900119860 + 119862T119862 = 0 (29)




(30)

Complexity 9



0119909 (119905) 119909T (119905) 119889119905 (31)


119864 = trace (119866119888) = 119899sum119894=1

120590119894 (32)




119866119888 = 119866119900 =[[[[[[[


0 0 0 sdot sdot sdot 120590119899

]]]]]]]


(33)


119866119888 = 119872119866119888119872T119866119900 = (119872minus1)T 119866119888119872minus1 (34)



(119905) = 119860119891 (119905) + 119861119906 (119905) 119910 (119905) = 119862119891 (119905) + 119863119906 (119905) (35)







1 (119905) = 119860111198911 (119905) + 119860121198912 (119905) + 1198611119906 (119905)2 (119905) = 119860211198911 (119905) + 119860221198912 (119905) + 1198612119906 (119905) 119910 (119905) = 11986211198911 (119905) + 11986221198912 (119905) + 119863119906 (119905)

119866119888 = 119866119900 = Σ(38)


10 Complexity




1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)


1198612 + 119862T2119880 = 0 (40)


1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)



119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)



First Start






Finally End

4 Case Study


Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 3












4 Complexity

LongPhu

0 + j0

2942 + j1279

2942 + j1278

minus606 minus j938

2303 + j168

minus5385 + j798

359 + j1297

1295

minus j2

973

8299 + j2641

5642 + j125

227 + j305 5534 minus j62

1200 + j56

2400 + j9876

4955 minus j687

900 + j3765

600 + j1594

1000 + j132

1860 + j1080600 + j1836

600 + j2988

764 minus j1175

720 + j3788

335 + j169

4812 + j4836924 minus j556

5657 + j03

720 + j888

5367 minus j1454

3352 minus j142

1492 minus j314 2446 + j82

2402 + j1358

2276 minus j725

2822 minus j123182 + j2722

13816 + j6534

319 + j5338

754 minus j2314

11446 minus j3809

7096 minus j3679

13711 minus j7312

281 + j1917

10167 minus j2432

1079

4 minus

j520

4574 minus j2212

13056 + j1896

583 + j124

10594 + j353

1546 minus j3265371 minus j5563

5493 + j3645

6528 + j3692

8812 + j1088

8416 + j303

646

9 + j1

266

4042 + j1491

8728 + j2066

855 + j2012

8444 + j4532

15564 + j2107561 + j623

730 + j516 minus932 + j16

6264 + j1372

831 + j1152

14766 + j2381

999

6 + j4

52

DakNong

ThanhMy

DocSoi

DucHoa

PhuLamMyTho

Pleiku

CauBong NhaBe

OMonDuyenHai

PhuMy

SongMay

DiLinh

DaNang

TanDinh

ThuongTin

PhoNoi

VungAng

HoaBinh

NhoQuanPitoong

VietTriHiepHoa

ThangLongQuangNinh

MongDuongSonLa

2729 + j1124

1168 minus j47

8255 minus j5352

1079

4 minus

j520

204 + j2716HaTinh

7219 minus j1846

7219 minus j1846

598 minus j923

598 minus j923

1028

2 minus

j97

8

minus8 minus j1227

1200 minus j1458

1022 + j345

1215

minus j4

28


VinhTan

519 + j741

855 minus j423

5083 kV

5022 kV

4924 kV

5114 kV

5104 kV

5021 kV

5008 kV

5129 kV

5132 kV4993 kV

4923 kV

5094 kV4888 kV

491 kV

5088 kV

503 kV

5059 kV

5167 kV

504 kV

5019 kV

518 kV

5018 kV

5033 kV522 kV

5046 kV

5071 kV

5052 kV 5027 kV

5138 kV

5122 kV

5019 kV

5009 kV






119894 = 1 119898 119895 = 1 119899

(2)

where













Complexity 5

Thyristors

TCSC

Controller


Ikt Itk

L

C

Zline

(a)

Ikt ItkminusjXtcsc

Xnew_line

Rline + jXline

(b)

Inductive region

Capacitive region

1284

90 110 120 180100 150 160 170130 140

120572

minus10

minus5

0

5

10

15

Xtcsc

(c)



[[[[

Δsys00]]]]= [[[[


Δ119883sysΔ119910sys1Δ119910sys2]]]]

+ [[[119864sys00]]]Δ119880sys

(3)


1015840sys11198621015840sys2]


(4)

119864sys = [[[119864sys100]]] (5)



where



(7)



6 Complexity

1205900

PlineK

1

1 + sT1

sTw

1 + sTw

1 + sT2

1 + sT3

++

Δ120590Xmin

Xmax

1

1 + sTtcsc

OutputXtcsc








= 119883119862 + 119883119871119862120587 1198702 = 41198832119871119862120587119883119871

(9)





(10)




(11)




(12)




(13)

Complexity 7


[[[[[[

Δsys

Δtcsc00

]]]]]]

=[[[[[[[


]]]]]]]

[[[[[[


]]]]]]

+ [[[[[[

119864sys-tcsc000

]]]]]]Δ119880sys-tcsc

(14)



]




119864new = [119864sys-tcsc 0 0 0]T

(15)





(119905) = 119860119909 (119905) + 119861119906 (119905) 119910 (119905) = 119862119909 (119905) + 119863119906 (119905) (17)

where


311 Controllability




int0minus119879119906 (119905)2 119889119905

subject to (119905) = 119860119909 (119905) + 119861119906 (119905) 119909 (minus119879) = 0119909 (0) = 1199090

(18)


119871119888 (1199090) = 12119909T0119866minus1119888 (119879) 1199090 (19)





8 Complexity


119866119888 (119879) = int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905 (21)




119871119900 (1199090) = 12 int0

minus119879

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905subject to 119909 (0) = 1199090

119906 equiv 0(22)


119871119900 (1199090) = 12119909T0119866minus1119900 1199090 (23)



nabla =[[[[[[[[[[[

1198621198621198601198621198602

119862119860119899minus1

]]]]]]]]]]]

(24)


119866119900 (119879) = int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905 (25)



12 int0

minusinfin119906 (119905)2 119889119905

119871119900 = 12 intinfin

0

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905 119909 (0) = 1199090 119906 equiv 0(26)



int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905


int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905

(27)


119889119866119888 (0 119905)119889119905 = 119860119866119888 (0 119905) + 119866119888 (0 119905) 119860T + 119861119861T119889119866119900 (0 119905)119889119905 = 119860T119866119900 (0 119905) + 119866119900 (0 119905) 119860 + 119862T119862

(28)


119860119866119888 + 119866119888119860T + 119861119861T = 0119860119879119866119900 + 119866119900119860 + 119862T119862 = 0 (29)




(30)

Complexity 9



0119909 (119905) 119909T (119905) 119889119905 (31)


119864 = trace (119866119888) = 119899sum119894=1

120590119894 (32)




119866119888 = 119866119900 =[[[[[[[


0 0 0 sdot sdot sdot 120590119899

]]]]]]]


(33)


119866119888 = 119872119866119888119872T119866119900 = (119872minus1)T 119866119888119872minus1 (34)



(119905) = 119860119891 (119905) + 119861119906 (119905) 119910 (119905) = 119862119891 (119905) + 119863119906 (119905) (35)







1 (119905) = 119860111198911 (119905) + 119860121198912 (119905) + 1198611119906 (119905)2 (119905) = 119860211198911 (119905) + 119860221198912 (119905) + 1198612119906 (119905) 119910 (119905) = 11986211198911 (119905) + 11986221198912 (119905) + 119863119906 (119905)

119866119888 = 119866119900 = Σ(38)


10 Complexity




1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)


1198612 + 119862T2119880 = 0 (40)


1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)



119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)



First Start






Finally End

4 Case Study


Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 Complexity

LongPhu

0 + j0

2942 + j1279

2942 + j1278

minus606 minus j938

2303 + j168

minus5385 + j798

359 + j1297

1295

minus j2

973

8299 + j2641

5642 + j125

227 + j305 5534 minus j62

1200 + j56

2400 + j9876

4955 minus j687

900 + j3765

600 + j1594

1000 + j132

1860 + j1080600 + j1836

600 + j2988

764 minus j1175

720 + j3788

335 + j169

4812 + j4836924 minus j556

5657 + j03

720 + j888

5367 minus j1454

3352 minus j142

1492 minus j314 2446 + j82

2402 + j1358

2276 minus j725

2822 minus j123182 + j2722

13816 + j6534

319 + j5338

754 minus j2314

11446 minus j3809

7096 minus j3679

13711 minus j7312

281 + j1917

10167 minus j2432

1079

4 minus

j520

4574 minus j2212

13056 + j1896

583 + j124

10594 + j353

1546 minus j3265371 minus j5563

5493 + j3645

6528 + j3692

8812 + j1088

8416 + j303

646

9 + j1

266

4042 + j1491

8728 + j2066

855 + j2012

8444 + j4532

15564 + j2107561 + j623

730 + j516 minus932 + j16

6264 + j1372

831 + j1152

14766 + j2381

999

6 + j4

52

DakNong

ThanhMy

DocSoi

DucHoa

PhuLamMyTho

Pleiku

CauBong NhaBe

OMonDuyenHai

PhuMy

SongMay

DiLinh

DaNang

TanDinh

ThuongTin

PhoNoi

VungAng

HoaBinh

NhoQuanPitoong

VietTriHiepHoa

ThangLongQuangNinh

MongDuongSonLa

2729 + j1124

1168 minus j47

8255 minus j5352

1079

4 minus

j520

204 + j2716HaTinh

7219 minus j1846

7219 minus j1846

598 minus j923

598 minus j923

1028

2 minus

j97

8

minus8 minus j1227

1200 minus j1458

1022 + j345

1215

minus j4

28


VinhTan

519 + j741

855 minus j423

5083 kV

5022 kV

4924 kV

5114 kV

5104 kV

5021 kV

5008 kV

5129 kV

5132 kV4993 kV

4923 kV

5094 kV4888 kV

491 kV

5088 kV

503 kV

5059 kV

5167 kV

504 kV

5019 kV

518 kV

5018 kV

5033 kV522 kV

5046 kV

5071 kV

5052 kV 5027 kV

5138 kV

5122 kV

5019 kV

5009 kV






119894 = 1 119898 119895 = 1 119899

(2)

where













Complexity 5

Thyristors

TCSC

Controller


Ikt Itk

L

C

Zline

(a)

Ikt ItkminusjXtcsc

Xnew_line

Rline + jXline

(b)

Inductive region

Capacitive region

1284

90 110 120 180100 150 160 170130 140

120572

minus10

minus5

0

5

10

15

Xtcsc

(c)



[[[[

Δsys00]]]]= [[[[


Δ119883sysΔ119910sys1Δ119910sys2]]]]

+ [[[119864sys00]]]Δ119880sys

(3)


1015840sys11198621015840sys2]


(4)

119864sys = [[[119864sys100]]] (5)



where



(7)



6 Complexity

1205900

PlineK

1

1 + sT1

sTw

1 + sTw

1 + sT2

1 + sT3

++

Δ120590Xmin

Xmax

1

1 + sTtcsc

OutputXtcsc








= 119883119862 + 119883119871119862120587 1198702 = 41198832119871119862120587119883119871

(9)





(10)




(11)




(12)




(13)

Complexity 7


[[[[[[

Δsys

Δtcsc00

]]]]]]

=[[[[[[[


]]]]]]]

[[[[[[


]]]]]]

+ [[[[[[

119864sys-tcsc000

]]]]]]Δ119880sys-tcsc

(14)



]




119864new = [119864sys-tcsc 0 0 0]T

(15)





(119905) = 119860119909 (119905) + 119861119906 (119905) 119910 (119905) = 119862119909 (119905) + 119863119906 (119905) (17)

where


311 Controllability




int0minus119879119906 (119905)2 119889119905

subject to (119905) = 119860119909 (119905) + 119861119906 (119905) 119909 (minus119879) = 0119909 (0) = 1199090

(18)


119871119888 (1199090) = 12119909T0119866minus1119888 (119879) 1199090 (19)





8 Complexity


119866119888 (119879) = int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905 (21)




119871119900 (1199090) = 12 int0

minus119879

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905subject to 119909 (0) = 1199090

119906 equiv 0(22)


119871119900 (1199090) = 12119909T0119866minus1119900 1199090 (23)



nabla =[[[[[[[[[[[

1198621198621198601198621198602

119862119860119899minus1

]]]]]]]]]]]

(24)


119866119900 (119879) = int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905 (25)



12 int0

minusinfin119906 (119905)2 119889119905

119871119900 = 12 intinfin

0

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905 119909 (0) = 1199090 119906 equiv 0(26)



int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905


int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905

(27)


119889119866119888 (0 119905)119889119905 = 119860119866119888 (0 119905) + 119866119888 (0 119905) 119860T + 119861119861T119889119866119900 (0 119905)119889119905 = 119860T119866119900 (0 119905) + 119866119900 (0 119905) 119860 + 119862T119862

(28)


119860119866119888 + 119866119888119860T + 119861119861T = 0119860119879119866119900 + 119866119900119860 + 119862T119862 = 0 (29)




(30)

Complexity 9



0119909 (119905) 119909T (119905) 119889119905 (31)


119864 = trace (119866119888) = 119899sum119894=1

120590119894 (32)




119866119888 = 119866119900 =[[[[[[[


0 0 0 sdot sdot sdot 120590119899

]]]]]]]


(33)


119866119888 = 119872119866119888119872T119866119900 = (119872minus1)T 119866119888119872minus1 (34)



(119905) = 119860119891 (119905) + 119861119906 (119905) 119910 (119905) = 119862119891 (119905) + 119863119906 (119905) (35)







1 (119905) = 119860111198911 (119905) + 119860121198912 (119905) + 1198611119906 (119905)2 (119905) = 119860211198911 (119905) + 119860221198912 (119905) + 1198612119906 (119905) 119910 (119905) = 11986211198911 (119905) + 11986221198912 (119905) + 119863119906 (119905)

119866119888 = 119866119900 = Σ(38)


10 Complexity




1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)


1198612 + 119862T2119880 = 0 (40)


1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)



119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)



First Start






Finally End

4 Case Study


Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 5

Thyristors

TCSC

Controller


Ikt Itk

L

C

Zline

(a)

Ikt ItkminusjXtcsc

Xnew_line

Rline + jXline

(b)

Inductive region

Capacitive region

1284

90 110 120 180100 150 160 170130 140

120572

minus10

minus5

0

5

10

15

Xtcsc

(c)



[[[[

Δsys00]]]]= [[[[


Δ119883sysΔ119910sys1Δ119910sys2]]]]

+ [[[119864sys00]]]Δ119880sys

(3)


1015840sys11198621015840sys2]


(4)

119864sys = [[[119864sys100]]] (5)



where



(7)



6 Complexity

1205900

PlineK

1

1 + sT1

sTw

1 + sTw

1 + sT2

1 + sT3

++

Δ120590Xmin

Xmax

1

1 + sTtcsc

OutputXtcsc








= 119883119862 + 119883119871119862120587 1198702 = 41198832119871119862120587119883119871

(9)





(10)




(11)




(12)




(13)

Complexity 7


[[[[[[

Δsys

Δtcsc00

]]]]]]

=[[[[[[[


]]]]]]]

[[[[[[


]]]]]]

+ [[[[[[

119864sys-tcsc000

]]]]]]Δ119880sys-tcsc

(14)



]




119864new = [119864sys-tcsc 0 0 0]T

(15)





(119905) = 119860119909 (119905) + 119861119906 (119905) 119910 (119905) = 119862119909 (119905) + 119863119906 (119905) (17)

where


311 Controllability




int0minus119879119906 (119905)2 119889119905

subject to (119905) = 119860119909 (119905) + 119861119906 (119905) 119909 (minus119879) = 0119909 (0) = 1199090

(18)


119871119888 (1199090) = 12119909T0119866minus1119888 (119879) 1199090 (19)





8 Complexity


119866119888 (119879) = int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905 (21)




119871119900 (1199090) = 12 int0

minus119879

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905subject to 119909 (0) = 1199090

119906 equiv 0(22)


119871119900 (1199090) = 12119909T0119866minus1119900 1199090 (23)



nabla =[[[[[[[[[[[

1198621198621198601198621198602

119862119860119899minus1

]]]]]]]]]]]

(24)


119866119900 (119879) = int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905 (25)



12 int0

minusinfin119906 (119905)2 119889119905

119871119900 = 12 intinfin

0

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905 119909 (0) = 1199090 119906 equiv 0(26)



int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905


int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905

(27)


119889119866119888 (0 119905)119889119905 = 119860119866119888 (0 119905) + 119866119888 (0 119905) 119860T + 119861119861T119889119866119900 (0 119905)119889119905 = 119860T119866119900 (0 119905) + 119866119900 (0 119905) 119860 + 119862T119862

(28)


119860119866119888 + 119866119888119860T + 119861119861T = 0119860119879119866119900 + 119866119900119860 + 119862T119862 = 0 (29)




(30)

Complexity 9



0119909 (119905) 119909T (119905) 119889119905 (31)


119864 = trace (119866119888) = 119899sum119894=1

120590119894 (32)




119866119888 = 119866119900 =[[[[[[[


0 0 0 sdot sdot sdot 120590119899

]]]]]]]


(33)


119866119888 = 119872119866119888119872T119866119900 = (119872minus1)T 119866119888119872minus1 (34)



(119905) = 119860119891 (119905) + 119861119906 (119905) 119910 (119905) = 119862119891 (119905) + 119863119906 (119905) (35)







1 (119905) = 119860111198911 (119905) + 119860121198912 (119905) + 1198611119906 (119905)2 (119905) = 119860211198911 (119905) + 119860221198912 (119905) + 1198612119906 (119905) 119910 (119905) = 11986211198911 (119905) + 11986221198912 (119905) + 119863119906 (119905)

119866119888 = 119866119900 = Σ(38)


10 Complexity




1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)


1198612 + 119862T2119880 = 0 (40)


1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)



119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)



First Start






Finally End

4 Case Study


Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









6 Complexity

1205900

PlineK

1

1 + sT1

sTw

1 + sTw

1 + sT2

1 + sT3

++

Δ120590Xmin

Xmax

1

1 + sTtcsc

OutputXtcsc








= 119883119862 + 119883119871119862120587 1198702 = 41198832119871119862120587119883119871

(9)





(10)




(11)




(12)




(13)

Complexity 7


[[[[[[

Δsys

Δtcsc00

]]]]]]

=[[[[[[[


]]]]]]]

[[[[[[


]]]]]]

+ [[[[[[

119864sys-tcsc000

]]]]]]Δ119880sys-tcsc

(14)



]




119864new = [119864sys-tcsc 0 0 0]T

(15)





(119905) = 119860119909 (119905) + 119861119906 (119905) 119910 (119905) = 119862119909 (119905) + 119863119906 (119905) (17)

where


311 Controllability




int0minus119879119906 (119905)2 119889119905

subject to (119905) = 119860119909 (119905) + 119861119906 (119905) 119909 (minus119879) = 0119909 (0) = 1199090

(18)


119871119888 (1199090) = 12119909T0119866minus1119888 (119879) 1199090 (19)





8 Complexity


119866119888 (119879) = int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905 (21)




119871119900 (1199090) = 12 int0

minus119879

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905subject to 119909 (0) = 1199090

119906 equiv 0(22)


119871119900 (1199090) = 12119909T0119866minus1119900 1199090 (23)



nabla =[[[[[[[[[[[

1198621198621198601198621198602

119862119860119899minus1

]]]]]]]]]]]

(24)


119866119900 (119879) = int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905 (25)



12 int0

minusinfin119906 (119905)2 119889119905

119871119900 = 12 intinfin

0

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905 119909 (0) = 1199090 119906 equiv 0(26)



int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905


int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905

(27)


119889119866119888 (0 119905)119889119905 = 119860119866119888 (0 119905) + 119866119888 (0 119905) 119860T + 119861119861T119889119866119900 (0 119905)119889119905 = 119860T119866119900 (0 119905) + 119866119900 (0 119905) 119860 + 119862T119862

(28)


119860119866119888 + 119866119888119860T + 119861119861T = 0119860119879119866119900 + 119866119900119860 + 119862T119862 = 0 (29)




(30)

Complexity 9



0119909 (119905) 119909T (119905) 119889119905 (31)


119864 = trace (119866119888) = 119899sum119894=1

120590119894 (32)




119866119888 = 119866119900 =[[[[[[[


0 0 0 sdot sdot sdot 120590119899

]]]]]]]


(33)


119866119888 = 119872119866119888119872T119866119900 = (119872minus1)T 119866119888119872minus1 (34)



(119905) = 119860119891 (119905) + 119861119906 (119905) 119910 (119905) = 119862119891 (119905) + 119863119906 (119905) (35)







1 (119905) = 119860111198911 (119905) + 119860121198912 (119905) + 1198611119906 (119905)2 (119905) = 119860211198911 (119905) + 119860221198912 (119905) + 1198612119906 (119905) 119910 (119905) = 11986211198911 (119905) + 11986221198912 (119905) + 119863119906 (119905)

119866119888 = 119866119900 = Σ(38)


10 Complexity




1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)


1198612 + 119862T2119880 = 0 (40)


1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)



119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)



First Start






Finally End

4 Case Study


Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 7


[[[[[[

Δsys

Δtcsc00

]]]]]]

=[[[[[[[


]]]]]]]

[[[[[[


]]]]]]

+ [[[[[[

119864sys-tcsc000

]]]]]]Δ119880sys-tcsc

(14)



]




119864new = [119864sys-tcsc 0 0 0]T

(15)





(119905) = 119860119909 (119905) + 119861119906 (119905) 119910 (119905) = 119862119909 (119905) + 119863119906 (119905) (17)

where


311 Controllability




int0minus119879119906 (119905)2 119889119905

subject to (119905) = 119860119909 (119905) + 119861119906 (119905) 119909 (minus119879) = 0119909 (0) = 1199090

(18)


119871119888 (1199090) = 12119909T0119866minus1119888 (119879) 1199090 (19)





8 Complexity


119866119888 (119879) = int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905 (21)




119871119900 (1199090) = 12 int0

minus119879

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905subject to 119909 (0) = 1199090

119906 equiv 0(22)


119871119900 (1199090) = 12119909T0119866minus1119900 1199090 (23)



nabla =[[[[[[[[[[[

1198621198621198601198621198602

119862119860119899minus1

]]]]]]]]]]]

(24)


119866119900 (119879) = int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905 (25)



12 int0

minusinfin119906 (119905)2 119889119905

119871119900 = 12 intinfin

0

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905 119909 (0) = 1199090 119906 equiv 0(26)



int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905


int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905

(27)


119889119866119888 (0 119905)119889119905 = 119860119866119888 (0 119905) + 119866119888 (0 119905) 119860T + 119861119861T119889119866119900 (0 119905)119889119905 = 119860T119866119900 (0 119905) + 119866119900 (0 119905) 119860 + 119862T119862

(28)


119860119866119888 + 119866119888119860T + 119861119861T = 0119860119879119866119900 + 119866119900119860 + 119862T119862 = 0 (29)




(30)

Complexity 9



0119909 (119905) 119909T (119905) 119889119905 (31)


119864 = trace (119866119888) = 119899sum119894=1

120590119894 (32)




119866119888 = 119866119900 =[[[[[[[


0 0 0 sdot sdot sdot 120590119899

]]]]]]]


(33)


119866119888 = 119872119866119888119872T119866119900 = (119872minus1)T 119866119888119872minus1 (34)



(119905) = 119860119891 (119905) + 119861119906 (119905) 119910 (119905) = 119862119891 (119905) + 119863119906 (119905) (35)







1 (119905) = 119860111198911 (119905) + 119860121198912 (119905) + 1198611119906 (119905)2 (119905) = 119860211198911 (119905) + 119860221198912 (119905) + 1198612119906 (119905) 119910 (119905) = 11986211198911 (119905) + 11986221198912 (119905) + 119863119906 (119905)

119866119888 = 119866119900 = Σ(38)


10 Complexity




1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)


1198612 + 119862T2119880 = 0 (40)


1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)



119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)



First Start






Finally End

4 Case Study


Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









8 Complexity


119866119888 (119879) = int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905 (21)




119871119900 (1199090) = 12 int0

minus119879

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905subject to 119909 (0) = 1199090

119906 equiv 0(22)


119871119900 (1199090) = 12119909T0119866minus1119900 1199090 (23)



nabla =[[[[[[[[[[[

1198621198621198601198621198602

119862119860119899minus1

]]]]]]]]]]]

(24)


119866119900 (119879) = int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905 (25)



12 int0

minusinfin119906 (119905)2 119889119905

119871119900 = 12 intinfin

0

1003817100381710038171003817119910 (119905)10038171003817100381710038172 119889119905 119909 (0) = 1199090 119906 equiv 0(26)



int1198790exp (119860119905) 119861119861T exp (119860T119905) 119889119905


int1198790exp (119860T119905) 119862T119862 exp (119860119905) 119889119905

(27)


119889119866119888 (0 119905)119889119905 = 119860119866119888 (0 119905) + 119866119888 (0 119905) 119860T + 119861119861T119889119866119900 (0 119905)119889119905 = 119860T119866119900 (0 119905) + 119866119900 (0 119905) 119860 + 119862T119862

(28)


119860119866119888 + 119866119888119860T + 119861119861T = 0119860119879119866119900 + 119866119900119860 + 119862T119862 = 0 (29)




(30)

Complexity 9



0119909 (119905) 119909T (119905) 119889119905 (31)


119864 = trace (119866119888) = 119899sum119894=1

120590119894 (32)




119866119888 = 119866119900 =[[[[[[[


0 0 0 sdot sdot sdot 120590119899

]]]]]]]


(33)


119866119888 = 119872119866119888119872T119866119900 = (119872minus1)T 119866119888119872minus1 (34)



(119905) = 119860119891 (119905) + 119861119906 (119905) 119910 (119905) = 119862119891 (119905) + 119863119906 (119905) (35)







1 (119905) = 119860111198911 (119905) + 119860121198912 (119905) + 1198611119906 (119905)2 (119905) = 119860211198911 (119905) + 119860221198912 (119905) + 1198612119906 (119905) 119910 (119905) = 11986211198911 (119905) + 11986221198912 (119905) + 119863119906 (119905)

119866119888 = 119866119900 = Σ(38)


10 Complexity




1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)


1198612 + 119862T2119880 = 0 (40)


1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)



119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)



First Start






Finally End

4 Case Study


Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 9



0119909 (119905) 119909T (119905) 119889119905 (31)


119864 = trace (119866119888) = 119899sum119894=1

120590119894 (32)




119866119888 = 119866119900 =[[[[[[[


0 0 0 sdot sdot sdot 120590119899

]]]]]]]


(33)


119866119888 = 119872119866119888119872T119866119900 = (119872minus1)T 119866119888119872minus1 (34)



(119905) = 119860119891 (119905) + 119861119906 (119905) 119910 (119905) = 119862119891 (119905) + 119863119906 (119905) (35)







1 (119905) = 119860111198911 (119905) + 119860121198912 (119905) + 1198611119906 (119905)2 (119905) = 119860211198911 (119905) + 119860221198912 (119905) + 1198612119906 (119905) 119910 (119905) = 11986211198911 (119905) + 11986221198912 (119905) + 119863119906 (119905)

119866119888 = 119866119900 = Σ(38)


10 Complexity




1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)


1198612 + 119862T2119880 = 0 (40)


1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)



119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)



First Start






Finally End

4 Case Study


Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









10 Complexity




1119880119861T1) = Σminus1 (Σ11198611 + 120590119896+1119862T

1119880) = 1198621Σ1 + 120590119896+1119880119861T1 = 119863 minus 120590119896+1119880

Σ = Σ21 minus 1205902119896+1Ψ119903

(39)


1198612 + 119862T2119880 = 0 (40)


1003817100381710038171003817119879 (119904) minus 119879119896 (119904)1003817100381710038171003817infin = 120590119896+1 (41)



119860119896 = 1015840119861119896 = 1015840119862119896 = 1015840119863119896 = 1015840

(42)



First Start






Finally End

4 Case Study


Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 11

Start

Run steady state





Yes

No

End


Eqs (41) and (42)








line ith


Aij Bij Cij Dij

Eq (39)

Eq (35)

Eq (38)

(Ak)ij lt 0

j + 1

j le m

i + 1

i le n

i = n

j = m


Tk )ij

= 0

E = trace(Gc)ij

maxi=12n

(Ei)

(c) Solve B2 + CT

2 U = 0

(a) Calculate Σ2



T C

T and DT


998400 C

998400 and D

998400

then A998400 B

998400 C

998400 and D


998400T C

998400T and D

998400T

(j 1 divide m)

(i 1 divide n)


12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









12 Complexity


Casenumber




(2) NQ-HT 544 2


(4) VA-DN 64 2















Mag

nitu

de



05

101520


A11



PI = 119899119897sum119894=1

1205921198942120599 ( 119875119897119894119875max119897119894

)2120599 (43)






Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 13






Sing

ular

val

ues (

dB)



Frequency (radsec)

minus80

minus60

minus40

minus20

0

20

40

60

80

100


0

05

1

15

2

25

Mag

nitu

de

times104






14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









14 Complexity


Casenumber




PLei-DS DN-DS PLei-

DLPLei-CB

PLei-DakN

DakN-CB DL-TD






Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 15



Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


0 4 6 8 10 12 14 16 18 202Time (sec)


minus50

Relat

ive a

ngle

(deg

ree)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)


minus505

101520

Relat

ive a

ngle

(deg

ree)

(c)


22225

23235

24245

25255

26265

Relat

ive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









16 Complexity


minus10

minus5

0

5

10

15Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus400

minus200

0

200

400

600

800

1000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


0200400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(c)






Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 17


minus30

minus20

minus10

0

10

20

30

40

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)


minus60

minus50

minus40

minus30

minus20

minus10

0

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus1200

minus1000

minus800

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)


400600800

100012001400160018002000

Activ

e pow

er fl

ow (M

W)

2 4 6 8 10 12 14 16 18 200Time (sec)

(d)


0

001

002

003

004

005

006

TCSC

equi

vale

nt re

acta

nce (

pu

)

2 4 6 8 10 12 14 16 18 200Time (sec)

(e)


18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









18 Complexity


18

20

22

24

26

28

30

32Re

lativ

e ang

le (d

egre

e)

2 4 6 8 10 12 14 16 18 200Time (sec)

(a)



01020304050

Rela

tive a

ngle

(deg

ree)

2 4 6 8 10 12 14 16 18 200Time (sec)

(b)



2 4 6 8 10 12 14 16 18 200Time (sec)

minus17

minus16

minus15

minus14

minus13

minus12

minus11

Rela

tive a

ngle

(deg

ree)

(a)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus420

minus380

minus340

minus300

minus260

minus220

Activ

e pow

er fl

ow (M

W)

(b)


2 4 6 8 10 12 14 16 18 200Time (sec)

minus600

minus400

minus200

0

200

Activ

e pow

er fl

ow (M

W)

(c)





Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 19


5 Conclusions





Appendix


119866119888 = CTC (A1)


C119866119888CT = 119880Σ119880T (A2)



119872 = CT119880TΣminus14 (A3)

Disclosure


Competing Interests


Acknowledgments


References









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









20 Complexity

































Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Complexity 21



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









improving power system stability with gramian matrix-based

Documents