Research ArticleImprovement of Power Flow Calculation with OptimizationFactor Based on Current Injection Method
Lei Wang Chen Chen and Tao Shen
School of Automation Chongqing University Chongqing 400044 China
Correspondence should be addressed to Lei Wang leiwang08cqueducn
Received 4 June 2014 Accepted 9 July 2014 Published 24 July 2014
Academic Editor Rongni Yang
Copyright copy 2014 Lei Wang et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper presents an improvement in power flow calculation based on current injection method by introducing optimizationfactor In the method proposed by this paper the PQ buses are represented by current mismatches while the PV buses arerepresented by power mismatches It is different from the representations in conventional current injection power flow equationsBy using the combined power and current injection mismatches method the number of the equations required can be decreasedto only one for each PV bus The optimization factor is used to improve the iteration process and to ensure the effectiveness of theimproved method proposed when the system is ill-conditioned To verify the effectiveness of the method the IEEE test systems aretested by conventional current injectionmethod and the improvedmethod proposed separatelyThen the results are comparedThecomparisons show that the optimization factor improves the convergence character effectively especially that when the system is athigh loading level andRX ratio the iteration number is one or two times less than the conventional current injectionmethodWhenthe overloading condition of the system is serious the iteration number in this paper appears 4 times less than the conventionalcurrent injection method
1 Introduction
Power flow studies are necessary for planning operatingeconomic scheduling and other analysis such as transientstability voltage stability and contingency studies The tasksof power flow calculation are to solve the steady-state oper-ating conditions of power systems based on the operationmodes and the wiring of the systems Different power flowmethods have emerged and continue to be developed withthe changing needs
The conventional power flow solution comprises powerequations expressed in terms of rectangular or polar coor-dinates Many important contributions have been reportedin this area [1ndash3] To improve the convergence characteristicof Gauss-Seidel method the Newton-Raphson method wasintroduced The NR method was once considered the stateof the art power-flow technique and widely accepted inindustry applications However the main disadvantage of theNewton-Raphson method is the necessity for factorizing andupdating the Jacobian matrix during the iterative solution
process [4] To solve this problem FastDecoupled power flowmethod was proposed to speed up the iteration process oftheNRmethod and decrease the requiredminimummemorystorage Nevertheless the convergence rate of the decoupledNR is influenced by the ill-conditioned case such as when asystem has high ratio of line RX [5ndash8] Some other methodshave been presented in other forms such as the use ofsequence component frame [9ndash11] and the method based onthe loop frame of reference [12]
In [13] a new power flow method based on currentinjection was presented in which the current injection equa-tions are written in rectangular coordinates The Jacobianmatrix is composed of 6 times 6 block matrices and has thesame structure as the nodal admittance matrix By usingthis method the Jacobian matrix can be updated faster thanusing conventional NR power flow method in the case ofPQ buses However in the case of the existence of PV busesthe current injection mismatch power flow method waspresented to improve the convergence character of currentinjection method [14 15] The improvement in [16] increased
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 437567 7 pageshttpdxdoiorg1011552014437567
2 Discrete Dynamics in Nature and Society
the number of required equations to three for each PVbus The paper [17] presented a new representation of PVbuses in revised current injection power flow method toreduce the number of required equations and to improve theconvergence character of well- and ill-conditioned systemswhile the elements related to PV buses have to be changedduring the iteration process In [18] a development of powerflow calculation for ill-conditioned system was presentedThe fact that ldquothe Taylor series expansion of the load flowequations is expressed up to the third term completely andthe final term has the same form but different variable asthe first termrdquo was used in [18] Following the main ideaspresented in [18] which is called Iwamotorsquos method in [15]a new second order power flow method was proposed This
method is useful for heavily loaded and overloaded systemsas well as ill-conditioned distribution systems
This paper improved the current injection method pre-sented in [17] following the main idea in [18] The opti-mization presented in [18] is briefly described here firstly tomake the paper clearThen the improvedmethod results andconclusions are presented
2 Current Injection Power Flow
The basic current injection method proposed in [17] is usedin this paper The combined power and current injectionmismatches power flow formulation can be calculated fromthe following
[[[[[[[[[[[[[[[
[
Δ1198681198981
Δ1198681199031
Δ1198681198982
Δ1198681199032
Δ119875119896
Δ119868119898119899
Δ119868119903119899
]]]]]]]]]]]]]]]
]
=
[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
[
1205971198681198981
1205971198811199031
1205971198681198981
1205971198811198981
1205971198681198981
1205971198811199032
1205971198681198981
1205971198811198982
sdot sdot sdot1205971198681198981
120597120575119896
sdot sdot sdot1205971198681198981
120597119881119903119899
1205971198681198981
120597119881119898119899
1205971198681199031
1205971198811199031
1205971198681199031
1205971198811198981
1205971198681199031
1205971198811199032
1205971198681199031
1205971198811198982
sdot sdot sdot1205971198681199031
120597120575119896
sdot sdot sdot1205971198681199031
120597119881119903119899
1205971198681199031
120597119881119898119899
1205971198681198982
1205971198811199031
1205971198681198982
1205971198811198981
1205971198681198982
1205971198811199032
1205971198681198982
1205971198811198982
sdot sdot sdot1205971198681198982
120597120575119896
sdot sdot sdot1205971198681198982
120597119881119903119899
1205971198681198982
120597119881119898119899
1205971198681199032
1205971198811199031
1205971198681199032
1205971198811198981
1205971198681199032
1205971198811199032
1205971198681199032
1205971198811198982
sdot sdot sdot1205971198681199032
120597120575119896
sdot sdot sdot1205971198681199032
120597119881119903119899
1205971198681199032
120597119881119898119899
120597119875119896
1205971198811199031
120597119875119896
1205971198811198981
120597119875119896
1205971198811199032
120597119875119896
1205971198811198982
sdot sdot sdot120597119875119896
120597120575119896
sdot sdot sdot120597119875119896
120597119881119903119899
120597119875119896
120597119881119898119899
120597119868119898119899
1205971198811199031
120597119868119898119899
1205971198811198981
120597119868119898119899
1205971198811199032
120597119868119898119899
1205971198811198982
sdot sdot sdot120597119868119898119899
120597120575119896
sdot sdot sdot120597119868119898119899
120597119881119903119899
120597119868119898119899
120597119881119898119899
120597119868119903119899
1205971198811199031
120597119868119903119899
1205971198811198981
120597119868119903119899
1205971198811199032
120597119868119903119899
1205971198811198982
sdot sdot sdot120597119868119903119899
120597120575119896
sdot sdot sdot120597119868119903119899
120597119881119903119899
120597119868119903119899
120597119881119898119899
]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
]
[[[[[[[[[[[[[[[[[[[[[[
[
Δ1198811199031
Δ1198811198981
Δ1198811199032
Δ1198811198982
Δ120575119896
Δ119881119903119899
Δ119881119898119899
]]]]]]]]]]]]]]]]]]]]]]
]
(1)
where Δ119868119903119896 real parts of current mismatch at bus 119896 Δ119868
119898119896
imaginary parts of current mismatch at bus 119896 Δ119875119896 active
power mismatch at bus 119896 Δ119881119903119896 real voltage component
correction at bus 119896 Δ119881119898119896 imaginary voltage component
correction at bus 119896 and Δ120575119896 voltage angle correction at bus
119896
21 Equations for PQBuses Thecurrentmismatch for a givenbus 119896 is
Δ119868119896=119875119904119901
119896minus 119895119876119904119901
119896
119864lowast
119896
minus
119899
sum
119894=1
119884119896119894119864119894 (2)
where 119864lowast
119896 complex conjugated voltage phasor at bus 119896
119875119904119901
119896 119876119904119901
119896 specified active and reactive net powers at bus 119896
119884119896119894
= 119866119896119894+ 119895119861119896119894 bus admittance matrix element and 119864
10038161003816100381610038161198811198941003816100381610038161003816 (119866119896119894 cos 120575119896119894 + 119861119896119894 sin 120575119896119894) (10)
where 120575119896119894= 120575119896minus 120575119894 120575119896 voltage phase angle at bus 119896
In the Jacobianmatrix of (1) the elements related to busesof PV type are given as follows
The method presented in [18] is described briefly here tomake the improved method proposed in next part clearlyThe conventional power flow equations in the rectangularcoordinates are
119910119904= 119910 (119909) (13)
The Taylor series expansion of (13) turns out to be
119910119904= 119910 (119909
119890) + 119869Δ119909 + 119910 (Δ119909) (14)
where 119909119890 estimate of 119909 119869 Jacobian matrix and Δ119909 error
(correction vector)Moving all the right-hand side of (14) to the left-hand side
119910119904minus 119910 (119909
119890) minus 119869Δ119909 minus 119910 (Δ119909) = 0 (15)
To obtain the solution of this equation it denotes by 120583the step size optimization factor Applying this factor to thisequation then it follows that
119910119904minus 119910 (119909
119890) minus 119869120583Δ119909 minus 119910 (120583Δ119909) = 0 (16)
Or
119910119904minus 119910 (119909
119890) minus 120583119869Δ119909 minus 120583
2
119910 (Δ119909) = 0 (17)
It can be rewritten as
119886 + 120583119887 + 1205832
119888 = 0 (18)
where
119886 = 119910119904minus 119910 (119909
119890)
119887 = minus119869Δ119909 = minus119886
119888 = minus119910 (Δ119909)
(19)
The following cost function is considered to determine thevalue of the 120583 in a least squared sense
119865 =1
2
119899
sum
119894=1
(119886119894+ 120583119887119894+ 1205832
119888119894)2
997888rarr Minimize (20)
The optimization solution is given by
120597119865
120597120583= 0 (21)
4 Discrete Dynamics in Nature and Society
Namely
1198920+ 1198921120583 + 11989221205832
+ 11989231205833
= 0 (22)
where
1198920=
119899
sum
119894=1
(119886119894119887119894)
1198921=
119899
sum
119894=1
(1198872
119894+ 2119886119894119888119894)
1198922= 3
119899
sum
119894=1
(119887119894119888119894)
1198923= 2
119899
sum
119894=1
1198882
119894
(23)
The state vector for the next iteration is given by
119909119896+1
= 119909119896
+ 120583119896
Δ119909119896
(24)
4 Improvement in Current Injection Method
41 Parameters for PQ Buses The improved method is basedon the current injection method by defining a step sizeoptimization factor The equation (5) is the expression ofimaginary current injection related to bus 119896 Neglecting allterms of order higher than two we then get
Hence the following equations can be got based on the theoryin part 3
119886119868119898119896
= Δ119868119898119896
119887119868119898119896
= minus119886119868119898119896
119888119868119898119896
= minus1
2
119899
sum
119894=1
(1205972
119868119898119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119868119898119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119868119898119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119868119898119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894)
(26)
In a similar way the following equations can be obtained
119886119868119903119896
= Δ119868119903119896
119887119868119903119896
= minus119886119868119903119896
119888119868119903119896
= minus1
2
119899
sum
119894=1
(1205972
119868119903119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119868119903119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119868119903119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119868119903119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894)
(27)
Thus the parameters can be obtained as follows
119886119896= [Δ119868
119898119896Δ119868119903119896]119879
119887119896= minus119886119896
119888119896= [119888119868119898119896
119888119868119903119896
]119879
(28)
42 Parameters for PV Buses From (1) for a PV bus one has
119875119904119901
119896= 119875119896+120597119875119896
120597120575119896
+
119899
sum
119894=1
119894 =119896
(120597119875119896
120597119881119903119894
Δ119881119903119894+
120597119875119896
120597119881119898119894
Δ119881119898119894)
+1
2(1205972
119875119896
1205971205752
119896
Δ2
120575119896
+
119899
sum
119894=1
119894 =119896
(1205972
119875119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894
+1205972
119875119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119875119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119875119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894))
(29)
Discrete Dynamics in Nature and Society 5
0000001
000001
00001
0001
001
01
1
10
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 1 Iteration characteristics of IEEE 30-bus system usingconventional current injection method
Using the same method as parameters for PQ buses thefollowing equations can be obtained
119886119896= Δ119875119896
119887119896= minus119886119896
119888119896= minus
1
2(1205972
119875119896
1205971205752
119896
Δ2
120575119896
+
119899
sum
119894=1
119894 =119896
(1205972
119875119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119875119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119875119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119875119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894))
(30)
5 Results
This paper used MATLAB to implement the program ofpower flow method proposed IEEE 30-bus and 118-bus testsystem are tested here Firstly the validity of the improvedmethod proposed is proven Here the systems are tested assymmetrical systems Tables 1 and 2 show a part of the resultsThen the results obtained by conventional current injectionand the improved method proposed respectively are com-pared Figures 1 and 2 show the iteration characteristics ofIEEE 30-bus test system using the two different methods andFigures 3 and 4 show the results of IEEE 118-bus system in thesame way
0000001
000001
00001
0001
001
01
1
10
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 2 Iteration characteristics of IEEE 30-bus system using theimproved method proposed
0000001
000001
00001
0001
001
01
1
1 2 3 4 5
Absolutemism
atch
ΔPmaxΔQmax
Figure 3 Iteration characteristics of IEEE 118-bus system usingconventional current injection method
0000001
000001
00001
0001
001
01
1
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 4 Iteration characteristics of IEEE 118-bus system using theimproved method proposed
6 Discrete Dynamics in Nature and Society
Table 1 Parts of the iteration results of 30-bus system
Bus Type Voltage (pu) 120579119896(∘)
1 0 10500 000005 2 10100 minus6504413 2 10500 minus6329520 1 10183 minus9497429 1 09972 minus1064000mdashslack bus 1mdashPQ bus 2mdashPV bus
Table 2 Parts of the iteration results of 118-bus system
Table 3 Number of iterations for different 119877119883 ratios-30-bussystem
119877119883 ratioMethod
Conventional currentinjection method
The improvedmethod proposed
05 2 21 2 24 3 36 4 38 6 4
Table 4 Number of iterations for different loading levels-30-bussystem
LoadingMethod
Conventional currentinjection method
The improvedmethod proposed
1 2 24 3 26 5 310 8 415 No convergence 6
Tables 3 and 4 compare the conventional current injectionmethod and the improved method proposed here when thesystem has high RX ratios and compare the two methodswhen the system is operating in overloaded condition
(a) Validity of the Method Tables 1 and 2 show the resultsof the iteration and present the validity of the improvedmethod The voltage amplitudes and angles are reasonableand acceptable At the same time the feasibility of themethodis proved
(b) Convergence CharacteristicThe iteration convergence cri-terion for the program of the improved method proposed isthat themaximum active powermismatch and themaximumreactive powermismatch are both less than the toleranceTheresults presented in Figures 1ndash4 show that the convergencecharacteristic of the improved method proposed is betterthan the conventional current injection method
(c) High RX Ratio Many power flow methods always havebad convergence characteristic if the system has high RXratio In Table 3 the RX ratios are divided into five levelsWhen the system has high XR ratio both the two methodshave good convergence characteristic but with the increaseof RX ratio the advantage of the improvedmethod proposedbecomes obvious the iteration number is one or two timesless than the conventional current injection method
(d) OverloadingThe comparison of the iteration number fordifferent loading cases is shown in Table 4 In this case thesystem is in overloading condition In Table 4 five loadingcases are presented There has been little difference betweenthe two methods when the overloading of the system is notserious However when the condition becomes more seriousthe iteration number of the improved method proposedappears 4 times less than the conventional current injectionmethod Even more when the convergence characteristicof the improved method proposed is still acceptable whenoverloading is serious the conventional current injectionmethod shows no convergence
6 Conclusions
This paper has presented improvement in power flow calcu-lation based on current injection method by introducing anoptimization factor Unlike other current injection methodsthe PV buses are represented by power mismatches hereThe results have demonstrated the good performance ofthe improved method proposed in this paper By usingoptimization factor the iteration character got improvedEven when the system has high RX ratio or operates inoverloaded conditions good results could be got by using thismethod
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National Natu-ral Science Foundation of China (no 51205046) and theFundamental Research Funds for the Central Universities
Discrete Dynamics in Nature and Society 7
(nos CDJZR170008 106112014CDJZR175501) The construc-tive comments provided by the anonymous reviewers and theeditors are also greatly appreciated
References
[1] H Ying-Yi and W Fu-Ming ldquoDevelopment of three-phaseNewton optimal power flow for studying imbalancesecurity intransmission systemsrdquo Electric Power Systems Research vol 55no 1 pp 39ndash48 2000
[2] B Stott ldquoReview of load-flow calculationmethodsrdquo Proceedingsof the IEEE vol 62 no 7 pp 916ndash929 1974
[3] A G Exposito and E R Ramos ldquoAugmented rectangular loadflow modelrdquo IEEE Transactions on Power Systems vol 17 no 2pp 271ndash276 2002
[4] C Xing Steady-State Analysis of Power Systems vol 262 ChiaElectric Power Press 3rd edition 2007
[5] W-M Lin and J-H Teng ldquoThree-phase distribution networkfast-decoupled power flow solutionsrdquo International Journal ofElectrical Power and Energy System vol 22 no 5 pp 375ndash3802000
[6] V M da Costa M L de Oliveira and M R Guedes ldquoDevel-opments in the analysis of unbalanced three-phase power flowsolutionsrdquo International Journal of Electrical Power and EnergySystems vol 29 no 2 pp 175ndash182 2007
[7] Y Song X Li and W Cai ldquoAdaptive and fault-tolerantreactive power compensation in power systems via multilevelSTATCOMsrdquo International Journal of Innovative ComputingInformation and Control vol 9 no 8 pp 3403ndash3413 2013
[8] D Nlu and Y Wei ldquoA novel social-environmental-economicdispatch model for thermalwind power generation and appli-cationrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 7 pp 3005ndash3014 2013
[9] K L Lo and C Zhang ldquoDecomposed three-phase power flowsolution using the sequence component framerdquo IEE ProceedingsC Generation Transmission and Distribution vol 140 no 3 pp181ndash188 1993
[10] M Abdel-Akher K M Nor and A H A Rashid ldquoImprovedthree-phase power-flowmethods using sequence componentsrdquoIEEE Transactions on Power Systems vol 20 no 3 pp 1389ndash1397 2005
[11] I Ngamroo ldquoDynamic events analysis ofThailand andMalaysiapower systems by discrete wavelet decomposition and shortterm fourier transform based on GPS synchronized phasordatardquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 5 pp 2203ndash2228 2013
[12] T Chen and N Yang ldquoLoop frame of reference based three-phase power flow for unbalanced radial distribution systemsrdquoElectric Power Systems Research vol 80 no 7 pp 799ndash806 2010
[13] P A N Garcia J L R Pereira S Carneiro and V M DaCosta ldquoThree-phase power flow calculations using the currentinjection methodrdquo IEEE Transactions on Power Systems vol 15no 2 pp 508ndash514 2000
[14] P A N Garcia J L R Pereira S Carneiro JrM P Vinagre andF V Gomes ldquoImprovements in the representation of PV buseson three-phase distribution power flowrdquo IEEE Transactions onPower Delivery vol 19 no 2 pp 894ndash896 2004
[15] C A Ferreira and V M Da Costa ldquoA second order power flowbased on current injection equationsrdquo International Journal ofElectrical Power and Energy Systems vol 27 no 4 pp 254ndash2632005
[16] V M da Costa N Martins and J L R Pereira ldquoDevelopmentsin the newton raphsonpower flow formulation based on currentinjectionsrdquo IEEE Transactions on Power Systems vol 14 no 4pp 1320ndash1326 1999
[17] S KamelMAbdel-Akher andF Jurado ldquoImprovedNRcurrentinjection load flow using power mismatch representation ofPV busrdquo International Journal of Electrical Power and EnergySystems vol 53 no 1 pp 64ndash68 2013
[18] S Iwamoto and Y Tamura ldquoA load flow calculation methodfor ill-conditioned power systemsrdquo IEEE Transactions on PowerApparatus and Systems vol 100 no 4 pp 1736ndash1743 1979
the number of required equations to three for each PVbus The paper [17] presented a new representation of PVbuses in revised current injection power flow method toreduce the number of required equations and to improve theconvergence character of well- and ill-conditioned systemswhile the elements related to PV buses have to be changedduring the iteration process In [18] a development of powerflow calculation for ill-conditioned system was presentedThe fact that ldquothe Taylor series expansion of the load flowequations is expressed up to the third term completely andthe final term has the same form but different variable asthe first termrdquo was used in [18] Following the main ideaspresented in [18] which is called Iwamotorsquos method in [15]a new second order power flow method was proposed This
method is useful for heavily loaded and overloaded systemsas well as ill-conditioned distribution systems
This paper improved the current injection method pre-sented in [17] following the main idea in [18] The opti-mization presented in [18] is briefly described here firstly tomake the paper clearThen the improvedmethod results andconclusions are presented
2 Current Injection Power Flow
The basic current injection method proposed in [17] is usedin this paper The combined power and current injectionmismatches power flow formulation can be calculated fromthe following
[[[[[[[[[[[[[[[
[
Δ1198681198981
Δ1198681199031
Δ1198681198982
Δ1198681199032
Δ119875119896
Δ119868119898119899
Δ119868119903119899
]]]]]]]]]]]]]]]
]
=
[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
[
1205971198681198981
1205971198811199031
1205971198681198981
1205971198811198981
1205971198681198981
1205971198811199032
1205971198681198981
1205971198811198982
sdot sdot sdot1205971198681198981
120597120575119896
sdot sdot sdot1205971198681198981
120597119881119903119899
1205971198681198981
120597119881119898119899
1205971198681199031
1205971198811199031
1205971198681199031
1205971198811198981
1205971198681199031
1205971198811199032
1205971198681199031
1205971198811198982
sdot sdot sdot1205971198681199031
120597120575119896
sdot sdot sdot1205971198681199031
120597119881119903119899
1205971198681199031
120597119881119898119899
1205971198681198982
1205971198811199031
1205971198681198982
1205971198811198981
1205971198681198982
1205971198811199032
1205971198681198982
1205971198811198982
sdot sdot sdot1205971198681198982
120597120575119896
sdot sdot sdot1205971198681198982
120597119881119903119899
1205971198681198982
120597119881119898119899
1205971198681199032
1205971198811199031
1205971198681199032
1205971198811198981
1205971198681199032
1205971198811199032
1205971198681199032
1205971198811198982
sdot sdot sdot1205971198681199032
120597120575119896
sdot sdot sdot1205971198681199032
120597119881119903119899
1205971198681199032
120597119881119898119899
120597119875119896
1205971198811199031
120597119875119896
1205971198811198981
120597119875119896
1205971198811199032
120597119875119896
1205971198811198982
sdot sdot sdot120597119875119896
120597120575119896
sdot sdot sdot120597119875119896
120597119881119903119899
120597119875119896
120597119881119898119899
120597119868119898119899
1205971198811199031
120597119868119898119899
1205971198811198981
120597119868119898119899
1205971198811199032
120597119868119898119899
1205971198811198982
sdot sdot sdot120597119868119898119899
120597120575119896
sdot sdot sdot120597119868119898119899
120597119881119903119899
120597119868119898119899
120597119881119898119899
120597119868119903119899
1205971198811199031
120597119868119903119899
1205971198811198981
120597119868119903119899
1205971198811199032
120597119868119903119899
1205971198811198982
sdot sdot sdot120597119868119903119899
120597120575119896
sdot sdot sdot120597119868119903119899
120597119881119903119899
120597119868119903119899
120597119881119898119899
]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
]
[[[[[[[[[[[[[[[[[[[[[[
[
Δ1198811199031
Δ1198811198981
Δ1198811199032
Δ1198811198982
Δ120575119896
Δ119881119903119899
Δ119881119898119899
]]]]]]]]]]]]]]]]]]]]]]
]
(1)
where Δ119868119903119896 real parts of current mismatch at bus 119896 Δ119868
119898119896
imaginary parts of current mismatch at bus 119896 Δ119875119896 active
power mismatch at bus 119896 Δ119881119903119896 real voltage component
correction at bus 119896 Δ119881119898119896 imaginary voltage component
correction at bus 119896 and Δ120575119896 voltage angle correction at bus
119896
21 Equations for PQBuses Thecurrentmismatch for a givenbus 119896 is
Δ119868119896=119875119904119901
119896minus 119895119876119904119901
119896
119864lowast
119896
minus
119899
sum
119894=1
119884119896119894119864119894 (2)
where 119864lowast
119896 complex conjugated voltage phasor at bus 119896
119875119904119901
119896 119876119904119901
119896 specified active and reactive net powers at bus 119896
119884119896119894
= 119866119896119894+ 119895119861119896119894 bus admittance matrix element and 119864
10038161003816100381610038161198811198941003816100381610038161003816 (119866119896119894 cos 120575119896119894 + 119861119896119894 sin 120575119896119894) (10)
where 120575119896119894= 120575119896minus 120575119894 120575119896 voltage phase angle at bus 119896
In the Jacobianmatrix of (1) the elements related to busesof PV type are given as follows
The method presented in [18] is described briefly here tomake the improved method proposed in next part clearlyThe conventional power flow equations in the rectangularcoordinates are
119910119904= 119910 (119909) (13)
The Taylor series expansion of (13) turns out to be
119910119904= 119910 (119909
119890) + 119869Δ119909 + 119910 (Δ119909) (14)
where 119909119890 estimate of 119909 119869 Jacobian matrix and Δ119909 error
(correction vector)Moving all the right-hand side of (14) to the left-hand side
119910119904minus 119910 (119909
119890) minus 119869Δ119909 minus 119910 (Δ119909) = 0 (15)
To obtain the solution of this equation it denotes by 120583the step size optimization factor Applying this factor to thisequation then it follows that
119910119904minus 119910 (119909
119890) minus 119869120583Δ119909 minus 119910 (120583Δ119909) = 0 (16)
Or
119910119904minus 119910 (119909
119890) minus 120583119869Δ119909 minus 120583
2
119910 (Δ119909) = 0 (17)
It can be rewritten as
119886 + 120583119887 + 1205832
119888 = 0 (18)
where
119886 = 119910119904minus 119910 (119909
119890)
119887 = minus119869Δ119909 = minus119886
119888 = minus119910 (Δ119909)
(19)
The following cost function is considered to determine thevalue of the 120583 in a least squared sense
119865 =1
2
119899
sum
119894=1
(119886119894+ 120583119887119894+ 1205832
119888119894)2
997888rarr Minimize (20)
The optimization solution is given by
120597119865
120597120583= 0 (21)
4 Discrete Dynamics in Nature and Society
Namely
1198920+ 1198921120583 + 11989221205832
+ 11989231205833
= 0 (22)
where
1198920=
119899
sum
119894=1
(119886119894119887119894)
1198921=
119899
sum
119894=1
(1198872
119894+ 2119886119894119888119894)
1198922= 3
119899
sum
119894=1
(119887119894119888119894)
1198923= 2
119899
sum
119894=1
1198882
119894
(23)
The state vector for the next iteration is given by
119909119896+1
= 119909119896
+ 120583119896
Δ119909119896
(24)
4 Improvement in Current Injection Method
41 Parameters for PQ Buses The improved method is basedon the current injection method by defining a step sizeoptimization factor The equation (5) is the expression ofimaginary current injection related to bus 119896 Neglecting allterms of order higher than two we then get
Hence the following equations can be got based on the theoryin part 3
119886119868119898119896
= Δ119868119898119896
119887119868119898119896
= minus119886119868119898119896
119888119868119898119896
= minus1
2
119899
sum
119894=1
(1205972
119868119898119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119868119898119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119868119898119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119868119898119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894)
(26)
In a similar way the following equations can be obtained
119886119868119903119896
= Δ119868119903119896
119887119868119903119896
= minus119886119868119903119896
119888119868119903119896
= minus1
2
119899
sum
119894=1
(1205972
119868119903119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119868119903119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119868119903119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119868119903119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894)
(27)
Thus the parameters can be obtained as follows
119886119896= [Δ119868
119898119896Δ119868119903119896]119879
119887119896= minus119886119896
119888119896= [119888119868119898119896
119888119868119903119896
]119879
(28)
42 Parameters for PV Buses From (1) for a PV bus one has
119875119904119901
119896= 119875119896+120597119875119896
120597120575119896
+
119899
sum
119894=1
119894 =119896
(120597119875119896
120597119881119903119894
Δ119881119903119894+
120597119875119896
120597119881119898119894
Δ119881119898119894)
+1
2(1205972
119875119896
1205971205752
119896
Δ2
120575119896
+
119899
sum
119894=1
119894 =119896
(1205972
119875119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894
+1205972
119875119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119875119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119875119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894))
(29)
Discrete Dynamics in Nature and Society 5
0000001
000001
00001
0001
001
01
1
10
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 1 Iteration characteristics of IEEE 30-bus system usingconventional current injection method
Using the same method as parameters for PQ buses thefollowing equations can be obtained
119886119896= Δ119875119896
119887119896= minus119886119896
119888119896= minus
1
2(1205972
119875119896
1205971205752
119896
Δ2
120575119896
+
119899
sum
119894=1
119894 =119896
(1205972
119875119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119875119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119875119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119875119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894))
(30)
5 Results
This paper used MATLAB to implement the program ofpower flow method proposed IEEE 30-bus and 118-bus testsystem are tested here Firstly the validity of the improvedmethod proposed is proven Here the systems are tested assymmetrical systems Tables 1 and 2 show a part of the resultsThen the results obtained by conventional current injectionand the improved method proposed respectively are com-pared Figures 1 and 2 show the iteration characteristics ofIEEE 30-bus test system using the two different methods andFigures 3 and 4 show the results of IEEE 118-bus system in thesame way
0000001
000001
00001
0001
001
01
1
10
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 2 Iteration characteristics of IEEE 30-bus system using theimproved method proposed
0000001
000001
00001
0001
001
01
1
1 2 3 4 5
Absolutemism
atch
ΔPmaxΔQmax
Figure 3 Iteration characteristics of IEEE 118-bus system usingconventional current injection method
0000001
000001
00001
0001
001
01
1
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 4 Iteration characteristics of IEEE 118-bus system using theimproved method proposed
6 Discrete Dynamics in Nature and Society
Table 1 Parts of the iteration results of 30-bus system
Bus Type Voltage (pu) 120579119896(∘)
1 0 10500 000005 2 10100 minus6504413 2 10500 minus6329520 1 10183 minus9497429 1 09972 minus1064000mdashslack bus 1mdashPQ bus 2mdashPV bus
Table 2 Parts of the iteration results of 118-bus system
Table 3 Number of iterations for different 119877119883 ratios-30-bussystem
119877119883 ratioMethod
Conventional currentinjection method
The improvedmethod proposed
05 2 21 2 24 3 36 4 38 6 4
Table 4 Number of iterations for different loading levels-30-bussystem
LoadingMethod
Conventional currentinjection method
The improvedmethod proposed
1 2 24 3 26 5 310 8 415 No convergence 6
Tables 3 and 4 compare the conventional current injectionmethod and the improved method proposed here when thesystem has high RX ratios and compare the two methodswhen the system is operating in overloaded condition
(a) Validity of the Method Tables 1 and 2 show the resultsof the iteration and present the validity of the improvedmethod The voltage amplitudes and angles are reasonableand acceptable At the same time the feasibility of themethodis proved
(b) Convergence CharacteristicThe iteration convergence cri-terion for the program of the improved method proposed isthat themaximum active powermismatch and themaximumreactive powermismatch are both less than the toleranceTheresults presented in Figures 1ndash4 show that the convergencecharacteristic of the improved method proposed is betterthan the conventional current injection method
(c) High RX Ratio Many power flow methods always havebad convergence characteristic if the system has high RXratio In Table 3 the RX ratios are divided into five levelsWhen the system has high XR ratio both the two methodshave good convergence characteristic but with the increaseof RX ratio the advantage of the improvedmethod proposedbecomes obvious the iteration number is one or two timesless than the conventional current injection method
(d) OverloadingThe comparison of the iteration number fordifferent loading cases is shown in Table 4 In this case thesystem is in overloading condition In Table 4 five loadingcases are presented There has been little difference betweenthe two methods when the overloading of the system is notserious However when the condition becomes more seriousthe iteration number of the improved method proposedappears 4 times less than the conventional current injectionmethod Even more when the convergence characteristicof the improved method proposed is still acceptable whenoverloading is serious the conventional current injectionmethod shows no convergence
6 Conclusions
This paper has presented improvement in power flow calcu-lation based on current injection method by introducing anoptimization factor Unlike other current injection methodsthe PV buses are represented by power mismatches hereThe results have demonstrated the good performance ofthe improved method proposed in this paper By usingoptimization factor the iteration character got improvedEven when the system has high RX ratio or operates inoverloaded conditions good results could be got by using thismethod
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National Natu-ral Science Foundation of China (no 51205046) and theFundamental Research Funds for the Central Universities
Discrete Dynamics in Nature and Society 7
(nos CDJZR170008 106112014CDJZR175501) The construc-tive comments provided by the anonymous reviewers and theeditors are also greatly appreciated
References
[1] H Ying-Yi and W Fu-Ming ldquoDevelopment of three-phaseNewton optimal power flow for studying imbalancesecurity intransmission systemsrdquo Electric Power Systems Research vol 55no 1 pp 39ndash48 2000
[2] B Stott ldquoReview of load-flow calculationmethodsrdquo Proceedingsof the IEEE vol 62 no 7 pp 916ndash929 1974
[3] A G Exposito and E R Ramos ldquoAugmented rectangular loadflow modelrdquo IEEE Transactions on Power Systems vol 17 no 2pp 271ndash276 2002
[4] C Xing Steady-State Analysis of Power Systems vol 262 ChiaElectric Power Press 3rd edition 2007
[5] W-M Lin and J-H Teng ldquoThree-phase distribution networkfast-decoupled power flow solutionsrdquo International Journal ofElectrical Power and Energy System vol 22 no 5 pp 375ndash3802000
[6] V M da Costa M L de Oliveira and M R Guedes ldquoDevel-opments in the analysis of unbalanced three-phase power flowsolutionsrdquo International Journal of Electrical Power and EnergySystems vol 29 no 2 pp 175ndash182 2007
[7] Y Song X Li and W Cai ldquoAdaptive and fault-tolerantreactive power compensation in power systems via multilevelSTATCOMsrdquo International Journal of Innovative ComputingInformation and Control vol 9 no 8 pp 3403ndash3413 2013
[8] D Nlu and Y Wei ldquoA novel social-environmental-economicdispatch model for thermalwind power generation and appli-cationrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 7 pp 3005ndash3014 2013
[9] K L Lo and C Zhang ldquoDecomposed three-phase power flowsolution using the sequence component framerdquo IEE ProceedingsC Generation Transmission and Distribution vol 140 no 3 pp181ndash188 1993
[10] M Abdel-Akher K M Nor and A H A Rashid ldquoImprovedthree-phase power-flowmethods using sequence componentsrdquoIEEE Transactions on Power Systems vol 20 no 3 pp 1389ndash1397 2005
[11] I Ngamroo ldquoDynamic events analysis ofThailand andMalaysiapower systems by discrete wavelet decomposition and shortterm fourier transform based on GPS synchronized phasordatardquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 5 pp 2203ndash2228 2013
[12] T Chen and N Yang ldquoLoop frame of reference based three-phase power flow for unbalanced radial distribution systemsrdquoElectric Power Systems Research vol 80 no 7 pp 799ndash806 2010
[13] P A N Garcia J L R Pereira S Carneiro and V M DaCosta ldquoThree-phase power flow calculations using the currentinjection methodrdquo IEEE Transactions on Power Systems vol 15no 2 pp 508ndash514 2000
[14] P A N Garcia J L R Pereira S Carneiro JrM P Vinagre andF V Gomes ldquoImprovements in the representation of PV buseson three-phase distribution power flowrdquo IEEE Transactions onPower Delivery vol 19 no 2 pp 894ndash896 2004
[15] C A Ferreira and V M Da Costa ldquoA second order power flowbased on current injection equationsrdquo International Journal ofElectrical Power and Energy Systems vol 27 no 4 pp 254ndash2632005
[16] V M da Costa N Martins and J L R Pereira ldquoDevelopmentsin the newton raphsonpower flow formulation based on currentinjectionsrdquo IEEE Transactions on Power Systems vol 14 no 4pp 1320ndash1326 1999
[17] S KamelMAbdel-Akher andF Jurado ldquoImprovedNRcurrentinjection load flow using power mismatch representation ofPV busrdquo International Journal of Electrical Power and EnergySystems vol 53 no 1 pp 64ndash68 2013
[18] S Iwamoto and Y Tamura ldquoA load flow calculation methodfor ill-conditioned power systemsrdquo IEEE Transactions on PowerApparatus and Systems vol 100 no 4 pp 1736ndash1743 1979
10038161003816100381610038161198811198941003816100381610038161003816 (119866119896119894 cos 120575119896119894 + 119861119896119894 sin 120575119896119894) (10)
where 120575119896119894= 120575119896minus 120575119894 120575119896 voltage phase angle at bus 119896
In the Jacobianmatrix of (1) the elements related to busesof PV type are given as follows
The method presented in [18] is described briefly here tomake the improved method proposed in next part clearlyThe conventional power flow equations in the rectangularcoordinates are
119910119904= 119910 (119909) (13)
The Taylor series expansion of (13) turns out to be
119910119904= 119910 (119909
119890) + 119869Δ119909 + 119910 (Δ119909) (14)
where 119909119890 estimate of 119909 119869 Jacobian matrix and Δ119909 error
(correction vector)Moving all the right-hand side of (14) to the left-hand side
119910119904minus 119910 (119909
119890) minus 119869Δ119909 minus 119910 (Δ119909) = 0 (15)
To obtain the solution of this equation it denotes by 120583the step size optimization factor Applying this factor to thisequation then it follows that
119910119904minus 119910 (119909
119890) minus 119869120583Δ119909 minus 119910 (120583Δ119909) = 0 (16)
Or
119910119904minus 119910 (119909
119890) minus 120583119869Δ119909 minus 120583
2
119910 (Δ119909) = 0 (17)
It can be rewritten as
119886 + 120583119887 + 1205832
119888 = 0 (18)
where
119886 = 119910119904minus 119910 (119909
119890)
119887 = minus119869Δ119909 = minus119886
119888 = minus119910 (Δ119909)
(19)
The following cost function is considered to determine thevalue of the 120583 in a least squared sense
119865 =1
2
119899
sum
119894=1
(119886119894+ 120583119887119894+ 1205832
119888119894)2
997888rarr Minimize (20)
The optimization solution is given by
120597119865
120597120583= 0 (21)
4 Discrete Dynamics in Nature and Society
Namely
1198920+ 1198921120583 + 11989221205832
+ 11989231205833
= 0 (22)
where
1198920=
119899
sum
119894=1
(119886119894119887119894)
1198921=
119899
sum
119894=1
(1198872
119894+ 2119886119894119888119894)
1198922= 3
119899
sum
119894=1
(119887119894119888119894)
1198923= 2
119899
sum
119894=1
1198882
119894
(23)
The state vector for the next iteration is given by
119909119896+1
= 119909119896
+ 120583119896
Δ119909119896
(24)
4 Improvement in Current Injection Method
41 Parameters for PQ Buses The improved method is basedon the current injection method by defining a step sizeoptimization factor The equation (5) is the expression ofimaginary current injection related to bus 119896 Neglecting allterms of order higher than two we then get
Hence the following equations can be got based on the theoryin part 3
119886119868119898119896
= Δ119868119898119896
119887119868119898119896
= minus119886119868119898119896
119888119868119898119896
= minus1
2
119899
sum
119894=1
(1205972
119868119898119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119868119898119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119868119898119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119868119898119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894)
(26)
In a similar way the following equations can be obtained
119886119868119903119896
= Δ119868119903119896
119887119868119903119896
= minus119886119868119903119896
119888119868119903119896
= minus1
2
119899
sum
119894=1
(1205972
119868119903119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119868119903119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119868119903119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119868119903119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894)
(27)
Thus the parameters can be obtained as follows
119886119896= [Δ119868
119898119896Δ119868119903119896]119879
119887119896= minus119886119896
119888119896= [119888119868119898119896
119888119868119903119896
]119879
(28)
42 Parameters for PV Buses From (1) for a PV bus one has
119875119904119901
119896= 119875119896+120597119875119896
120597120575119896
+
119899
sum
119894=1
119894 =119896
(120597119875119896
120597119881119903119894
Δ119881119903119894+
120597119875119896
120597119881119898119894
Δ119881119898119894)
+1
2(1205972
119875119896
1205971205752
119896
Δ2
120575119896
+
119899
sum
119894=1
119894 =119896
(1205972
119875119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894
+1205972
119875119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119875119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119875119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894))
(29)
Discrete Dynamics in Nature and Society 5
0000001
000001
00001
0001
001
01
1
10
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 1 Iteration characteristics of IEEE 30-bus system usingconventional current injection method
Using the same method as parameters for PQ buses thefollowing equations can be obtained
119886119896= Δ119875119896
119887119896= minus119886119896
119888119896= minus
1
2(1205972
119875119896
1205971205752
119896
Δ2
120575119896
+
119899
sum
119894=1
119894 =119896
(1205972
119875119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119875119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119875119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119875119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894))
(30)
5 Results
This paper used MATLAB to implement the program ofpower flow method proposed IEEE 30-bus and 118-bus testsystem are tested here Firstly the validity of the improvedmethod proposed is proven Here the systems are tested assymmetrical systems Tables 1 and 2 show a part of the resultsThen the results obtained by conventional current injectionand the improved method proposed respectively are com-pared Figures 1 and 2 show the iteration characteristics ofIEEE 30-bus test system using the two different methods andFigures 3 and 4 show the results of IEEE 118-bus system in thesame way
0000001
000001
00001
0001
001
01
1
10
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 2 Iteration characteristics of IEEE 30-bus system using theimproved method proposed
0000001
000001
00001
0001
001
01
1
1 2 3 4 5
Absolutemism
atch
ΔPmaxΔQmax
Figure 3 Iteration characteristics of IEEE 118-bus system usingconventional current injection method
0000001
000001
00001
0001
001
01
1
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 4 Iteration characteristics of IEEE 118-bus system using theimproved method proposed
6 Discrete Dynamics in Nature and Society
Table 1 Parts of the iteration results of 30-bus system
Bus Type Voltage (pu) 120579119896(∘)
1 0 10500 000005 2 10100 minus6504413 2 10500 minus6329520 1 10183 minus9497429 1 09972 minus1064000mdashslack bus 1mdashPQ bus 2mdashPV bus
Table 2 Parts of the iteration results of 118-bus system
Table 3 Number of iterations for different 119877119883 ratios-30-bussystem
119877119883 ratioMethod
Conventional currentinjection method
The improvedmethod proposed
05 2 21 2 24 3 36 4 38 6 4
Table 4 Number of iterations for different loading levels-30-bussystem
LoadingMethod
Conventional currentinjection method
The improvedmethod proposed
1 2 24 3 26 5 310 8 415 No convergence 6
Tables 3 and 4 compare the conventional current injectionmethod and the improved method proposed here when thesystem has high RX ratios and compare the two methodswhen the system is operating in overloaded condition
(a) Validity of the Method Tables 1 and 2 show the resultsof the iteration and present the validity of the improvedmethod The voltage amplitudes and angles are reasonableand acceptable At the same time the feasibility of themethodis proved
(b) Convergence CharacteristicThe iteration convergence cri-terion for the program of the improved method proposed isthat themaximum active powermismatch and themaximumreactive powermismatch are both less than the toleranceTheresults presented in Figures 1ndash4 show that the convergencecharacteristic of the improved method proposed is betterthan the conventional current injection method
(c) High RX Ratio Many power flow methods always havebad convergence characteristic if the system has high RXratio In Table 3 the RX ratios are divided into five levelsWhen the system has high XR ratio both the two methodshave good convergence characteristic but with the increaseof RX ratio the advantage of the improvedmethod proposedbecomes obvious the iteration number is one or two timesless than the conventional current injection method
(d) OverloadingThe comparison of the iteration number fordifferent loading cases is shown in Table 4 In this case thesystem is in overloading condition In Table 4 five loadingcases are presented There has been little difference betweenthe two methods when the overloading of the system is notserious However when the condition becomes more seriousthe iteration number of the improved method proposedappears 4 times less than the conventional current injectionmethod Even more when the convergence characteristicof the improved method proposed is still acceptable whenoverloading is serious the conventional current injectionmethod shows no convergence
6 Conclusions
This paper has presented improvement in power flow calcu-lation based on current injection method by introducing anoptimization factor Unlike other current injection methodsthe PV buses are represented by power mismatches hereThe results have demonstrated the good performance ofthe improved method proposed in this paper By usingoptimization factor the iteration character got improvedEven when the system has high RX ratio or operates inoverloaded conditions good results could be got by using thismethod
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National Natu-ral Science Foundation of China (no 51205046) and theFundamental Research Funds for the Central Universities
Discrete Dynamics in Nature and Society 7
(nos CDJZR170008 106112014CDJZR175501) The construc-tive comments provided by the anonymous reviewers and theeditors are also greatly appreciated
References
[1] H Ying-Yi and W Fu-Ming ldquoDevelopment of three-phaseNewton optimal power flow for studying imbalancesecurity intransmission systemsrdquo Electric Power Systems Research vol 55no 1 pp 39ndash48 2000
[2] B Stott ldquoReview of load-flow calculationmethodsrdquo Proceedingsof the IEEE vol 62 no 7 pp 916ndash929 1974
[3] A G Exposito and E R Ramos ldquoAugmented rectangular loadflow modelrdquo IEEE Transactions on Power Systems vol 17 no 2pp 271ndash276 2002
[4] C Xing Steady-State Analysis of Power Systems vol 262 ChiaElectric Power Press 3rd edition 2007
[5] W-M Lin and J-H Teng ldquoThree-phase distribution networkfast-decoupled power flow solutionsrdquo International Journal ofElectrical Power and Energy System vol 22 no 5 pp 375ndash3802000
[6] V M da Costa M L de Oliveira and M R Guedes ldquoDevel-opments in the analysis of unbalanced three-phase power flowsolutionsrdquo International Journal of Electrical Power and EnergySystems vol 29 no 2 pp 175ndash182 2007
[7] Y Song X Li and W Cai ldquoAdaptive and fault-tolerantreactive power compensation in power systems via multilevelSTATCOMsrdquo International Journal of Innovative ComputingInformation and Control vol 9 no 8 pp 3403ndash3413 2013
[8] D Nlu and Y Wei ldquoA novel social-environmental-economicdispatch model for thermalwind power generation and appli-cationrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 7 pp 3005ndash3014 2013
[9] K L Lo and C Zhang ldquoDecomposed three-phase power flowsolution using the sequence component framerdquo IEE ProceedingsC Generation Transmission and Distribution vol 140 no 3 pp181ndash188 1993
[10] M Abdel-Akher K M Nor and A H A Rashid ldquoImprovedthree-phase power-flowmethods using sequence componentsrdquoIEEE Transactions on Power Systems vol 20 no 3 pp 1389ndash1397 2005
[11] I Ngamroo ldquoDynamic events analysis ofThailand andMalaysiapower systems by discrete wavelet decomposition and shortterm fourier transform based on GPS synchronized phasordatardquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 5 pp 2203ndash2228 2013
[12] T Chen and N Yang ldquoLoop frame of reference based three-phase power flow for unbalanced radial distribution systemsrdquoElectric Power Systems Research vol 80 no 7 pp 799ndash806 2010
[13] P A N Garcia J L R Pereira S Carneiro and V M DaCosta ldquoThree-phase power flow calculations using the currentinjection methodrdquo IEEE Transactions on Power Systems vol 15no 2 pp 508ndash514 2000
[14] P A N Garcia J L R Pereira S Carneiro JrM P Vinagre andF V Gomes ldquoImprovements in the representation of PV buseson three-phase distribution power flowrdquo IEEE Transactions onPower Delivery vol 19 no 2 pp 894ndash896 2004
[15] C A Ferreira and V M Da Costa ldquoA second order power flowbased on current injection equationsrdquo International Journal ofElectrical Power and Energy Systems vol 27 no 4 pp 254ndash2632005
[16] V M da Costa N Martins and J L R Pereira ldquoDevelopmentsin the newton raphsonpower flow formulation based on currentinjectionsrdquo IEEE Transactions on Power Systems vol 14 no 4pp 1320ndash1326 1999
[17] S KamelMAbdel-Akher andF Jurado ldquoImprovedNRcurrentinjection load flow using power mismatch representation ofPV busrdquo International Journal of Electrical Power and EnergySystems vol 53 no 1 pp 64ndash68 2013
[18] S Iwamoto and Y Tamura ldquoA load flow calculation methodfor ill-conditioned power systemsrdquo IEEE Transactions on PowerApparatus and Systems vol 100 no 4 pp 1736ndash1743 1979
The state vector for the next iteration is given by
119909119896+1
= 119909119896
+ 120583119896
Δ119909119896
(24)
4 Improvement in Current Injection Method
41 Parameters for PQ Buses The improved method is basedon the current injection method by defining a step sizeoptimization factor The equation (5) is the expression ofimaginary current injection related to bus 119896 Neglecting allterms of order higher than two we then get
Hence the following equations can be got based on the theoryin part 3
119886119868119898119896
= Δ119868119898119896
119887119868119898119896
= minus119886119868119898119896
119888119868119898119896
= minus1
2
119899
sum
119894=1
(1205972
119868119898119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119868119898119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119868119898119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119868119898119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894)
(26)
In a similar way the following equations can be obtained
119886119868119903119896
= Δ119868119903119896
119887119868119903119896
= minus119886119868119903119896
119888119868119903119896
= minus1
2
119899
sum
119894=1
(1205972
119868119903119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119868119903119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119868119903119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119868119903119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894)
(27)
Thus the parameters can be obtained as follows
119886119896= [Δ119868
119898119896Δ119868119903119896]119879
119887119896= minus119886119896
119888119896= [119888119868119898119896
119888119868119903119896
]119879
(28)
42 Parameters for PV Buses From (1) for a PV bus one has
119875119904119901
119896= 119875119896+120597119875119896
120597120575119896
+
119899
sum
119894=1
119894 =119896
(120597119875119896
120597119881119903119894
Δ119881119903119894+
120597119875119896
120597119881119898119894
Δ119881119898119894)
+1
2(1205972
119875119896
1205971205752
119896
Δ2
120575119896
+
119899
sum
119894=1
119894 =119896
(1205972
119875119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894
+1205972
119875119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119875119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119875119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894))
(29)
Discrete Dynamics in Nature and Society 5
0000001
000001
00001
0001
001
01
1
10
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 1 Iteration characteristics of IEEE 30-bus system usingconventional current injection method
Using the same method as parameters for PQ buses thefollowing equations can be obtained
119886119896= Δ119875119896
119887119896= minus119886119896
119888119896= minus
1
2(1205972
119875119896
1205971205752
119896
Δ2
120575119896
+
119899
sum
119894=1
119894 =119896
(1205972
119875119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119875119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119875119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119875119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894))
(30)
5 Results
This paper used MATLAB to implement the program ofpower flow method proposed IEEE 30-bus and 118-bus testsystem are tested here Firstly the validity of the improvedmethod proposed is proven Here the systems are tested assymmetrical systems Tables 1 and 2 show a part of the resultsThen the results obtained by conventional current injectionand the improved method proposed respectively are com-pared Figures 1 and 2 show the iteration characteristics ofIEEE 30-bus test system using the two different methods andFigures 3 and 4 show the results of IEEE 118-bus system in thesame way
0000001
000001
00001
0001
001
01
1
10
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 2 Iteration characteristics of IEEE 30-bus system using theimproved method proposed
0000001
000001
00001
0001
001
01
1
1 2 3 4 5
Absolutemism
atch
ΔPmaxΔQmax
Figure 3 Iteration characteristics of IEEE 118-bus system usingconventional current injection method
0000001
000001
00001
0001
001
01
1
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 4 Iteration characteristics of IEEE 118-bus system using theimproved method proposed
6 Discrete Dynamics in Nature and Society
Table 1 Parts of the iteration results of 30-bus system
Bus Type Voltage (pu) 120579119896(∘)
1 0 10500 000005 2 10100 minus6504413 2 10500 minus6329520 1 10183 minus9497429 1 09972 minus1064000mdashslack bus 1mdashPQ bus 2mdashPV bus
Table 2 Parts of the iteration results of 118-bus system
Table 3 Number of iterations for different 119877119883 ratios-30-bussystem
119877119883 ratioMethod
Conventional currentinjection method
The improvedmethod proposed
05 2 21 2 24 3 36 4 38 6 4
Table 4 Number of iterations for different loading levels-30-bussystem
LoadingMethod
Conventional currentinjection method
The improvedmethod proposed
1 2 24 3 26 5 310 8 415 No convergence 6
Tables 3 and 4 compare the conventional current injectionmethod and the improved method proposed here when thesystem has high RX ratios and compare the two methodswhen the system is operating in overloaded condition
(a) Validity of the Method Tables 1 and 2 show the resultsof the iteration and present the validity of the improvedmethod The voltage amplitudes and angles are reasonableand acceptable At the same time the feasibility of themethodis proved
(b) Convergence CharacteristicThe iteration convergence cri-terion for the program of the improved method proposed isthat themaximum active powermismatch and themaximumreactive powermismatch are both less than the toleranceTheresults presented in Figures 1ndash4 show that the convergencecharacteristic of the improved method proposed is betterthan the conventional current injection method
(c) High RX Ratio Many power flow methods always havebad convergence characteristic if the system has high RXratio In Table 3 the RX ratios are divided into five levelsWhen the system has high XR ratio both the two methodshave good convergence characteristic but with the increaseof RX ratio the advantage of the improvedmethod proposedbecomes obvious the iteration number is one or two timesless than the conventional current injection method
(d) OverloadingThe comparison of the iteration number fordifferent loading cases is shown in Table 4 In this case thesystem is in overloading condition In Table 4 five loadingcases are presented There has been little difference betweenthe two methods when the overloading of the system is notserious However when the condition becomes more seriousthe iteration number of the improved method proposedappears 4 times less than the conventional current injectionmethod Even more when the convergence characteristicof the improved method proposed is still acceptable whenoverloading is serious the conventional current injectionmethod shows no convergence
6 Conclusions
This paper has presented improvement in power flow calcu-lation based on current injection method by introducing anoptimization factor Unlike other current injection methodsthe PV buses are represented by power mismatches hereThe results have demonstrated the good performance ofthe improved method proposed in this paper By usingoptimization factor the iteration character got improvedEven when the system has high RX ratio or operates inoverloaded conditions good results could be got by using thismethod
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National Natu-ral Science Foundation of China (no 51205046) and theFundamental Research Funds for the Central Universities
Discrete Dynamics in Nature and Society 7
(nos CDJZR170008 106112014CDJZR175501) The construc-tive comments provided by the anonymous reviewers and theeditors are also greatly appreciated
References
[1] H Ying-Yi and W Fu-Ming ldquoDevelopment of three-phaseNewton optimal power flow for studying imbalancesecurity intransmission systemsrdquo Electric Power Systems Research vol 55no 1 pp 39ndash48 2000
[2] B Stott ldquoReview of load-flow calculationmethodsrdquo Proceedingsof the IEEE vol 62 no 7 pp 916ndash929 1974
[3] A G Exposito and E R Ramos ldquoAugmented rectangular loadflow modelrdquo IEEE Transactions on Power Systems vol 17 no 2pp 271ndash276 2002
[4] C Xing Steady-State Analysis of Power Systems vol 262 ChiaElectric Power Press 3rd edition 2007
[5] W-M Lin and J-H Teng ldquoThree-phase distribution networkfast-decoupled power flow solutionsrdquo International Journal ofElectrical Power and Energy System vol 22 no 5 pp 375ndash3802000
[6] V M da Costa M L de Oliveira and M R Guedes ldquoDevel-opments in the analysis of unbalanced three-phase power flowsolutionsrdquo International Journal of Electrical Power and EnergySystems vol 29 no 2 pp 175ndash182 2007
[7] Y Song X Li and W Cai ldquoAdaptive and fault-tolerantreactive power compensation in power systems via multilevelSTATCOMsrdquo International Journal of Innovative ComputingInformation and Control vol 9 no 8 pp 3403ndash3413 2013
[8] D Nlu and Y Wei ldquoA novel social-environmental-economicdispatch model for thermalwind power generation and appli-cationrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 7 pp 3005ndash3014 2013
[9] K L Lo and C Zhang ldquoDecomposed three-phase power flowsolution using the sequence component framerdquo IEE ProceedingsC Generation Transmission and Distribution vol 140 no 3 pp181ndash188 1993
[10] M Abdel-Akher K M Nor and A H A Rashid ldquoImprovedthree-phase power-flowmethods using sequence componentsrdquoIEEE Transactions on Power Systems vol 20 no 3 pp 1389ndash1397 2005
[11] I Ngamroo ldquoDynamic events analysis ofThailand andMalaysiapower systems by discrete wavelet decomposition and shortterm fourier transform based on GPS synchronized phasordatardquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 5 pp 2203ndash2228 2013
[12] T Chen and N Yang ldquoLoop frame of reference based three-phase power flow for unbalanced radial distribution systemsrdquoElectric Power Systems Research vol 80 no 7 pp 799ndash806 2010
[13] P A N Garcia J L R Pereira S Carneiro and V M DaCosta ldquoThree-phase power flow calculations using the currentinjection methodrdquo IEEE Transactions on Power Systems vol 15no 2 pp 508ndash514 2000
[14] P A N Garcia J L R Pereira S Carneiro JrM P Vinagre andF V Gomes ldquoImprovements in the representation of PV buseson three-phase distribution power flowrdquo IEEE Transactions onPower Delivery vol 19 no 2 pp 894ndash896 2004
[15] C A Ferreira and V M Da Costa ldquoA second order power flowbased on current injection equationsrdquo International Journal ofElectrical Power and Energy Systems vol 27 no 4 pp 254ndash2632005
[16] V M da Costa N Martins and J L R Pereira ldquoDevelopmentsin the newton raphsonpower flow formulation based on currentinjectionsrdquo IEEE Transactions on Power Systems vol 14 no 4pp 1320ndash1326 1999
[17] S KamelMAbdel-Akher andF Jurado ldquoImprovedNRcurrentinjection load flow using power mismatch representation ofPV busrdquo International Journal of Electrical Power and EnergySystems vol 53 no 1 pp 64ndash68 2013
[18] S Iwamoto and Y Tamura ldquoA load flow calculation methodfor ill-conditioned power systemsrdquo IEEE Transactions on PowerApparatus and Systems vol 100 no 4 pp 1736ndash1743 1979
Figure 1 Iteration characteristics of IEEE 30-bus system usingconventional current injection method
Using the same method as parameters for PQ buses thefollowing equations can be obtained
119886119896= Δ119875119896
119887119896= minus119886119896
119888119896= minus
1
2(1205972
119875119896
1205971205752
119896
Δ2
120575119896
+
119899
sum
119894=1
119894 =119896
(1205972
119875119896
120597119881119903119896120597119881119903119894
Δ119881119903119896Δ119881119903119894+
1205972
119875119896
120597119881119903119896120597119881119898119894
Δ119881119903119896Δ119881119898119894
+1205972
119875119896
120597119881119898119896120597119881119903119894
Δ119881119898119896Δ119881119903119894
+1205972
119875119896
120597119881119898119896120597119881119898119894
Δ119881119898119896Δ119881119898119894))
(30)
5 Results
This paper used MATLAB to implement the program ofpower flow method proposed IEEE 30-bus and 118-bus testsystem are tested here Firstly the validity of the improvedmethod proposed is proven Here the systems are tested assymmetrical systems Tables 1 and 2 show a part of the resultsThen the results obtained by conventional current injectionand the improved method proposed respectively are com-pared Figures 1 and 2 show the iteration characteristics ofIEEE 30-bus test system using the two different methods andFigures 3 and 4 show the results of IEEE 118-bus system in thesame way
0000001
000001
00001
0001
001
01
1
10
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 2 Iteration characteristics of IEEE 30-bus system using theimproved method proposed
0000001
000001
00001
0001
001
01
1
1 2 3 4 5
Absolutemism
atch
ΔPmaxΔQmax
Figure 3 Iteration characteristics of IEEE 118-bus system usingconventional current injection method
0000001
000001
00001
0001
001
01
1
1 2 3 4
Absolutemism
atch
ΔPmaxΔQmax
Figure 4 Iteration characteristics of IEEE 118-bus system using theimproved method proposed
6 Discrete Dynamics in Nature and Society
Table 1 Parts of the iteration results of 30-bus system
Bus Type Voltage (pu) 120579119896(∘)
1 0 10500 000005 2 10100 minus6504413 2 10500 minus6329520 1 10183 minus9497429 1 09972 minus1064000mdashslack bus 1mdashPQ bus 2mdashPV bus
Table 2 Parts of the iteration results of 118-bus system
Table 3 Number of iterations for different 119877119883 ratios-30-bussystem
119877119883 ratioMethod
Conventional currentinjection method
The improvedmethod proposed
05 2 21 2 24 3 36 4 38 6 4
Table 4 Number of iterations for different loading levels-30-bussystem
LoadingMethod
Conventional currentinjection method
The improvedmethod proposed
1 2 24 3 26 5 310 8 415 No convergence 6
Tables 3 and 4 compare the conventional current injectionmethod and the improved method proposed here when thesystem has high RX ratios and compare the two methodswhen the system is operating in overloaded condition
(a) Validity of the Method Tables 1 and 2 show the resultsof the iteration and present the validity of the improvedmethod The voltage amplitudes and angles are reasonableand acceptable At the same time the feasibility of themethodis proved
(b) Convergence CharacteristicThe iteration convergence cri-terion for the program of the improved method proposed isthat themaximum active powermismatch and themaximumreactive powermismatch are both less than the toleranceTheresults presented in Figures 1ndash4 show that the convergencecharacteristic of the improved method proposed is betterthan the conventional current injection method
(c) High RX Ratio Many power flow methods always havebad convergence characteristic if the system has high RXratio In Table 3 the RX ratios are divided into five levelsWhen the system has high XR ratio both the two methodshave good convergence characteristic but with the increaseof RX ratio the advantage of the improvedmethod proposedbecomes obvious the iteration number is one or two timesless than the conventional current injection method
(d) OverloadingThe comparison of the iteration number fordifferent loading cases is shown in Table 4 In this case thesystem is in overloading condition In Table 4 five loadingcases are presented There has been little difference betweenthe two methods when the overloading of the system is notserious However when the condition becomes more seriousthe iteration number of the improved method proposedappears 4 times less than the conventional current injectionmethod Even more when the convergence characteristicof the improved method proposed is still acceptable whenoverloading is serious the conventional current injectionmethod shows no convergence
6 Conclusions
This paper has presented improvement in power flow calcu-lation based on current injection method by introducing anoptimization factor Unlike other current injection methodsthe PV buses are represented by power mismatches hereThe results have demonstrated the good performance ofthe improved method proposed in this paper By usingoptimization factor the iteration character got improvedEven when the system has high RX ratio or operates inoverloaded conditions good results could be got by using thismethod
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National Natu-ral Science Foundation of China (no 51205046) and theFundamental Research Funds for the Central Universities
Discrete Dynamics in Nature and Society 7
(nos CDJZR170008 106112014CDJZR175501) The construc-tive comments provided by the anonymous reviewers and theeditors are also greatly appreciated
References
[1] H Ying-Yi and W Fu-Ming ldquoDevelopment of three-phaseNewton optimal power flow for studying imbalancesecurity intransmission systemsrdquo Electric Power Systems Research vol 55no 1 pp 39ndash48 2000
[2] B Stott ldquoReview of load-flow calculationmethodsrdquo Proceedingsof the IEEE vol 62 no 7 pp 916ndash929 1974
[3] A G Exposito and E R Ramos ldquoAugmented rectangular loadflow modelrdquo IEEE Transactions on Power Systems vol 17 no 2pp 271ndash276 2002
[4] C Xing Steady-State Analysis of Power Systems vol 262 ChiaElectric Power Press 3rd edition 2007
[5] W-M Lin and J-H Teng ldquoThree-phase distribution networkfast-decoupled power flow solutionsrdquo International Journal ofElectrical Power and Energy System vol 22 no 5 pp 375ndash3802000
[6] V M da Costa M L de Oliveira and M R Guedes ldquoDevel-opments in the analysis of unbalanced three-phase power flowsolutionsrdquo International Journal of Electrical Power and EnergySystems vol 29 no 2 pp 175ndash182 2007
[7] Y Song X Li and W Cai ldquoAdaptive and fault-tolerantreactive power compensation in power systems via multilevelSTATCOMsrdquo International Journal of Innovative ComputingInformation and Control vol 9 no 8 pp 3403ndash3413 2013
[8] D Nlu and Y Wei ldquoA novel social-environmental-economicdispatch model for thermalwind power generation and appli-cationrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 7 pp 3005ndash3014 2013
[9] K L Lo and C Zhang ldquoDecomposed three-phase power flowsolution using the sequence component framerdquo IEE ProceedingsC Generation Transmission and Distribution vol 140 no 3 pp181ndash188 1993
[10] M Abdel-Akher K M Nor and A H A Rashid ldquoImprovedthree-phase power-flowmethods using sequence componentsrdquoIEEE Transactions on Power Systems vol 20 no 3 pp 1389ndash1397 2005
[11] I Ngamroo ldquoDynamic events analysis ofThailand andMalaysiapower systems by discrete wavelet decomposition and shortterm fourier transform based on GPS synchronized phasordatardquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 5 pp 2203ndash2228 2013
[12] T Chen and N Yang ldquoLoop frame of reference based three-phase power flow for unbalanced radial distribution systemsrdquoElectric Power Systems Research vol 80 no 7 pp 799ndash806 2010
[13] P A N Garcia J L R Pereira S Carneiro and V M DaCosta ldquoThree-phase power flow calculations using the currentinjection methodrdquo IEEE Transactions on Power Systems vol 15no 2 pp 508ndash514 2000
[14] P A N Garcia J L R Pereira S Carneiro JrM P Vinagre andF V Gomes ldquoImprovements in the representation of PV buseson three-phase distribution power flowrdquo IEEE Transactions onPower Delivery vol 19 no 2 pp 894ndash896 2004
[15] C A Ferreira and V M Da Costa ldquoA second order power flowbased on current injection equationsrdquo International Journal ofElectrical Power and Energy Systems vol 27 no 4 pp 254ndash2632005
[16] V M da Costa N Martins and J L R Pereira ldquoDevelopmentsin the newton raphsonpower flow formulation based on currentinjectionsrdquo IEEE Transactions on Power Systems vol 14 no 4pp 1320ndash1326 1999
[17] S KamelMAbdel-Akher andF Jurado ldquoImprovedNRcurrentinjection load flow using power mismatch representation ofPV busrdquo International Journal of Electrical Power and EnergySystems vol 53 no 1 pp 64ndash68 2013
[18] S Iwamoto and Y Tamura ldquoA load flow calculation methodfor ill-conditioned power systemsrdquo IEEE Transactions on PowerApparatus and Systems vol 100 no 4 pp 1736ndash1743 1979
Table 3 Number of iterations for different 119877119883 ratios-30-bussystem
119877119883 ratioMethod
Conventional currentinjection method
The improvedmethod proposed
05 2 21 2 24 3 36 4 38 6 4
Table 4 Number of iterations for different loading levels-30-bussystem
LoadingMethod
Conventional currentinjection method
The improvedmethod proposed
1 2 24 3 26 5 310 8 415 No convergence 6
Tables 3 and 4 compare the conventional current injectionmethod and the improved method proposed here when thesystem has high RX ratios and compare the two methodswhen the system is operating in overloaded condition
(a) Validity of the Method Tables 1 and 2 show the resultsof the iteration and present the validity of the improvedmethod The voltage amplitudes and angles are reasonableand acceptable At the same time the feasibility of themethodis proved
(b) Convergence CharacteristicThe iteration convergence cri-terion for the program of the improved method proposed isthat themaximum active powermismatch and themaximumreactive powermismatch are both less than the toleranceTheresults presented in Figures 1ndash4 show that the convergencecharacteristic of the improved method proposed is betterthan the conventional current injection method
(c) High RX Ratio Many power flow methods always havebad convergence characteristic if the system has high RXratio In Table 3 the RX ratios are divided into five levelsWhen the system has high XR ratio both the two methodshave good convergence characteristic but with the increaseof RX ratio the advantage of the improvedmethod proposedbecomes obvious the iteration number is one or two timesless than the conventional current injection method
(d) OverloadingThe comparison of the iteration number fordifferent loading cases is shown in Table 4 In this case thesystem is in overloading condition In Table 4 five loadingcases are presented There has been little difference betweenthe two methods when the overloading of the system is notserious However when the condition becomes more seriousthe iteration number of the improved method proposedappears 4 times less than the conventional current injectionmethod Even more when the convergence characteristicof the improved method proposed is still acceptable whenoverloading is serious the conventional current injectionmethod shows no convergence
6 Conclusions
This paper has presented improvement in power flow calcu-lation based on current injection method by introducing anoptimization factor Unlike other current injection methodsthe PV buses are represented by power mismatches hereThe results have demonstrated the good performance ofthe improved method proposed in this paper By usingoptimization factor the iteration character got improvedEven when the system has high RX ratio or operates inoverloaded conditions good results could be got by using thismethod
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National Natu-ral Science Foundation of China (no 51205046) and theFundamental Research Funds for the Central Universities
Discrete Dynamics in Nature and Society 7
(nos CDJZR170008 106112014CDJZR175501) The construc-tive comments provided by the anonymous reviewers and theeditors are also greatly appreciated
References
[1] H Ying-Yi and W Fu-Ming ldquoDevelopment of three-phaseNewton optimal power flow for studying imbalancesecurity intransmission systemsrdquo Electric Power Systems Research vol 55no 1 pp 39ndash48 2000
[2] B Stott ldquoReview of load-flow calculationmethodsrdquo Proceedingsof the IEEE vol 62 no 7 pp 916ndash929 1974
[3] A G Exposito and E R Ramos ldquoAugmented rectangular loadflow modelrdquo IEEE Transactions on Power Systems vol 17 no 2pp 271ndash276 2002
[4] C Xing Steady-State Analysis of Power Systems vol 262 ChiaElectric Power Press 3rd edition 2007
[5] W-M Lin and J-H Teng ldquoThree-phase distribution networkfast-decoupled power flow solutionsrdquo International Journal ofElectrical Power and Energy System vol 22 no 5 pp 375ndash3802000
[6] V M da Costa M L de Oliveira and M R Guedes ldquoDevel-opments in the analysis of unbalanced three-phase power flowsolutionsrdquo International Journal of Electrical Power and EnergySystems vol 29 no 2 pp 175ndash182 2007
[7] Y Song X Li and W Cai ldquoAdaptive and fault-tolerantreactive power compensation in power systems via multilevelSTATCOMsrdquo International Journal of Innovative ComputingInformation and Control vol 9 no 8 pp 3403ndash3413 2013
[8] D Nlu and Y Wei ldquoA novel social-environmental-economicdispatch model for thermalwind power generation and appli-cationrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 7 pp 3005ndash3014 2013
[9] K L Lo and C Zhang ldquoDecomposed three-phase power flowsolution using the sequence component framerdquo IEE ProceedingsC Generation Transmission and Distribution vol 140 no 3 pp181ndash188 1993
[10] M Abdel-Akher K M Nor and A H A Rashid ldquoImprovedthree-phase power-flowmethods using sequence componentsrdquoIEEE Transactions on Power Systems vol 20 no 3 pp 1389ndash1397 2005
[11] I Ngamroo ldquoDynamic events analysis ofThailand andMalaysiapower systems by discrete wavelet decomposition and shortterm fourier transform based on GPS synchronized phasordatardquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 5 pp 2203ndash2228 2013
[12] T Chen and N Yang ldquoLoop frame of reference based three-phase power flow for unbalanced radial distribution systemsrdquoElectric Power Systems Research vol 80 no 7 pp 799ndash806 2010
[13] P A N Garcia J L R Pereira S Carneiro and V M DaCosta ldquoThree-phase power flow calculations using the currentinjection methodrdquo IEEE Transactions on Power Systems vol 15no 2 pp 508ndash514 2000
[14] P A N Garcia J L R Pereira S Carneiro JrM P Vinagre andF V Gomes ldquoImprovements in the representation of PV buseson three-phase distribution power flowrdquo IEEE Transactions onPower Delivery vol 19 no 2 pp 894ndash896 2004
[15] C A Ferreira and V M Da Costa ldquoA second order power flowbased on current injection equationsrdquo International Journal ofElectrical Power and Energy Systems vol 27 no 4 pp 254ndash2632005
[16] V M da Costa N Martins and J L R Pereira ldquoDevelopmentsin the newton raphsonpower flow formulation based on currentinjectionsrdquo IEEE Transactions on Power Systems vol 14 no 4pp 1320ndash1326 1999
[17] S KamelMAbdel-Akher andF Jurado ldquoImprovedNRcurrentinjection load flow using power mismatch representation ofPV busrdquo International Journal of Electrical Power and EnergySystems vol 53 no 1 pp 64ndash68 2013
[18] S Iwamoto and Y Tamura ldquoA load flow calculation methodfor ill-conditioned power systemsrdquo IEEE Transactions on PowerApparatus and Systems vol 100 no 4 pp 1736ndash1743 1979
(nos CDJZR170008 106112014CDJZR175501) The construc-tive comments provided by the anonymous reviewers and theeditors are also greatly appreciated
References
[1] H Ying-Yi and W Fu-Ming ldquoDevelopment of three-phaseNewton optimal power flow for studying imbalancesecurity intransmission systemsrdquo Electric Power Systems Research vol 55no 1 pp 39ndash48 2000
[2] B Stott ldquoReview of load-flow calculationmethodsrdquo Proceedingsof the IEEE vol 62 no 7 pp 916ndash929 1974
[3] A G Exposito and E R Ramos ldquoAugmented rectangular loadflow modelrdquo IEEE Transactions on Power Systems vol 17 no 2pp 271ndash276 2002
[4] C Xing Steady-State Analysis of Power Systems vol 262 ChiaElectric Power Press 3rd edition 2007
[5] W-M Lin and J-H Teng ldquoThree-phase distribution networkfast-decoupled power flow solutionsrdquo International Journal ofElectrical Power and Energy System vol 22 no 5 pp 375ndash3802000
[6] V M da Costa M L de Oliveira and M R Guedes ldquoDevel-opments in the analysis of unbalanced three-phase power flowsolutionsrdquo International Journal of Electrical Power and EnergySystems vol 29 no 2 pp 175ndash182 2007
[7] Y Song X Li and W Cai ldquoAdaptive and fault-tolerantreactive power compensation in power systems via multilevelSTATCOMsrdquo International Journal of Innovative ComputingInformation and Control vol 9 no 8 pp 3403ndash3413 2013
[8] D Nlu and Y Wei ldquoA novel social-environmental-economicdispatch model for thermalwind power generation and appli-cationrdquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 7 pp 3005ndash3014 2013
[9] K L Lo and C Zhang ldquoDecomposed three-phase power flowsolution using the sequence component framerdquo IEE ProceedingsC Generation Transmission and Distribution vol 140 no 3 pp181ndash188 1993
[10] M Abdel-Akher K M Nor and A H A Rashid ldquoImprovedthree-phase power-flowmethods using sequence componentsrdquoIEEE Transactions on Power Systems vol 20 no 3 pp 1389ndash1397 2005
[11] I Ngamroo ldquoDynamic events analysis ofThailand andMalaysiapower systems by discrete wavelet decomposition and shortterm fourier transform based on GPS synchronized phasordatardquo International Journal of Innovative Computing Informa-tion and Control vol 9 no 5 pp 2203ndash2228 2013
[12] T Chen and N Yang ldquoLoop frame of reference based three-phase power flow for unbalanced radial distribution systemsrdquoElectric Power Systems Research vol 80 no 7 pp 799ndash806 2010
[13] P A N Garcia J L R Pereira S Carneiro and V M DaCosta ldquoThree-phase power flow calculations using the currentinjection methodrdquo IEEE Transactions on Power Systems vol 15no 2 pp 508ndash514 2000
[14] P A N Garcia J L R Pereira S Carneiro JrM P Vinagre andF V Gomes ldquoImprovements in the representation of PV buseson three-phase distribution power flowrdquo IEEE Transactions onPower Delivery vol 19 no 2 pp 894ndash896 2004
[15] C A Ferreira and V M Da Costa ldquoA second order power flowbased on current injection equationsrdquo International Journal ofElectrical Power and Energy Systems vol 27 no 4 pp 254ndash2632005
[16] V M da Costa N Martins and J L R Pereira ldquoDevelopmentsin the newton raphsonpower flow formulation based on currentinjectionsrdquo IEEE Transactions on Power Systems vol 14 no 4pp 1320ndash1326 1999
[17] S KamelMAbdel-Akher andF Jurado ldquoImprovedNRcurrentinjection load flow using power mismatch representation ofPV busrdquo International Journal of Electrical Power and EnergySystems vol 53 no 1 pp 64ndash68 2013
[18] S Iwamoto and Y Tamura ldquoA load flow calculation methodfor ill-conditioned power systemsrdquo IEEE Transactions on PowerApparatus and Systems vol 100 no 4 pp 1736ndash1743 1979
