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Electric Power Systems Research 84 (2012) 90– 99

Contents lists available at SciVerse ScienceDirect

Electric Power Systems Research

jou rn al h om epa ge: www.elsev ier .com/ locate /epsr

n efficient power flow algorithm for weakly meshed distribution systems

ary Changa,∗, Shou-Yung Chua,b, Ming-Fong Hsua, Ching-Sheng Chuanga, Hung-Lu Wanga

Department of Electrical Engineering, National Chung Cheng University, Chia-Yi 621, Taiwan, ROCTaiwan Power Company, Taipei, 106, Taiwan ROC

r t i c l e i n f o

rticle history:eceived 31 March 2011eceived in revised form 3 August 2011ccepted 12 October 2011vailable online 9 November 2011

eywords:ower floworward and backward sweep

a b s t r a c t

This paper proposes an efficient backward and forward sweep algorithm for the three-phase power flowanalysis of weakly meshed distribution system. In the backward sweep, the Kirchhoff’s circuit laws arefirstly used to calculate each line current and the upstream bus voltage of each line or a transformerbranch. Next, the linear proportional principle is applied to find the ratios of the specified (or new) andthe calculated bus voltages of the decomposed real and imaginary parts of the network. The compensationmethod is used to break meshes and to calculate the current injections at each end bus created by breakingthe mesh. A useful bus indexing scheme is also proposed to determine the updated voltages at the junctionand/or terminal busses for the six decomposed networks in the forward sweep. The procedure repeats

istribution systemadial network

and stops after the voltage mismatch of each pair of end busses of the opened mesh and the mismatch ofthe calculated and specified voltages at the substation bus are less than the predefined threshold value.The proposed method is tested by three IEEE benchmark systems with meshes, and with default systemdata and different system conditions. Results show that the proposed algorithm is efficient, accurate,

with

and robust in comparing

networks.

. Introduction

The goal of the distribution system power flow function iso study the distribution networks under various loading condi-ions and configurations. Provided with bus voltage magnitudesnd phase angles output from the power flow function, one canerive more information for the distribution network, includingeal and reactive power flow in each line, line section power loss,nd the total real and reactive power at each bus. Distribution sys-em power flow study is the backbone for distribution automation.herefore, an accurate, robust and computationally efficient distri-ution power flow tool is highly demanded.

To date, most widely adopted three-phase power flow compu-ational algorithms can be grouped into two general classes. Onelass is based on the requirements of power balance equationsith or without their derivatives, such as Newton–Raphson andauss–Seidel types and their derivatives [1–7]. The other is basedn the applications of the Kirchhoff’s voltage and current laws (KVLnd KCL); the ladder network and the backward/forward (BW/FW)weep methods and their derivates fall into this class [8–12].iterature survey shows that comparisons between different dis-

ribution system power flow methods have been made [13,14];owever, most of the aforementioned approaches have their lim-

tations. Some of the reviewed approaches only can be applied to

∗ Corresponding author. Tel.: +886 5 2428302; fax: +886 5 2720862.E-mail address: wchang@ee.ccu.edu.tw (G. Chang).

378-7796/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2011.10.005

the commonly used backward/forward sweep method for weakly meshed

© 2011 Elsevier B.V. All rights reserved.

single-phase distribution networks, while others are with slow con-vergence speeds. Among the reviewed methods, the BW/FW sweepmethods possess better performances under different system con-ditions. In [15] the authors had presented the power flow analysismethod for the radial network based on the decomposition of thenetwork with only considering constant-impedance loads in themodel. In this paper, the authors extend their previous work for theweakly meshed network with incorporating the real and imaginarydecomposition of the network, the linear proportional principle,the compensation method [8], and the new bus indexing schemewhile taking three different load models (i.e. constant PQ, constantimpedance, and constant current) into account. The convergencecharacteristic of the proposed method is then illustrated and threeIEEE benchmark distribution systems are extensively tested withvarious conditions. Results tested on the benchmark systems withmeshes show that the proposed algorithm is accurate, robust, andcomputationally efficient in comparing with the classic BW/FWsweep method [8]. To have a fair comparison of the solution accu-racy, Appendix A also includes the three IEEE benchmark systemswithout meshes under full load tested by the proposed, the ladderiteration [10], and the classic FW/BW sweep methods.

2. Real and imaginary decomposition of series componentsin a distribution system

In a distribution system, the network components mainlyinclude two parts: shunt and series elements. Shunt elementsinclude spot loads, distributed loads, and capacitor banks and

G. Chang et al. / Electric Power Syst

dsFut

V

I

weeir

aaz

iIiqn

2t

mg

iiiciitdl

Fig. 1. A generalized series component between two busses.

ecide the current magnitude and phase angle for each bus. Lineegments and transformers are categorized as series components.ig. 1 shows a general series element model. In this model, thepstream bus voltage and line current can be expressed by a func-ion of the downstream bus voltage and line current given by

ABC = A · Vabc + B · Iabc (1)

ABC = C · Vabc + D · Iabc (2)

here VABC is the vector of the three-phase voltages at the sendingnd; Vabc is the vector of the three-phase voltages at the receivingnd; IABC is the vector of the three-phase line currents at the send-ng end; Iabc is the vector of the three-phase line currents at theeceiving end.

In (1) and (2), A, B, C, and D are given in (3) and (4), where And D are 3 × 3 constant matrices, B is a 3 × 3 impedance matrix,nd C is a 3 × 3 null matrix [9]. The impedance elements of B areuv = ruv + jxuv, where u, v = 1, 2, 3.

The KVL applied to a series component is composed of thempedance, voltages, and currents in complex-number quantities.f each complex-number quantity is decomposed into the real andmaginary parts, the original KVL equation in complex-numberuantities can be transformed into two separate equations in real-umber quantities.

.1. Voltage and current equations of distribution lines andransformers

Figs. 2 and 3 show the general model of a distribution line seg-ent containing series impedance and shunt admittance and the

eneral model of a distribution transformer.For a line segment associated with (1) and (2), A and D are

dentity matrices, B is an impedance matrix associated with linempedances, and C is a null matrix. It is noted that the line modelsnclude single-phase, two-phase, and three-phase for wye and deltaonnections when considering the effects of line coupling and earthn the distribution system [10,16]. For a distribution transformer, Cs a null matrix and the elements of A, B, and D are dependent onhe transformer connection types. Based on the real and imaginary

ecompositions, the voltage and current equations of a distribution

ine segment or a distribution transformer can be rewritten by (3)

Fig. 2. Exact distribution line model.

ems Research 84 (2012) 90– 99 91

and (4).[Vr

AN + jV iAN

VrBN + jV i

BNVr

CN + jV iCN

]=

[˛11 ˛12 ˛13˛21 ˛22 ˛23˛31 ˛32 ˛33

] [Vr

an + jV ian

Vrbn

+ jV ibn

Vrcn + jV i

cn

]

+[

r11 + jx11 r12 + jx12 r13 + jx13r21 + jx21 r22 + jx22 r23 + jx23r31 + jx31 r32 + jx32 r33 + jx33

] [Ira + jIi

a

Irb

+ jIib

Irc + jIi

c

](3)

[IrA + jIi

AIrB + jIi

BIrC + jIi

C

]=

[d11 d12 d13d21 d22 d23d31 d32 d33

] [Ira + jIi

a

Irb

+ jIib

Irc + jIi

c

](4)

By separating the real and imaginary parts of (3) and (4), the volt-age and current equations for a series component in the distributionsystem in matrix forms can be further expressed by (5)–(8) as fol-lows:

VrABC = A · Vr

abc + Br · Irabc − Bi · Ii

abc (5)

ViABC = A · Vi

abc + Br · Iiabc + Bi · Ir

abc (6)

IrABC = D · Ir

abc (7)

IiABC = D · Ii

abc (8)

where VrABC , Vi

ABC , Vrabc , Vi

abc , IrABC , Ii

ABC , Irabc , Ii

abc , Br, and Bi are asso-ciated matrices for the real and imaginary components of voltages,currents, and coefficients.

3. Solution algorithm

3.1. Linear proportional principle

Consider an M-bus resistive network shown in Fig. 4 with aspecified constant voltage, Vsp

1 , at bus 1. Before employing thebackward sweep to calculate each bus voltage, an initial voltageat the terminal bus, V0

M , is firstly assumed. During the backwardsweep, the current flowing through the terminal bus M and theline current between busses M and M − 1 are obtained according toIM = V0

M/RM and IM−1,M = IM, respectively. The voltage at bus M − 1is Vcal

M−1 = IM−1,M × RM−1,M + V0M and the current flowing through

RM−1 can be expressed as IM−1 = VcalM−1/RM−1. The voltage at any

other bus m becomes

Vcalm = Im,m+1 × Rm,m+1 + Vcal

m+1, m = 1, 2, . . . , M − 2 (9)

where Im = Vcalm /Rm and Im,m+1 = Im+1 + Im+1,m+2. Following this pro-

cedure, the voltage at bus 1, Vcal1 , can be calculated. Since there is a

specified voltage at bus 1, the ratio of the specified voltage to thecalculated voltage becomes

K = Vsp1

Vcal1

(10)

Because Fig. 4 is a linear passive network, the final solution of eachbus voltage, Vm, of (11) then can be determined by the calculatedvoltage at each bus multiplying by the ratio of (10) based on thelinear proportional principle.

Vm = K · Vcalm , m = 2, . . . , M (11)

3.2. Bus indexing scheme

Fig. 5 shows a more complicated distribution network. By

employing the proposed bus indexing scheme, the junction (i.e.branching node) or terminal bus locations at branches or at lateralscan be quickly found. Since the radial system can be treated as amain feeder with laterals, these laterals may also have sub-laterals.

92 G. Chang et al. / Electric Power Systems Research 84 (2012) 90– 99

Fig. 3. Generalized distribution transformer model.

s resi

Tdnbtifabbb0qi

3

dAtcca

Fig. 4. An M-bu

he proposed bus indexing scheme includes three indices and isenoted by (l, m, n). The first index, l, indicates the upstream branchumber of the bus. The second index, m, indicates the branch num-er where the bus locates at its from-end. The third index, n, ishe bus number among all ordered busses at the same branch. Fornstance, the index of bus B2 is denoted by (1, 2, 2), where the maineeder is designated as the first branch (i.e. branch 1 or branch A)nd bus B2 is located at the from-end of the second branch (i.e.ranch B, which is named branch 2), the bus number of B2 is 2 onranch B. In the same manner, the substation bus, A1, is denotedy (0, 1, 1), where the upstream branch of the substation bus is. With adopting the proposed bus indexing scheme, it allows touickly determine the junction or the terminal bus of the system

n the proposed forward sweep described below.

.3. Proposed forward sweep procedure for the radial network

The distribution network shown in Fig. 5 is also used toemonstrate the proposed radial power flow solution procedure.ssume that initial voltages are given for all terminal and junc-

ion busses and the substation bus (i.e. bus A1) is given a specifiedomplex-number (i.e. phasor) voltage. In the backward sweep, allalculations are in phasor forms. Except that the terminal bus volt-ge being the initial value, each upstream bus voltage is calculated

A5A4A3A2A1

B1B2

B3

C2C1

D

D

F1

G

H1

H2

Fig. 5. A more complicated network for

stive network.

according to its adjacent calculated downstream bus voltages anddownstream branch currents. For instance, there are three branchesemanated from bus A5; that is, there are three calculated volt-ages at bus A5 and they depend on the currents of branches A5–B1,A5–C1, A5–H1 and the downstream voltages at busses B1, C1, andH1, respectively. For bus A4, the calculated voltage is determinedby the initial or pre-updated voltage at bus A5. Since there is a spec-ified voltage at bus A1, the associated ratios of the specified voltageto the calculated voltage at bus A1 for the real and the imaginaryparts can be obtained by (12) and (13), respectively.

KiA1

=Vi,sp

A1

Vi,calA1

(12)

KrA1

=Vr,sp

A1

Vr,calA1

(13)

In the forward sweep, the updated voltage, either for the real orthe imaginary part of the network, at each downstream bus can beobtained according to the linear proportional principle. Then, the

updated voltage at each bus located on branch A can be determinedby the calculated voltage multiplied by the ratio of (12) and (13).For instance, the updated voltage at the junction bus A5 is equal tothe initial voltage (for the first forward sweep) or the pre-updated

C3

1

2

D7D6D5D4D3

E3E2E1

F2

G21

I2I1

J1

J2

J3

bus indexing scheme illustration.

r Syst

vvtbfaidtnabusss

K

K

Izto

3d

icbtsleum

1

2

3

4

5

G. Chang et al. / Electric Powe

oltage multiplied by the ratio of (12) and (13). After the updatedoltage at bus A5 is obtained, the new voltage can be regarded ashe specified voltage at A5. Then, the updated voltages of thoseusses located at three branches emanated from bus A5 can beound by each calculated bus voltage multiplied by the ratio of (14)nd (15), respectively, where w indicates the connecting branchndex. Therefore, the bus voltage update procedure goes through allownstream busses to obtain the new voltages. By taking advan-age of the linear property of the decomposed networks, it onlyeeds to find the voltage at each junction or terminal bus. The realnd imaginary parts of the voltages at each junction and terminalusses are then combined to obtain the phasor voltages and to besed in the subsequent backward sweep procedure. Therefore, theolution time is significantly reduced by using the new bus indexingcheme to map the junction or terminal busses during the forwardweep:

iw =

Vi,newA5

Vi,calA5,w

, w = B, C, H (14)

rw =

Vr,newA5

Vr,calA5,w

, w = B, C, H (15)

t is noted that, to avoid null imaginary part in the calculation, theero reference angle of phase A is shifted by 45◦ and so do the otherwo phases. After performing the proposed load flow analysis, thebtained phase angle of each bus voltage is shifted by −45◦.

.4. Proposed solution procedure for the weakly meshedistribution network

In the proposed method, the compensation method describedn [8] is adopted to break meshes and to calculate the injectionurrent at each end bus associated with the opened mesh. Theackward sweep of the proposed method is the same as that ofhe ladder iteration approach of [10] and the new bus indexingcheme is employed in the forward sweep for each decomposedinear network and to obtain the updated three-phase voltages atach junction or terminal bus. The iteration repeats and terminatesntil the convergence criteria are met. The following summarizesajor steps of the solution procedure:

. Input the system and load data that includes spot and dis-tributed loads with different constant-PQ, constant-impedanceand constant-current models. Specify threshold values for con-vergence requirement and the maximum iteration number forthe proposed algorithm.

. Open meshes and build the breakpoint impedance matrix. Ini-tialize the current injections associated with each opened mesh.

. Assign the specified substation voltage and set initial flat three-phase voltages at each junction bus and the terminal bus.

. Sort the distances from each bus to the substation and prioritizethe busses according to the sorted distances. Set the index foreach bus according to the proposed bus indexing scheme. Setiteration count, k = 1, and start the backward sweep procedurefrom the farthest terminal bus.

. Backward sweep: employing KCL and KVL to find the calcu-lated voltage of each upstream bus of (16) and the line currentof (17). Record the calculated voltages at the junction bussescorresponding to each branching feeder or lateral:

cal,m cal,n m,n

VABC = A · Vabc

+ B · Iabc

(16)

Im,nabc

= Iload,nabc

+∑q ∈ C

Iqabc

(17)

ems Research 84 (2012) 90– 99 93

In (16), m is the upstream bus index and n is the downstreambus index located on the branch between busses m and n. In(17), the first term at the right-hand side is the calculated loadcurrent based on different load models and the second term isthe sum of branch currents from the set of line sections, C, whereq is the line section index.

6. Check if the iteration count is greater than 1 and is less thanthe maximum iteration number. For the first iteration, proceedto next step. Stop the procedure if the maximum iteration num-ber is reached. Otherwise, calculate the voltage mismatch at eachpair of end busses associated with the opened mesh and the mis-match of the specified and calculated voltages at the substationbus. If all mismatches are less than the specified threshold value,the solution procedure stops; otherwise, proceed to next step.

7. Forward sweep: using linear proportional principle and the busindexing scheme to find the updated real and imaginary phasevoltages at each junction or terminal bus by the correspondingratios given in (18) and (19), where Kr

jun,ph,kand Ki

jun,ph,kare given

in (20) and (21) for the three phases and ph, h = 1, 2, . . ., H, indi-cates the terminal bus or the junction bus corresponding to eachof its branching feeders or laterals, jun stands for the substationor junction bus, and where H is the total number of branchingfeeders and laterals at that junction bus. The updated real andimaginary three-phase voltages are given in (22) and (23). Theforward sweep continues until all terminal busses are reachedand the corresponding phasor voltages are determined:

Vrph,k = Kr

jun,ph,k · Vr,calph,k

, k = a, b, c (18)

Viph,k = Ki

jun,ph,k · Vi,calph,k

, k = a, b, c (19)

Krjun,ph,k =

Vr,spjun,k

Vr,caljun,ph,k

, k = a, b, c (20)

Kijun,ph,k =

Vi,spjun,k

V i,caljun,ph,k

, k = a, b, c (21)

Vrabc = Kr

abc · Vr,calabc

(22)

Viabc = Ki

abc · Vi,calabc

(23)

where Krabc and Ki

abc are vectors of the ratios associated withthree phases for the real and imaginary parts.

8. Calculate the current injections at end busses associated witheach opened mesh and update the currents. Return to step 5.

To have a better understanding of the proposed solution proce-dure, Appendix B illustrates a 7-node system with a step-by-stepprocedure for the distribution power flow calculation.

3.5. Graphical illustration of convergence trend

Consider one phase of a radial distribution system shown inFig. 6. Assigning the specified substation voltage, Vsp

1 , and the initialvoltage to be VM,ini at the end bus M, the calculated voltage pha-sor at each network bus can be obtained in the backward sweepand the updated real and imaginary decomposition voltages at thejunction and terminal busses are found in the new forward sweepprocedure. The convergence criterion of the proposed algorithm isthat the maximum mismatch of the specified and the calculatedvoltages at the substation bus is less than a predefined value. Afterthe substation voltage converges at the hth iteration, the general

form of the final voltage at any bus n can be expressed by

V (h)n = Vr,sp

1

Vr,cal(h)1

· Vr,cal(h)n + j

V i,sp1

Vi,cal(h)1

· Vi,cal(h)n (24)

94 G. Chang et al. / Electric Power Systems Research 84 (2012) 90– 99

1V

12Z 23Z MMZ ,1−

MLZ ,

12I2I

23I3I

34IMI

MV3V2V

1−MI

1−MV

MMI ,1−

...

... 1, −MLZ3,LZ2,LZ

system

wnsf

|

BcTrmwi

s

1

2

Fig. 6. A radial distribution

here n = 1, 2, 3, . . ., M. Vr,cal(h)n and Vi,cal(h)

n are the real and imagi-ary parts of the calculated voltage obtained at the hth backwardweep for bus n. The voltage mismatch at any bus n then can beound by

�V (h)n | = |V (h)

n − Vcal(h)n | =

∣∣∣∣∣[

Vr,sp1 · Vr,cal(h)

n

Vr,cal(h)1

− Vr,cal(h)n

]

+j

[Vi,sp

1 · Vi,cal(h)n

Vi,cal(k)1

− Vi,cal(h)n

]∣∣∣∣∣=

∣∣∣∣∣[

Vr,sp1

Vr,cal(h)1

− 1

]· Vr,cal(h)

n + j

[Vi,sp

1

Vi,cal(h)1

− 1

]· Vi,cal(h)

n

∣∣∣∣∣ (25)

y observing (25), it is found that, as the substation voltageonverges, the voltages at downstream busses also converge.he proposed distribution system power flow algorithm can beegarded as a variance of the classic backward/forward sweepethod. The detailed convergence analysis for the classic back-ard/forward sweep method and its variances also can be found

n [17–19].The following statements describe the convergence trend, as

hown in Fig. 7, of the proposed solution algorithm:

. In Fig. 6, if Vsp1 < Vcal(l)

1 at the lth iteration, l = 1, 2, . . ., h − 1, theratio of the specified voltage to the calculated voltage at thesubstation bus is less than 1. The updated voltage at each down-stream bus, M, will be updated toward a smaller value. On thecontrary, if Vsp

1 > Vcal(l)1 at the lth iteration, the ratio of the spec-

ified voltage to the calculated voltage at the substation bus islarger than 1. The voltage at each downstream bus, M, will beupdated toward a larger value.

. The voltage update procedure in the forward sweep of the pro-

posed algorithm is accelerated by the proposed bus indexingscheme applying to the three-phase decomposed real and imag-inary networks. That is, the bus voltage is updated layer by layerin the solution algorithm which is not the same as the classic

Fig. 7. Convergence diagram.

for convergence analysis.

backward/forward sweep method being updated bus by bus. Theproposed method provides computational advantages over theclassic backward/forward sweep method and is extremely fastin the distribution systems with less junction busses or networklayers.

3. In the backward sweep of the proposed algorithm, the conver-gence of the calculated substation voltage depends on the lineimpedance, the apparent power of the load, and the terminalbus voltage. Because the terminal bus voltage is updated towardconvergence by the ratio of the specified voltage to the calcu-lated voltage at the substation, the calculated substation voltagealways converges to the specified value, as indicated in Fig. 7.

4. Case study

The proposed algorithm has been tested by three IEEE dis-tribution benchmark systems on a Pentium-4 PC with 2.4 GHzCPU [20]. The three test feeders possess different line models(overhead lines and underground cables, including line coupling),different transformer connections, and unbalanced spot and dis-tributed loads consisting of constant-PQ, constant-impedance, andconstant-current loads. It is noted that voltage regulators in thesestudy benchmark systems are omitted. Since analyzing the powerflow problem of any system with voltage regulators, the power flowanalysis without voltage regulator must be performed first. Then,the tap of voltage regulator can be determined with consideringdetailed regulator models [10].

In the study, the convergence tolerance for bus voltage is0.001 p.u. and is defined by |V1,spec − V1,cal|/|V1,spec|. The maxi-mum iteration number is 100. Study cases include the three IEEEbenchmark systems with meshes at different loadings. Resultsobtained are also compared with the classic backward/forwardsweep method [8]. Table 1 shows the execution time and itera-tion numbers for the three benchmark systems without and withmeshes. The absolute maximum differences of the voltage magni-tudes and the phase angles at each phase between the proposedand the classic methods are shown in Table 2.

The effects of different loadings on convergence are given inTables 3–6, where the notation X indicates the divergent case.Due to the space limitation, only test results for 34- and 123-busbenchmark systems are shown. Tables 7 and 8 list the comparedresults for different system loadings with respect to different lineR/X ratios for the 34-bus and 123-bus radial systems, respectively.These tables show that the convergence performance of proposedmethod is superior to the classic FW/BW sweep method and is veryrobust. For instance, as shown in Tables 3 and 5, even though thesystems are heavily loaded, the proposed method takes the leastnumber of iterations and minimum solution time to converge. It isalso noted that, in Table 3, the 34-bus system possesses very longfeeder and is lightly loaded. Because of the length of the feeder

and the unbalanced loading, such system could cause the con-vergence problem in the high loads for the classic FW/BW sweepmethod; however, the proposed method converges with the lin-ear characteristic. Though Table 5 shows that the performance for

G. Chang et al. / Electric Power Systems Research 84 (2012) 90– 99 95

Table 1Number of iteration and execution time for three benchmark systems at full load (P + jQ).

Benchmark system Mesh number FW/BW sweep [8] Proposed method

Number of iterations Time (s) Number of iterations Time (s)

13-Bus0 4 0.0376 3 0.03122 7 0.0936 4 0.0830

34-Bus0 4 0.1470 2 0.06543 6 0.4160 3 0.2080

123-Bus0 4 0.9124 2 0.40046 3 1.1930 3 0.9063

Table 2Absolute maximum difference of bus voltages and phase angles at each phase between proposed and classic BW/FW sweep methods for three benchmark systems at fullload.

Test system Mesh number Absolute maximum difference of voltage magnitudes (p.u.) Absolute maximum difference of phase angles (◦)

Ph. A Ph. B Ph. C Ph. A Ph. B Ph. C

13-Bus0 0.0002 0.0001 0.0002 0.0140 0.0120 0.01512 0.0008 0.0048 0.0007 0.1221 0.0819 0.2230

34-Bus0 0.0005 0.0004 0.0010 0.1029 0.2561 0.08853 0.0056 0.0067 0.0016 0.5302 0.3718 0.2484

123-Bus0 0.0005 0.0003 0.0004 0.0133 0.0054 0.00466 0.0003 0.0006 0.0003 1.5399 0.0734 1.4556

Table 3System loading vs. convergence for the 34-bus system.

34-Bus system loading FW/BW sweep Proposed method

Number of iterations Time (s) Number of iterations Time (s)

P + jQ (full load) 6 0.416 4 0.08301.5(P + jQ) 40 4.7700 3 0.36002(P + jQ) X X 5 0.67602.5(P + jQ) X X 5 0.7030

Table 4Absolute maximum difference of bus voltages and phase angles at each phase between proposed and classic BW/FW sweep methods for the 34-bus system at differentloadings.

34-Bus system loading Absolute maximum difference of voltage magnitudes (p.u.) Absolute maximum difference of phase angles (◦)

Ph. A Ph. B Ph. C Ph. A Ph. B Ph. C

P + jQ 0.0056 0.0067 0.0016 0.5302 0.3718 0.24841.5(P + jQ) 0.0048 0.0080 0.0078 0.6626 0.8626 0.1768

Table 5System loading vs. convergence for the 123-bus system.

123-Bus system loading FW/BW sweep Proposed method

Number of iterations Time (s) Number of iterations Time (s)

P + jQ 3 1.1930 3 0.90631.5(P + jQ) 3 2.0840 3 1.80702(P + jQ) 4 3.1040 4 2.31242.5(P + jQ) 5 3.4400 4 2.4400

Table 6Absolute maximum difference of bus voltages and phase angles at each phase between proposed and classic BW/FW sweep methods for the 123-bus system at differentloadings.

123-Bus system loading Absolute maximum difference of voltage magnitudes (p.u.) Absolute maximum difference of phase angles (◦)

Ph. A Ph. B Ph. C Ph. A Ph. B Ph. C

P + jQ 0.0018 0.0034 0.0017 1.5399 0.0734 1.45561.5(P + jQ) 0.0052 0.0050 0.0025 2.2598 0.1519 2.20322(P + jQ) 0.0033 0.0100 0.0045 4.9499 0.2006 4.64842.5(P + jQ) 0.0097 0.0130 0.0054 6.3255 0.3963 6.0420

96 G. Chang et al. / Electric Power Systems Research 84 (2012) 90– 99

Table 7Line R/X ratio vs. convergence for the 34-bus system.

System loading Line R/X ratio FW/BW sweep Proposed method

Number of iterations Time (s) Number of iterations Time (s)

0.5(P + jQ)

0.5 3 0.1096 2 0.07501 3 0.1096 2 0.07481.5 3 0.1090 2 0.07502 4 0.1440 2 0.07182.5 5 0.1778 3 0.1158

(P + jQ)

0.5 4 0.1530 2 0.06561 4 0.1470 2 0.06541.5 4 0.1498 3 0.10922 4 0.1406 3 0.10962.5 5 0.1844 3 0.1092

1.5(P + jQ)

0.5 4 0.1402 3 0.10961 5 0.1782 3 0.11261.5 7 0.2500 3 0.10982 6 0.2124 3 0.10962.5 X X 3 0.1096

2(P + jQ)

0.5 5 0.1782 3 0.10941 X X 3 0.11261.5 X X 3 0.10982 X X 3 0.10962.5 X X 4 0.1530

2.5(P + jQ)

0.5 6 0.2124 3 0.11261 X X 3 0.11861.5 X X 3 0.10902 X X 4 0.15282.5 X X 5 0.1970

titetH

TL

he 123-bus system seems to be marginally better in the number ofterations for the proposed method, the reason may be that the sys-

em is with too many short branches. Since the proposed methodmploys the backward sweep to update both voltages and currents,he proposed method does not show its superiorities in this case.owever, the proposed method is more computationally efficient

able 8ine R/X ratio vs. convergence for the 123-bus system.

System loading Line R/X ratio FW/BW sweep

Number of iterations

0.5(P + jQ)

0.5 4

1 4

1.5 4

2 4

2.5 4

(P + jQ)

0.5 4

1 4

1.5 4

2 4

2.5 4

1.5(P + jQ)

0.5 4

1 4

1.5 5

2 5

2.5 5

2(P + jQ)

0.5 4

1 5

1.5 5

2 5

2.5 5

2.5(P + jQ)

0.5 5

1 6

1.5 6

2 6

2.5 X

than the classic approach in much less CPU time required achievingconvergence for the 123-bus system.

By observing Tables 4 and 6, the calculated absolute maxi-mum differences of voltage magnitudes and phase angles betweenthe proposed and classic methods are (0.008 p.u., 0.8626◦) and(0.013 p.u., 6.3255◦) for different system loadings, respectively.

Proposed method

Time (s) Number of iterations Time (s)

0.9126 2 0.39700.9188 2 0.39660.9156 2 0.40430.9158 2 0.40660.9126 2 0.3968

0.9060 2 0.40020.9124 2 0.40040.9094 2 0.40300.9124 3 0.65340.9124 3 0.6562

0.9094 3 0.66260.9158 3 0.65621.1408 3 0.65921.1408 3 0.65321.1408 3 0.6564

0.9126 3 0.65321.1500 3 0.65621.1432 3 0.65361.1468 3 0.65321.1404 4 0.91261.1408 4 0.90921.3622 4 0.90941.3688 4 0.90941.3624 4 0.9030X 5 1.1688

G. Chang et al. / Electric Power Systems Research 84 (2012) 90– 99 97

Table 9Comparison of iteration number and execution time for the three benchmark test systems.

IEEE benchmark system FW/BW sweep [8] Ladder iteration [10] Proposed method

Iteration number Time (s) Iteration number Time (s) Iteration number Time (s)

13-Bus 4 0.0376 3 0.0314 3 0.031234-Bus 4 0.1470 4 0.1562 2 0.0654123-Bus 4 0.9124 3 0.7096 2 0.4004

Table 10Comparison of maximum absolute differences of bus voltage magnitude and phase angle for the three benchmark test systems.

Max. difference of voltage magnitudes (p.u.) Max. difference of voltage phase angles (◦)

Aa Ba Ca Aa Ba Ca

Proposed vs. BW/FW sweep [8]13-Bus 0.0004 0.0002 0.0000 0.01 0.01 0.0234-Bus 0.0009 0.0004 0.0008 0.10 0.24 0.10

Proposed vs. ladder iteration [10]123-Bus 0.0004 0.0003 0.0005 0.02 0.01 0.0113-Bus 0.0000 0.0001 0.0005 0.014 0.012 0.015

0.0009 0.103 0.256 0.0890.0004 0.013 0.005 0.005

Ot

5

tscittbbAspbpigaf

Am

Tbtmtsmcc

A

ts

Fig. 8. 7-Node illustrative system.

Table 11Line data of the 7-node illustrative system.

Section Length (mile) Impedance (�)

1–2 1 0.1 + j0.2 = 0.2236∠63.44◦

2–3 2 0.2 + j0.4 = 0.4472∠63.44◦

3–4 3 0.3 + j0.6 = 0.6708∠63.44◦

4–5 3 0.3 + j0.6 = 0.6708∠63.44◦

4–6 2 0.2 + j0.4 = 0.4472∠63.44◦

6–7 2 0.2 + j0.4 = 0.4472∠63.44◦

Table 12Bus load data of the 7-node illustrative system.

Bus no. Load (kW + jkVAR)

2 40 + j30 = 50∠36.87◦

3 80 + j60 = 100∠36.87◦

34-Bus 0.0005 0.0004

123-Bus 0.0004 0.0003

a Phase.

verall, the solution accuracy of the proposed method is well main-ained as that of the classic approach.

. Conclusions

In this paper, an efficient decomposition-based method forhree-phase distribution power flow analysis of weakly meshedystems is proposed. The advantages of the proposed method overlassic approaches in that, in the analysis, the distribution networks decomposed into real and imaginary parts in conjunction withhe new bus indexing scheme in the forward sweep to increasehe computational efficiency. The proposed solution algorithm haseen described in details and extensively tested by three IEEEenchmark distribution systems with different system conditions.

small illustrative example is also provided in the appendix tohow the detailed solution procedure. Results obtained by the pro-osed method are compared with those obtained by the classicackward/forward sweep and ladder iteration approaches. The pro-osed method has been shown to be superior in the number of

terations, computationally efficient, and the robustness of conver-ence in comparing with the classic approach, while the solutionccuracy is well maintained. The proposed method can be appliedor providing near real-time three-phase power flow analysis.

ppendix A. Comparison of solution accuracy for the threeethods

To have a fair comparison of the solution accuracy,ables 9 and 10 are included in the appendix, where the three IEEEenchmark systems without meshes under full load are tested byhe proposed, the ladder iteration [10], and the FW/BW sweep

ethods. Table 9 shows the number of iterations and executionime required to achieve convergence. Table 10 lists the relativeolution accuracies based on the comparison between the threeethods. It is observed that the converged solutions are very

lose to each other, except that the proposed method is moreomputationally efficient.

ppendix B. A simple case illustration

The simple 7-node system shown in Fig. 8 is used to illustratehe proposed algorithm for distribution load flow calculation. Theystem data are given in Tables 11 and 12. The bus index obtained

4 40 + j30 = 50∠36.87◦

5 80 + j60 = 100∠36.87◦

6 40 + j30 = 50∠36.87◦

7 80 + j60 = 100∠36.87◦

98 G. Chang et al. / Electric Power Syst

Table 13Bus index of the 7-node illustrative system.

Bus no. Index

1 (0, 1, 1)2 (0, 1, 2)3 (0, 1, 3)4 (0, 1, 4)5 (1, 2, 1)6 (1, 3, 1)

alibostt4svmbfc

Aeroaov

1

TI

7 (1, 3, 2)

ccording to the bus indexing scheme described in Section 3.2 isisted in Table 13. For all busses of branch 1 (busses 1–4), the firstndex is 0, because there is no branch in the upstream branch ofranch 1. Since branch 1 is the main branch in the system, the sec-nd index is 1. The third one is defined by the distance between theubstation and the associated bus. In step (1) of the Forward Sweep,he updated voltage of each bus on branch 1 can be obtained byhe calculated voltages (busses 2 and 3) or the initial voltage (bus) multiplied by the ratios of real and imaginary part of the sub-tation voltage. In the classic BW/FW sweep method, the updatedoltage is calculated for each bus consecutively, while the proposedethod is more computationally efficient by only finding the last

us in the branch to obtain the updated voltage of that bus. There-ore, the updated voltage of bus 4 is obtained without the need ofalculating the updated voltages of busses 2 and 3.

For convenience, the calculations are only performed for phase. Calculations for the other two phases are similar, except thatach phase angle is shifted by 120◦. The specified voltage of theoot node (i.e. the substation, bus 1) is 1000∠0◦ V. For conveniencef calculations, the substation bus voltage is initially shifted by 45◦

nd becomes 1000∠45◦. After convergence is met, the phase anglef each bus voltage is shifted back by −45◦. Table 14 lists initialoltages at three other busses.

. Backward sweep:(1) Starting at node 5, calculate the nodal current (i.e. load cur-

rent):

I5 = 80 + j601000∠45◦ × 1000 = 100∠−8.13◦ = 99−j14.14 = I5−4

This current is also the line section current between nodes(i.e. busses) 5 and 4, I5–4.

(2) Using the current I5 to calculate the node voltage at bus4 (i.e. the calculated voltage of bus 4 is obtained from thevoltage and current of bus 5):

V4(5 cal) = 1000∠45◦ + (100∠ − 8.13◦) × (0.3 + j0.6)

= 745.3 + j762.26 = 1066.07∠45.645◦

(3) Selecting node 7, calculate the nodal current (i.e. load cur-rent):

I7 = 80+j601000∠45◦ × 1000 = 100∠−8.13◦ = 99−j14.14 = I7−6

(4) Using current I7 to find the calculated voltage of bus 6:

V6(7 cal) = 1000∠45◦ + (100∠ − 8.13◦) × (0.2 + j0.4)

= 732.57 + j743.88 = 1044.04∠45.44◦

able 14nitial bus voltages assigned for the 7-node illustrative system.

Bus no. Voltage (V)

4 1000∠45◦

5 1000∠45◦

7 1000∠45◦

ems Research 84 (2012) 90– 99

(5) Using calculated voltage of bus 6 to obtain the nodal currentof bus 6:

I6 = 40 + j301044.04∠45.44◦ × 1000 = 47.89∠ − 8.57◦

= 47.36 − j7.14

(6) Applying the KCL to determine current flowing from bus 6to bus 4:

I6−4 = I6 + I7−6 = (47.36 − j7.14) + (99 − j14.14)

= 146.36 − j21.28 = 147.9∠ − 8.27◦

(7) Using the calculated voltage of bus 6 and line section cur-rent to compute the calculated voltage of bus 4:

V4(6 cal) =1044.04∠45.44◦+(147.9∠−8.27◦) × (0.2 + j0.4)

= 770.35+j798.16 = 1109.28∠46.02◦

(8) Employing the initial voltage at bus 4 to find the load cur-rent:

I4 = 40 + j301000∠45◦ × 1000 = 50∠ − 8.13◦ = 49.5 − j7.07

(9) Using the KCL to obtain the current of line sections 3–4:

I4−3 = I4 + I5−4 + I6−4 = (49.5 − j7.07) + (99 − j14.14)

+(146.36 − j21.28) = 294.86 − j42.49 = 297.9∠ − 8.2◦

(10) Finding the calculated voltage of bus 3 by the line sectioncurrent I3-4 and the initial voltage of bus 4:

V3(4 cal) = 1000∠45◦ + (297.9∠ − 8.2◦) × (0.3 + j0.6)

= 821.07 + j871.26 = 1197.18∠46.7◦

(11) Using the calculated voltage V3(4 cal) to find the load currentof bus 3:

I3 = 80 + j601197.18∠46.7◦ × 1000 = 83.53∠ − 9.83◦

= 82.3 − j14.26

(12) Employing the KCL to obtain I3–2:

I3−2 = I3 + I4−3 = (82.3 − j14.26) + (294.86 − j42.49)

= 377.16 − j56.75 = 381.41∠ − 8.56◦

(13) Using the line section current I4–3 and the load current ofbus 3 to find the calculated voltage V2(3 cal):

V2(3 cal) =1197.18∠46.7◦+(381.41∠−8.56◦) × (0.2+ j0.4)

= 919.22 + j1010.76 = 1366.24∠47.72◦

(14) Finding the load current of bus 2:

I2 = 40 + j301366.24∠47.72◦ × 1000 = 36.6∠ − 10.85◦

= 35.95 − j6.89

(15) Calculating the line section current between busses 2 and3:

I2−1 = I2 + I3−2 = (35.95 − j6.89) + (377.16 − j56.75)

= 413.11 − j63.46 = 417.96∠ − 8.73◦

(16) Obtaining the calculated voltage of the root bus at the sub-station:

V1(2 cal) =1366.24∠47.72◦+ (417.96∠−8.73◦)×(0.1+j0.2)

= 973.22+j1087.04 = 1459∠48.16◦ = V1(cal)

(17) Check the voltage mismatch between the specified and cal-culated voltages at the root bus, except for the first iteration.

r Syst

2

07

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systems, Int. J. Electr. Power 22 (2000) 521–530.

G. Chang et al. / Electric Powe

If the mismatch is less than the specified convergence value,the solution procedure stops. Otherwise, proceed to nextstep.

. Forward sweep:(1) Using the specified voltage V1(spec.) and the calculated voltage

V1(2 cal) at the root bus to find the ratios of real and imaginaryparts for branch 1. Then, employ these two ratios to updatethe voltage of each bus locating on branch 1:specified voltage : V1(spec.) = 1000∠45◦ = 707.107 + j707.1calculated voltage : V1(2 cal) = 973.22 + j1087.04

real part ratio : Kr1 = 707.107

973.22= 0.7266

imaginary part ratio : Ki1 = 707.107

1087.04= 0.6505

Updating the voltage at bus 2:Vupdated

2-real= 919.22 × 0.7266 = 667.91

Vupdated2-image

= 1010.76 × 0.6505 = 657.5

Vupdated2 = 667.91 + j657.5 = 937.24∠44.55◦V

Updating the voltage at bus 3:Vupdated

3-real= 821.07 × 0.7266 = 596.59

Vupdated3-image

= 871.26 × 0.6505 = 566.75

Vupdated3 = 596.59 + j566.75 = 822.88∠43.53◦V

Updating the voltage at bus 4:Vupdated

4-real= 707.11 × 0.7266 = 513.79

Vupdated4-image

= 707.11 × 0.6505 = 459.98

Vupdated4 = 513.79 + j459.98 = 689.61∠41.84◦V

For this procedure, one can neglect finding the updated volt-ages of busses 2 and 3. Because the updated voltage of bus 4is obtained by the initial voltage of bus 4 multiplied by theratios of the substation, Kr

1 and Ki1. This procedure is differ-

ent from those of the classic BW/FW sweep method and it ismore computationally efficient.

(2) By using the updated voltage at bus 4 and the calculated volt-age V4(5 cal) to find the ratios of real and imaginary parts forupdating the voltage at bus 5:

real part ratio : Kr4−5 = 513.79

745.3= 0.6894

imaginary part ratio : Ki4−5 = 459.98

762.26= 0.6034

Updating the voltage at bus 5:Vupdated

5-real= 707.11 × 0.6894 = 487.48

Vupdated5-image

= 707.11 × 0.6034 = 426.67

Vupdated5 = 487.48 + j426.67 = 647.83∠41.19◦V

(3) Using the same way of step (2) to get the ratios and updatethe voltages for busses 6 and 7:

513.79

real part ratio : Kr

4−6 =770.35

= 0.6669

imaginary part ratio : Ki4−6 = 459.98

798.16= 0.5763

Updated voltage of bus 6:

[

[

ems Research 84 (2012) 90– 99 99

Vupdated6-real

= 732.57 × 0.6669 = 488.6

Vupdated6-image

= 743.88 × 0.5763 = 428.7

Vupdated6 = 488.6 + j428.7 = 650∠41.26◦V

Updated voltage of bus 7:Vupdated

7-real= 707.11 × 0.6669 = 471.57

Vupdated7-image

= 707.11 × 0.5763 = 407.51

Vupdated7 = 471.57 + j428.7 = 637.31∠42.27◦V

3. End of the first iteration, proceed to next iteration.

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