power flow analysis

International Journal of Advances in Engineering & Technology, March 2012.

©IJAET ISSN: 2231-1963

514 Vol. 3, Issue 1, pp. 514-524

POWER FLOW ANALYSIS OF THREE PHASE UNBALANCED

RADIAL DISTRIBUTION SYSTEM

Puthireddy Umapathi Reddy1, Sirigiri Sivanagaraju

2, Prabandhamkam

Sangameswararaju3

1Department of Electrical and Electronics Engineering, Sree Vidyanikethan Engineering

College, Tirupati-517102, India. 2Department of Electrical and Electronics Engineering, Jawaharlal Nehru Technological

University College of Engineering Kakinada, Kakinada-533 003, India. 3Department of Electrical and Electronics Engineering, Sri Venkateswara University College

of Engineering, Tirupati-517502, India.

ABSTRACT This paper provides a new approach for power flow and modeling analysis of three phase unbalanced radial

distribution systems (URDS) using the simple forward/backward sweep-based algorithm. A three phase load

flow solution is proposed considering voltage regulator and transformer with detailed load modeling, for the

transformer modeling symmetrical components theory is used and zero sequence-voltage and-current updating

for the sweep-based methods is shown. The validity and effectiveness of the proposed method is demonstrated by

a simple 19-bus unbalanced system for grounded wye-delta and delta grounded wye transformer connections.

Results are in agreements with the literature and show that the proposed model is valid and reliable.

KEYWORDS: Distribution System, Forward-Backward sweep-based methods, Three-phase load model

analysis, Power flow analysis.

I. INTRODUCTION

Power distribution systems have different characteristics from transmission systems [1],[2].They are

characterized as Radial/weakly meshed structures, Unbalanced networks/loads: single, double and

three phase loads, High resistance/reactance(R/X) ratio of the lines, Extremely large number of

branches/nodes, Shunt capacitor banks and distribution transformers, Low voltage levels compared

with those of transmission systems and distributed generators[3].[4]. Because of the inherent

unbalanced nature of the power distribution system, each bus may be having loads that can be three-

phase grounded wye or ungrounded delta connected, two-phase grounded or single-phase grounded

[5]. The unbalanced nature of power distribution systems requires special three phase component and

system models [6]. The operation and planning studies of distribution system requires a steady state

conditions of system can be obtained from the load flow solution[7],[8].The efficiency of the entire

process depends heavily on the efficiency and capability of the load flow program used for this

purpose[9].

Most of the researchers presented techniques, especially to obtain the load flow solution of

distribution networks [10],[11] have proposed a load flow solution method by writing an algebraic

equation for bus voltage magnitude. However this method is suitable for single-phase analysis [12]. A

few researchers have proposed load flow solution techniques[17] to analyze unbalanced distribution

networks [13],[14] have formulated load flow problem as a set of non linear power mismatch

equations as a function of the bus voltages. These equations have been solved by Newton’s method

[15],[16] have proposed three phase power flow algorithm based on the forward and back word walk

along the network. The method considers some aspects of three phase modelling of branches and

detailed load modelling [18],[19]. P. Aravindhababu, proposed a method [20], A new fast decoupled



515 Vol. 3, Issue 1, pp. 514-524

power flow method for distribution systems. Improvements in the representation of PV buses on

three-phase distribution power flow [21] are proposed. A new approach have given [22], [23] for

three-Phase Fast Decoupled Load Flow for Unbalanced Distribution Systems. T.H. Chen, N.C. Yang,

proposed three-phase power-flow by direct ZBR method for unbalanced radial distribution systems

[24]. A Simple and Direct Approach for Unbalanced Radial Distribution System three phase Load

Flow Solution [25] have been explained. The significance of this power flow analysis is to apply load

flow data for capacitor placement, network reconfiguration, voltage regulator placement etc. in

URDS.

This paper presents an algorithm for solving load model and power flow analysis of three-phase

unbalanced radial distribution systems. The algorithm is capable of solving for systems with many

feeders emanating from grid substation with large number of nodes and branches. This paper

considers all types of load modelling i.e distribution system line model, line shunt admittance model

distributed load model, capacitor model, transformer modeling, Forward-Backward Sweep (FBS) load

flow, algorithm for load flow, results and discussion, conclusions and references. Based on the

proposed algorithm, a computer program has been developed using MATLAB and results are

presented for typical network of 19-node URDS.

II. MODELING OF UNBALANCED RADIAL DISTRIBUTION SYSTEM

Radial distribution system can be modeled as a network of buses connected by distribution lines,

switches or transformers. Each bus may also have a corresponding load, shunt capacitor and/or co-

generator connected to it. This model can be represented by a radial interconnection of copies of the

basic building block shown in Figure 1. Since a given branch may be single-phase, two-phase, or

three-phase, each of the labeled quantities is respectively a complex scalar, a 2 × 1, or a 3 × 1

complex vector. The model consist of distribution line with are without voltage regulator or Switch or

Transformer.

Figure 1. Basic building block of unbalanced radial distribution system inclusion of all models

2.1 Distribution system line model

For the analysis of power transmission line, two fundamental assumptions are made, namely: Three-

phase currents are balanced and Transposition of the conductors to achieve balanced line parameters.

A general representation of a distribution system with N conductors can be formulated by resorting to

the Carson’s equations, leading to a N× N primitive impedance matrix. The standard method used to

form this matrix is the Kron reduction, based on the Kirchhoff’s laws. For instance a four-wire

grounded star connected overhead distribution line shown in figure 2 results in a 4× 4 impedance

matrix.

The corresponding equations are



516 Vol. 3, Issue 1, pp. 514-524

Figure 2. Model of the three-phase four wire distribution line

(1)

It can be represented in matrix form as

+=nj

Innj

znqV

npV

abcjI

Tnj

z

nj

zabcj

ZabcqV

abcpV

(2)

If the neutral is grounded, the voltage npV and n

qV can be considered to be equal. From the lst row of

eqn. (2) it is possible to obtain

abcjI

Tnj

z1nn

jzn

jI

−−=

(3)

and substituting eqn.(3) into eqn. (2), the final form corresponding to the Kron’s reduction becomes

abcjIabc

jZeabc

qVabcpV += (4)

Where

=−

−=

ccj

zecbj

zecaj

ze

bcj

zebbj

zebaj

ze

acj

zeabj

zeaaj

ze

Tnj

z1nn

jznj

zabcj

Zabcj

Ze (5)

abcjI is the Current vector through line between nodes p and q can be equal to the sum of the load

currents of all the nodes beyond line between node p and q plus the sum of the charging currents of all

the buses beyond line between node p and q, of each phase.

Therefore the bus q voltage can be computed when we know the bus p voltage, mathematically, by

rewriting eqn. (4)

ac

jz

bn

jz

an

jz

b

jI

c

jI

n

jI

Bus q a

jI Bus p

apV

} cn

jz

} bc

jz

} ab

jzbb

jz

cc

jz

nn

jz

aa

jz

bpV

cpV c

qV

bqV

aqV

a•

b•

c•

n•

•a

•b

•c

•n

+

=

n

j

c

j

b

j

a

j

nn

j

nc

j

nb

j

na

j

cn

j

cc

j

cb

j

ca

j

bn

j

bc

j

bb

j

ba

j

an

j

ac

j

ab

j

aa

j

n

q

c

q

b

q

a

q

n

p

c

p

b

p

a

p

I

I

I

I

zzzz

zzzz

zzzz

zzzz

V

V

V

V

V

V

V

V



517 Vol. 3, Issue 1, pp. 514-524

−=

cjI

bjI

ajI

ccj

zecbj

zecaj

ze

bcj

zebbj

zebaj

ze

acj

zeabj

zeaaj

ze

cpV

bpV

apV

cqV

bqV

aqV

(6)

2.2 Line shunt admittance model

These current injections for representing line charging, which should be added to the respective

compensation current injections at nodes p and q, are given by

( )( )

( )

++−

++−

++−

=

c

q

b

q

a

q

cc

j

cb

j

ca

j

cb

j

ca

j

bc

j

bc

j

bb

j

ba

j

ba

j

ac

j

ab

j

ac

j

ab

j

aa

j

c

q

b

q

a

q

V

V

V

yyyyy

yyyyy

yyyyy

Ish

Ish

Ish

2

1 (7)

2.3 Distribution System Load Model

Constant Power: Real and reactive power injections at the node are kept constant. This load

corresponds to the traditional PQ approximation in single-phase analysis.

Constant Impedance: These types of loads are useful to model large industrial loads. The impedance

of the load is calculated by the specified real and reactive power at nominal voltage and is kept

constant.

Constant Current: The magnitude of the load current is calculated by the specified real and reactive

power at nominal voltage and is kept constant.

2.3.1 Distributed load model Sometimes the primary feeder supplies loads through distribution transformers tapped at various

locations along line section. If every load point is modeled as a node then there are a large number of

nodes in the system. So these loads are represented as lumped loads. At one fourth length of line from

sending node, where two thirds of the load is connected. For this a dummy node is created. One third

loads is connected at the receiving node.

In the unbalanced distribution system, loads can be uniformly distributed along a line. When the loads

are uniformly distributed it is not necessary to model each and every load in order to determine the

voltage drop from the source end to the last loads.

2.4 Capacitor model

Shunt capacitor banks are commonly used in distribution systems to help in voltage regulation and to

provide reactive power support. The capacitor banks are modeled as constant susceptances connected

in either star or delta. Similar to the load model, all capacitor banks are modeled as three-phase banks

with the currents of the missing phases set to zero for single-phase and two-phase banks.

2.5 Transformer modeling

Three-phase transformer is represented by two blocks shown in Figure 3. One block represents the per

unit leakage admittance matrix abcTY and the other block models the core loss as a function of

voltage on the secondary side.



518 Vol. 3, Issue 1, pp. 514-524

Figure 3. General Three-phase Transformer Model

Now that ′abc

SPY is not singular, the non zero sequence components of the voltages on the primary

side can be determined by

′ ′−

′′= abc

SVabc

SSYabc

sI

1-abcSPYabc

PV

(8)

Similar results can be obtained for forward sweep calculation

″ ″−

″″= abc

PVabc

SPYabc

sI

1-abcSSYabc

SV (9)

Where″abc

SV is the nonzero sequence component of abc

SV ,

″abcSSY is same as abc

SSY , except

htat the last row is replaced with [1 1 1 ], ″abc

sI and

″abcSPY are obtained by setting the elements in

the last row of abcSI and abc

SPY to 0, respectively. Once the nonzero-sequence components of

abcPV or abc

SV are calculated, zero-sequence components are added to them to form the line-to-

neutral voltages so that the forward/backward sweep procedure can continued.

III. FORWARD - BACKWARD SWEEP (FBS) LOAD FLOW METHOD

3.1 Backward Sweep:

The purpose of the backward sweep is to update branch currents in each section, by considering the

previous iteration voltages at each node. During backward propagation voltage values are held

constant at the values obtained in the forward path and updated branch currents are transmitted

backward along the feeder using backward path. Backward sweep starts from extreme end branch and

proceeds along the forward path.

Figure 4.Single phase line section with load connected at node q between phase ‘a’& neutral n.

Figure 4 shows phase a of a three-phase system where lines between nodes p and q feed the node q

and all the other lines connecting node q draw current from line between node p and q.

a

qIL

ajI a

jI

p

aqV

aajze

aqS

q

aqIC



519 Vol. 3, Issue 1, pp. 514-524

During this propagation different load currents and capacitor currents (if exist) are calculated using

mathematical models of loads and capacitors presented in section 2.1.

The line charging currents of all the branches are added to the load current. Figure 5 shown a branch

‘j’ of the distribution network, connected between two nodes p and q and M sub-laterals are connected

to it. The parent branch current feeds the load at the th

q node and the sub-laterals connected to the

parent branch. This current can be calculated using Eqn. (10).

Figure 5. Branch j of distribution network connected to M sub-laterals

∑∈

∑∈

−++=

Mm Mm

1k

mabcqV

k

mabcshY

kabcmI

kabcqIL

kabcjI (10)

Where

k

mabcshY is the half line shunt admittance of the branch in k

th iteration.

kabcjI is the branch current vector in line section j in k

th iteration.

kabcmI is the current vector in branch m before updating in k

th iteration.

1k

mabcqV

− is the voltage vector of the branch m in (k-1)

th iteration.

M represents the set of line sections connected to jth branch

If capacitor bank is placed at the receiving end of the branch then capacitor current should also be

included. Table 1. shows a mathematical Models of different loads(star & delta connected) which

gives constant power, constant impedance and constant current. Another advantage of the proposed

method is all the data is stored in vector form, thus saving an amount of computer memory. The

proposed method finds extensive use in network reconfiguration, capacitor placement and voltage

regulator placement studies. Table 1. Mathematical Models of different loads



520 Vol. 3, Issue 1, pp. 514-524

3.2 Forward Sweep

The purpose of the forward sweep is to calculate the voltages at each node starting from the source

node. The source node voltage is set as 1.0 per unit and other node voltages are calculated as

−+=

kabcjI

kabcpV

abcshYabc

jZe

kabcpV

kabcqV (11)

Where

kabcpV

kabcqV , are the voltage vectors of phases for p

th and q

th nodes respectively in k

th

iteration.

=

ccj

zecbj

zecaj

ze

bcj

zebbj

zebaj

ze

acj

zeabj

zeaaj

ze

abcj

Ze

kabcjI is the current vector in j

th branch in k

th iteration.

These calculations will be carried out till the voltage at each bus is within the specified limits.

Therefore the real and reactive power losses in the line between nodes p and q may be written as:

*)abcj

)(IabcqV

abcp(V

abcj

S −= (12)

Where

abcj

S is a vector of power loss with three, two or single phase

abcpV and abc

qV are voltage vector of three phases at nodes p and q

abcjI is the branch current vector of three phases for the section connected in between p

th and q

th

node

3.3 Forward Backward sweep method algorithm

Step 1: Read input data regarding the unbalanced radial distribution system.

Step 2: Determine forward Backward propagation paths.

Step 3: Initialize the voltage magnitude at all nodes as 1 p.u and voltage angles to be 00, -120

0,

and 1200

for phase A, phase B and phase C respectively.

Step 4: Determine forward Backward propagation paths.

Step 5: Initialize the voltage magnitude at all nodes as 1 p.u and voltage angles to be 00, -120

0,

and 1200

for phase A, phase B and phase C respectively.

Step 6: Set iteration count k=1 and∈ = 0.0001

Step 7: Calculate load currents and capacitor currents(if exist) at all nodes.

Step 8: Calculate the branch currents using eqn. (10) in the backward sweep.

Step 9: Calculate node voltages using eqn. (11) in the forward sweep

Step10: Check for the convergence, if the difference between the voltage magnitudes in two

consecutive iterations is less than ∈ then go to step 9 else set k=k+1 and go to step 5.

Step11: Calculate real and reactive power loss in each branch.

Step12: Print voltages and power losses at each node.

Step13: Stop.



521 Vol. 3, Issue 1, pp. 514-524

IV. RESULTS AND DISCUSSION

This method computes the power flow solution for its given radial network with its loadings

and illustrated with 19 node test system of unbalanced radial distribution system. The outcome

of this paper is to apply load flow data for capacitor placement, network reconfiguration, voltage

regulator placement etc. in URDS are very useful.

4.1 Example – 1

A 19 node unbalanced radial distribution system is shown in Figure 6. The line and load data are

given in [23]. For the load flow the base voltage and base MVA are chosen as 11 kV and 1000 kVA

respectively. Table2: Voltage and Phase angles of 19 node URDS

Figure 6. Single line diagram of 19 node URDS

Node

No.

Existing method [23] Proposed method

Phase A Phase B Phase C Phase A Phase B Phase C

aV

(p.u)

aV∠

deg

bV

(p.u)

bV∠

deg

cV

(p.u)

cV∠

deg

aV

(p.u)

aV∠

deg

bV

(p.u)

bV∠

deg

cV

(p.u)

cV∠

deg

1 1.0000 0.00 1.0000 -120.06 1.0000 120.06 1.0000 0.00 1.0000 -120.00 1.0000 120.00

2 0.9875 0.01 0.9891 -120.04 0.9880 120.11 0.9874 0.01 0.9890 -119.98 0.9878 120.05

3 0.9854 0.00 0.9887 -120.04 0.9863 120.14 0.9854 0.00 0.9885 -119.98 0.9862 120.06

4 0.9824 0.03 0.9839 -120.02 0.9830 120.12 0.9823 0.03 0.9838 -119.97 0.9829 120.06

5 0.9820 0.03 0.9837 -120.03 0.9828 120.12 0.9820 0.03 0.9836 -119.97 0.9826 120.07

6 0.9793 0.04 0.9808 -120.02 0.9801 120.13 0.9791 0.04 0.9805 -119.96 0.9799 120.07

7 0.9786 0.04 0.9803 -120.02 0.9796 120.13 0.9786 0.04 0.9801 -119.96 0.9794 120.08

8 0.9728 0.06 0.9738 -120.00 0.9735 120.14 0.9727 0.06 0.9737 -119.94 0.9733 120.08

9 0.9659 0.08 0.9660 -119.97 0.9657 120.14 0.9657 0.08 0.9658 -119.91 0.9656 120.09

10 0.9560 0.10 0.9555 -119.93 0.9550 120.16 0.9562 0.09 0.9552 -119.86 0.9548 120.09

11 0.9550 0.10 0.9543 -119.92 0.9533 120.17 0.9548 0.10 0.9543 -119.86 0.9533 120.10

12 0.9548 0.11 0.9538 -119.92 0.9536 120.16 0.9547 0.11 0.9536 -119.87 0.9535 120.10

13 0.9544 0.10 0.9534 -119.90 0.9521 120.17 0.9544 0.10 0.9535 -119.85 0.9521 120.11

14 0.9545 0.10 0.9539 -119.91 0.9528 120.17 0.9543 0.10 0.9537 -119.86 0.9528 120.11

15 0.9526 0.11 0.9510 -119.91 0.9512 120.15 0.9526 0.11 0.9510 -119.83 0.9511 120.12

16 0.9535 0.13 0.9514 -119.91 0.9522 120.15 0.9533 0.13 0.9514 -119.86 0.9521 120.10

17 0.9536 0.10 0.9533 -119.91 0.9522 120.16 0.9534 0.10 0.9531 -119.90 0.9519 120.11

18 0.9537 0.10 0.9531 -119.92 0.9522 120.16 0.9536 0.10 0.9530 -119.82 0.9520 120.10

19 0.9516 0.13 0.9498 -119.91 0.9505 120.16 0.9515 0.13 0.9496 -119.86 0.9503 120.10



522 Vol. 3, Issue 1, pp. 514-524

Table3: Active and Reactive Power flows of 19 node URDS

Table4: Summary of test results of 19 node URDS

Voltage profile with comparison of the proposed method with existing method and active and reactive

Power flows of 19 node URDS are given in table 2 and 3. Voltage variation is given in table 2, which

gives better magnitudes are obtained in proposed method. The active power flow gives higher power

flow capacity with proposed method shown in Table 3. The Table 4 gives summary of test results for

19 node unbalanced radial distribution systems. From table 4 it has been observed that the minimum

voltage in phases A, B, C is 0.9515, 0.9496 and 0.9502 at node 19. The maximum percentage voltage

regulation in phases A, B and C are 4.82%, 5.01% and 4.93%. The total active power loss in phases of

A, B and C are 4.34, 4.42 and 4.54 kW and the total reactive power loss in phases of A, B and C are

1.95,1.90 and 1.94 kVA respectively. The real power losses in phases A, B and C are 3.64%, 3.96%

and 3.78% and the reactive power losses are 3.32%, 3.34% and 3.33% of their total loads. The

solution is converged in 4 iterations and time taken is 0.00645 seconds for 19 node URDS. The

proposed method is capable of solving for systems with many feeders emanating from grid substation

with large number of nodes compared with the existing method [23] and results are found satisfactory.

V. CONCLUSIONS

In this paper, a simple algorithm has been presented to solve power flow and load modeling i.e

distribution system line model, line shunt admittance model, Distributed load model, capacitor model

and transformer modeling of unbalanced radial distribution networks. The proposed method has good

Description Phase A Phase B Phase C

Minimum Voltage 0.9515 0.9496 0.9503

Max. Voltage regulation (%) 4.82 5.01 4.93

Total Active Power Loss (kW) 4.34 4.42 4.54

Total Reactive Power Loss (kVAr) 1.95 1.90 1.94

Total Active Power Demand (kW) 126.32 116.14 123.17

Total Reactive Power Demand (kVAr) 61.12 56.13 59.65

Total Feeder Capacity (kVA) 140.23 129.21 136.67



523 Vol. 3, Issue 1, pp. 514-524

convergence property for practical distribution networks with practical R/X ratio. Computationally,

this method is extremely efficient; as it solves simple algebraic recursive equations for voltage

phasers and another advantage is all the data is stored in vector form, thus saving computer memory.

The Forward-Backward Sweep (FBS) algorithm is capable of solving for systems with many feeders

emanating from grid substation with large number of nodes and branches. A computer program has

been developed using MATLAB and results are presented for typical network of 19-node URDS.

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Authors

P.UMAPATHI REDDY: He Received B.E from Andra University and M.Tech.,(Electrical

Power Systems) from Jawaharlal Nehru Technological University, Anantapur, India in 1998

and 2004 respectively, Now he is pursuing Ph.D. degree. Currently he is with Department of

Electrical and Electronics Engineering, Sree Vidyanikethan Engineering College, Tirupati,

India. His research interest includes Power distribution Systems and Power System operation

and control. He is Life Member of Indian Society for Technical Education.

S.Sivanaga Raju: He received B.E from Andra University and M.Tech.degree in 2000 from

IIT, Kharagpur and did his Ph.D from Jawaharlal Nehru Technological University,

Anantapur, India in 2004. He is presently working as Associate professor in J.N.T.U.College

of Engineering Kakinada,(Autonomous) Kakinada, Andrapradesh, India. He received two

national awards (Pandit Madan Mohan Malaviya memorial Prize and best paper prize award

from the Institute of Engineers (India) for the year 2003-04. He is referee for IEEE journals.

He has around 75 National and International journals in his credit. His research interest

includes Power distribution Automation and Power System operation and control.

P. Sangameswara Raju: He is presently working as professor in S.V.U. College

Engineering, Tirupati. Obtained his diploma and B.Tech in electrical Engineering, M.Tech in

power system operation and control and Ph.d in S. V. University, Tirupati. is areas of interest

are power system operation, planning and application of fuzzy logic to power system,

application of power system like non-linear controllers.

power flow analysis

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