Download power flow analysis of three phase unbalanced

International Journal of Advances in Engineering & Technology, March 2012.
©IJAET
ISSN: 2231-1963
POWER FLOW ANALYSIS OF THREE PHASE UNBALANCED
RADIAL DISTRIBUTION SYSTEM
Puthireddy Umapathi Reddy1, Sirigiri Sivanagaraju2, Prabandhamkam
Sangameswararaju3
1
Department of Electrical and Electronics Engineering, Sree Vidyanikethan Engineering
College, Tirupati-517102, India.
2
Department of Electrical and Electronics Engineering, Jawaharlal Nehru Technological
University College of Engineering Kakinada, Kakinada-533 003, India.
3
Department 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 threephase 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
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Vol. 3, Issue 1, pp. 514-524
International Journal of Advances in Engineering & Technology, March 2012.
©IJAET
ISSN: 2231-1963
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 cogenerator 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: Threephase 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
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Vol. 3, Issue 1, pp. 514-524
International Journal of Advances in Engineering & Technology, March 2012.
©IJAET
ISSN: 2231-1963
Bus p
z
a•
b•
I bj
V pa
V pb
c•
Bus q
I aj
I
V pc
c
j
I nj
n•
aa
j
z bb
j
z ccj
z
nn
j
}z
}z
}z
•a
ab
j
z ac
j
bc
j
cn
j
z bn
j
z
Vqa
an
j
Vqc
Vqb
•b
•c
•n
Figure 2. Model of the three-phase four wire distribution line
 V pa 
 V qa 
 z aaj
  I aj 
z abj
z acj
z an
j
 b 
 b 
 ba
 

z bbj
z bcj
z bnj   I bj 
V p 
V q 
z j
 c  =  c  +  ca
 

(1)
z cbj
z ccj
z cnj   I cj 
V p 
V q 
z j
 n 
 n 
 na
nb
nc
nn  
n 
z j
z j
z j   I j 
 V p 
 V q 
 z j
It can be represented in matrix form as
 abc z n   abc 
V abc  V abc  Z j
j  I j

p
q

=
+
(2)
 

 n
  n
  nT
n
nn

I
V
V


z
 p
  q
 z j
j   j


n and V n can be considered to be equal. From the lst row of
If the neutral is grounded, the voltage V p
q
eqn. (2) it is possible to obtain
−1 nT abc
I nj = −z nn
zj Ij
j
(3)
and substituting eqn.(3) into eqn. (2), the final form corresponding to the Kron’s reduction becomes
abc = V abc + Ze abc I abc
Vp
q
j
j
(4)
Where
ze aa ze ab ze ac 
 j
j
j 


abc − z n z nn −1z n T = ze ba ze bb ze bc 
Ze abc
=
Z
j
j
j j
j
j
j 
 j
 ca
cb
cc 
ze j ze j ze j 


(5)
I abc
is the Current vector through line between nodes p and q can be equal to the sum of the load
j
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)
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Vol. 3, Issue 1, pp. 514-524
International Journal of Advances in Engineering & Technology, March 2012.
©IJAET
ISSN: 2231-1963
a  ze aa ze ab ze ac 
V qa  V p
 j
j
j 
 
 


 b   b   ba
bb ze bc 
V
=
V
−
ze
ze
 q   p  j
j
j 
 c   c  

V q  V p  ze ca ze cb ze cc 
j
j
j


I a 
 j 
 b
I j 
 
I c 
 j 
(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
(
ab
ac
 Ishqa 
− y aa
j + yj + yj
 b 1
y ba
 Ishq  = 
j
2
 c

ca
yj

 Ishq 
)
y ab
j
(
bb
bc
− y ba
j + yj + yj
y cbj
y ac
j
)
y bcj
(
− y caj + y cbj + y ccj
)





Vqa 
 b
Vq 
 c
Vq 
(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 Y abc and the other block models the core loss as a function of
T
voltage on the secondary side.
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Vol. 3, Issue 1, pp. 514-524
International Journal of Advances in Engineering & Technology, March 2012.
©IJAET
ISSN: 2231-1963
Figure 3. General Three-phase Transformer Model
′
Now that Y abc is not singular, the non zero sequence components of the voltages on the primary
SP
side can be determined by
′
′ 
V Pabc = Y abc 
 SP 
-1
′
 abc ′
I
− Y abc V abc
SS
S
 s

 
Similar results can be obtained for forward sweep calculation
″
″ 
VSabc = Y abc 
 SS


-1





(8)
″
 abc ″
I
− Y abc V abc
SP
P
 s





(9)
″
″
Where V abc is the nonzero sequence component of V abc , Y abc is same as Y abc , except
S
S
SS
SS
″
″
htat the last row is replaced with [1 1 1 ], I abc and Y abc are obtained by setting the elements in
s
SP
the last row of I abc and Y abc to 0, respectively. Once the nonzero-sequence components of
S
SP
abc or V abc are calculated, zero-sequence components are added to them to form the line-toVP
S
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.
p
I aj
I aj
zeaa
j
Vqa
q
ILaq
S qa
ICqa
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.
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Vol. 3, Issue 1, pp. 514-524
International Journal of Advances in Engineering & Technology, March 2012.
©IJAET
ISSN: 2231-1963
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
q
th
to it. The parent branch current feeds the load at the
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
k
abc k +
I abc
=
IL
q
j
abc
∑ Im
m ∈M
k
+
∑
m ∈M
 abc k
Y
 sh m





 abc k − 1 
V

 q

m


(10)
Where
k
is the half line shunt admittance of the branch in kth iteration.
Y abc
sh m
k
is the branch current vector in line section j in kth iteration.
I abc
j
abc k is the current vector in branch m before updating in kth iteration.
Im
k −1
is the voltage vector of the branch m in (k-1)th iteration.
Vqabc
m
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
519
Vol. 3, Issue 1, pp. 514-524
International Journal of Advances in Engineering & Technology, March 2012.
©IJAET
ISSN: 2231-1963
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


k
abc k + Ze abc Y abc  V abc k  − I abc k 
Vqabc = V p
(11)




j
j
sh   p





Where
Vqabc
k
abc k are the voltage vectors of phases for pth and qth nodes respectively in kth
,Vp
iteration.
ze aa
 j

abc
Ze j
= ze ba
j

ca
 ze
 j
I abc
j
k
ze ab
j
bb
ze j
ze cb
j
ze ac 
j

bc
ze j 


ze cc
j 
is the current vector in jth branch in kth 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:
abc − V abc )(I abc )*
S abc
= (V p
q
j
j
(12)
Where
S abc
is a vector of power loss with three, two or single phase
j
abc and V abc are voltage vector of three phases at nodes p and q
Vp
q
I abc
is the branch current vector of three phases for the section connected in between pth and qth
j
node
3.3 Forward Backward sweep method algorithm
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
Step 7:
Step 8:
Step 9:
Step10:
Step11:
Step12:
Step13:
520
Read input data regarding the unbalanced radial distribution system.
Determine forward Backward propagation paths.
Initialize the voltage magnitude at all nodes as 1 p.u and voltage angles to be 00, -1200,
and 1200 for phase A, phase B and phase C respectively.
Determine forward Backward propagation paths.
Initialize the voltage magnitude at all nodes as 1 p.u and voltage angles to be 00, -1200,
and 1200 for phase A, phase B and phase C respectively.
Set iteration count k=1 and ∈ = 0.0001
Calculate load currents and capacitor currents(if exist) at all nodes.
Calculate the branch currents using eqn. (10) in the backward sweep.
Calculate node voltages using eqn. (11) in the forward sweep
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.
Calculate real and reactive power loss in each branch.
Print voltages and power losses at each node.
Stop.
Vol. 3, Issue 1, pp. 514-524
International Journal of Advances in Engineering & Technology, March 2012.
©IJAET
ISSN: 2231-1963
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
Existing method [23]
Phase A
Node
No.
Phase B
Proposed method
Phase C
Phase A
Phase B
Phase C
Va
∠V a
Vb
∠Vb
Vc
∠Vc
Va
∠V a
Vb
∠Vb
Vc
∠ Vc
(p.u)
deg
(p.u)
deg
(p.u)
deg
(p.u)
deg
(p.u)
deg
(p.u)
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
Figure 6. Single line diagram of 19 node URDS
521
Vol. 3, Issue 1, pp. 514-524
International Journal of Advances in Engineering & Technology, March 2012.
©IJAET
ISSN: 2231-1963
Table3: Active and Reactive Power flows of 19 node URDS
Table4: Summary of test results of 19 node URDS
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
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
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ISSN: 2231-1963
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.
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