Download International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.9, pp. 1539-1544

International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.9, pp. 1539-1544
ISSN 2078-2365
http://www.ieejournal.com/
Backward/Forward Sweep Based
Distribution
Load Flow Method
S. Sunisith1, K. Meena2
[email protected], [email protected]

Abstract—The function of an electric power system is to
connect the power station to the consumer loads by means of
inter connected system of transmission and distribution
networks. Therefore, an electric power system consists of three
principal components: The power station, the transmission
lines and the distribution system. The power flow method is a
fundamental tool in application software for distribution
management system. In this paper, a method to solve the
distribution power flow problem has been introduced. The
reason, why the convergence of these widely used methods
deteriorates when the network becomes radial, is also well
analyzed. Subsequently, a theoretical formulation of
backward/forward sweep distribution load flow method is
described. This proposed method has clear theory foundation
and takes full advantage of the radial structure of distribution
systems. The numerical test proved that, this method is very
robust and has excellent convergence characteristics.
Index Terms—Backward/Forward Sweep, Distribution load
flow method.
I. CONNECTION SCHEMES OF DISTRIBUTION
SYSTEM
All distribution of electrical energy is done by constant
voltage system. In practice, the following distribution circuits
are generally used
i) Radial System:
In this system, separate feeders radiate from a single
substation and feed the distributors at one end only. Fig. 1.1(i)
shows a single line diagram of a radial system for dc
distribution where a feeder OC supplies a distributor AB at
point A. Obviously, the distributor is fed at one end only i.e.,
point A is this case. Fig. 1.1(ii) shows a single line diagram of
radial system for ac distribution. The radial system is
employed only when power is generated at low voltage and
the substation is located at the centre of the load.
Fig. 1.1 (i) & (ii) Radial System
This is the simplest distribution circuit and has the lowest
initial cost. However, it suffers from the following drawbacks
a) The end of the distributor nearest to the feeding
point will be heavily loaded.
b) The consumers are dependent on a single feeder and
single distributor. Therefore, any fault on the feeder
or distributor cuts off supply to the
c) Consumers who are on the side of the fault away
from the substation.
d) The consumers at the distant end of the distributor
would be subject to serious voltage fluctuations
when the load on the distributor changes. Due to
these limitations, this system is used for short
distances only.
ii) Ring main system:
In this system, the primaries of distribution
transformers form a loop. The loop circuit starts from the
substation bus-bars, makes a loop through the area to be
served, and returns to the substation. Fig. 1.2 shows the
single line diagram of ring main system for ac
distribution where substation supplies to the closed
feeder LMNOPQRS. The distributors are tapped from
different points M, O and Q of the feeder through
distribution transformers. The ring main system has the
following advantages
a) There are less voltage fluctuations at consumer’s
terminals.
b) The system is very reliable as each distributor is fed
via two feeders. In the event of fault on any section of the
feeder, the continuity of supply is maintained. For example,
1539
S. Sunisith et. al.,
Backward/Forward Sweep Based Distribution Load Flow Method
International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.9, pp. 1539-1544
ISSN 2078-2365
http://www.ieejournal.com/
suppose that fault occurs at any point F of section SLM of the
feeder. Then section SLM of the feeder can be isolated for
repairs and at the same time continuity of supply is maintained
to all the consumers via the feeder SRQPONM.
Fig.1.2 Ring main System
(iii) Interconnected system: When the feeder ring is
energized by two or more than two generating stations or
substations, it is called inter-connected system. Fig. 1.3
shows the single line diagram of interconnected system
where the closed feeder ring ABCD is supplied by two
substations S1 and S2 at points D and C respectively.
Distributors are connected to points O, P, Q and R of the
feeder ring through distribution transformers.
performance of a power system and for analyzing the
effectiveness of alternative plans for system expansion to
meet increased load demand. These analyses require the
calculation of numerous load flows for both normal and
emergency operating conditions.
The load flow problem consists of the calculation of
power flows, and voltages of a network for specified terminal
or bus conditions. A single phase representation is adequate
since power systems are usually balanced. Associated with
each bus are four quantities: the real and reactive power, the
voltage magnitude and the phase angle. Three types of buses
are represented in the load flow calculation and at a bus, two
of the four quantities are specified. It is necessary to select one
bus, called the slack bus, to provide the additional real and
reactive power to supply the transmission losses. At this bus
the voltage magnitude and phase angle are specified. The
remaining buses of the system are designated either as voltage
controlled buses or load buses. The real power and voltage
magnitudes are specified at a voltage controlled bus. The real
and reactive powers are specified at a load bus.
The static load flow equations are given by
Pp 
  Ep Eq Ypq cos(pq  q  p)
n
q 1
Qp  -
  Ep Eq Ypq sin(pq  q  p)
n
q 1
The above equations are said to be non linear in
nature because of involvement of trigonometric terms. Direct
solution is not possible, we need to apply iterative Techniques
to solve the equations.
Those are
i.
Gauss -Seidal Method
ii.
Newton- Raphson Method
iii.
Newton’s Decoupled Method
iv.
Fast Decoupled Method.
Fig. 1.3 Inter Connected System
The interconnected system has the following
advantages:
(a) It increases the service reliability.
(b) Any area fed from one generating station during peak
load hours can be fed from the other generating station.
This reduces reserve power capacity and increases
efficiency of the system.
II. LOAD FLOW STUDIES
Load flow calculations provide power flows and voltages
for a specified power system subject to the regulating
capability of generators, condensers and tap changing under
load transformers as well as specified net interchange
between individual operating systems. This information is
essential for the continuous evolution of the current
III. BACKWARD/FORWARD SWEEP BASED
DISTRIBUTION LOAD FLOW METHOD
Our studies have shown that, typically, only a few
iterations were required for the solution of distribution
networks using this power flow solution technique. For the
weakly meshed transmission networks the number of
iterations was higher, due to the additional nonlinearities
introduced by generator buses (PV nodes). In all the cases
studied this power flow technique was significantly more
efficient than the Newton-Raphson power flow algorithm
while converging to the same solution.
In this chapter a brief discussion of Breadth First Search
method and its applications are provided. Then we emphasize
the application of the backward/forward sweep based
1540
S. Sunisith et. al.,
Backward/Forward Sweep Based Distribution Load Flow Method
International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.9, pp. 1539-1544
ISSN 2078-2365
http://www.ieejournal.com/
distribution load flow method to the distribution networks and
few practical considerations concludes the Paper.
In our algorithm, regardless of its original topology, the
distribution network is first converted to a radial network.
Hence, an efficient algorithm for the solution of radial
networks is crucial to the viability of the overall solution
method. The solution method used for radial distribution
networks is based on the direct application of the KVL and
KCL. For our implementation, we developed a node and
branch oriented approaches using an efficient numbering
scheme to enhance the numerical performance of the solution
method. We first describe this node numbering scheme.
branches connected to the root node, the current in branch
L, JL is calculated as
JL(k)= - IL2(k) + ∑ (currents in branches
emanating from node L2)
L=b, b-1 ,…….,1
Where IL2(k) is the current injection at node L2. This
is the direct application of the KCL.
IV. BREADTH FIRST SEARCH METHOD (BFS)
In graph theory breadth-first search (BFS) is a graph search
algorithm that begins at the root node and explores all the
neighboring nodes. Then for each of those nearest nodes, it
explores their unexplored neighbor nodes, and so on, until it
finds the goal.
Fig. 5.1 single line diagram of IEEE 15 bus system using BFS and Branch
numbering scheme
Fig. 4.1 Example of breadth first search method
BFS is a uniformed search method that aims to
expand and examine all nodes of a graph systematically in
search of a solution. In other words, it exhaustively searches
the entire graph without considering the goal until it finds it. It
does not use a heuristic.
V. SOLUTION METHODOLOGY
Given the voltage at the root node and assuming a
flat profile for the initial voltages at all other nodes, the
iterative solution algorithm consists of three steps
1. Nodal current calculation: At iteration k, the nodal
current injection, Ii(k), at network node i is calculated as,
Ii(k) = (Si / Vi(k-1))* - YiVi(k-1)
i=1,2,…….,n
where ,Vi(k-l) is the voltage at node i calculated
during the (k-l)th iteration and Si is the specified power
injection at node i. Yi is the sum of all the shunt elements at
the node i.
2. Backward sweep: At iteration k, starting from the
branches in the last layer and moving towards the
For IEEE 15-Bus system the branch currents from the Fig. 5.1
are calculated as given below
J[14]= -I[15];
J[13]= -I[14]+(J[14]);
J[12]= -I[13];
J[11]= -I[12];
J[10]= -I[11];
J[9]= -I[10];
J[8]= -I[9]+(J[13]);
J[7]= -I[8]+(J[10]+J[11]+J[12]);
J[6]= -I[7];
J[5]= -I[6];
J[4]= -I[5]+(J[9]);
J[3]= -I[4]+(J[7]+J[8]);
J[2]= -I[3]+(J[5]+J[6]);
J[1]= -I[2]+(J[2]+J[3]+J[4]);
3. Forward sweep: Nodal voltages are updated in a forward
sweep starting from branches in the first layer toward
those in the last. For each branch, L, the voltage at node
L2 is calculated using the updated voltage at node L1 and
the branch current calculated in the preceding backward
sweep.
1541
S. Sunisith et. al.,
Backward/Forward Sweep Based Distribution Load Flow Method
International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.9, pp. 1539-1544
ISSN 2078-2365
http://www.ieejournal.com/
VII. FLOW CHART
VL2(k) = VL1(k) – ZL JL(k)
L=1,2,….,b
Where, ZL is the series impedance of branch L. This is the
direct application of the KVL..
For IEEE 15 bus system the nodal voltages from Fig. 5.1 are
calculated as given below
v[2]=v[1]-(z[1]*J[1]);
v[3]=v[2]-(z[2]*J[2]);
v[4]=v[2]-(z[3]*J[3]);
v[5]=v[2]-(z[4]*J[4]);
v[6]=v[3]-(z[5]*J[5]);
v[7]=v[3]-(z[6]*J[6]);
v[8]=v[4]-(z[7]*J[7]);
v[9]=v[4]-(z[8]*J[8]);
v[10]=v[5]-(z[9]*J[9]);
v[11]=v[8]-(z[10]*J[10]);
v[12]=v[8]-(z[11]*J[11]);
v[13]=v[8]-(z[12]*J[12]);
v[14]=v[9]-(z[13]*J[13]);
v[15]=v[14]-(z[14]*J[14]);
Steps 1, 2 and 3 are repeated until convergence is achieved.
VI. CONVERGENCE CRITERION
We used the maximum real and reactive power
mismatches at the network nodes as our convergence
criterion. As described in the solution method, the nodal
current injections, at iteration k, are calculated using the
scheduled nodal power injections and node voltages from the
previous iteration. The node voltages at the same iteration are
then calculated using these nodal current injections. Here, the
power injection for node i at kth iteration, Si(k) is calculated as
Si(k) = Vi(k)(Ii(k))* - Yi │Vi(k)│2 --------(6.1)
The real and reactive power mismatches at bus i are
then calculated as
∆Pi(k) = Re [ Si(k) – Si ]
∆Qi(k) = Im [Si(k) – Si ]
VIII. ALGORITHM FOR BACKWARD/FORWARD
SWEEP BASED DISTRIBUTION LOAD FLOW
METHOD
Step 1: Read power system data, i.e., no. of buses, no. of lines,
slack bus, base kV, base kVA, bus data, line data.
Step 2: Starting from the root node, number the nodes and
branches in the network by using Breadth First Search method
and Branch Numbering Scheme respectively.
Step 3: Calculate the injected powers, i.e.,
Pinj(i) = Pgen(i) – Pload(i)
Qinj(i) = Qgen(i) – Qload(i)
,
i=1,2,……,n
i=1,2,………,n
Step 4: Set iteration count, k=1.
Step 5:Set convergence є=0.001, ∆Pmax=0.0and Qmax=0.0.
Step 6: Calculate nodal current injection at network node ‘i’
as
Ii(k) = ( Si / Vi(k-1) )* - YiVi(k-1)
i=1,2,…….,n
1542
S. Sunisith et. al.,
Backward/Forward Sweep Based Distribution Load Flow Method
International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.9, pp. 1539-1544
ISSN 2078-2365
http://www.ieejournal.com/
Step 7: Backward sweep: Calculate current in the branch L,
JL as
JL(k)= - IL2(k) + ∑ (currents in branches
emanating from node L2)
L=b, b-1 ,…….,1
Step 8: Forward sweep: Calculate the voltage at node L2 as
VL2 = VL1(k) – ZL JL(k)
L=1,2,….,b
Step 9: Calculate the power injection at node ‘i’ as
Si(k) = Vi(k)(Ii(k))* - Yi │Vi(k)│2
Step 10: Calculate real and reactive power mismatches as
∆Pi(k) = Re [ Si(k) – Si ]
∆Qi(k) = Im [Si(k) – Si ]
Fig. 9.1 single line diagram of IEEE 15 bus system using BFS and Branch
numbering scheme
i=1,2,………,n
Step 11: Check
∆Pi(k) > ∆Pmax, then set ∆Pmax=∆Pi(k)
Test results of backward/forward sweep based distribution
load flow method
1. The no. of iterations taken for convergence = 3
2. Convergence criteria = 0.001
(ii) IEEE 31-BUS DISTRIBUTION SYSTEM
∆Qi(k) > ∆Qmax , then set ∆Qmax=∆Qi(k)
Step 12: If ∆Pmax <= є and ∆Qmax <= є , then go to step 14
Else go to step 13
Step 13: Set k=k+1 and go to step 4.
Step 14: Print that problem is converged in ‘k’ iterations.
Step 15: Stop
IX. TEST RESULTS
No. of buses = 31
No. of lines = 30
Substation bus = 1
Base kV = 23
Base kVA = 100
(i) IEEE 15- BUS DISTRIBUTION SYSTEM
No. of buses = 15
No. of lines = 14
Sub station bus = 1
Base kV = 11
Base kVA = 100
Fig. 9.2 single line diagram of IEEE 31 bus system using BFS and Branch
numbering scheme
1543
S. Sunisith et. al.,
Backward/Forward Sweep Based Distribution Load Flow Method
International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.9, pp. 1539-1544
ISSN 2078-2365
http://www.ieejournal.com/
Test results of backward/forward sweep based distribution
load flow method
1. The no. of iterations taken for convergence = 5
2. Convergence criteria = 0.001
(iii) IEEE 69-BUS DISTRIBUTION SYSTEM
No. of buses = 69
No. of lines = 68
Sub station bus = 1
Base kV = 12.66
Base kVA = 1000
Test results of backward/forward sweep based distribution
load flow method
1. The no. of iterations taken for convergence = 3
2. Convergence criteria = 0.001
(iv) IEEE 85-BUS DISTRIBUTION SYSTEM
No. of buses = 85
No. of lines = 84
Sub station bus = 1
Base kV = 11
Base kVA = 100
Test results of backward/forward sweep based distribution
load flow method
1. The no. of iterations taken for convergence = 3
2. Convergence criteria = 0.001
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2078-2365.
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[7] “Computer Methods in Power System Analysis” by Glenn
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ROHIT MEHTA.
[9] “Electric Power Distribution” by A S PABLA.
X. CONCLUSION
Firstly, a mass of methods to solve the radial
distribution power flow problem are introduced in this paper.
Subsequently, the reason, why the convergence of these
widely used methods deteriorates when the network is radial,
is well analyzed. In this case, an improved solution is needed
to deal with radial network. So, a theoretical formulation of
backward/forward sweep distribution load flow method is
developed in this paper. Furthermore, the implementation of
this method in distribution management system remains very
straightforward. This method takes full advantage of the
radial structure of distribution systems, to achieve high speed,
robust convergence and low memory requirements. The
numerical tests proved that, this method is very efficient for
radial distribution network. This method is also applicable to
weakly meshed networks with compensation based power
flow method.
REFERENCES
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S. Sunisith et. al.,
Backward/Forward Sweep Based Distribution Load Flow Method