 Open Access
 Total Downloads : 1669
 Authors : S Hussain Mohisin, Dr V.Ganesh
 Paper ID : IJERTV2IS121281
 Volume & Issue : Volume 02, Issue 12 (December 2013)
 Published (First Online): 27122013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Load Flow Solution U sing Simplified NewtonRaphson Method
S Hussain Mohisin Dr V. Ganesh
PG Scholar, JNTUAC Pulivendula Associate professor, JNTUAC Pulivendula
Abstract
The power flow analysis is of great importance in planning and designing for the future expansion of power systems as well as in determining the best operation of existing systems. There exist two widelyused numerical methods (the GaussSeidel: GS and the NewtonRaphson: NR) to solve this problem and therefore referred to as the GS and the NR powerflow solution methods, respectively. Although the standard NewtonRaphson (NR) method is the most powerful algorithm for the power flow analysis in electric power systems, the calculation of Jacobian matrix derivatives involves high computational time. The proposed method presents a simplified NewtonRaphson power flow solution method to simplify overall equation complexity and computation time. The simplified NewtonRaphson method employs nonlinear current mismatch equations instead of the commonly used power mismatch equations. Numerical results are presented with 5bus test system and IEEE 30bus test system and compared with standards NR method.

Introduction
The main function of electric power systems is to deliver electric energy to its loads sufficiently, efficiently and economically. The steadystate performances of an interconnected power system during normal operation can be analyzed based on nonlinear nodal analysis to form power flow equations and must be solved by some efficient iterative methods [19]. Power flow analysis is commonly used as a part of power system operation and planning. Since AC powerflow solution methods were first developed over half a century ago, there exist two widelyused numerical methods (the
GaussSeidel (GS) and the NewtonRaphson (NR) to solve this problem and therefore referred to as the GS and the NR powerflow solution methods, respectively. As broadly known, the NR method has been successfully developed and accepted as the most powerful algorithm for the power flow analysis in electric power systems. In largescale power systems containing several
hundred or up to thousand buses, the standard NR method gives a slow execution time due to a large updated Jacobian matrix that needs to be recalculated and factorized at each iteration [10, 11]. Consequently, the decoupled and fast de coupled power flow versions [12, 13] were released. Hence, the powerflow solution can be obtained faster. This method is very useful in practical power system analyses, e.g. contingency analysis, online power flow control, etc. [4, 14].
Having a long history of development gives power flow algorithms a vast number and a various kind of applications. Enhancing the algorithm efficiency of power flow calculation has been carried out in many different approaches. Network partition technique can separate a whole power system into subsystems, therefore power flow solution of the complete system can be obtained by direct coupling of solutions from separate subsystems [15] based on the GS method. This concept is very useful to parallelize power flow algorithms in order to implement a parallel and sequential power flow performing on a computer cluster whether the GS method or some other numerical methods such as successive over relaxation (SOR) method is used as the main solver [16, 17]. In some point of view, an initial guest solution of power flow calculation is one of key factors that cause slow computation.
In [18, 19], an initial linear solution based on the decoupling principle of real and reactive power decomposition was utilized as the starting point to the power flow calculation. In addition, there are some modified versions of the NR power flow method to handle illconditioned power systems [20, 21]. The calculation algorithm has been continually developed by several researchers across the world. A complex form of the power flow calculation was introduced for a three phase unsymmetrical powerflow solution [22, 23]. Powerflow solutions based on a local search method were claimed [24] to be robust and be applicable to those cases in which conventional power flow method failed. Due to advancement of FACTS technology, the power flow equations were modified and rewritten into currentinjected forms for the incorporation of FACTS devices and any kind of control strategy [25, 26]. Moreover, the
study of powerflow solution methods for particular applications, e.g. economic dispatch [27], optimal power flow [28], FACTS devices [29, 30], and
AC/DC power systems [31], was reported.
The current balance equation at bus I is
n
n
Ii Yik Vk
k1
n
Over several decades, electrical power systems
(Ig,i Id,i ) Yik Vk 0
k1
(1)
have been characterized using the nodal analysis to solve for a set of voltage solutions. In general, electrical demands are defined in constant power. This leads to nonlinearity of nodal voltage
equations. To date, the standard NR power flow
In practice, loads in electrical power systems are in form of powers, therefore it is convenient to rewrite eqn. (1) into a function of powers as follows.
S S
S S
* n
g,i d,i
method is one of the most powerful algorithms,
Fi
V
V
i

Yik Vk 0
k1
(2)
which has long history of development, and is widely used to develop commercial powerflow solution software. Although the standard NR power
Define Fi = Gi + jHi be the current mismatch at bus
i,
V V ; V V
flow method is very efficient and commonly used
i i i
k k k
for the power flow calculation in several power system textbooks [19], to formulate iterative Jacobian updating matrix equations requires complicated formulae and long expressions. In this
Yij Yij ij is
Sg,i Sd,i Ssch,i Ssch,i i
Expressing eqn. (2) in polar form,
paper, the iterative NR method is still employed as *
n
n
the main solution framework. The essential
Ssch ,i i
Y
V 0
(3)
difference is that the proposed algorithm is to find roots of the current mismatch equations instead of
Vi i
ik
k1
ik k k
those of the power mismatch equations. This
Separating the real and imaginary parts,
approach can simplify a very long and complicated mathematical formula to a very simplistic and short mathematical expression. With this simplification, reduction of the overall execution time is expected. To achieve this goal, expressions to obtain

sch ,i cos n Y V cos(
S
S
i
i
V
V
i i ik k
i i ik k
i k1
S
S
V
V

sch ,i sin n Y V sin(
i
i
ik k ) 0
(4)
) 0
elements of Jacobian updating matrix formulae must be derived.
i i ik k
i k1
ik k
(5)
In this paper presents the formulation of the proposed NR power flow problem. Derivation of the Jacobian updating matrix elements is included and the floatingpoint operation counting to evaluate its computational effort. Numerical examples are selected to observe the effectiveness of the proposed method.
Eqns. (4) and (5) constitute a set of nonlinear algebraic equations in terms of the inependent variables, voltage magnitude in per unit and phase angle in radians. Expanding eqns. (4) and (5) in Taylors series about the initial estimates and neglecting all higher order terms results a set of linear equations. In short form, it can be written as
G
G
G
V


Formulation of proposed simplified power flow solution
H H
H V
V
The power flow problem is a zerofinding problem
G A A
2
2
to determine voltage solutions of nonlinear power
1
(6)
mismatch equations [1]. If alternative nonlinear current mismatch equations are selected and used as functions of estimating roots. Given that an n bus power system, which bus number 1 is assigned to be a slack bus of constant voltage magnitude and
zero phase angle. Considering the ith bus, current
balance equations characterizing this bus can be expressed as follows. In this method, the set of nonlinear equations are formulated based on current mismatch equations. The mathematical equations for simplified NewtonRaphson method are as follows [34].
H A3 A4 V
The elements of sub matrices A1, A2, A3, and A4 can be derived in the similar manner as jacobian matrix of the standard NR method, which are the partial derivates of eqns. (4) and (5) with respect to and
V .
The equations are summarized as, the diagonal and the offdiagonal elements of A1 are
Gi
i
sch ,i sin
S
S
Vi
(7)

V Y sin
ii i
k
k
ik
ik
Jacobian updating step dominates the overall execution time. In general, the time consumed to perform multiplication and division is about the same, but is larger than addition and subtraction. Hence, the operation counting of addition FLOPs is
Gi
k
Vk
Yik
sin
(8)
negligible. Throughout this paper, FLOPs always
i i i ii
i i i ii
means the multiplication FLOPs for short and it is employed to evaluate the computational effort of
The diagonal and the offdiagonal elements of A2
S
S
are
the proposed algorithm. The amount of FLOPs required by each method to formulate Jacobian
V
V
Gi
sch ,i cos
Y
cos
(9)
matrices is summarized in Table 1 and where O(n)
Vi
Gi
Vk
2
i
Yik
cos

i ii
k
k

i
(10)
means terms of order n.
Table 1 Number of FLOPs
ik
ik
Submatrix
Number of FLOPs
Standard NR
Proposed NR
J1
Diagonal Offdiagonal Total
3(n1)
4(n2)
3n2+O(n)
2(n2)
3n2+O(n)
6(n2)
J2
Diagonal Offdiagonal Total
2(n1)+3
4(n2)
2n2+O(n)
(n2)
2n2+O(n)
5(n2)
J3
Diagonal Offdiagonal Total
3(n1)
4(n2)
3n2+O(n)
2(n2)
3n2+O(n)
6(n2)
J4
Diagonal Offdiagonal Total
2 (n1)+3
4(n2)
2n2+O(n)
(n2)
2n2+O(n)
5(n2)
Overall
10n2+O(n)
22(n2)
Submatrix
Number of FLOPs
Standard NR
Proposed NR
J1
Diagonal Offdiagonal Total
3(n1)
4(n2)
3n2+O(n)
2(n2)
3n2+O(n)
6(n2)
J2
Diagonal Offdiagonal Total
2(n1)+3
4(n2)
2n2+O(n)
(n2)
2n2+O(n)
5(n2)
J3
Diagonal Offdiagonal Total
3(n1)
4(n2)
3n2+O(n)
2(n2)
3n2+O(n)
6(n2)
J4
Diagonal Offdiagonal Total
2 (n1)+3
4(n2)
2n2+O(n)
(n2)
2n2+O(n)
5(n2)
Overall
10n2+O(n)
22(n2)
The diagonal and the offdiagonal elements of A3
are
Hi Ssch ,i
i Vi
cos i i Vi Yii cos ii i
(11)
Hi V Y cos
(12)
k
k ik
ik k
The diagonal and the offdiagonal elements of A4
S
S
are
V
V
Hi
sch ,i sin
Y
sin
(13)
Vi
Hi
Vk
2
i
Yik
sin

i ii
k
k
ik
ik

i
(14)
As a total number of buses n gets larger, the number of FLOPs grows quadratically in the standard NR method. Interestingly, the FLOP number required by the proposed NR method is
The new estimates for bus voltages are
( t1) ( t) ( t)
(15)
linearly proportional to the total number of buses n. Fig. 1 shows the amount of FLOPs required by the
i i i
V( t1) V( t) V( t)
(16)
two methods.
i i i
N
N
The process is continued until the current mismatch
9 Number of FLOPs per iteration to update the Jacobian matrix
M
M
( t ) i
and
( t ) i
are less than the specified accuracy,
10
Standard NR
i.e.,
G( t) & H( t)
(17)
10 Proposed NR
8
8
7
i i 10
To compare the effectiveness of the proposed NR method against the standard NR method, expressions of the Jacobian matrix elements of A1, A2, A3 and A4, the calculated real and imaginary current matrix elements of G and H, and the calculated real and reactive power matrix elements of Pcal and Qcal need to be evaluated using the floating point operation.
6
10
FLOPs
FLOPs
5
10
4
10
3
10
2
10
1
10
0 1 2 3 4
10 10
10 10 10
Total Number of Buses


FLOPs Evolution
The execution time of the power flow calculation depends on the amount of floatingpoint operations (FLOPs) [34, 35]. Assume that other steps of the two NR methods are exactly the same, therefore the
Fig. 1 Number of FLOPS per iteration to update the Jacobian Matrix

Results and Analysis
The effectiveness of the simplified Newton Raphson power flow method was tested against 5 bus [2] and 30bus [1] IEEE test systems. Each individual test was performed by using Intel i5 Processer in which the power flow programs were coded in MATLAB [35. From the computer simulation, the voltage solution of each test case was calculated. Both NR power flow methods used here took 1106 perunit as the termination criteria for the maximum allowable voltage tolerance.
The 5bus power systems voltages and line losses are calculated using proposed simplified NR method. The obtained results are compared with the solution of existing standard NR method, has shown in the Table 2 and Table 3 respectively and observed that the results are nearly matched. The 30bus systems voltages of standard NR method and simplified NR method are given in Table 4.
Table 2 Voltages for the 5Bus System
Bus No
Standard NR
Proposed NR
Voltage
Voltage
V p.u.
Angle Deg.
V p.u.
Angle Deg.
1
1.0600
0.0000
1.0600
0.0000
2
1.0000
2.0612
1.0000
2.0502
3
0.9872
4.6367
0.9872
4.6286
4
0.9841
4.9570
0.9841
4.9483
5
0.9717
5.7649
0.9717
5.7547
Table 3 Line flows and line losses details for the 5 Bus System
From Bus
To Bus
Standard NR
Proposed NR
Line Losses
Line Losses
MW
MVAr
MW
MVAr
1
2
2.486
1.087
2.479
1.065
1
3
1.518
0.692
1.515
0.701
2
3
0.360
2.871
0.360
2.869
2
4
0.461
2.554
0.462
2.552
2
5
1.215
0.729
1.215
0.730
3
4
0.040
1.823
0.040
1.823
4
5
0.043
4.652
0.043
4.653
Total
6.122
10.777
6.114
10.803
For 5bus system the power mismatch is 9.82099e010 and number of iterations is 4 in standard NR method where as in current mismatch is 4.17729e007 and number of iterations is 5 in proposed NR method. For 30bus system the power mismatch is 4.6806e008 and number of iterations is 4 in standard NR method whereas in current mismatch is 1.16066e007 and number of iterations is 8 in proposed NR method. The fig. 2 and fig. 3 shows the power and current mismatches with respect to standard NR method and proposed simplified NR method
Table 4 Voltages for the 30Bus System
Bus No
Standard NR
Proposed NR
Voltage
Voltage
V p.u.
Angle Deg.
V p.u.
Angle Deg.
1
1.0600
0.0000
1.0600
0.0000
2
1.0430
5.3504
1.0430
5.0522
3
1.0205
7.5309
1.0210
7.1807
4
1.0115
9.2830
1.0120
8.8467
5
1.0100
14.1684
1.0100
13.4622
6
1.0100
11.0625
1.0103
10.5093
7
1.0022
12.8651
1.0024
12.2493
8
1.0100
11.8154
1.0100
11.1283
9
1.0499
14.1031
1.0501
13.5661
10
1.0432
15.6944
1.0434
15.1660
11
1.0820
14.1031
1.0820
13.5661
12
1.0565
14.9577
1.0566
14.4731
13
1.0710
14.9577
1.0710
14.4731
14
1.0415
15.8500
1.0416
15.3593
15
1.0367
15.9373
1.0369
15.4407
16
1.0432
15.5301
1.0434
15.0270
17
1.0382
15.8606
1.0384
15.3399
18
1.0268
16.5480
1.0270
16.0400
19
1.0241
16.7193
1.0243
16.2044
20
1.0281
16.5205
1.0283
16.0022
21
1.0308
16.1386
1.0310
15.6101
22
1.0313
16.1243
1.0316
15.5959
23
1.0259
16.3233
1.0261
15.8134
25
1.0199
16.4931
1.0201
15.9651
26
1.0162
16.0730
1.0164
15.5237
27
0.9985
16.4936
0.9987
15.9441
28
1.0066
11.6871
1.0067
11.1059
29
1.0025
16.7844
1.0029
16.2214
30
0.9911
17.6688
0.9914
17.1052
Fig. 2 Power and Current Mismatches for 5Bus Power System
Table 4.7 is the summary of the effectiveness of the proposed method by giving the required iteration and calculation time in comparison with those of the standard NR method. It notes that, in Table 5, SNR and PNR denote the standard NR method and the proposed NR method, respectively.
Fig. 3 Power and Current Mismatches for 30Bus Power System
Table 5 Simulation result for required iteration and time of computation
Test system
Method
Required Iterations
Execution time (s)
Calculating time ratio
5Bus
SNR
4 0.0231
1.5197
PNR
5
0.0152
–
30Bus
SNR
4
0.1686
1.3255
PNR
8
0.1272
–
From the Table 5, the PNR method spends shorter calculation times for all test cases even the though the test cases iteration high in PNR method compared with SNR. Undoubtedly, the PNR method is faster for these two test cases. Since the PNR method takes less requirement of re calculation in its Jacobian matrix per iteration, the calculation time ratios for these three test cases are remarkably larger with a factor of 1.5197 and 1.3255 respectively.

Conclusions
Power flow calculation is one of the most essential parts in electric power system operation in order to analyze, simulate, design and control the steady state system performances properly. Although there exist several powerful power flow solvers based on the standard NR method, their problem formulation gives complication due to the need to calculate derivatives in the Jacobian matrix. The proposed method uses nonlinear current mismatch equations instead of the commonlyused power mismatches to simplify overall equation complexity. With performance evaluation found in session 3, a total number of operations required by the proposed NR method is linearly proportional to the size of the Jacobian matrix, while that of the standard NR method is quadratic. This means that the
calculation time of the standard NR method increases more rapidly as a total bus number increases than that of the proposed NR method does. From this advantage, the calculation time consumed by the proposed NR method is expected to be less than that of the standard one. This can leads to improvement of powerflow software development in fast computational speed and less memory usage.

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Bibliography

S.Hussain Mohisin, PG scholar at JNTUAC, pulivendula.

Dr.V.Ganesh, presently working as a associate professor at JNTUAC, pulivendula. Previously he worked as a head of the department for electrical and electronics engineering at JNTUAC, pulivendula for about 4 years. Presently he is guiding 7 p.hd scholars. His areas of interest are electrical distribution system, FACTs devices, renewable energy sources etc.