 Open Access
 Total Downloads : 1364
 Authors : Ayoade Benson Ogundare
 Paper ID : IJERTV2IS2605
 Volume & Issue : Volume 02, Issue 02 (February 2013)
 Published (First Online): 28022013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Application of Sparsity Characteristics of Power Systems to AC PowerFlow Modelling and Simulation
Ayoade Benson OGUNDARE
Lagos State Polytechnic, Ikorodu, Lagos
Abstract
Power flow is the basic tool for power system analysis which reveals the system operation in a steadystate mode for evaluation of the power system operations. Unfortunately, the accuracy, simulation time, computer storage size and convergence of any model used depend largely on the size of the bus admittance matrix of the system under study.
This paper, therefore, presents the application of power system sparsity characteristic to powerflow modelling and simulations to develop a model for storing the bus admittance matrix of power systems. In this paper a powerful MATLAB simulation algorithms have been developed that are capable of modeling large complex systems using sparsity characteristic of the power systems.
The validity of this proposed method using MATLAB 2009a is tested using a 5bus, 6bus and the standard IEEE 14bus, 24bus and 30bus systems. The results are presented in graphical forms and discussed. The proposed method shows that as the system is increasing in size, the percentage of stored bus admittance elements decreases. Thus, an appreciable reduction in the computer memory required to store the bus admittance matrix and in turns reduces the overall simulation time.
Keywords: Powerflow, Steadystate, Sparsity characteristic, Admittance.
1. Introduction
As every nation approach an industrialized community, the rate at which the power demand increase is unpredictable if not properly analyzed [1, 2]. This calls for an accurate analysis and modelling of the steady state of the given system or network. As such, powerflow analysis is becoming the most important tool for evaluating power system performance under a steady state operation in recent times [2, 3]. This evaluation of the system reveals the parameters of the system under a steadystate mode or operation. Power flow analysis is
also an important tool during planning stages of new power system or addition to existing ones like adding new generator sites, meeting increase load demand and locating new transmission sites [4]. In power flow analysis, some power system parameters are important in determining the system performance. These parameters include voltage magnitude and angle at every bus of the system, real and reactive power injections at all the buses and power flows through interconnecting power channels, reactive power flows along the transmission lines, real and reactive power losses along the transmission lines and total losses [3]. In the study of electric power systems, several different researches have been carried out by different researchers using different methods. The most commonly used methods for power flow study are include GaussSeidel, Newton Raphson and Fast decoupled. Among all these methods, NewtonRaphson is found to be the most widely deployed model in power systems applications [5, 2]. However, these conventional power flow algorithms are not efficient due to the challenges they encountered, which in most cases, results to divergence of the algorithms. These challenges include singularity of the Jacobian matrix most especially in NewtonRaphson method, large memory capacity for storing the elements of the Jacobian and the bus admittance matrix, convergence error or problem, high or large simulation time etc.
To combat the problems inherent in divergence algorithm, various modifications have been used by different researchers in the past [5]. This paper provides an excellent method of using sparsity characteristics of the power systems. Several different approaches have been applied to improve the numerical behaviour of Newton Raphson which is normally applied to a large scale power system. However, all these algorithms can still be improved if the sparsity characteristic of the system is explored.
All practical power systems have majority or larger percentage of their buses not connected through transmission lines [4]. This characteristic is explored in this paper such that only nonzero elements are stored and thereby resulting to savings in computer memory (the CPU time per iteration is made to be relatively small). The savings in computer memory is very important when
dealing with large practical power systems to reduce the computation time [6] by reducing large memory required for the storage of the elements of bus admittance matrix. Until recently, most industry effort and interest has been devoted to the algorithms for reducing the space occupied when storing the elements of Jacobian and bus admittance matrices by the computer memory. The efficient Newton Raphson method for large power system has not been fully deployed to practical power system as the sparsity characteristic of the system has not been successfully used due to the repetitive solution of a large set of linear equations required in load flow problem which is one of the most time consuming parts of large power systems simulations. The major advantages of the application of this work are found in the areas such as existence of solution, uniqueness of solution convergence to the solution and speed of convergence.
In reference [2], fast NewtonRaphson method, through the application of preconditioning method in the Generalized Minimal Residual method (GMRES), called NewtonKrylov method is proposed. Validation of this proposed method showed that the convergence characteristics of the conventional Newton model are well preserved and the simulation time is reduced tremendously.
Seki, K in reference [5], in his work considered the analysis of no converge networks using loadflow program with complex number state variables. This work
validation of the proposed method are the line parameters for a 5bus, 6bus, 14bus, 24bus and 30bus system.
Interconnections among power systems result in extremely large networks [2, 4]. This complexity in power system structure increases the computation and simulation times [6]. There are two major way of tackling the long simulation and computational times. Either by developing a more efficient sparsity computational model or by using a method called equivalence models which can be used to reduce the size of the given system to its thevenin equivalent model. This paper focuses on the first approach.
2. Proposed Sparsity Technique
In large power systems, each bus is connected to only a small number of other buses. Therefore, bus admittance matrix of a large power system is very sparse. This means that the bus admittance matrix will contain larger percentage of zeros as compared to the nonzero elements. This characteristic feature shows a considerable reduction in the storage handling of the computer which indicates a substantial improvement compared to the work reported in literature. This sparsity feature of Ybus matrix also extends to Jacobian matrix. According to reference [8], Sparsity can be simply defined to indicate the absence of certain problem interconnections. Mathematically, the sparsity of
an n n matrix is given by reference [4] as
shows that since the NewtonRaphson powerflow equations is differential and does not satisfy Cauchy Riemma conditions, it cannot be used to predict the
Sparsity =
Total no of
zero elements 100%
n2
voltage collapse behavior of any given system with complex number state variables. A new model based on the GaussSeidel iteration method which does not
In a large power system such as the ones considered in this work,sparsity may be as high as 97% . Though Ybus
calculate the differential values of the power equations.
is sparse,
Zbus
is full. This sparsity is employed in this
However, the main roadblock is that, this method is found to be slower than the conventional NewtonRaphson model and it is difficult for the unstable voltage solutions of the network to be identified.
Gill, P. E. et al in reference [7] presented the reviews on applications of sparsity to optimization techniques in power systems. It is shown that, this method is of great importance, especially, in large practical power systems, to determine suitable finitedifference vector of Hessian matrix which is based on quasiNewton approximation.
Sparse matrix solution in optimal direct current loadflow using Crout technique is proposed in reference [4]. This algorithm is based on the Crout Method for solving load flow of sparsed power systems.
For decades, the commercially available softwares in use for the simulation of practical power systems include PSS/E, EuroStag, Simpow, CYME, PowerWorld and Neplan. In this paper, MATLAB 2009 is used for the simulation of the proposed model and the results are compared to validate the model. The data used for the
work to ensure that only the nonzero elements are stored and the full characteristic of the original matrix is not lost.
3. Model Validation
The model proposed in this work is validated using the line parameters shown in the tables 1 to 6 below.
Table 1. Line data for a 3bus system
Bus 
Bus No. 
R (pu) 
X (pu) 
B (pu) 
Tap setting 
1 
2 
0.02 
0.04 
0 
1 
1 
3 
0.01 
0.03 
0 
1 
2 
3 
0.0125 
0.025 
0 
1 
Table 2. Line data for 5bus system
Bus 
Bus No. 
R (pu) 
X (pu) 
B (pu) 
Tap setting 
1 
2 
0.020 
0.060 
0.030 
1.0 
1 
3 
0.080 
0.240 
0.025 
1.0 
2 
3 
0.060 
0.180 
0.020 
1.0 
2 
4 
0.060 
0.180 
0.020 
1.0 
2 
5 
0.040 
0.120 
0.015 
1.0 
3 
4 
0.010 
0.030 
0.010 
1.0 
4 
5 
0.080 
0.240 
0.025 
1.0 
Table 3. Line data for a 6bus system
Bus 
Bus No. 
R (pu) 
X (pu) 
B (pu) 
Tap setting 
1 
4 
0.035 
0.225 
0.0065 
1.0 
1 
5 
0.025 
0.105 
0.0045 
1.0 
1 
6 
0.040 
0.215 
0.0055 
1.0 
2 
4 
0.000 
0.035 
0.0000 
1.0 
3 
5 
0.000 
0.042 
0.0000 
1.0 
4 
6 
0.028 
0.125 
0.0035 
1.0 
5 
6 
0.026 
0.175 
0.0300 
1.0 
Bus 
Bus No. 
R (pu) 
X (pu) 
B (pu) 
Tap setting 
1 
2 
0.01938 
0.05917 
0.0528 
1.0 
1 
5 
0.05403 
0.22304 
0.0492 
1.0 
2 
3 
0.04699 
0.19797 
0.0438 
1.0 
2 
4 
0.05811 
0.17632 
0.0374 
1.0 
2 
5 
0.05695 
0.017388 
0.0340 
1.0 
3 
4 
0.06701 
0.17103 
0.0346 
1.0 
4 
5 
0.01335 
0.04211 
0.0128 
1.0 
4 
7 
0.00 
0.20912 
0.00 
0.978 
4 
9 
0.00 
0.55618 
0.00 
0.969 
5 
6 
0.00 
0.25202 
0.00 
0.932 
6 
11 
0.09498 
0.1989 
0.00 
1.0 
6 
12 
0.12291 
0.25581 
0.00 
1.0 
6 
13 
0.06615 
0.13027 
0.00 
1.0 
7 
8 
0.00 
0.17615 
0.00 
1.0 
7 
9 
0.00 
0.11001 
0.00 
1.0 
8 
10 
0.03181 
0.08450 
0.00 
1.0 
8 
14 
0.12711 
0.27038 
0.00 
1.0 
10 
11 
0.08205 
0.19207 
0.00 
1.0 
12 
13 
0.22092 
0.19988 
0.00 
1.0 
13 
14 
0.17093 
0.34802 
0.00 
1.0 
4
Bus 
Bus No. 
R (pu) 
X (pu) 
B (pu) 
Tap setting 
1 
2 
0.01938 
0.05917 
0.0528 
1.0 
1 
5 
0.05403 
0.22304 
0.0492 
1.0 
2 
3 
0.04699 
0.19797 
0.0438 
1.0 
2 
4 
0.05811 
0.17632 
0.0374 
1.0 
2 
5 
0.05695 
0.017388 
0.0340 
1.0 
3 
4 
0.06701 
0.17103 
0.0346 
1.0 
4 
5 
0.01335 
0.04211 
0.0128 
1.0 
7 
0.00 
0.20912 
0.00 
0.978 

4 
9 
0.00 
0.55618 
0.00 
0.969 
5 
6 
0.00 
0.25202 
0.00 
0.932 
6 
11 
0.09498 
0.1989 
0.00 
1.0 
6 
12 
0.12291 
0.25581 
0.00 
1.0 
6 
13 
0.06615 
0.13027 
0.00 
1.0 
7 
8 
0.00 
0.17615 
0.00 
1.0 
7 
9 
0.00 
0.11001 
0.00 
1.0 
8 
10 
0.03181 
0.08450 
0.00 
1.0 
8 
14 
0.12711 
0.27038 
0.00 
1.0 
10 
11 
0.08205 
0.19207 
0.00 
1.0 
12 
13 
0.22092 
0.19988 
0.00 
1.0 
13 
14 
0.17093 
0.34802 
0.00 
1.0 
Table 4. Line data for a 14bus system
Table 5. Line data for a 24bus system
Bus 
Bus No. 
R (pu) 
X (pu) 
B (pu) 
Tap setting 
3 
1 
0.0006 
0.0044 
0.0295 
1.0 
4 
5 
0.0007 
0.0050 
0.0333 
1.0 
1 
5 
0.0023 
0.0176 
0.1176 
1.0 
5 
8 
0.0110 
0.0828 
0.5500 
1.0 
5 
9 
0.0054 
0.0405 
0.2669 
1.0 
5 
10 
0.0099 
0.0745 
0.4949 
1.0 
6 
8 
0.0077 
0.0576 
0.3830 
1.0 
2 
8 
0.0043 
0.0317 
0.2101 
1.0 
2 
7 
0.0012 
0.0089 
0.0589 
1.0 
7 
24 
0.0025 
0.0186 
0.1237 
1.0 
8 
14 
0.0054 
0.0405 
0.2691 
1.0 
8 
10 
0.0098 
0.0742 
0.4930 
1.0 
8 
24 
0.0020 
0.0148 
0.0982 
1.0 
9 
10 
0.0045 
0.0340 
0.2257 
1.0 
15 
21 
0.0122 
0.0916 
0.0689 
1.0 
10 
17 
0.0061 
0.0461 
0.3064 
1.0 
11 
12 
0.0010 
0.0074 
0.0491 
1.0 
12 
14 
0.0060 
0.0455 
0.3025 
1.0 
13 
14 
0.0036 
0.0272 
0.1807 
1.0 
16 
19 
0.0118 
0.0887 
0.5892 
1.0 
17 
18 
0.0002 
0.0020 
0.0098 
1.0 
17 
23 
0.0096 
0.0721 
0.4793 
1.0 
17 
21 
0.0032 
0.0239 
0.1589 
1.0 
19 
20 
0.0081 
0.609 
0.4046 
1.0 
20 
22 
0.0090 
0.0680 
0.4516 
1.0 
20 
23 
0.0038 
0.0284 
0.1886 
1.0 
Table 6. Line data for a 30bus system
Bus 
Bus No. 
R (pu) 
X (pu) 
B (pu) 
Tap setting 
1 
2 
0.0192 
0.0575 
0.02640 
1.0 
1 
3 
0.0452 
0.1852 
0.02040 
1.0 
2 
4 
0.0570 
0.1737 
0.01840 
1.0 
3 
4 
0.0132 
0.0379 
0.00420 
1.0 
2 
5 
0.0472 
0.1983 
0.02090 
1.0 
2 
6 
0.0581 
0.1763 
0.01870 
1.0 
4 
6 
0.0119 
0.0414 
0.00450 
1.0 
5 
7 
0.0460 
0.1160 
0.01020 
1.0 
6 
7 
0.0267 
0.0820 
0.00850 
1.0 
6 
8 
0.0120 
0.0420 
0.00450 
1.0 
6 
9 
0.0 
0.2080 
0.0 
0.978 
6 
10 
0.0 
0.5560 
0.0 
0.969 
9 
11 
0.0 
0.2080 
0.0 
1.0 
9 
10 
0.0 
0.1100 
0.0 
1.0 
4 
12 
0.0 
0.2560 
0.0 
0.932 
12 
13 
0.0 
0.1400 
0.0 
1.0 
Bus 
Bus No. 
R (pu) 
X (pu) 
B (pu) 
Tap setting 
12 
14 
0.1231 
0.2559 
0.0 
1.0 
12 
15 
0.0662 
0.1304 
0.0 
1.0 
12 
16 
0.0945 
0.1987 
0.0 
1.0 
14 
15 
0.220 
0.9117 
0.0 
1.0 
16 
17 
0.0824 
0.1923 
0.0 
1.0 
15 
18 
0.1073 
0.2185 
0.0 
1.0 
18 
19 
0.0639 
0.1292 
0.0 
1.0 
19 
20 
0.0340 
0.0680 
0.0 
1.0 
10 
20 
0.0936 
0.2090 
0.0 
1.0 
10 
17 
0.0324 
0.0845 
0.0 
1.0 
10 
21 
0.0348 
0.0749 
0.0 
1.0 
10 
22 
0.0727 
0.1499 
0.0 
1.0 
21 
22 
0.0116 
0.0236 
0.0 
1.0 
15 
23 
0.1000 
0.2020 
0.0 
1.0 
22 
24 
0.1150 
0.1790 
0.0 
1.0 
23 
24 
0.1320 
0.2700 
0.0 
1.0 
24 
25 
0.1885 
0.3292 
0.0 
1.0 
25 
26 
0.2544 
0.3800 
0.0 
1.0 
25 
27 
0.1093 
0.2087 
0.0 
1.0 
28 
27 
0.0 
0.3960 
0.0 
0.968 
27 
29 
0.2198 
0.4153 
0.0 
1.0 
27 
30 
0.3202 
0.6027 
0.0 
1.0 
29 
30 
0.2399 
0.4533 
0.0 
1.0 
8 
28 
0.0636 
0.2000 
0.0214 
1.0 
6 
28 
0.0169 
0.0599 
0.065 
1.0 
Bus 
Bus No. 
R (pu) 
X (pu) 
B (pu) 
Tap setting 
12 
14 
0.1231 
0.2559 
0.0 
1.0 
12 
15 
0.0662 
0.1304 
0.0 
1.0 
12 
16 
0.0945 
0.1987 
0.0 
1.0 
14 
15 
0.2210 
0.9117 
0.0 
1.0 
16 
17 
0.0824 
0.1923 
0.0 
1.0 
15 
18 
0.1073 
0.2185 
0.0 
1.0 
18 
19 
0.0639 
0.1292 
0.0 
1.0 
19 
20 
0.0340 
0.0680 
0.0 
1.0 
10 
20 
0.0936 
0.2090 
0.0 
1.0 
10 
17 
0.0324 
0.0845 
0.0 
1.0 
10 
21 
0.0348 
0.0749 
0.0 
1.0 
10 
22 
0.0727 
0.1499 
0.0 
1.0 
21 
22 
0.0116 
0.0236 
0.0 
1.0 
15 
23 
0.1000 
0.2020 
0.0 
1.0 
22 
24 
0.1150 
0.1790 
0.0 
1.0 
23 
24 
0.1320 
0.2700 
0.0 
1.0 
24 
25 
0.1885 
0.3292 
0.0 
1.0 
25 
26 
0.2544 
0.3800 
0.0 
1.0 
25 
27 
0.1093 
0.2087 
0.0 
1.0 
28 
27 
0.0 
0.3960 
0.0 
0.968 
27 
29 
0.2198 
0.4153 
0.0 
1.0 
27 
30 
0.3202 
0.6027 
0.0 
1.0 
29 
30 
0.2399 
0.4533 
0.0 
1.0 
8 
28 
0.0636 
0.2000 
0.0214 
1.0 
6 
28 
0.0169 
0.0599 
0.065 
1.0 
0
0.5
1
1.5
2
2.5
3
3.5
4
Y Sparse Symmetric Matrix
0 0.5 1 1.5 2 2.5 3 3.5 4
nonzeros=9 (100.000%)
Figure 1. Sparse matrix for a 3bus system
Y Sparse Symmetric Matrix
0
1
2
3
4
5
4. Simulation Results and Discussion
The results of the simulation, using the proposed technique, validated using the line data of tables 1 to 6, are presented in figures 1 to 6. These results can be summarised as shown in table 7.
Table 7. Summary of the simulation results
6
0 1 2 3 4 5 6
nonzeros=19 (76.000%)
Figure 2. Sparse matrix for a 5bus system
Y Sparse Symmetric Matrix
0
1
2
3
4
5
S/N 
No of Buses 
Percentage of stored elements 
1 
3 
100.000 
2 
5 
76.000 
3 
6 
55.556 
4 
14 
27.551 
5 
24 
13.194 
6 
30 
12.444 
S/N 
No of Buses 
Percentage of stored elements 
1 
3 
100.000 
2 
5 
76.000 
3 
6 
55.556 
4 
14 
27.551 
5 
24 
13.194 
6 
30 
12.444 
6
70 1 2 3 4 5 6 7
nonzeros=20 (55.556%)
Figure 3. Sparse matrix for a 6bus system
Y Sparse Symmetric Matrix
0
5
10
15
0 5 10 15
nozeros=54 (27.551%)
Figure 4. Sparse matrix for a 14bus system
Y Sparse Symmetric Matrix
0
5
10
15
20
25
0 5 10 15 20 25
nonzeros=76 (13.194%)
Figure 5. Sparse matrix for a 24bus system
Y Sparse Symmetric Matrix
0
5
10
15
20
25
30
0 5 10 15 20 25 30
nonzeros=112 (12.444%)
different networks have been presented in both tabular and graphical forms as case studies. The result of the simulation revealed that the sparsity technique applications to large power systems reduces the computer memory required for the storage of the elements of the bus admittance matrix which consequently leads to savings in memory capability required for the simulation.
6. References

G. Anupam, DC power Flow Based Contingency Analysis Using Graphics Processing Unit, A Thesis Submitted to University of Drexel, for the award of M.SC Electrical Engineering, March 2010, pp. 172.

AmhrizPbrez, H,.Acha,E and .FuerteEsquivel,C.R, Advanced SVC Models for NewtonRaphson Load Flow and Newton Optimal Power Flow Studies, IEEE transactions on power systems, Vol. 15, No. 1, February 2000.

A.B Ogundare and A.S Alayande, Load,flow analysis and simulation using NewtonRaphson iterative methods, Global journal of Engg. & Tech., Vol. 2, No. 4, pp. 561568, 2009.

K. S. Saha and B.C. Roy, Sparse Matrix Solution in Optimal D.C LoadFlow by Crowt Method, International Journal of Recent Trends in Engineering, Vol. 2, No.7, November 2009.

K. Seki, Analysis of No Converge Networks using the Load Flow Program with Complex Number State Variables, Transmission & Distribution Conference and Exhibition, IEEE/ PES, Asia Pacific, Vol. 2, Pp 1124 1127, 610 Oct. 2002.

T. Udomsak, P. Padej and K. Thanatchai, Application of Load Transfer Technique for Distribution Power Flow, World Academy of Science, Engineering and Technology, 23 2008.

P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, Sparse Matrix Method In Optimization, Siam Journal, Science and Statistics Computation. Vol. 5, No. 3, September 1984.

W. Tinney and S. Meyer, Solution of Large Sparse Systems by Ordered Triangular Factorization, I.E.E.E. Transactions On Automatic Control, Vol. AC18 , No 4, August 1973.
Figure 6. Sparse matrix for a 30bus system
The output of the MATLAB simulation for various line data from different networks are displayed in figures 1 to 6 above.
5. Conclusion
Various aspects of the applications of sparsity technique in power systems have been extensively studied. An efficient algorithm for the modelling and simulation of the sparsity model has been presented and tested. The results of the simulation for various line parameters from