 Open Access
 Authors : Mazen M. Abouelezz , I. O. Habiballah
 Paper ID : IJERTV10IS120058
 Volume & Issue : Volume 10, Issue 12 (December 2021)
 Published (First Online): 16122021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Comparative Review Between Bus Building Algorithms for Impedance Bus Matrix
Mazen M. Abouelezz Master Student,
King Fahad University for petroleum and minerals
I.O. Habiballah Associate Professor,
King Fahad University for petroleum and minerals
AbstractThis paper will present different proposed algorithms to build the impedance matrix and optimization method to reduce computation time by using genetic algorithm. Then the advantages and disadvantages of the algorithms are discussed.
Index: Zbus, Building algorithm, Optimization for Zbus, power system.
I. INTRODUCTION
Power systems/networks is the backbone of power distribution to houses, industry and services. This rises the priority of a way to analyze the network for load flow or fault analysis. These two analyses require fast computation for the rapid increase in the network, hence the need of a better algorithm to reach the solution of the network. Impedance matrix or admittance matrix are the main ways to do these analyses. The two matrices are inverse of each other, so building one of them then find inverse is enough to have both. The impedance matrix provides more information and needed for the short circuit and unsymmetrical fault analysis. The zero, positive and negative sequence are using the Zbus matrix. A building algorithm or an optimization to current algorithm to build the Impedance matrix is needed to reduce the time to finish the analysis.
This paper investigates the different algorithms and optimizations done to reduce time to build the Zbus matrix and highlight the advantages and disadvantages.
The paper is structured as follows: In Section II, literature
and general to be used even in the presence of CCVS. [5] proposed a way to integrate Genetic Algorithms (GAs) in finding the Zbus matrix. This introduction of GA is to be able to find the best path in a system to minimize the calculation of Zbus matrix. [5] tested on a 14bus matrix to show the impact. Also, provided a theorem in reducing the number of calculation (time needed to find Zbus). [6] investigated the impedance matrix required computation for phase transposition for short transmission lines. [6] showed multiple study cases of each of approximated Zbus and Kron reduced impedance matrix, respectively. The results of the case studies shows that the ground resistivity has a significate effect on approximation error. The work of [7] consists of programmable strategy (code) that takes an input file describing the system layout and its values. Then the code calculates the Zbus by the general simple building Algorithm, afterwards calculates Ybus by inverse of Zbus. Also, [6] and [7] highlighted that finding the Zbus is very important to calculate Ybus.

CONCEPT OF ZBUS MATRIX
The Zmatrix is the relation between a given bus injection current and the bus voltage given as (1):
[] = []/[] (1)Where the diagonal and offdiagonal elements of the matrix are as follows in (2) and (3), respectively:
review on work about Zbus matrix. Section III states the theory and equations of the Zbus matrix. Section IV presents two building algorithms and an optimization method using genetic algorithm to reduce the time needed to find Zbus. Section V gives general observations on each algorithm and optimization. Finally, Section VI provides conclusion.
=
=
(2)
(3)
II. LITERATURE REVIEW
In the literature there is many works and investigation made to find the best way to reach the Zbus matrix and its applications in the power flow field. One method to build Z matrix is proposed by [1][3]. Where they looked at the Tree Set and LinkSet of a network to develop the branch current vs injection current of the bus matrix and the branch current vs bus voltage matrix (BCIC) and (VBC), respectively. Then the

ZBUS BUILDING ALGORITHMS AND OPTIMIZATIONS
This section shows the different algorithms and methods to obtain or reduce computation of Zbus:

Direct Building Algorithm
The direct method is a sequential and recursive process where the Zbus matrix building follows:
= [
= [
,
two matrices are multiplied and Kron reduction is applied to
, ( )
+
] (4)
the multiplication result to obtain Zmatrix. The work of [2]
and [3] is improved by [4], which revised the calculation if there was a Current Control Voltage Source (CCVS) on one or more nodes in the system. This made the method more robust
Where this algorithm divides the addition of new values into four cases as shown in [8],
2 0
1
1
1
1
1 2
4 4
5 0 0
0
0
1
1 5
Case 3: Addition of a transmission line connecting two existing [6] [0 0 0 0 0 1] [6]
2 0
1
1
1
1
1 2
4 4
5 0 0
0
0
1
1 5
Case 3: Addition of a transmission line connecting two existing [6] [0 0 0 0 0 1] [6]
Case 1: Addition of a line between the reference and a new bus
Case 2: Addition of a transmission line between an existing bus and a new bus
1
3 =
1 1 1 1 1 1
0 0 1 1 1 1
0 0 0 1 1 1
1
3
(9)
buses
Case 4: Addition of a line connecting the reference to an existing bus
Cases 3 and 4 are a loop closure since the connect to an existing bus, in these cases an intermediate matrix is introduced
Taking into consideration the effect of LS as expressed in (10):
1 = 1 + 7
3 = 3 8 (10)
, ,
6
= 6
7
= [
] (5)
, , () + ( ) 2( ) +
The matrix in (9) becomes as follow:
Then is found from by applying bus elimination
1
1 1
+ 7
method.
2
1
2
1
1
1
1
1
0
1
1
1
1
0
0
1
1
1
0
0
0
1
1
0
0
0
0
1
[0 0
0
0
0
1
1
1
1
1
0
1
/td>
1
1
1
0
0
1
1
1
0
0
0
1
1
0
0
0
0
1
[0 0
0
0
0
3 =
1 3 8
(11)

Treeset and Linkset algorithm
This algorithm proposed by [1][3] uses the tree set (TS) and the corresponding links (LS) of the system to obtain two
4
5 [6]
1 4
1 5
1] [6 7]
matrices BCIC and VBC, respectively. Then develop the bus matrix from these two matrices. First TS and LS contains
The matrix in (11) can be rewritten as follows:
buses currents from [Case1, Case2] and buses currents from [Case3, Case4], respectively. A 6bus system is shown in Fig.1
1 1
1 1
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
1
0
0
1
1
1
1
0
0
0
1
1
1
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
1
0
0
1
1
1
1
0
0
0
1
1
1
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
2 0
1 2
3 0
1
3
4 = 0
0 4
(11)
5 0
0 5
6
7 [8]
0
0
[ 0 [] = []0 6
0 7
1 ] [ 8 ]
] (12)
[Afterwards the development of VBC of TS is done through:
Fig 1 A 6Bus example used for the algorithm
6
6 = 0
=1
(13)
From Fig.1, TS and LS would be as follows:
= [1, 2, 3, 4, 5, 6] (6)
= [7, 8] (7)
The relation between the line current and the current injected
Which the equation in (13) can be rewritten as follows:
1
0
1
0
0
0
0
0
1
1
2
0
0
0
0
2 0
3 = 0 1
0
0
2
0
3
4 0 1
2
3
4
0
0 4
1
0
1
0
0
0
0
0
1
1
2
0
0
0
0
2 0
3 = 0 1
0
0
2
0
3
4 0 1
2
3
4
0
0 4
2 3 (14)
to a bus is expressed as follows:
6
5
[6]0 [0]
1 2 3 4 5 0 5 [1 2 3 4 5 6] [6]
= ( = 1,2,3,4,5,6)
=
(8)
Then by adding the effect of LS into (14) yields the following:
Where is the line current and is the injected current. The matrix describing the relation in (8) is shown in (9):
1
2
3
1 0 0 0 0 0 0 0
1
2
0
0
0
0 0 0
0
0
0 0 0
1
2
0
0
0
0 0 0
0
0
0 0 0
1 2 3
1
2
3
= Replacement time needed
= Subtraction time needed
4 0 0 0 0
4 =
1 2 3
4 (15)
5
1 2 3 4 5 0 0 0
5
= Addition time needed
6
1 2 3 4 5 6 0 0
6
0
0 2
3
4
5
6 7 0 7
= Division time needed
[ 0 ] [12
3 0 0
0 0 8
] [8]
= Multiplication time needed
[ 0 ] = [] [] (16)
Using the equation from (19)(22) the cost/fitness function is defined as follows:
Next The Zbus matrix of the system is found as follows:
[] = [][] 0] [
[() =
=1
(23)
11 12
For the selection of the chromosome to propagate the
= [ ] [
] (17)
21 22
tournament without replacement method is used. This method
eliminates the low fitness valued chromosomes.
Finally, Kron reduction is applied to (17) yields that the matrix to be:
[] = [11 12 (22 )121 ][]The Cross over between the chromosomes of different parents is not used in [5].
The mutation in genetic algorithm is used by [5]. To ensure the produced generation, that have a mutation still satisfies the two
[]
requirements of the first generation, the following steps are
=
[ ] (18)proposed:
This work is improved by [4] as they proposed a solution for systems containing CCVS. The line/branch having CCVS with a control parameter, that control parameter is multiplied by column having CCVS in it. On the other hand, the matrix of BCIC is the same.

Genetic algorithm to optimize path
This optimization proposed by [5] is using genetic algorithm to find the optimal path to calcuate the Zbus matrix to minimize the number of computations needed.
The first generation of the genetic algorithm is randomly generated, but should have these two requirements:

The first element (gen) of each chromosome should be a generator that is added between a new bus and the reference bus

Each chromosome should meet the condition that each bus must have an indirect or direct connection to the reference bus.

After producing the first generation a cost function is defined to evaluate all the chromosomes of the current generation. [5] proposed the following equation (19) (22) representing the 4 cases of bus addition, respectively.
= ( + 1)2 (19)
= ( + 1)2 + ( 20)
= ( + 2)2 + 2 + + 2 ( 21)
= ( + 1)2 + ( + 1)2 + 2 +
Step 1: x and y are randomly generated in the range [2, L]. Step 2: The gene at y is saved in temporary variable
Step 3: the genes from x to y1 are shifted one position to the right
Step 4: The gene saved in temporary is then saved at position x
Finally, the time needed to get the Zbus matrix is the time needed to finish the algorithm in addition to the time to build the matrix. According to the results of [5] the time to finish the algorithm and find the optimal path is very minimal and negligible compared to the time needed to build the matrix. The minimum time needed for random path to build Zbus and finding optimal path then building Zbus according to
() are 7500 and 6000, respectively.


OBSERVATIONS ON EACH METHOD
The direct method algorithm can be used to program a general code as done by [7], but the algorithm suffers from needing more memory and computational power compared to the other two methods. The tree set algorithm results in [2] shows that it needs less memory and computational power compared to the direct method as the system/network gets bigger, where the comparison of proposed algorithm of 4bus system and 16bus system are 0.42 and 0.2 of the amount of computations needed of the direct method, respectively. Then the method is improved to include CCVS by [4], making the algorithm more appealing. Finally, the use genetic algorithm to find optimal path proved that it has effect on computation as it reduced to time needed to calculate the Zbus matrix by 20%.
+ 2
( 22)
But genetic algorithm in [5] is only applied on the direct building algorithm. It would be beneficial to apply it on other
Where is the order of the Zbus matrix and
algorithms as well. But the genetic algorithm suffers from the need to be applied from scratch with any addition of line or bus into the network.

CONCLUSION
Different Algorithms were discussed, the direct Zbus matrix building algorithm and the treeset and linkset algorithm. The direct method provides is more general and easier to program, but the treeset is faster method that is not recursive and can be generalized to include branches with CCVS. Applying genetic algorithm to the direct building method produce good results in improving the algorithm efficiency by 20%.

ACKNOWLEDGMENTS
The authors acknowledge the support of King Fahd University of Petroleum and Minerals in conducting this research work.

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