Designs of Fast Decoupled Load Flow for Study Purpose

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Energy Procedia 17 (2012) 127 – 133

2012 International Conference on Future Electrical Power and Energy Systems

Designs of Fast Decoupled Load Flow for Study Purpose Yubin Yao1 ,Mingwu Li2 1

Marine Engineering College, Dalian Maritime University, Dalian, China 2 Yantai Hi-yearning Electronic-Technology Co. Ltd, Yantai, China

Abstract The fast decoupled load flow (FDLF) program which employing sparse matrix techniques needs less computer storage and running time, but makes it harder by modifying the program to adapt to further study such as power system security and continuation power flow. To meet study purposes a less complicated but fast FDLF algorithm is presented. A great deal of unnecessary computation is exempted in the algorithm simply by adding some selection actions in the procedure of Gauss elimination. The presented method needs a little more running time than the one using sparse matrix techniques, but it is much faster than the one without sparse matrix techniques. And it runs as fast as the practical Newton power flow with sparse matrix techniques does. So the presented method can satisfy the needs for study purpose. Experimental results validate the practicability of the presented method. by by Elsevier Ltd. Selection and/or peer-review under responsibility of Hainan University. © 2012 2011Published Published Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer] Keywords: Fast decoupled load flow; running time; sparsity; Gauss elimination

1. Introduction The power flow (or load flow) predicts all flows and voltages in the network when given the status of generators and loads. It is the tool most heavily used by power engineers [1]. The principal information obtained from a power flow study is the magnitude and phase angle of the voltage at each bus and the real and reactive power flows in equipment such as transmission lines and transformers in a power system under balanced three-phase steady state conditions [2-3]. As a by-product of this calculation, some other data such as equipment losses can be computed. The power flow is basic tool for some static power system analysis such as power system security, optimum power flow, state estimation, continuation power flow [4]. Several methods are applied to power flow study. Among of these methods, Newton method [5] and fast decupled load flow (FDLF) [6-7] are widely used for their good convergence, fast calculation speed and modest computer storage request. Comparing with the Newton method, The FDLF method is much faster. A typical power system has an average of fewer than three branches (either transmission lines or transformers) connected to each bus. As such, the admittance matrix is highly sparse matrix, of which each row has an average of fewer than four non-zero elements, one off-diagonal for each branch and the diagonal. The coefficient matrices of the correction equation for the Newton method and the FDLF are also sparse matrix. So the power flow programs can employ sparse matrix techniques to reduce computer storage and time requirements [8]. The sparse matrix techniques made it possible to apply the Newton

1876-6102 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Hainan University. doi:10.1016/j.egypro.2012.02.073

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method to systems of arbitrary size, to attain for the first time both speed and excellent convergence characteristics [1]. The sparse techniques include compact storage and compact computation of sparse matrix, and reordering of buses to avoid fill-in of the matrix during Gauss elimination. The compact storage of matrices makes it inconvenient to search and modify the elements of the matrix, such the program is difficult to modify to adapt to further study such as power system security which is based on power flow study. To solve this problem, an FDLF algorithm for study purpose is designed in this paper. The presented method is easy to adapt to other purpose since compact storage of matrices is not used, and fast because a great deal of unnecessary computation is exempted in the algorithm by some slight but significant modifications. The results of a practical power system validate the practicability of the presented method. 2. Type Sparse Matrices in the FDLF Method The admittance matrix and coefficient matrix of the correction equation in FDLF program are highly sparse. Typically each row of the admittance matrix has an average of fewer than four non-zero elements. A. The Bus Admittance Matrix Systematic formulation of equations determined at the nodes of a circuit by applying Kirchhoff’s current law is the basis for various computer solutions of power system problems. The standard form of nodal equations is as ollows:

ª Y11 Y12 «Y « 21 Y22 « Y31 Y32 « # « # «¬YN 1 YN 2

Y13 Y23

" Y1N º ª V1 º " Y2 N »» ««V2 »» " Y3 N » « V3 » »« » # »« # » " YNN »¼ «¬VN »¼

Y33 # YN 3

ª I1 º «I » « 2» « I3 » « » «# » «¬ I N »¼

(1)

or in matrix notation is:

YV

I

(2)

where Y is the N h N bus admittance matrix, V is the column vector of N bus phase voltages, and I is the column vector of N phase currents. The diagonal element Yii of the bus admittance matrix is the self admittance of bus i, which is sum of all the admittances connected to bus i as follows:

Yii

¦y

ik

(3)

ki

where yik is the admittance between buses i and k, and k  i denotes the bus k is connected to bus i. The off-diagonal element Yij of the bus admittance matrix is the mutual admittance between buses i and j, which is negative of the sum of all the admittances between buses i and j as follows:

Yij

¦ y ij

The admittance matrix has following characteristics:

(4)

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(a) It is sparse with the degree of sparsity increasing with the network size. When there has not a branch between buses i and j, Yij is zero, and need not be stored. (b) It is symmetrical, if there are no phase-shifting transformers. The lower triangular matrix need not be stored. For compact storage of a matrix, we can use the following three arrays: (a) Value -- is to store the non-zero elements of the matrix. (b) Col -- is the column number of each non-zero element. (c) Row -- is the location in array Value for the first non-zero element in each row of the matrix. Using above three arrays to store a matrix, it is difficult to add, delete or modify an element in the matrix. If 8 bytes of computer storage are used for each double floating-point number and 4 bytes for each integer, the total number of bytes for a matrix with compact storage is:

S

8m  4m  4n

(5)

where n is the number of rows of the matrix, m is the number of non-zero elements of the matrix. The element of the admittance matrix is complex number, needs two double floating-point numbers each, but shares same column number, So for a power system with N buses and M branches, the total number of bytes for the admittance matrix with compact storage is:

S

16( M  N )  4( M  N )  4 N

(6)

B. The Coefficient Matrices of Equations The equations for the FDLF method are as follows:

'P / V

B' 'T

(7)

'Q / V

B" 'V

(8)

where ǻP/V and ǻQ/V are respectively the column vector of active and reactive power mismatches divided by voltage magnitude, ǻV and ǻș are the column vector of voltage magnitude and phase angle corrections respectively, Bƍ is an (n – 1) × (n – 1) matrix for all buses except the slack bus and BƎ is an m × m matrix for PQ buses, n is the number of buses in the system, and m is the number of PQ buses. There is no equation in (7) for the slack bus whose phase angle is specified and active power is unspecified, so the dimension of Bƍ reduces to (n – 1) which is the number of buses except the slack bus. There are no equations in (8) for the slack bus and PV buses whose voltage magnitude is specified and reactive power is unspecified, so equations (8) only include equations for PQ buses and the dimension of BƎ reduces to m which is the number of PQ buses. The matrices are from the network susceptance matrix B, but series resistances, shunt reactances, offnominal ratio transformer taps are omitted from Bƍ. And phase-shifter effects are omitted from BƎ. The matrices Bƍ and BƎ are both real and sparse. C. Solutions to Linear Equations by Gauss Elimination Consider the following set of linear algebraic equations in matrix notation: (9) AX B where A is the N h N square matrix, X and B are N vectors. Given A and B, the solution X can obtained by Gauss elimination. There are three procedures to solve linear equations by Gauss elimination: Gauss elimination, forward substitution and back substitution. 1) Gauss elimination: When xk is eliminated from the ith equation, the new element of the coefficient matrix is:

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Yubin Yao and Mingwu Li / Energy Procedia 17 (2012) 127 – 133

aij( k )

aij( k 1)  aik( k 1) ˜ akj( k 1) / akk( k 1)

(10)

j k  1,", n, i k  1,", n where the superscript (k) denotes step k of Gauss elimination. 2) Forward substitution: When xk is eliminated from the ith equation, the new element of the righthand vector is:

bi( k )

bi( k 1)  aik( k 1) ˜ bk( k 1) / akk( k 1) i

k  1,", n

(11)

3) Back substitution: The solution xi by back substitution is as follow:

xi

(bi( i 1) 

n

¦a

( i 1) ij

˜ x j ) / aii( i 1) i

n  1,",1

(12)

j i 1

3. Design of Power Flow for Study Purpose Since the admittance matrix, the coefficient matrices of equations and the matrices after Gauss elimination are all highly sparse matrices, employing sparse matrix techniques in power flow programs can greatly reduce computer storage and time requirements. Sparse matrix techniques are a standard feather of today’s power flow programs in practice. But the programs with sparse matrix techniques are very difficult to program, debug and modify. That is chief disadvantage to the researcher who uses power flow programs for study purpose because he frequently modifies the programs to adapt to further study such as power system security, optimum power flow, continuation power flow, etc.. To solve this problem, an FDLF algorithm for study purpose is presented in this paper. C. Design for Coefficient Matrices of Equations The dimension of coefficient matrices Bƍ and BƎ is less than that of the admittance matrix B. B is an n × n matrix, while Bƍ is an (n – 1) × (n – 1) matrix and BƎ is an m × m matrix. For the convenience of programming and storage- saving, the matrices are generally storied by their actual size in many programs which sparse matrix techniques are not employed. That is, Bƍ is storied by an (n – 1) × (n – 1) array and BƎ is storied by an m × m array. To meet this arrangement, some FDLF programs require that PQ buses are numbered before PV buses and the slack bus is numbered as the last number n, which restricts the flexibility of input data and imposes on users a tedious task of rearranging the data for the program. Some FDLF programs satisfy the requirement of data arrangement by renumbering buses in program, which releases users’ burden for rearranging input data but makes the program complex and difficult to modify for further study. Another shortcut is that the program can only deal one slack bus in the system. To make the program easy to use and modify, coefficient matrices Bƍ and BƎ in this paper are both stored with an n × n arrays. All elements in rows and columns of Bƍ are zero for slack buses, and the elements in rows and columns of BƎ are zero for slack buses and PV buses. To design matrices in this way enhance the flexibility of input data and allow multiply slack buses in the network. All these flexibility are guaranteed by the reasonable design on the procedure of solutions to linear equations. D. Design for the Procedure of Solutions to Linear Equations Because coefficient matrices Bƍ and BƎ are both stored with an n × n arrays in the program, there are many rows whose elements are all zero. Those rows can skip by selection actions. Design in this way the program can deal with multiple slack buses in the system and allow buses numbered arbitrarily. From (10), it can find that when aik is zero, the computation to eliminate xk from the ith equation is unnecessary. So before such action, it is necessary to evaluate that whether aik is zero. The flow chart of Gauss elimination to linear equations is shown in Fig. 1.

Yubin Yao and Mingwu Li / Energy Procedia 17 (2012) 127 – 133

Floating-point numbers are represented only approximately by most computers [9]. So we substitute aij  0 with | aij | > İ in Fig. 1, İ is a specified small positive number. start k=1 | akk | > İ ?

N

Y i=k+1 i>n?

i=i+1 N

Y

N | aik | > İ ? Y j=k+1

Y

j>n? N aij = aij – aik akj / akk j=j+1

k=k+1 Y

k
Figure 1 Flow chart of Gauss elimination to linear equations.

4.

Case Study

A real network named Northeast network in China are used to testify our conclusions. There are 445 buses and 544 branches (include transmission lines and transformers) in it, which is large enough for study purpose. The specified convergence tolerance is 0.0001. The power flow study on this network was carried out by four different methods: Method 1 (M1): The FDLF method without sparse matrix techniques. Method 2 (M2): The recommended FDLF method, which sparse matrix techniques are not applied, but unnecessary computation is exempted in the algorithm by adding some selection actions. Method 3 (M3): The FDLF method with sparse matrix techniques. Method 4 (M4): The Newton method with sparse matrix techniques.

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The study is carried out on a personal computer with an Intel Pentium process of 1.10 GHz. The results are listed in Table I, the computer storage of M1 and M2 includes storage for the admittance matrix and for the coefficient matrices; the computer storage of M3 and M4 includes storage for the admittance matrix, the coefficient matrices, and the matrices for Gauss elimination. TABLE I Computer Storage and Running Time Required by Different Methods Methods

Storage (B)

Running Time (s)

M1 M2 M3 M4

4752600 4752600 52804 151284

0.422 0.156 0.063 0.156

The FDLF program without sparse matrix techniques consumes much more computer time than the one with sparse matrix techniques. The running time can be evidently reduced just by adding some selection actions in the program. The running time for this example reduces to 0.156s which is same as that of the Newton method with sparse matrix techniques, and is acceptable. Sparse matrix techniques can greatly save computer storage. The storage requirements of the presented method are 90 times of that of the program with sparse matrix techniques, but storage requirements of less than 5 MB is not too much to a modern computer with memory of several GB. 5. Conclusion The power flow is not only a tool tells operating states of the system but also a basic tool for some static power system analysis such as power system security, optimum power flow, state estimation, continuation power flow. For researchers who use power flow program as tool for further study, the maintainability rather than the storage requirements and running time is a matter of the utmost concern. Such simple and fast power flow program is very helpful to the researchers. Sparse matrix techniques which make the program complicate are not suitable for study purpose, and not applied in the presented FDLF algorithm. The power flow program can speed up by exempting some unnecessary computation in the procedure of Gauss elimination simply by adding some adequate selection actions. Study on a large practical network shows that though minor modification is made, the improvement in running time is obviously. Acknowledgment This work was supported in part by NSFC, P. R. China under Grant 61074017, and by the Education Department of Liaoning Provine, China under Grant 2008085. References [1]

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