Improvement of Sparse Computation Application in Power System Short Circuit Study

Transcription

1 Volume 44, Number 1, Improvement o Sparse Computation Application in Power System Short Circuit Study A. MEGA *, M. BELKACEMI * and J.M. KAUFFMANN ** * Research Laboratory LEB, L2ES Department o Electrical Engineering University o Batna Batna 05000, Algeria ** Université Franche-Comte, L2ES LRE T_31 UTBM Bât F - F Belort Abstract: This paper presents a contribution that enhances the eiciency o sparsity techniques or solving some problems in large scale power system. Alterations are introduced on storage scheme and nodes ordering in reerences [8,9] to led a reduction o the computation time and space memory. The ast sparse vector approach (FSV) is presented and implemented in a short circuit program to compute ault currents and ault bus voltages. It is based on a ormulation which computes only the required column o the Z impedance matrix (or some elements o the column) or any particular type o ault. The tests carried out on several power systems ranging rom the IEEE 30 bus to a 1084 bus test systems show that the proposed approach gives an important reduction o the computation time. 1. INTRODUCTION Most o the power network computer applications are ormulated as a set o sparse linear equations. The main computational burden, particularly in on-line network calculations, resides in the solution o these network matrix equations. Sparse matrix techniques, whose very purpose is computational eiciency, are the dominant actors in the choice o the methods. The basic order-actorize-solve sparsity techniques based on either the triangular actorization LDL T or the bi-actorization LR methods have been in use in power system sotware or decades now and are still in production use [1,2]. However, many alterations and reining techniques that oer considerable improvements in eiciency in appropriate applications continue to be introduced by researchers. Partial matrix reactorization has been used in cases where only some elements o the system matrix need to be recalculated [3]. Reerence [4] developed an heuristic ordering algorithm which minimizes the ill-ins and path lengths in partial reactorization. In some applications, important savings are also available by exploring the advantages o the sparse vector techniques which avoid the unnecessary computations when solving sets o sparse linear equations characterized by a sparse unknown vector and/or a sparse constant vector x/b [5] and using the so called ast-orward and ast-backward substitutions [6]. The ast-orward/ast-backward eature can be urther explored to speed up calculations in problems requiring just a single value or a single solution such as in ault studies.

2 4 ACTA ELECTROTEHNICA This paper presents a contribution on improving the eiciency o sparse vector techniques by perorming a special node ordering to avoid the need o diagonal multiplications and the ast orward substitutions in computing any column o Z matrix. I some elements o the columns are wanted, a part o the backward substitution operations can be also avoided. The approach is named the ast sparse vector (FSV) and is implemented in a short circuit program to compute ault currents (or short circuit capacities) and ault bus voltages. In the ollowing section a brie review o the short circuit equations is given [7,10]. 2. SHORT CIRCUIT STUDY Short circuit studies and ault analysis are thereore very important since they provide data such as voltages and currents during and ater the dierent types o aults which are necessary in designing and monitoring the protective schemes o the power system in both planning and operation stages. The usual short circuit computations are concerned with three phase aults and single phase-to-ground aults as they are considered to be the severest and the most requent. Supposing the ault location to be at a real bus marked q and the preault bus voltage is 1.0 p.u., then the ollowing short circuit equations are obtained by using Thevenin s theorem (Figure 1): 1 Iqq = (1) Z I qq qq M 1 q Y Mi, q I Mi, q M 2 M i Figure 1. Short Circuit Currents or Bus Fault. Z iq Vi = 1 Z qq (2) i = 1,, n where n is the number o buses, q the aulted bus, I the ault current (or the ault level), qq Vi the ault voltage o bus i, Zqq the impedance matrix diagonal element corresponding to the aulted bus and Ziq the impedance matrix element located in line i and column q. The ault current in a transmission line ij is given by: I = V V. y (3) ij [ i j ] ij where y ij is the admittance o the line connected between the buses i and j. For each ault on a bus location q, the short circuit current and the ault bus voltages are calculated using the above equations. It should be noted here that or each ault at bus location q, the corresponding q th column element o the Z impedance matrix needs to be computed by inverting the Y network admittance matrix. In the short circuit study it is sometimes necessary to know currents (or short circuit capacities) o lines connected to the aulted bus. The nodes connected to q are called M i. The three-phase short-circuit capacity o the aulted node q, S qq and the short-circuit capacity S Mi,q in the related branch M i -q are given by the ollowing equations: S qq =M VA / Z qq (4) S Mi,q = S qq (Z qq Z Mi,q )Y Mi,q S qq and S Mi,q are expressed in MVA and M VA is the system power reerence. It should be noticed, however, that not all the elements o q column are needed, only those which correspond to the same position o non-zero elements o the Y-matrix Thus the FSV aproach presents an eicient technique in order to calculate only the necessary elements Z qq and Z Mi,q. It appears then that the main computation burden resides in the storage and inversion o the network admittance matrix. To overcome this drawback, next sections present a sparse storage scheme or the Y matrix and its actor

3 Volume 44, Number 1, matrices. The ast sparse vector approach uses this sparsity based matrix storage scheme to compute only the required column o the Z impedance matrix or any ault location by perorming a special node ordering and avoiding the usual solution steps. 3. STORAGE SCHEME In order to exploit the beneits o the symmetry and the sparse nature o the power network Y admittance matrix, a pact matrix storage scheme in which only the non-zero elements are retained is employed. For this purpose, it is necessary to introduce tables o indexing inormation to identiy the non-zero elements and to acilitate their addressing. The storage strategy used is the so-called linked lists, which make provision or the structural modiications which are expected as a result o the matrix processing [8]. It consists o two parts: a complex vector to hold the numerical values o the admittance matrix elements and an overhead storage comprising three integer vectors used as pointers. In this scheme, only the non-zero elements are stored. The same vectors are used to store the actor matrices produced by the actorisation process o the original matrix. Considering n as the number o network buses and nz the number o non-zero elements and non-diagonal o the Y matrix. The used vectors in the storage scheme are deined as ollows: LCOL(n) is a vector o n elements giving the element number o the irst non-zero element in each column. NOZE(n) is a vector o n elements representing the number o the non-zero elements and non-diagonal in each column. ITAG(nz) is the vector o nz elements pointing to the row numbers o the nonzero elements. LNXT(nz) is the vector o nz elements giving the column number o the next non-zero element. I the element is the last one in the column, the corresponding LNXT entree is equal to zero. CE(nz) is a vector containing the numerical values o non-zero element and non-diagonal. D(n) is a vector containing the numerical values o diagonal elements. In order to reduce the space memory, the diagonal elements are stored in D vector, because each element needs only one index and the D vector can be transormed in a solution vector. 4. SPARSE COMPUTATION The Z matrix is usually obtained by inverting the Y matrix. To compute any particular Z matrix column q required or short circuit studies, reerence [9] applied the sparse matrix techniques to solve the network set o equation Yz = I by exploring the sparse vector eature o the constant vector I and thus perorming a ast orward and a ast backward substitutions. The Y matrix is actorized to the LDL T orm, z is the solution vector containing the q th column o the Z matrix and I is a singleton vector having the q th element equal to 1 and the remaining elements are zero. This paper is an enhancement to the above work leading to a aster solution which generates the relevant Z matrix elements required or any ault location. The Yz = I set o equations are solved by perorming a special node ordering, avoiding the orward substitution and keeping only ew backward substitution operations. To illustrate this FSV approach, let the set equations Yz = I be written in terms o the actorized orm as: LDL T z = I (5) The solution o (5) is given by z = ( LT ) 1D 1L 1 I It can be obtained in three steps as ollows: z' = L 1I D' = D z = 1 z' ( LT ) 1D' (6)

4 6 ACTA ELECTROTEHNICA (6) symbolizes the ast orward substitution, the diagonal multiplication and the ast backward substitution respectively. I during the symbolic actorization stage, the aulted bus q is put in the last position that is q = n, then the singleton I becomes: I = [ 0,0,,0,1] T and the above solution steps (6) reduces to z' = I D' = D 1 z = ( LT ) I= [0,0, 1 D',0, d 1] T n (7) d n is the n th element o the diagonal matrix actor D. It is clear rom the above equations that the determination o the unique non-zero element o the two singletons z and D is straightorward and does not require any computation eort to be made. Thereore the ast orward substitution and the diagonal multiplication computations are totally eliminated. I only the ault current at a bus n is required in the short circuit analysis, then only the diagonal element Znn needs to be computed and thereore, the three solution steps (6) reduce to a single arithmetic operation: Z nn = 1/ dn. (8) To calculate some elements o the columns we use the Fast Back Substitution. I the number m o the elements Z iq to be calculated is smaller than n, we classiy the nodes i in order to have the last positions beore q. the number o operations (6) will be minimized to m(m+1)/2 multiplications and m(m-1)/2 additions. In the case o computing the short-circuit capacities o the related branches o the aulted node, the operation number will be 18 in maximum (or 4 nodes connected to the aulted bus which is a maximum or a classical network coniguration) and it is independent o the size o the network. Thereore, the proposed aster sparse vector solution which generates the relevant Z matrix elements required or any ault location is obtained by using the last equation o (6) and in some cases only ew operations are used TEST RESULTS The FSV approach has been implemented into a short circuit program written in Fortran language under Windows 98 and has been used to compute ault currents, ault bus voltages and short circuit capacities o the related lines o aulted node as expressed by equations (1), (2) and (4). The program is run on a Pentium II, 200MHz. To evaluate the perormances o the proposed FSV approach, series o tests have been carried out on several power systems ranging rom the IEEE 30 bus test system to a 1084 bus test system. The main characteristics o these systems are shown in Table 1. Table 1. Test systems characteristics. Test system Buses Branches Generators IEEE bus 59-bus IEEE bus 685- bus bus bus Table 2 shows the operations counts o the proposed ast sparse vector approach (FSV) compared to those o the sparse vector short circuit program (SV) o reerence [9]. The operations considered are the multiplications and additions required to compute a Z matrix column corresponding to a given aulted bus. Table 2. Operations counts. Test system SV methods FSV methods Ratio FSV/SV IEEE bus 59-bus IEEE bus 685-bus

5 Volume 44, Number 1, bus bus Figure 2 shows the computing times as a unction o the size o the test systems or computing a Z matrix column corresponding to a given ault location using the FSV and SV methods. The computing time considered here is the time needed or each method to construct the network Y matrix, computing its L and D actor matrices, computing the Z matrix column and calculating the ault current and the ault bus voltages or a given ault location CPU time Number o buses Analysis o table 2 shows clearly that the ratio o the number o operations o the proposed FSV approach to that o the SV method is around 0.4 or all the test systems. This means that or computing the Z matrix column or a ault at a given bus, 60% o the computational eort is earned by the use o the FSV approach. Figure 2 demonstrates the superiority and the potential o the proposed FSV approach illustrated by the important reduction in the computing time due to the elimination o all ast orward substitutions, the diagonal multiplications and part o the ast backward substitution operations. It should be noted that i the ault bus voltages are not needed then the three solution steps are eliminated and the ault current or the ault level o the corresponding aulted bus is calculated by a simple use o equations (1) and (8). FSV SV-SC Figure 2. CPU time (sec): SV and FSV methods 6. CONCLUSIONS Reined solutions o power systems sparse linear equations using sparse vector techniques have been the main ocus o this paper. A ast sparse vector approach named FSV has been ormulated and tested or computing the relevant elements o the Z matrix required to determine the ault current or the ault level and the ault bus voltages o a power system network. It has been shown that the proposed FSV approach has the ollowing properties: - Reduction by 60% o the computational eort compared to the sparse vector based short circuit program SV. - Reduction o the space memory by using sparsity techniques based on a speciic storage scheme and on the use o dynamical management. - avoiding the complex algorithms induced by the ast orward and ast backward substitution. - Limiting to a maximum o 18 operations independently o the size o the network, or the short circuit capacities o aulted node and related lines. Further work would be devoted to developing similar sparse reined solutions o other power system problems such as transient stability. REFERENCES [1] B. Stott and O. Alsaç, An Overview o Sparse Matrix Techniques or On-Line network Application, IFAC Power Systems and Power Plant Control, pp.19-25, Beijing, [2] W.F. Tinney and J.W. Waler, Direct Solutions o Sparse Network Equations by Optimally Ordered triangular Factorization, Proc. IEEE, Vol. 55, pp , Nov [3] S.M. Chang and V. Brandwajn, Partial Reactorization, IEEE Trans. on Power Systems, Vol. PWRS-1, No. 1, pp , February [4] R. Betancourt, An Eicient Heuristic Ordering Algorithm or Partial matrix reactorization, IEEE Trans. on Power

6 8 ACTA ELECTROTEHNICA Systems, Vol. PWRS-3, No. 3, pp , August [5] W.F. Tinney, V. Brandwajn and S.M. Chan, Sparse Vector Methods, IEEE Trans. on Power Systems, Vol. PAS-104, No. 2, pp , February [6] A.R. Basso, C.R. Minussi and A. Padilha, Fast-Forward/Fast-Backward Subsitutions on Vector Computers, IEEE Trans. on Power Systems, Vol. 14, No. 4, pp , November [7] O.L. Elgerd, Electrical Energy Systems Theory, McGraw-Hill Edition, [8] K. Zollenkop, Bi-actorization Basic Computational Algorithm and Programming Techniques, Large Sparse Sets o Linear Equations, Edited by J.K. Reid, Academic Press, 1971, pp [9] M. Belkacemi, N. Aloui, L. Benarhi, M. Ghezaili, Sparsity Techniques Application in Power System Short Circuit Computation, 3 rd Regional CIGRE Conerence, May 1999, Doha, Qatar. [10] G.W.Stagg and A.H.Elabiad, Computer Methods In Power System Analysis, McGraw-Hill Edition, 1983.