Classification of networks based on inherent structural characteristics

Transcription

1 Classification of networks based on inherent structural characteristics Tajudeen H. Sikiru, Adisa A. Jimoh, Yskandar Hamam, John T. Agee and Roger Ceschi Department of Electrical Engineering, Tshwane University of Technology, South Africa and LISV of UVSQ, France. Department of Electrical Engineering, Tshwane University of Technology, South Africa. ESIEE Paris, France and FSATI at Tshwane University of Technology, South Africa. LISV of UVSQ and ESME Sudria, France. Abstract This paper seeks to identify and classify power system networks based on their inherent definitive structural property. The participation between the load buses is observable from the eigenvectors obtained from a Schur complement of the Y-admittance matrix. This is used to classify power system networks as either topologically strong or weak. Results show that in a topologically weak network, load voltages are below the nominal voltage limit. Conversely, in a topologically strong network they are above the nominal limit. I. INTRODUCTION The main purpose of transmission networks is to convey power from generating stations to load centres. Every transmission network has unique characteristics arising from the manner in which lines are interconnected and the impedance values of such lines [1]. It is important to identify the effect of these two factors on the operational behaviour of a power system network. Since, a transmission network cannot be loaded beyond its stability limit [2]. The direct consequence is that transmission network determines the amount of power that could be transported from generating stations, even if the generating stations have higher supply capacity [3]. Hence, the level of load demand that could be met in a power system network directly depends on the inherent structural characteristics [4] [7] of the network. These characteristics do not only affect the amount of transferable power, it also determines the locations and sizes of the other network devices such as reactive power compensators [6], harmonic filters [5], and new generator location [7] that could boost the operational efficiency of a network. This paper discusses the identification and classification of power system networks based on their structural characteristics. For the remainder of the paper, section II presents the inherent structural characteristics and classification of networks, while section III presents and discusses two numerical examples. Section IV concludes the paper. II. INHERENT STRUCTURAL CHARACTERISTICS AND CLASSIFICATION OF NETWORKS A. The inherent structural characteristics of networks Power system operations are represented by two variable quantities, current and voltage, which are interrelated through the network structure called the Y-admittance matrix. The Y-admittance matrix captures the nature of the structural interconnection of buses and the value of the line impedance between them [6], [7]. The current flow in the network and the bus voltages are governed by circuit theorem, mathematically represented as I = Y V (1) where Y 1 = Z, Y is admittance, while Z is the impedance of the network. Suppose the Y-admittance matrix is partitioned as [ ] YGG Y GL Y = (2) Y LG Y LL where the Y-admittance matrix is a square matrix with dimension (G + L) (G + L), Y GG is a square matrix of dimension G G containing the connectivity to generator buses. Y GL is a G L matrix relating the generator to load buses. Y LG is the transpose of Y GL. Y LL is a square matrix of dimension L L containing the connectivity to load buses. G is the number of generator buses and L is the number of load buses in the network respectively. The matrix Y GG is a diagonally dominant matrix [8] since the diagonal elements of this matrix are a ii G a ij for all i = 1, 2,, G (3) j=1 j i Due to this property, matrix Y GG is non-singular and invertible [8], [9] /12/$31.00 c 2012 IEEE

2 Therefore, the Schur complement [10], [11] of Y GG in Y from (2) is C LL = Y LL Y LG Y 1 GG Y GL (4) The relationship between this Schur complement [10] and the Y-admittance matrix is det Y = det Y GG det C LL (5) Matrix C LL is equivalent to the load admittance matrix where the influence of generator buses has been eliminated. It contains only structural characteristics of load buses. Since generators are the sources of active power in a network, generator voltages are presumed constant and known, because they are easily controllable. The voltages of interest and the most critical are those of load voltages in power system analysis [1]. The relationship between the load structural characteristics and the load voltages becomes obvious from expanding (1) as [ ] [ ] [ ] IG YGG Y GL VG = (6) I L Y LG Y LL The algebraic manipulation of (6) gives V L [V L ] = [C LL ] 1 [I L W LG I G ] (7) Where W LG = Y LG Y 1 GG. The load structural characteristics (C LL ) is inversely related to the load voltages as shown by (7). The effect of the structural interconnection between load buses and the network load voltages could only become clearer by quantifying the structural impact of the load buses on the load voltages. To achieve this, the eigenvalue decomposition [12] is used to analyze the structural impact of the load buses as C LL = MRM = n m i µ i m i (8) i=1 Where M is an orthonormal matrix with corresponding m i eigenvectors. The diagonal matrix R has the eigenvalues µ i as its diagonal elements. The inverse of matrix C LL is C 1 LL = MR 1 M = Substituting (9) into (7) yields [V L ] = [ n i=1 n i=1 m i m i µ i (9) m i m i µ i ][I L W LG I G ] (10) The effect of the interconnectivity and line impedance values between load buses and their effect on the overall load voltages are revealed in (10) through the reciprocal relationship between the bus eigenvalues and the bus voltages. Buses associated with the smallest eigenvalues have the most contributively effect on the load voltages based on the reciprocal relationship presented in (10). B. Classification of networks Line impedance is made up of resistive and reactive components. The reactive component of transmission lines contributes to the reactive power flow in the network [13]. Reactive component may consist of inductive and capacitive properties. The voltages at the load buses depend on the adequacy of reactive power in the network. Transmission networks with sufficient reactive property may have bus voltages above the nominal voltage limit of 1.0 p.u. On the other hand, shortage of this property may cause the bus voltages to be below the nominal voltage limit [14]. In an ideal network, where there is adequate reactive element in the network and the total impedances of the transmission network and the load are exactly matched, the voltages on all the buses will be at 1.0 p.u (the nominal voltage value) [1]. However, in reality this is not achievable because of environmental factors that limit the expansion of the network, scarce resources and unforeseen load growth. The best structural design of a network that could be hoped for is the one that allows the load voltages to be within reasonable range in respect to the nominal voltage limit. Depending on the topological structure of a network and its line impedances, a network acquires certain definitive property that determines voltages at the load buses. Since, load voltages are not allowed to deviate significantly from the nominal value because of power quality and stability issues, the total impedance of loads in a network must be matched with that of the transmission network for maximum power to be transferable to the load [15]. The satisfaction of this circuit theorem condition, limits the maximum loading a transmission network can support. Thus, the definitive property a network acquires based on its topological structure is unique. As such, a network could have its load voltages to be generally above the nominal limit, which we termed topological strong network or below the nominal limit, which we define as topologically weak network. As load increases or reduces in a network, the effect is a corresponding reduction or increase in load voltages. However, a network will retain its definitive property except there is a significant change in the network topology. Hence, a network that is topologically strong or weak will still exhibit the same characteristics despite a small variation in loading, since the impedances of the transmission network and load are expected to be matched to ensure operational efficiency of the network. This definitive property is inherent in all power system networks. The structural characteristics of load buses captured by the matrix C LL could be used to determine the definitive property of networks, since each bus is associated with an eigenvalue based on the electrical distance between the load buses. The association of a unique eigenvector to a load bus corresponding to the smallest eigenvalue, we have termed the participation between load buses. For a network where the buses are far away from one another, the absolute value of the smallest eigenvalue will be zero (i.e. µ n = 0 to a precision of 10 4 ). This indicates a network with less reactive element and consequently less reactive power support. The eigenvectors (m i )

3 corresponding to the smallest eigenvalue will have the same value, indicating the non-participation electromagnetically between the buses. This characteristic is typical of topologically weak networks. On the other hand, when the absolute value of the smallest eigenvalue is greater than zero (i.e. µ n > 0 to a precision of 10 4 ), the corresponding absolute eigenvectors will have different values. This characterised a topologically strong network, where the electrical distance between load buses is small. The next section presents two numerical examples, one each for topologically weak and strong network respectively. III. NUMERICAL EXAMPLES AND DISCUSSION A. IEEE 30 bus network The first numerical example to be considered in this section is the IEEE 30 bus network. The single line diagram is shown in Fig. 1. The absolute value of the smallest eigenvalue for this test network is less than the precision defined, hence it is zero, the corresponding eigenvectors associated with this eigenvalue are the same as shown in Fig. 2. The unchanging nature of the eigenvectors associated with the load buses shows the non-participation between these buses as presented in Fig. 2. This is due to the large electrical distances between the load buses as captured by the Y-admittance matrix. The data required to build the Y-admittance matrix for this test network are shown in the appendix. The nonparticipation between the load buses and the fact that the smallest eigenvalue is zero, indicate that this test network is a topologically weak network. As such, its load bus voltages are expected to be generally below the nominal limit of 1.0 p.u. Fig. 3 shows the voltage profile for this test network and all the load voltages are well below the nominal limit. In order to illustrate the non-effect of loading on the unique definitive property of a network, which is indeed inherent to a network due to its topological characteristics, a 50% reduction in loading at all the buses was carried out; the new network bus voltages are shown in Fig. 3. Even with a 50% reduction in total loading of the network, the bus voltages are all still below the nominal voltage limit. This network is indeed a topologically weak network. Fig. 1. IEEE 30 bus network Fig. 2. Eigenvectors corresponding to the smallest eigenvalue of IEEE 30 bus network B. Southwest 40 bus network In the case of the second numerical example, we will consider the Southwest England 40 bus network. The single line diagram for this network is shown in Fig. 4. The smallest eigenvalue (in absolute value) is greater that the precision defined for this test network and it is The eigenvectors corresponding to this smallest eigenvalue is shown in Fig. 5. The eigenvectors associated with each load buses have varying numerical values as seen in Fig. 5 and hence unique for each load bus. This property arises from the small electrical distances between the load buses and the strong structural tie between them. The transmission network data for this test network are shown in the appendix. The varying numerical values of each eigenvector indicates Fig. 3. Voltage profiles of IEEE 30 bus network

4 Fig. 6. Voltage profiles of Southwest England 40 bus network Fig. 4. Southwest England 40 bus network operational efficiency of the network could be more effective, if the structural characteristics of the network are taking into consideration in locating these devices. Fig. 5. Eigenvectors corresponding to the smallest eigenvalue for the Southwest England 40 bus network that this test network is a topologically strong network. The voltage profiles of the network at 100% loading are shown in Fig. 6 and are above the nominal voltage limit. A 50% increase in loading level shows that a couple of voltages are still above the nominal voltage limit of 1.0 p.u, even for such a high increase in loading level. This test network is indeed topologically strong to carry such a level of loading and still have a couple of voltages above the nominal limit. The advantages of having a simple approach of classifying network are twofold. Firstly, in the planning phase of a network, the behavioural characteristics of each proposed design for a network may be quickly compared without running repetitive load flow studies. This advantage comes handy for a relatively large network, where many alternative paths for connecting load buses may need to be evaluated before a suitable design is selected. Secondly, during the operational phase of a network, it may be necessary to improve the functionality of a network due to expansion constraints. In this case, the locations of network devices that could improve the IV. CONCLUSION This paper has demonstrated the identification and classification of power system networks using the definitive structural property of the network. Power system networks are broadly classified into topologically strong or weak networks. Topologically strong networks have load voltages above the nominal voltage limit, whereas topologically weak networks have load voltages below this limit. These classifications are based on the participation of load buses observable from the eigenvectors of a Schur complement of the Y-admittance matrix termed the structural characteristics impact of load buses. REFERENCES [1] A. v. Meier, Electric power systems: a conceptual introduction. New Jersey: John Wiley & Sons, Inc., [2] P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, N. Hatziargyriou, D. Hill, A. Stankovic, and C. Taylor, Definition and classification of power system stability ieee/cigre joint task force on stability terms and definitions, IEEE Transactions on Power Systems, vol. 19, no. 3, pp , [3] O. O. Obadina and G. Berg, Var planning for power system security, IEEE Transactions on Power Systems, vol. 4, no. 2, pp , [4] M. A. Laughton and M. A. El-Iskandarani, On the inherent network structure, in Proceedings 6th PSCC, 1978, pp [5] G. Carpinelli, A. Russo, M. Russo, and P. Verde, Inherent structure theory of network for power system harmonics, IEE Proceedings- Generation, Transmission and Distribution, vol. 145, no. 2, pp , [6] J. R. Macedo Jr, J. W. Resende, and M. I. Samesima, The inherent structure theory of networks and admittance matrix sparsity relationship, in IEEE 10th International conference on hamonics and quality of power, 2002, pp [7] T. H. Sikiru, A. A. Jimoh, and J. T. Agee, Optimal location of network devices using a novel inherent network topology based technique, in IEEE AFRICON 2011, Livingstone, Zambia, September 2011, pp [8] J. Liu, J. Li, Z. Huang, and X. Kong, Some properties of schur complements and diagonal-schur complements of diagonally dominant matrices, Linear Algebra and Its Applications, vol. 428, no. 4, pp , 2008.

5 [9] J. Liu and F. Zhang, Disc separation of the schur complement of diagonally dominant matrices and determinantal bounds, SIAM journal on matrix analysis and applications, vol. 3, pp , [10] R. W. Cottle, Manifestations of the schur complement, Linear Algebra and Its Applications, vol. 8, no. 3, pp , [11] D. Carlson, What are schur complements, anyway? Linear Algebra and Its Applications, vol. 74, pp , [12] G. H. Golub and C. F. Van Loan, Matrix computations, 3rd ed. Baltimore: Johns Hopkins University Press, [13] D. Thukaram and C. Vyjayanthi, Relative electrical distance concept for evaluation of network reactive power and loss contributions in a deregulated system, IET Generation, Transmission and Distribution, vol. 3, pp , [14] A. Chakrabarti, D. P. Kothari, A. K. Mukhopadhyay, and A. De, An introduction to reactive power control and voltage stability in power transmission systems. New Delhi: PHI Learning Pvt. Ltd., [15] O. O. Obadina and G. J. Berg, Determination of voltage stability limit in multimachine power systems, IEEE Transactions on Power Systems, vol. 3, no. 4, pp , APPENDIX In this section, the transmission network parameters for IEEE 30 and Southwest England 40 bus networks are shown in the tables below. TABLE I TRANSMISSION LINE PARAMETERS FOR IEEE 30 BUS NETWORK S/N From To R(p.u) X(p.u) TABLE II TRANSMISSION LINE PARAMETERS FOR IEEE 30 BUS NETWORK CONTINUED S/N From To R(p.u) X(p.u) Transformers parameters TABLE III LINE PARAMETERS FOR SOUTHWEST ENGLAND 40 BUS NETWORK S/N From To R(p.u) X(p.u) B(p.u)

6 TABLE IV LINE PARAMETERS FOR SOUTHWEST ENGLAND 40 BUS NETWORK CONTINUED S/N From To R(p.u) X(p.u) B(p.u) TABLE V TRANSFORMER PARAMETERS FOR SOUTHWEST ENGLAND 40 BUS NETWORK S/N From To R(p.u) X(p.u)