IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 4, NOVEMBER 2011 1905 Reduced-Order Transfer Matrices From RLC Network Descriptor Models of Electric Power Grids Francisco Damasceno Freitas, Senior Member, IEEE, Nelson Martins, Fellow, IEEE, Sergio Luis Varricchio, Senior Member, IEEE, Joost Rommes, and Franklin C. Véliz Abstract—This paper compares the computational perfor- mances of four model order reduction methods applied to large-scale electric power RLC networks transfer functions with many resonant peaks. Two of these methods require the state-space or descriptor model of the system, while the third requires only its frequency response data. The fourth method is proposed in this paper, being a combination of two of the previous methods. The methods were assessed for their ability to reduce eight test systems, either of the single-input single-output (SISO) or mul- tiple-input multiple-output (MIMO) type. The results indicate that the reduced models obtained, of much smaller dimension, reproduce the dynamic behaviors of the original test systems over an ample range of frequencies with high accuracy. Index Terms—Descriptor systems, dominant poles, domi- nant subspaces, eigenvalues, frequency response, index-2 DAE, large-scale systems, low rank Gramians, passivity, reduced models, resonant peaks, RLC circuits, singular systems, transfer function, vector fitting. I. INTRODUCTION M ODEL order reduction (MOR) has been widely used in scalar and multivariable control system applications [1], vibration analysis of large mechanical structures, and VLSI cir- cuit design [2]. Network equivalents are used in power system electromagnetic transient studies [3], as well as in real-time sim- ulators [4] and harmonic distortion analysis [5]. Power system models requiring detailed modeling include those having a large number of resonant peaks in their frequency responses, over a wide range of frequencies [6]. Detailed representations of every component in a large-scale system yield more accurate results but require excessive CPU time, calling therefore for the further development of model order reduction techniques. A good reduced-order model (ROM) should have a reduced state-space vector and be able to reproduce the simulated input- Manuscript received December 31, 2009; revised January 13, 2010, October 05, 2010, and January 12, 2011; accepted March 02, 2011. Date of publication May 12, 2011; date of current version October 21, 2011. The work of F. D. Freitas was supported in part by FINATEC. Paper no. TPWRS-01021-2009. F. D. Freitas is with the Department of Electrical Engineering, University of Brasilia, Brasilia, DF, CEP:70910-900, Brazil (e-mail: ffreitas@ene.unb.br). N. Martins and S. L. Varricchio are with CEPEL, Rio de Janeiro, RJ, CEP:20001-970, Brazil (e-mail: nelson@cepel.br; slv@cepel.br). J. Rommes is with NXP Semiconductors, Central R&D, High Tech Campus 46, 5656 AE Eindhoven, The Netherlands (e-mail: joost.rommes@nxp.com). F. C. Véliz is with the Pontific Catholic University (PUC), Rio de Janeiro, RJ, Brazil (e-mail: franklin@cepel.br). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2011.2136442 output response of the original system with the desired accu- racy while requiring a considerably smaller computational ef- fort. Also, the CPU time needed to generate the reduced-order model should not be excessive, so that the whole ROM utiliza- tion process is cost-effective. The main contributions of this paper are: 1) the performances of three existing MOR methods are compared for large power system RLC models; 2) a combination of two methods is proposed, generating a new MOR method, which is also tested against the previous three methods; 3) a simple and effective choice for the alternating direction implicit (ADI) parameters of the low-rank Choleski factor (LRCF) method is presented, when applied to RLC circuits; 4) tests including time and frequency responses as well as spectral (eigenvalue) plots are shown; 5) full data on a 34-bus subtransmission system from practice, together with eight related descriptor system models, are provided in this paper and online, for data exchange in the electromagnetic transients as well as the MOR communities. The first method is the LRCF method, described in [7] and [8]. The second method is based on the computation of dominant subspaces (Subspace Accelerated Dominant Pole Algorithm and Subspace Accelerated MIMO Dominant Pole Algorithm (SADPA [9] and SAMDP [10], respectively)). The third method is vector fitting (VF) [11]–[13], which allows the computation of the reduced model directly from the (finely enough) dis- cretized values of the frequency response of the original system. The fourth method is a new hybrid method, which combines the use of the LRCF method with the VF method, yielding both computational and qualitative gains, allowing transformation of nonpassive ROMs obtained by LRCF into passive ROMs. All transmission lines considered in this paper are modeled by ladder networks, comprised of cascaded RLC PI-circuits, having fixed parameters. Representation of transmission lines by large numbers of RLC PI-circuits results in many resonant peaks in the system’s frequency response, spread over a large frequency range. The formulation and studies of frequency-dependent pa- rameter transmission lines can be found in several classical references [14]–[16] and selected reprints [17]. SISO modal equivalents for -domain models of power systems having long transmission lines with frequency-dependent parameters are described in [18]. Multi-port frequency-dependent equiv- alents, involving some approximations but given in terms of a passive RLC circuit, which are frequently used in practical EMTP studies, are described in [3]. The terms multi-port and MIMO are used interchangeably in this paper. The frequency dependence of longitudinal parameters is also considered in 0885-8950/$26.00 © 2011 IEEE