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Harmonic Domain Dynamic Transfer Function of a Nonlinear Time-Periodic Network

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Harmonic Domain Dynamic Transfer Function of a Nonlinear Time-Periodic Network IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 4, OCTOBER 2003 1433 Harmonic Domain Dynamic Transfer Function of a Nonlinear Time-Periodic Network Taku Noda, Member, IEEE, Adam Semlyen, Life Fellow, IEEE, and Reza Iravani, Senior Member, IEEE Abstract—This paper presents a new concept called harmonic operating in its periodic steady state, in terms of the frequency domain dynamic transfer function (HDDTF), which characterizes response of perturbations of harmonic components. The net- the dynamics of a nonlinear, time-periodic network as seen from work is seen from a port or multiple ports, and the nonlinear a port (or multiple ports), in terms of the frequency response of harmonic perturbations superimposed on its underlying periodic state equations that describe the given network are linearized steady state. It pertains to the transient behavior superimposed on about the periodic steady state. The HDDTF is then calculated the steady state. The HDDTF is a transfer-function matrix @ A from the linearized state equations with periodic coefficients. relating the vectors of harmonic domain input and output endowed If the port voltages and currents are selected as the input and with -domain properties. Because the network can contain sat- the output of the state equations, respectively, such an HDDTF urable (nonlinear) elements and periodically-switching (time-pe- riodic) power electronics components, the HDDTF may be used may be called a harmonic domain dynamic admittance matrix for the analysis of power quality problems. It may also serve for (HDDAM), and it is compatible with admittance- and conduc- the identification of a reduced-order dynamic equivalent of a non- tance-based methods widely used in power system analysis. The linear, time-periodic network to be used in time-domain transient concept of the HDDTF can be considered as an extension of simulations. The HDDTF is obtained by linearization, about the its static case, which is used for steady-state analysis [3], [4]. periodic steady state, of the nonlinear state equations describing a given network. Following the derivation of the HDDTF, a modal The HDDTF includes the information not only of steady state analysis to characterize the HDDTF by its diagonalization is pre- but also of the frequency response of harmonic perturbations sented. Two test systems are used to produce numerical examples. in addition to the static case. Thus, equivalently, the HDDTF Index Terms—Dynamics, electromagnetic transient analysis, characterizes the transient behavior superimposed on the pe- large-scale systems, nonlinear circuits, periodic functions, power riodic steady state in the time domain. In short, the HDDTF electronics, power quality, power system harmonics, saturable is a transfer-function matrix relating the vectors of har- cores, transfer functions. monic domain input and output endowed with -domain prop- erties. This paper also presents a modal decomposition analysis I. INTRODUCTION to characterize the HDDTF by its diagonalization. The modal domain equations give simpler and more tangible interpretation ARMONIC sources have a primordial role in the opera- of the HDDTF. H tion of interconnected power systems and lead to impor- In HDDTF applications, the network can contain saturable tant power quality assessment issues [1], [2]. Such sources are (nonlinear) elements and periodically-switching (time-periodic) usually nonlinear or time-periodic system components; the non- power electronics components. Therefore, the HDDTF may be linear components include saturable elements such as the mag- used for the analysis of power quality problems. It can also be netizing circuits of transformers, and the time-periodic elements used for the identification of a reduced-order dynamic equiva- include periodically-switching power electronics converters. It lent of a nonlinear, time-periodic network for time domain tran- is often useful to characterize the dynamics of a network in a sient simulations as presented by the authors [5]. In this paper, compact form, and transfer functions like impedance and ad- a simple illustrative network and a relatively large network are mittance matrices are used to this end for linear time-invariant used to provide numerical examples. networks. However, if the network is nonlinear and time-peri- odic, such an entity is, to our knowledge, not available at this II. DERIVATION OF HDDTF time. In this paper, a concept called harmonic domain dynamic A. Nonlinear Time-Periodic Network transfer function (HDDTF) is introduced. The HDDTF charac- Let us consider a nonlinear time-periodic network, operating terizes the dynamics of a nonlinear and time-periodic network, in periodic steady state with the base frequency , as seen from a port or from multiple ports as shown in Fig. 1. The number of Manuscript received February 17, 2002. This work was supported by the Nat- state variables in the network is , and the number of terminals ural Sciences and Engineering Research Council (NSERC) of Canada. (phases) in the port(s) is . The network consists of linear T. Noda is with Electrical Insulation Department, Central Research time-invariant , , components, nonlinear , , compo- Institute of Electric Power Industry, Tokyo 201-8511, Japan (e-mail: [email protected]). He is currently a Visiting Scientist at the University of nents, periodically operating switches, and independent voltage Toronto, ON, Canada. and current sources with periodic waveforms. The operating pe- A. Semlyen and R. Iravani are with the Department of Electrical and riod of the switches and the sources is , Computer Engineering, University of Toronto, Toronto, ON, M5S 3G4, Canada (e-mail: [email protected]; [email protected]). and therefore, the waveforms of all voltages and currents are Digital Object Identifier 10.1109/TPWRD.2003.817788 -periodic. To obtain the underlying periodic steady state, the 0885-8977/03$17.00 © 2003 IEEE 1434 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 4, OCTOBER 2003 The Fourier series of in (3) is (7) and the same applies to the other coefficient matrices , , and in (3) and (4). Substituting (5)–(7) into (3) and (4) gives (a) (b) Fig. 1. Nonlinear time-periodic network, operating in its periodic steady state. (a) Single-port case. (b) Multi-port case. part of the system outside the objective network must be in- cluded in the calculation, and the steady-state solution is as- sumed to be available at the outset from a harmonic power flow solution. A recent paper [6] reviews many of the best harmonic power flow solution methods presently available. The state equations of the objective network, as seen from the port(s), can be derived in the following form [7]: (8) (1) (2) where are state, input, output vectors, respectively, and , nonlinear algebraic functions. B. Harmonic Domain State Equations Let us assume that a disturbance takes place outside the net- work causing excursions of , , from the steady state. The ex- cursions are denoted as , , , and assumed small enough to allow the linearization: (9) (3) Note that the common exponential term has cancelled out. We now focus on the first summation regarding on the (4) right-hand side of (8). It is expanded in ascending order with Because the coefficient matrices , , , and are -peri- respect to the harmonics: odic due to the linearization, the above equations are the -pe- riodic linear state equations representing the network. If we regard the network together with outside elements as one closed system, the transition matrix of the whole system is expressed as based on Floquet theory [8]–[10], where is a -periodic and a constant matrix. (10) A particular mode of the transition matrix takes the simple form , where is the complex frequency and a -periodic By neglecting the -th and higher harmonics of and also vector [10]. Therefore, all variables , , and vary, for neglecting -th and higher harmonic components of , the one mode, in unison according to the exponential modal pattern, coefficients of the exponential terms can be expressed by the and can be replaced by its -periodic part multiplied following matrix notation (for ): by an exponential function of time: . Expressing the -periodic part by its Fourier series, we obtain . (5) The same applies to and . The time derivative of (5) is given by . (6) (11) NODA et al.: HARMONIC DOMAIN DYNAMIC TRANSFER FUNCTION OF A NONLINEAR TIME-PERIODIC NETWORK 1435 If is a scalar, then the above defined is a Hermitian Toeplitz three-phase case is used here for illustration. The input vector matrix. If is a matrix, then is a “block” Hermitian Toeplitz takes the form in the time domain, matrix, consisting of the Fourier coefficients of . Defining and in the harmonic domain the harmonics of as are arranged as (12) the size of is by , and its nonzero bandwidth is . The harmonic domain state vector, consisting of the Fourier co- efficients of , is denoted by [to be distin- guished from its time domain counterpart in (3) and (4)]. Ap- plying this matrix notation to all terms on the right-hand sides of (8) and (9), we obtain the harmonic domain state equations (19) (13) The same applies to the output vector . (14) If the port voltages and currents are chosen as and respec- tively, such an HDDTF is a harmonic domain dynamic admit- where , are the harmonic domain input and tance matrix (HDDAM), which is compatible with admittance- output vectors, and is the matrix of dynamic differentiation and conductance-based methods widely used in power system with respect to time, in the harmonic domain, defined by analysis. (15) III. MODAL ANALYSIS OF HDDTF with and is the by Once the by HDDTF matrix has been calculated, identity matrix. The above definition of includes not only the results should be interpreted in a meaningful way so as to the harmonic frequency but also the complex frequency facilitate any useful application of the obtained information. A to express the dynamics of the network, and thus, this is an first and simple description of the results is of course directly extension of the static differentiation matrix available by representing the individual elements of de- (16) fined in (18).
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