Frequency-domain modeling of time-periodic switched electrical networks: A review

1. Introduction

Time-domain (TD) modeling and simulation of power systems that include periodically switched converters generally achieve high accuracy when using different existent platforms from the electromagnetic transient (EMT) program family [1]. This accuracy comes at a cost of relatively long simulation times, together with the need of a sophisticated user. Meanwhile, fixed-step TD simulation has its own limitations in treating switching converters for different cases.

A well-established alternative for a full-fledged TD simulation of switched devices is the time-averaging models [2], [3]. Nonetheless, averaged models do not capture high-frequency ripples due to periodically switched system’s dynamics.

Moreover, the increased use of distributed energy resources, together with the increased reliance on HVDC, are pushing converters with different topologies at different voltage levels of power systems. This trend requires a straightforward approach for switched network analysis. FD approaches where phasorquantities of the same frequency are used to describe the system states have always proved to be an effective and a useful tool for simplifying the analysis of electric networks. Even with the presence of nonlinear elements including switched components, harmonics enable the use of phasor domain approaches albeit some changes and maneuvers to allow for adopting the same well established simple concepts, e.g., impedance, transfer functions, etc.

Tough TD simulation is more advantageous than FD in analyzing nonlinear systems, the inability to practically address FD representation of converters deprived power system analysts from a host of practical tools for design and analysis, e.g., stability tools, load flow and steady state analysis tools, etc.

This paper provides a brief discussion of these efforts, limited to the application of power converters modeling. In modeling converters in FD, three main directions exist at present. The first is Floquet theory, where a transformation is considered to mask the TP nature of a system. The system is considered in Floquet domain as a time-invariant system subject to the same concepts and analyses of such systems. The second is Harmonic Domain Dynamic Transfer Functions. This approach considers relationships among the frequency spectra vectors of input and output signals. The third approach relies on the use of Equivalent Signal theory to describe the behavior of a converter/time-periodic network by means of synthetic signals intersecting with the actual signals at the switching instances. This paper compares these three mathematical tools from the points of view of: (i) computational effort for model preparation, (ii) possible applications to modern power networks, and (iii) suitability for integration with present network analysis tools.

2. Floquet theory

2.1. Concept

Solution of dynamic systems is sometimes simplified when modeled in alternative coordinate frames. Floquet theory provides a robust transformation, via the system’s transition matrix, to convert a linear time-periodic (LTP) system into a system with linear time-invariant (LTI) state matrix [4], [5]. A main feature of Floquet transformation relies on its ability to provide zero/pole location on the complex plane. The potentiality of such feature is that it can be used for parameter design and stability assessment. Although Floquet method has been originally conceived for LTP networks, it can readily be extended to linearized versions of nonlinear/switched systems.

Despite their periodic nature, analysis of power systems has rarely been performed by using Floquet coordinates. This is due to the fact that for large systems the computation of the transition matrix becomes difficult. Nevertheless, for switched devices such complexity is substantially diminished. Hence, a potential application of Floquet-based models is in the area of distributed generation where converters with different switching frequencies are usually involved.

2.2. Mathematical foundation

An n-dimensional LTP system, with period T, may be expressed as(1) $\begin{matrix} \dot{x} (t) = A (t) x (t) + B (t) u (t), \\ y (t) = C (t) x (t) + D (t) u (t), \end{matrix}$ where variables and matrices are of appropriate dimensions. Note that, in (1), we have assumed that state, input, forward, and feedback matrices are all T-periodic. System (1) may be transformed, via Floquet transformation, to new coordinates, yielding(2) $\begin{matrix} \dot{v} (t) = Qv (t) + \bar{B} (t) u (t), \\ y (t) = \bar{C} (t) v (t) + D (t) u (t), \end{matrix}$ where the coordinates transformation, given by (3a), is provided by the T-periodic matrix P(t).(3a) $x (t) = P (t) v (t) .$

From (2), (3a):(3b) $\begin{matrix} \bar{B} (t) = P^{- 1} (t) B (t), \\ \bar{C} (t) = C (t) P (t) . \end{matrix}$ Q in (2) represents a constant matrix and its eigenvalues define system’s stability properties; $\bar{B} (t) and \bar{C} (t)$ are periodic under the transformation (3a).

It can be proved that(4) $P (t) = Φ (t) \exp (- Qt),$ where $Φ (t)$ represents the transition matrix of system (1) and satisfies:(5) $\dot{Φ} (t, t_{o}) = A (t) Φ (t, t_{o}), with Φ (t_{o}, t_{o}) = I_{d},$ where Id represents an identity matrix. Also, we have:(6) $Φ (t + T, t_{o}) = Φ (t, t_{o}) \exp (QT) .$

2.3. Publications in power systems analysis utilizing Floquet theory

The theory proposed by Floquet [4], [5] was originally applied to control design of LTP systems [6], focusing on simple mathematical models. Characterization of zero/pole map location [7] and stability analysis of DC-AC inverters [8] have been a major research area with Floquet theory as an auxiliary tool. Based on the fundamentals in [9], a recent line of research is the application of Floquet theory to modeling of power converters [10], [11]. The fact that most electronic devices rely on pulse-width modulation (PWM) techniques has facilitated computation of transition matrices. The fact that the transition matrix is expressed as the product of a periodic matrix and an exponential matrixpermits to model an LTP system in Floquet coordinates aimed at transient and steady-state computations. However, transient studies have been rarely researched. A case study involving transient and steady states is presented next.

2.4. Example

A high-voltage direct current (HVDC) system, depicted in Fig. 1, is simulated via TD numerical solution of: (a) system (1), labeled as TD, and (b) its counterpart in Floquet coordinates, system (2), labeled as Floquet. Both rectifier and inverter components consist of two three-level, three-phase, neutral-point clamped (NPC) voltage source converters (VSCs).

For illustration purposes, the transient scenario consists of starting-up of the HVDC system of Fig. 1; that is, open-loop operation. Fig. 2 presents the transient voltage at capacitor terminals at the left-hand-side of the DC-link where a very close agreement between TD and Floquet can be seen in the enclosed part. Table 1 contains the employed simulation data. Although Floquet-based models are not conceived to compete with TD models, Table 1 shows that Floquet-based simulation is able to use a time-step 20 times larger than TD as matrix Qremains constant for the complete observation time. Hence, for this open-loop scenario, Floquet results in 5.44 times faster simulation than TD.

Table 1. TD and Floquet-based model simulation data comparison.

Empty Cell	TD	Floquet
Δt	10 μs	200 μs
Simulation time frame	0.5 s
CPU time	3.0884 s	0.5676 s

CPU times have been highlighted in bold type letter.

With respect to FD characterization, Fig. 3 depicts the poles location of the HVDC system, which is readily available by Floquet method via matrix Q; stability may be observed from this characterization.

3. Harmonic domain dynamic transfer function [12]

3.1. Concept

A Harmonic Domain Dynamic Transfer Function, referred to as HDDTF hereafter, characterizes the dynamics of a nonlinear and TP network seen from a port or a multi-terminal port in terms of the frequency response of harmonic perturbations superimposed on an underlying periodic steady-state. An HDDTF is a transfer-function matrix relating the vectors of harmonic domain input and output endowed with Laplace s-domain properties.

3.2. Mathematical foundation

Consider a network operating in a periodic steady-state whose base angular frequency is $ω_{0}$ . The network consists of linear and nonlinear R, L, and Ccomponents, voltage and current sources, and switches. The sources and the switches operate with the period $T = 2 π / ω_{0}$ and thus defining the network as T-periodic. The state equations of the network seen from one port can be expressed in the following form:(7) $\begin{matrix} {\dot{x}}_{t} = f (x_{t}, u_{t}), \\ y_{t} = g (x_{t}, u_{t}), \end{matrix}$ where xt, ut, and yt are the vectors of state variables, inputs, and outputs in the TD, respectively. The lengths of ut and yt are the same as the number of terminals of the port, and that of xt is the number of state variables in the network. Also, f and g represent nonlinear vector functions.

Assume that a disturbance takes place outside the port, and the excursion of xt, ut, and yt are denoted by Δxt, Δut, and Δyt, respectively. If (7) is linearized around the steady-state, one obtains(8) $\begin{matrix} Δ {\dot{x}}_{t} = A_{t} Δ x_{t} + B_{t} Δ u_{t}, \\ Δ y_{t} = C_{t} Δ x_{t} + D_{t} Δ u_{t}, \end{matrix}$ where(9) $A_{t} = \frac{\partial f}{\partial x_{t}}, B_{t} = \frac{\partial f}{\partial u_{t}}, C_{t} = \frac{\partial g}{\partial x_{t}}, D_{t} = \frac{\partial g}{\partial u_{t}},$ are T-periodic matrices. According to the Floquet theory mentioned in the previous section, the Fourier series of Δxt is expressed by(10) $Δ x_{t} = \sum_{h} Δ x_{h} e^{(s + jh ω_{0}) t},$ where ∑h denotes the sum for h from −∞ to +∞, and the same applies to Δut and Δyt. The time derivative of (10) is(11) $Δ {\dot{x}}_{t} = e^{st} \sum_{h} (s + jh ω_{0}) Δ x_{h} e^{jh ω_{0} t} .$

The Fourier series of At is expressed by(12) $A_{t} = \sum_{k} A_{k} e^{jk ω_{0} t},$ where ∑k denotes the sum for k from −∞ to +∞, and the same applies to Bt, Ct, and Dt. Substituting these Fourier series into (8) gives the following linearized state equations in the harmonic domain (for details see [12]).(13) $\begin{matrix} S Δ x = A Δ x + B Δ u, \\ Δ y = C Δ x + D Δ u . \end{matrix}$

Let us assume that there exist N state variables in the network, M terminals in the port, and we consider up to the Kth harmonics. The size of Δx is N(2K + 1) and its elements are the Fourier coefficients of Δxt. The size of Δu and Δy is M(2N + 1), and their elements are the Fourier coefficients of Δut and Δyt, respectively. Typical harmonic domain arrangements of Δx, Δu, and Δy are presented in (14).(14) $Δ x = [\begin{matrix} Δ x_{- (N - 1)} \\ ⋮ \\ Δ x_{- 1} \\ Δ x_{0} \\ Δ x_{1} \\ ⋮ \\ Δ x_{N - 1} \end{matrix}], Δ u = [\begin{matrix} Δ u_{- (N - 1)} \\ ⋮ \\ Δ u_{- 1} \\ Δ u_{0} \\ Δ u_{1} \\ ⋮ \\ Δ u_{N - 1} \end{matrix}], Δ y = [\begin{matrix} Δ y_{- (N - 1)} \\ ⋮ \\ Δ y_{- 1} \\ Δ y_{0} \\ Δ y_{1} \\ ⋮ \\ Δ y_{N - 1} \end{matrix}] .$

The dimension of S, A, B, C, and D is 2N + 1 by 2N + 1. The time derivative operator S is the diagonal matrix given by(15) $S = [\begin{matrix} {s - j (N - 1) ω_{0}} I \\ ⋱ \\ sI \\ ⋱ \\ {s + j (N - 1) ω_{0}} I \end{matrix}],$ where I represents an identity matrix whose size is M by M. The elements of A, see (16), are the Fourier coefficients of At.(16) $A = [\begin{matrix} A_{0} & A_{- 1} & A_{- 2} \\ A_{1} & A_{0} & A_{- 1} \\ A_{2} & A_{1} & A_{0} \\ ⋱ \\ A_{0} & A_{- 1} & A_{- 2} \\ A_{1} & A_{0} & A_{- 1} \\ A_{2} & A_{1} & A_{0} \end{matrix}] .$

Note that matrix A is a block Hermitian Toeplitz-type matrix. The same applies to matrices B, C, and D.

Substituting the first equation of (13) into the second equation gives the following input-output relationship.(17) $Δ y = {C {(S - A)}^{- 1} B + D} Δ u .$

Finally, from (17), the HDDTF of the network is given by(18) $H (s) = C {(S - A)}^{- 1} B + D .$

3.3. Publications in power systems analysis utilizing HDDTFs

HDDTF accounts for the coupling among different harmonic components in input and output signals. One of the early applications is in [13], where a model was developed for a diode converter. The concept of harmonic transfer functions was used to develop a model for Thyristor controlled rectifiers in [14], though for a limited in number of harmonics. HDDTF was used in [15] to identify a reduced order dynamic equivalent with the structure of a companion-form matrix.

3.4. Example

Fig. 4 presents a nonlinear and TP network. The nonlinear inductance LNrepresents the magnetizing reactance of a transformer; the nonlinear resistance RL is a thyristor-controlled, thus T-periodic, load which is 100 Ω when −2π/3 < ωt < −π/3 and π/3 < ωt < 2π/3 and 200 Ω otherwise. Fig. 5 shows some elements of the HDDTF of the network. The diagonal elements (0, 0), (1, 1), and (5, 5) show the admittances for the DC, the fundamental, and the 5th harmonic components, respectively. The off-diagonal element (2, 4) gives the transfer admittance from the fourth voltage harmonic to the second current harmonic component. Using the HDDTF, the frequency responses among different harmonic orders can be obtained.

4. Equivalent signal theory

4.1. Concept

The equivalent signal theory permits to represent a sampled signal by a synthetic continuous function [16]. The outcome of the equivalent signal concept is the ability to generate FD transfer functions between sampled-input and sampled-output. These transfer functions are named Generalized Transfer Functions (GTFs) [17], [18], [19], [20], [21]. The method has been mainly applied to switched networks.

Similar to traditional transfer functions, GTFs permit to characterize a switched device in FD via an input/output equivalent signal’s relationship. As GTFs involve s-domain and sampled (z-domain) signals simultaneously, they are able to provide zeros/poles location for a switched system. This feature provides system designers of an educated guess of the actual system performance. A drawback of GTFs is that they generate a family of N equivalent signals for an N-switching phase system.

4.2. Mathematical foundation

Consider a momentarily switched system with a switching period T and a duty cycle d, driven by input x(t) and providing output y(t). The GTF representing such system is given by [16](19) ${GTF}_{d} = L {y_{d}^{e} (t)} / L {x^{e} (t)},$ where L represents the Laplace transform operator. In (19), $y_{d}^{e} (t)$ [and also $x^{e} (t)$ ] denotes equivalent signal of the discrete sequence $y {kT + dT}, k = \dots, - 1, 0, 1, \dots$ , noting that y(t) can be sampled at instants kT + dT that correspond to the starting of a switching phase, at the end of a switching phase, or at any arbitrary point after the end of a switching phase. The equivalent signal $y_{d}^{e} (t)$ : (a) represents interpolation of the sequence $y {kT + dT}, k = \dots, - 1, 0, 1, \dots$ , being the same as the instantaneous output at sampling points kT + dT, and (b) has frequency components within the spectrum of x(t). Furthermore, $y_{d}^{e} (t)$ is identical to y(t) if the Nyquist’s criterion is satisfied [16]. The subscript in $y_{d}^{e} (t)$ means that construction of equivalent signals depends on parameter d; thus, a family of envelopes of $y_{d}^{e} (t)$ is generated via GTFs.

4.3. Publications in power systems analysis utilizing the approach

The importance of GTFs relies in the fact that they can simultaneously describe continuous-time (s-domain) and discrete-time (z-domain) input/output characteristics of a switched network. Two-phase case of switched circuits has been described as application of GTFs theory in specialized literature [17], [18], [19], [20]. This is due to mathematical complexity on obtaining the GTFs. A great deal of effort has been dedicated to addressing the numerical task of simulating GTFs in semi-symbolic ways.

4.4. Example

This example illustrates the GTF computation of an HBC under PWM scheme[21], as depicted in Fig. 6. The following parameters are used: L = 690 μH, R = 5 mΩ, vin = 1200 V, vs = 400 V, modulating signal with amplitude of 0.9, switching frequency fs = 1620 Hz (Ts = 617 μs), and zero initial conditions.