1 Introduction

There is a surge of interest in academic and industrial research on DC grids mainly due to their well-known advantages over AC grids in a number of applications [1, 2]. DC distributions are being developed for applications such as transportation [3, 4], commercial and industrial buildings [5, 6], data centers [7], generating stations [8], and integration of distributed resources [9].

Power electronics interfaces play a key role in DC grids [10]. The three-phase dual active bridge (3p-DAB) DC–DC converter topology is widely considered to meet various requirements such as capability of bidirectional power flow, galvanic isolation, high efficiency and high power density [11,12,13]. The 3p-DAB also allows the use of different winding connections among which the Y−Δ type connection has been shown to provide increased performance [14].

Small-signal frequency-domain modeling of converters is important in controller and filter design as well as the first stability assessments [15]. As for traditional AC grids, the integration of power electronics in DC grids also requires time-domain system-level studies such as voltage regulation performance with load variations, transient stability, and protection coordination [16,17,18,19]. Since the behavior of a converter depends on both its topology and controller, the closed-loop modeling of converters is necessary for system-level studies [20].

Each topology has its own particularities which should be considered in the development process of simulation models. Large-signal modeling of 3p-DAB converters is not trivial because of its DC-AC-DC structure, three-phase transformer connections and higher number of switches compared to other isolated bidirectional DC–DC converter topologies [21]. The methodology presented in [22] is the first attempt to reduce the computation time for system-level studies with 3p-DAB in electromagnetic transient type (EMT-type) programs. A discretized switching function (SWF) model in the αβ frame is implemented. It reduces the number of electrical nodes while preserving the switching effect. However, it still requires time steps in the range of 1/1000th of the switching period to maintain numerical stability. The reduction in the computation time compared to the ideal model approach is also not shown.

The ideal model approach is in fact well-known for the analysis of 3p-DAB converters and other topologies because its implementation using ideal switches is straightforward in circuit simulation tools. Ideal model is a good reference for model validation [19, 23]. However, because the circuit topology is time-variant, and the switching effect is modeled, the computation time is demanding for system-level studies.

The SWF method [24] is proposed as an alternative methodology to ideal model in which the converter circuit is replaced by controlled voltage and current sources. Since the circuit topology is time-invariant, the SWF method reduces the computation time [17]. However, since it keeps the switching effect, small time steps are still required. Averaged models are well-known for neglecting the switching effect and allowing the use of larger simulation time steps which significantly reduce the computation time [25, 26].

In this paper, a generalized averaging model (GAM) of the Y−Δ type 3p-DAB converter is developed, implemented and compared with the ideal, SWF [24] and state-space averaging (SSA) [23, 25] methods [23, 25] in electromagnetic transient program (EMTP) [27] as shown in Fig. 1. In Fig. 1, simulation approaches indicated with a star * are compared in this paper. The ideal model is used as a reference. The main contribution of this paper is to provide a solution to the challenges in developing and implementing a computational efficient and accurate model that enables time-domain system-level simulation of DC grids with 3p-DAB converters. By comparing the performance of the developed GAM with alternative modeling techniques, this paper also provides a complete comprehensive understanding of the advantages and limitations of the GAM approach for the 3p-DAB.

Fig. 1
figure 1

Classification of simulation approaches for time-domain simulation of 3p-DAB converter

Full size image

The paper is organized as follows. In Section 2, the operating principles of the 3p-DAB converter are introduced. In Section 3, the development of the GAM model and its generalization to other transformer connections are presented. In Section 4, the performance of the GAM model is analyzed in terms of transient response, computation time and frequency response. A large-scale test system is also implemented to validate the performance of the GAM model for fault current and transient recovery voltage analysis.

2 Operating principle of 3p-DAB converter

The 3p-DAB converter topology is shown in Fig. 2a, where vi is the input voltage; vo is the output voltage; ii is the input current; io is the output current; \(\delta\) is the control phase-shift. Note that, throughout this paper, small letters are used for time-domain variables. Capital letters will be used for steady-state values. The phase-shift is denoted with \(\delta = \delta (t)\) and its steady-state value with D.

Fig. 2
figure 2

3p-DAB converter

Full size image

The 3p-DAB is composed of two three-phase bridges connected through a three-phase medium-frequency transformer. It is most commonly operated using single phase-shift (SPS) control method with a fixed switching frequency \(\omega_{\text{s}} = 2{\pi} f_{\text{s}}\), and a 180° (or 6-step) modulation scheme [14, 28]. Filter capacitors Ci and Co are necessary to reduce the voltage ripple at \(6f_{\text{s}}\) which is induced by both bridges. Their equivalent series resistances (ESRs), RCi and RCo, are also included because of their importance in stability analysis [29].

The phase-shift \(\delta\) between the gating signals applied to the two bridges is the variable used to regulate the voltage of either one of the two ports. For example, in forward operation (0 ≤ \(\delta\) ≤ \({\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0pt} 2}\)), the active power is transferred from the input bridge to the output bridge such that vo is regulated to reach its reference value \(V_{\text{o}}^{\text{ref}}\) shown in Fig. 2b. Reverse operation consists in reversing the sign of \(\delta\) and regulating vi to reach its reference value \(V_{\text{i}}^{\text{ref}} .\) In Fig. 2b, Gc is the transfer function of the closed-loop controller and Gf is the transfer function of the output voltage measurement filter. The current source iload is a load perturbation superimposed on the resistive load R and is used to define the converter output impedance Zo(s) in Section 4.5.

In steady state, the relationships between the input and output voltages for the Y−Δ type 3p-DAB converter are given as follows.

$$V_{\text{o}} = \left\{ {\begin{array}{*{20}l} {\frac{mRD}{{\omega_{\text{s}} L_{\text{s}} }}V_{\text{i}} } \hfill & {0 < D \le \frac{\uppi}{6}} \hfill \\ {\left( {\frac{mR}{{\omega_{\text{s}} L_{\text{s}} }}\left[ {\frac{3}{2}\left( {D - \frac{{D^{2} }}{\uppi}} \right) - \frac{\uppi}{24}} \right]} \right)V_{\text{i}} } \hfill & {\frac{\uppi}{6} \le D \le \frac{\uppi}{2}} \hfill \\ \end{array} } \right.$$
(1)

where Vi and Vo are the steady state input and output voltages; R is the load resistance; m is the transformer winding ratio (\(m = {M \mathord{\left/ {\vphantom {M {\sqrt 3 }}} \right. \kern-0pt} {\sqrt 3 }}\)); Ls is the total transformer primary referred leakage inductance for which balanced operation is assumed (LS = LA = LB = LC).

3 Generalized averaging model

The GAM method, also known as dynamic phasor [26], is a recognized methodology for the analysis of power electronics converters [30,31,32,33,34]. The idea is to decompose the converter state variables and its switching functions using a kth coefficient Fourier series. For the 3p-DAB, it is proposed here to use only the index k = 0 for the DC voltages, and the index k = ± 1 for both the transformer AC currents and switching function components at the switching frequency \(\omega_{\text{s}}\). Higher harmonics are not considered because the three-phase structure leads to a next harmonic at k = 6 which can generally be neglected in system-level studies. This will be shown in Section 4.1.

Seen from the DC grids, the GAM method leads to the equivalent index-0 model shown in Fig. 3. In Fig. 3, diodes Di and Do are used to model the clamping effect of the three-phase bridge freewheeling diodes on negative voltage at the DC ports. Their importance is showed through an example in Section 4.3. The index-0 model is governed by a system of equations which is function of the index-1 model. The index-1 model includes the AC dynamic of both the transformer currents and the switching functions shown in Fig. 4.

Fig. 3
figure 3

Equivalent GAM of 3p-DAB seen from DC grids

Full size image
Fig. 4
figure 4

Switching functions \(s_{1}\)and \(s_{1}^{'}\) of phase A for Y−Δ type (− 30°) transformer

Full size image

The large-signal model of the 3p-DAB converter is given here using the following state-space notation:

$$\frac{\text{d}}{{{\text{d}}t}}\left\langle \varvec{x} \right\rangle = \varvec{A}\left\langle \varvec{x} \right\rangle + \varvec{B}\left\langle \varvec{u} \right\rangle$$
(2)
$$\left\langle \varvec{y} \right\rangle = \varvec{C}\left\langle \varvec{x} \right\rangle + \varvec{D}\left\langle \varvec{u} \right\rangle$$
(3)

where \(\left\langle \varvec{x} \right\rangle\), \(\left\langle \varvec{u} \right\rangle\), \(\left\langle \varvec{y} \right\rangle\) are the vectors of averaged state-variables, inputs and outputs respectively:

$$\left\langle \varvec{x} \right\rangle = \left[ {\begin{array}{*{20}c} {\left\langle {i_{\text{A}} } \right\rangle_{1}^{\text{R}} } & {\left\langle {i_{\text{A}} } \right\rangle_{1}^{\text{I}} } & {\left\langle {i_{\text{B}} } \right\rangle_{1}^{\text{R}} } & {\left\langle {i_{\text{B}} } \right\rangle_{1}^{\text{I}} } & {\left\langle {i_{\text{C}} } \right\rangle_{1}^{\text{R}} } & {\left\langle {i_{\text{C}} } \right\rangle_{1}^{\text{I}} } \\ \end{array} } \right]^{\text{T}}$$
(4)
$$\left\langle \varvec{u} \right\rangle = \left[ {\begin{array}{*{20}c} {\left\langle {v_{\text{i}} } \right\rangle_{0} } & {\left\langle {v_{\text{o}} } \right\rangle_{0} } \\ \end{array} } \right]^{\text{T}}$$
(5)
$$\;\left\langle \varvec{y} \right\rangle = \left[ {\begin{array}{*{20}c} {\left\langle {i_{\text{i}} } \right\rangle_{0} } & {\left\langle {i_{\text{o}} } \right\rangle_{0} } \\ \end{array} } \right]^{\text{T}}$$
(6)

where the superscripts R and I denote the real and imaginary components of the averaged state-variables. The definition of the system matrices \(\varvec{A}\), \(\varvec{B}\), \(\varvec{C}\) and \(\varvec{D}\) which are time-dependent, requires to first write the index-0 input and output currents in terms of both the transformer primary currents, and the input and output bridge switching functions. The switching functions for the input bridge \(s_{1}\), \(s_{3}\), and \(s_{5}\), and the output bridge \(s_{1}^{'}\), \(s_{3}^{'}\), \(s_{5}^{'}\) are defined by the gating signals coming from the output of the modulator as shown in Fig. 4. For phase B, the switching functions \(s_{3}\) and \(s_{3}^{'}\) are − 120° phase-shifted from phase A. For phase C, the switching functions \(s_{5}\) and \(s_{5}^{'}\) are + 120° phase-shifted from phase A.

The input current is given by:

$$i_{\text{i}} = s_{1} i_{\text{A}} + s_{3} i_{\text{B}} + s_{5} i_{\text{C}}$$
(7)

While (7) is valid for any transformer connection, the winding connection is included along with its ratio m in the output current equation. For the Y−Δ type (− 30°) connection, it leads to:

$$i_{\text{o}} = m\left[ {\left( {s_{1}^{\prime } - s_{3}^{\prime } } \right)i_{\text{A}} + \left( {s_{3}^{\prime } - s_{5}^{\prime } } \right)i_{\text{B}} + \left( {s_{5}^{\prime } - s_{1}^{\prime } } \right)i_{\text{C}} } \right]$$
(8)

where \(s_{1}^{\prime }\), \(s_{3}^{\prime }\), \(s_{5}^{\prime }\) are dependent on the instantaneous value of the control phase-shift \(\delta = \delta (t)\). To evaluate (7) and (8), the following primary transformer current equations should be solved:

$$\frac{{{\text{d}}i_{\text{A}} }}{{{\text{d}}t}} = \left( {{1 \mathord{\left/ {\vphantom {1 {L_{\text{A}} }}} \right. \kern-0pt} {L_{\text{A}} }}} \right)\left[ {\left( {{{\left( {2s_{1} - s_{3} - s_{5} } \right)} \mathord{\left/ {\vphantom {{\left( {2s_{1} - s_{3} - s_{5} } \right)} 3}} \right. \kern-0pt} 3}} \right)v_{\text{i}} - m\left( {s_{1}^{\prime } - s_{3}^{\prime } } \right)v_{\text{o}} } \right]$$
(9)
$$\frac{{{\text{d}}i_{\text{B}} }}{{{\text{d}}t}} = \left( {{1 \mathord{\left/ {\vphantom {1 {L_{\text{B}} }}} \right. \kern-0pt} {L_{\text{B}} }}} \right)\left[ {\left( {{{\left( {2s_{3} - s_{1} - s_{5} } \right)} \mathord{\left/ {\vphantom {{\left( {2s_{3} - s_{1} - s_{5} } \right)} 3}} \right. \kern-0pt} 3}} \right)v_{\text{i}} - m\left( {s_{3}^{\prime } - s_{5}^{\prime } } \right)v_{\text{o}} } \right]$$
(10)
$$\frac{{{\text{d}}i_{\text{C}} }}{{{\text{d}}t}} = \left( {{1 \mathord{\left/ {\vphantom {1 {L_{\text{C}} }}} \right. \kern-0pt} {L_{\text{C}} }}} \right)\left[ {\left( {{{\left( {2s_{5} - s_{1} - s_{3} } \right)} \mathord{\left/ {\vphantom {{\left( {2s_{5} - s_{1} - s_{3} } \right)} 3}} \right. \kern-0pt} 3}} \right)v_{\text{i}} - m\left( {s_{5}^{\prime } - s_{1}^{\prime } } \right)v_{\text{o}} } \right]$$
(11)

Equations (7)–(11) are averaged using Fourier series properties [26, 35, 36]. The resulting system of equations defined by (2)–(6) and (12)–(18) is implemented in EMTP by the control blocks, and solved using [37]. The calculated values of the input current \(\left\langle {i_{\text{i}} } \right\rangle_{0}\) and the output current \(\left\langle {i_{\text{o}} } \right\rangle_{0}\) are used to control the current sources of Fig. 3. The system matrices are given by:

$$\left\{ {\begin{array}{*{20}l} {\varvec{A} = \left[ {\begin{array}{*{20}c} 0 & {\omega_{\text{s}} } & 0 & 0 & 0 & 0 \\ { - \omega_{\text{s}} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\omega_{\text{s}} } & 0 & 0 \\ 0 & 0 & { - \omega_{\text{s}} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\omega_{\text{s}} } \\ 0 & 0 & 0 & 0 & { - \omega_{\text{s}} } & 0 \\ \end{array} } \right]} \hfill \\ {\varvec{B}\;{ = }\;\left[ {\begin{array}{*{20}c} {B_{11} } & {B_{12} } \\ {B_{21} } & {B_{22} } \\ {B_{31} } & {B_{32} } \\ {B_{41} } & {B_{42} } \\ {B_{51} } & {B_{52} } \\ {B_{61} } & {B_{62} } \\ \end{array} } \right]} \hfill \\ {\varvec{C}\;{ = }\;\left[ {\begin{array}{*{20}c} {C_{11} } & {C_{12} } & {C_{13} } & {C_{14} } & {C_{15} } & {C_{16} } \\ {C_{21} } & {C_{22} } & {C_{23} } & {C_{24} } & {C_{25} } & {C_{26} } \\ \end{array} } \right]\;} \hfill \\ {\varvec{D}\;{ = }\;\left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ \end{array} } \right]} \hfill \\ \end{array} } \right.$$
(12)

with

$$\left\{ \begin{aligned} B_{12} { = }\frac{m}{{L_{\text{A}} }}\left( {\left\langle {s_{3}^{\prime } } \right\rangle_{1}^{\text{R}} - \left\langle {s_{1}^{\prime } } \right\rangle_{1}^{\text{R}} } \right) \hfill \\ B_{22} = \frac{m}{{L_{\text{A}} }}\left( {\left\langle {s_{3}^{\prime } } \right\rangle_{1}^{\text{I}} - \left\langle {s_{1}^{\prime } } \right\rangle_{1}^{\text{I}} } \right) \hfill \\ B_{32} { = }\frac{m}{{L_{\text{B}} }}\left( {\left\langle {s_{5}^{\prime } } \right\rangle_{1}^{\text{R}} - \left\langle {s_{3}^{\prime } } \right\rangle_{1}^{\text{R}} } \right) \hfill \\ B_{42} = \frac{m}{{L_{\text{B}} }}\left( {\left\langle {s_{5}^{\prime } } \right\rangle_{1}^{\text{I}} - \left\langle {s_{3}^{\prime } } \right\rangle_{1}^{\text{I}} } \right) \hfill \\ B_{52} { = }\frac{m}{{L_{\text{C}} }}\left( {\left\langle {s_{1}^{\prime } } \right\rangle_{1}^{\text{R}} - \left\langle {s_{5}^{\prime } } \right\rangle_{1}^{\text{R}} } \right) \hfill \\ B_{62} = \frac{m}{{L_{\text{C}} }}\left( {\left\langle {s_{1}^{\prime } } \right\rangle_{1}^{\text{I}} - \left\langle {s_{5}^{\prime } } \right\rangle_{1}^{\text{I}} } \right) \hfill \\ \end{aligned} \right.$$
(13)
$$\left\{ \begin{aligned} B_{11} { = }\frac{1}{{3L_{\text{A}} }}\left[ {2\left\langle {s_{1} } \right\rangle_{1}^{\text{R}} - \left\langle {s_{3} } \right\rangle_{1}^{\text{R}} - \left\langle {s_{5} } \right\rangle_{1}^{\text{R}} } \right] \hfill \\ B_{21} { = }\frac{1}{{3L_{\text{A}} }}\left[ {2\left\langle {s_{1} } \right\rangle_{1}^{\text{I}} - \left\langle {{\text{s}}_{3} } \right\rangle_{1}^{\text{I}} - \left\langle {s_{5} } \right\rangle_{1}^{\text{I}} } \right] \hfill \\ B_{31} { = }\frac{1}{{3L_{\text{B}} }}\left[ {2\left\langle {s_{3} } \right\rangle_{1}^{\text{R}} - \left\langle {s_{1} } \right\rangle_{1}^{\text{R}} - \left\langle {s_{5} } \right\rangle_{1}^{\text{R}} } \right] \hfill \\ B_{41} { = }\frac{1}{{3L_{\text{B}} }}\left[ {2\left\langle {s_{3} } \right\rangle_{1}^{\text{I}} - \left\langle {s_{1} } \right\rangle_{1}^{\text{I}} - \left\langle {s_{5} } \right\rangle_{1}^{\text{I}} } \right] \hfill \\ B_{51} { = }\frac{1}{{3L_{\text{C}} }}\left[ {2\left\langle {s_{5} } \right\rangle_{1}^{\text{R}} - \left\langle {s_{1} } \right\rangle_{1}^{\text{R}} - \left\langle {s_{3} } \right\rangle_{1}^{\text{R}} } \right] \hfill \\ B_{61} { = }\frac{1}{{3L_{\text{C}} }}\left[ {2\left\langle {s_{5} } \right\rangle_{1}^{\text{I}} - \left\langle {s_{1} } \right\rangle_{1}^{\text{I}} - \left\langle {s_{3} } \right\rangle_{1}^{\text{I}} } \right] \hfill \\ \end{aligned} \right.$$
(14)
$$\left\{ \begin{aligned} C_{21} { = }2m\left( {\left\langle {s_{1}^{\prime } } \right\rangle_{1}^{\text{R}} - \left\langle {s_{3}^{\prime } } \right\rangle_{1}^{\text{R}} } \right) \hfill \\ C_{22} = 2m\left( {\left\langle {s_{1}^{\prime } } \right\rangle_{1}^{\text{I}} - \left\langle {s_{3}^{\prime } } \right\rangle_{1}^{\text{I}} } \right) \hfill \\ C_{23} { = }2m\left( {\left\langle {s_{3}^{\prime } } \right\rangle_{1}^{\text{R}} - \left\langle {s_{5}^{\prime } } \right\rangle_{1}^{\text{R}} } \right) \hfill \\ C_{24} = 2m\left( {\left\langle {s_{3}^{\prime } } \right\rangle_{1}^{\text{I}} - \left\langle {s_{5}^{\prime } } \right\rangle_{1}^{\text{I}} } \right) \hfill \\ C_{25} { = }2m\left( {\left\langle {s_{5}^{\prime } } \right\rangle_{1}^{\text{R}} - \left\langle {s_{1}^{\prime } } \right\rangle_{1}^{\text{R}} } \right) \hfill \\ C_{26} = 2m\left( {\left\langle {s_{5}^{\prime } } \right\rangle_{1}^{\text{I}} - \left\langle {s_{1}^{\prime } } \right\rangle_{1}^{\text{I}} } \right) \hfill \\ \end{aligned} \right.$$
(15)
$$\left\{ \begin{aligned} C_{11} { = }2\left\langle {s_{1} } \right\rangle_{1}^{\text{R}} \hfill \\ C_{12} = 2\left\langle {s_{1} } \right\rangle_{1}^{\text{I}} \hfill \\ C_{13} { = }2\left\langle {s_{3} } \right\rangle_{1}^{\text{R}} \hfill \\ C_{14} { = }2\left\langle {s_{3} } \right\rangle_{1}^{\text{I}} \hfill \\ C_{15} = 2\left\langle {s_{5} } \right\rangle_{1}^{\text{R}} \hfill \\ C_{16} { = }2\left\langle {s_{5} } \right\rangle_{1}^{\text{I}} \hfill \\ \end{aligned} \right.$$
(16)

For the input bridge, the averaged switching functions are constant because they do not depend on the control phase-shift. They are given by:

$$\left\{ \begin{aligned} \left\langle {s_{1} } \right\rangle_{1} = \frac{1}{\uppi}{\text{e}}^{{ - {\text{j}}\frac{\uppi}{2}}} \hfill \\ \left\langle {s_{3} } \right\rangle_{1} = \frac{1}{\uppi}{\text{e}}^{{{\text{j}}\frac{\uppi}{6}}} \hfill \\ \left\langle {s_{5} } \right\rangle_{1} = \frac{1}{\uppi}{\text{e}}^{{{\text{j}}\frac{{ 5\uppi}}{6}}} \hfill \\ \end{aligned} \right.$$
(17)

For the output bridge, they depend on the instantaneous value of the control phase-shift \(\delta = \delta (t)\) and are referenced with respect to the input bridge:

$$\left\{ \begin{aligned} \left\langle {s_{1}^{\prime } } \right\rangle_{1} = \left\langle {s_{1} } \right\rangle_{1} {\text{e}}^{{ - {\text{j}}\left( {\frac{\uppi}{6} + \delta } \right)}} \hfill \\ \left\langle {s_{5}^{\prime } } \right\rangle_{1} = \left\langle {s_{5} } \right\rangle_{1} {\text{e}}^{{ - {\text{j}}\left( {\frac{\uppi}{6} + \delta } \right)}} \hfill \\ \left\langle {s_{3}^{\prime } } \right\rangle_{1} = \left\langle {s_{3} } \right\rangle_{1} {\text{e}}^{{ - {\text{j}}\left( {\frac{\uppi}{6} + \delta } \right)}} \hfill \\ \end{aligned} \right.$$
(18)

4 Performance comparison

4.1 Closed-loop time-domain transient response

The transient response of the GAM model is first compared with equivalent closed-loop ideal, SWF and SSA models in EMTP. The converter parameters are given in Table 1.

Table 1 3p-DAB converter parameters
Full size table

Large-signal step function perturbations are applied and the resulting output voltage \(v_{\text{o}}\) is shown in Figs. 5, 6, 7 and 8. The results are analyzed in Section 4.6. In Fig. 5, the integration time step is \(\Delta t = 0.01\;\upmu{\text{s}}\) for the reference (Ref.) and SWF models, and \(\Delta t = 0.1\;\upmu{\text{s}}\) for the GAM and SSA models. In Fig. 6, it shows the converter output voltage when the input voltage is stepped from 800 V to 400 V. It is observed that the SWF and GAM models consider the transient oscillations at the switching frequency fs. In Fig. 7, converter output voltage when the load is stepped from high-load condition (R = 0.94 Ω) to no-load condition. It is concluded that all the models consider the transient oscillations at the cross-over frequency fφm. In Fig. 8, it is shown that only the SWF model includes the ripple at 6fs. The component at 6fs is negligible in large-signal studies because the ripple at 6fs is generally significantly attenuated by Co.

Fig. 5
figure 5

Simulation results for complete simulation run

Full size image
Fig. 6
figure 6

Zoom #1 on simulation results

Full size image
Fig. 7
figure 7

Zoom #2 on simulation results

Full size image
Fig. 8
figure 8

Zoom #3 on simulation results

Full size image

4.2 Simulation timing analysis

When performing system-level studies, the simulation time step \(\Delta t\) is generally selected based on the frequency spectra of the targeted power system studies. In the case of many power electronic converters, even for low-frequency phenomena, if the switching effect is modeled, it is necessary to reduce the time step \(\Delta t\) to typically 1/100th of the switching period \(1/f_{\text{s}}\) (rule of thumb). In the case of the 3p-DAB, since the ripple is at \(6f_{\text{s}}\), it requires having a time step smaller than around 1/600th of the switching period. This problem has been highlighted in [22] for the specific case of the 3p-DAB converter.

Since the computation time \(t_{\text{cpu}}\) highly depends on the integration time step \(\Delta t\), multiple simulation runs as in Fig. 5 have been performed to thoroughly compare the models. The results are shown in Fig. 9 and are further analyzed in Section 4.6. An Intel® Core™ processor i7-4700HQ (2.4 GHz) with 12 GB of RAM, and EMTP-RV software version 3.4 is used. The simulation time tsim = 60 ms.

Fig. 9
figure 9

Computation time for different integration time step \(\Delta t\) for simulation scenario of Fig. 5

Full size image

4.3 Transient short circuit with multiple 3p-DAB converters

A bolted fault of 20 ms duration is applied at the output of ten 3p-DAB converters connected in parallel through RL devices which model typical converter input filters and output cables. An output VI-characteristic with a 2% droop centered on \(V_{\text{o}}^{\text{nom}} = 37.5\;{\text{V,}}\) and a 10% maximum current limit \(I_{\text{o}}^{ \hbox{max} } = 30\;{\text{A}}\) is implemented to model a common converter fault protection scheme shown in Fig. 10 [4].

Fig. 10
figure 10

Internal VI-characteristic of each 3p-DAB converter

Full size image

The transient short circuit current for each model is shown in Fig. 11. The maximum integration time step obtained in Fig. 9 is used for each model, i.e., \(\Delta t = 0.06\;\upmu{\text{s}}\) for the Ref. model, \(\Delta t = 0.1\;\upmu{\text{s}}\) for the SWF model, and \(\Delta t = 1\;\upmu{\text{s}}\) for the GAM and SSA models. The converter is operated at Vs = 600 V and fs = 50 kHz.

Fig. 11
figure 11

Simulation results for a bolted fault at the output of ten 3p-DAB in parallel

Full size image

A breakdown analysis of the CPU effort is also shown in Fig. 12. The results are also analyzed together in Section 4.6.

Fig. 12
figure 12

CPU calculation effort of total computation time tcpu

Full size image

Moreover, diodes Di and Do are used to model the clamping effect of the 3p-DAB freewheeling diodes on negative voltage at the DC ports. These diodes are always present in practical applications. Their importance on system-level simulation accuracy is shown in Fig. 13 by comparing the ideal model results with the SWF, GAM and SSA model results without Di and Do. These diodes are often neglected but their impact may not be negligible as shown experimentally in [38].

Fig. 13
figure 13

Simulation results for a bolted fault at the output of ten 3p-DAB connected in parallel without Di and Do for the SWF, GAM and SSA models

Full size image

4.4 Validation of GAM for power system transient analysis in a large-scale test system

The effectiveness and accuracy of the GAM model to study large-scale systems are validated through a short circuit test case in which the transient fault current as well as the transient recovery voltages at fault clearing are analyzed.

A single line diagram of the test system is shown in Fig. 14. It is composed of a primary voltage ring bus at 750 V, two intermediate voltage bus at 380 V, and a low voltage ring bus at 110 V. The system is divided into ten identical regions with a total of 20 3p-DAB converters. All the converters are designed with an LC input filter. The parameters of the 3p-DAB converters used in this test case are provided in Table 2. Cables are modeled with equivalent RL circuits (not shown in Fig. 14). Detailed models of battery [4, 39], circuit breaker detection and arcing [4, 40], fuse detection and arcing [38], as well as metal oxide varistor (MOV) [38] are used. More details on the mathematical formulation and the implementation of each model can be found in the references provided above.

Fig. 14
figure 14

Large-scale test system for validation of GAM

Full size image
Table 2 Summary of 3p-DAB parameters for large-scale test system
Full size table

At t = 20 ms, a 1 mΩ fault is applied in region 6 of the 110 V bus shown in Fig. 14. The short circuit current and the sequence of events are shown in Fig. 15 while the transient recovery voltage (V5 in Fig. 14) is given in Fig. 16. These results confirm that the developed GAM can be used for accurate grid-level analyses such as fault current evaluation and transient recovery voltage assessment.

Fig. 15
figure 15

Analysis of transient fault current at fault application and during fault clearing

Full size image
Fig. 16
figure 16

Analysis of voltage during fault application and fault clearing

Full size image

The total simulation time tsim is 50 ms. For the reference model, the integration time step \(\Delta t\) is 0.05 μs, and the total computation time tcpu is 4565 s. This is the largest time step that can be used to maintain numerical stability. For the GAM, the integration time step \(\Delta t\) is increased to 1 μs such as the total computation time is significantly reduced to tcpu = 45 s.

4.5 Closed-loop frequency-domain response

As seen in Fig. 7, the main frequency component of the oscillations in the closed-loop response is close to the cross-over frequency of the designed PI controller, i.e. \(f_{{\varphi {\text{m}}}}\) = 1.5 kHz.

However, as previously mentioned, for system-level studies, the frequency content is a function of the aimed power system analysis. This means that, instead of being step functions, the large-signal perturbations may contain a large spectrum of frequencies.

Therefore, the precision of the models at different frequencies is assessed here. For system-level analysis, the input-to-output voltage relationship \(G_{\text{vg}}^{\text{c}} (s ) { = }\left. {(v_{\text{o}} (s)/v_{\text{i}} (s))} \right|_{{i_{\text{load}} (s) = 0}}\), the load rejection behavior (also known as back-current gain) \(G_{\text{load}}^{\text{c}} (s) = \left. {(i_{\text{i}} (s)/i_{\text{load}} (s))} \right|_{{v_{\text{i}} (s) = 0}}\), the input impedance \(Z_{\text{in}}^{\text{c}} (s) = \left. {(v_{\text{i}} (s)/i_{\text{i}} (s))} \right|_{{i_{\text{load}} (s) = 0}}\) and the output impedance \(Z_{\text{o}}^{\text{c}} (s) = \left. {-(v_{\text{o}} (s)/i_{\text{load}} (s))} \right|_{{v_{\text{i}} (s) = 0}}\) are the main closed-loop transfer functions of interest.

The transfer functions \(G_{\text{vg}}^{\text{c}}\) and \(G_{\text{load}}^{\text{c}}\) are particularly interesting because they show how perturbations at the input of the converter propagate at the output and vice-versa. The input and output impedances, defined by \(Z_{\text{in}}^{\text{c}}\) and \(Z_{\text{o}}^{\text{c}}\) respectively, are also relevant because, by definition, they show the local reaction of the converter to perturbations at either one of the ports.

The models are validated by applying sine wave perturbations at different frequencies in the simulation models with the parameters shown in Table 1. The perturbations frequency is varied from 100 Hz up to the switching frequency \(f_{\text{s}}\) = 50 kHz. The magnitude of the output signals is extracted by FFT analysis. The results are shown in Fig. 17 and further analyzed in Section 4.6. In Fig. 17, the converter is operated at the following operating point: Vi = 600 V, Vo = 37.5 V, Po = 1 kW.

Fig. 17
figure 17

Closed-loop frequency response

Full size image

4.6 Performance analysis

Based on the results obtained in Sections 4.1 to 4.5, the performance of the GAM is now compared with the implemented ideal, SWF and SSA models. The analysis mainly focuses on accuracy, frequency response as well as computational speed. The main outcomes regarding these three criteria are summarized in Table 3. It leads to the conclusion that the proposed GAM offers the best compromise in terms of accuracy, speed and wideband response for system-level analysis. Further details on the implementation of models are summarized in Table 4, while additional justifications are given below.

Table 3 Comparison summary of 3p-DAB models
Full size table
Table 4 Summary of 3p-DAB model implementation in EMTP
Full size table

By first analyzing the time-domain simulation results, it is concluded that all the models accurately represent the averaged dynamic of the converter at the cross-over frequency \(f_{{\varphi {\text{m}}}}\) in Fig. 7. As expected, similar to the SWF model, the GAM method also considers the transient component at the switching frequency \(f_{\text{s}} ,\) but it is not the case with SSA in Fig. 6. The SWF model also considers the output voltage ripple at \(6f_{\text{s}}\) (characteristic of the 3p-DAB) while both GAM and SSA neglect this component in Fig. 8. Even if the transient responses of all the models are very similar for step function perturbations in Figs. 6 and 7, it can also be seen from the frequency-domain analysis in Fig. 17 that, unlike the SWF model and the GAM, the SSA model is not accurate to reproduce the converter behavior for perturbations beyond 1/5 of \(f_{\text{s}}\). These conclusions are summarized in Table 3.

Furthermore, from Figs. 9 and 11, it is concluded that both GAM and SSA provide a significant increase in computation speed when compared to the reference and SWF models. This is mainly explained by two fundamental differences. The primary reason is the fact that they both allow the use of larger simulation time steps \(\Delta t,\) which significantly increases the computation speed. It is also explained, to a lesser extent, by the reduction in the size of the network main system of equations in Table 4, which, in this case, follows the modified-augmented-nodal analysis (MANA) formulation approach [27, 41].

However, for the same \(\Delta t,\) the benefit in computation speed in Fig. 9 of both the SSA and GAM methods compared to the reference model is much less significant than the reduction in the size of the network main system of equations in Table 4. This is because the size of the network main system of equations cannot be used alone to conclude on the relative simulation speed between models since control system equations are also solved separately from network equations in Table 4 [37, 41]. The computation burden of solving the larger non-linear and coupled set of control equations defined by (2)–(6) and (12)–(18) also explains why, for the same \(\Delta t,\) the computation time is higher with GAM as compared to SSA in Fig. 9.

Finally, it is also worth mentioning that the SWF, GAM and SSA models all lead to a reduced number of solution points (niter) in the time-domain as compared to the reference model in Table 4. In fact, for the simplest 3p-DAB model representation, the number of solution points for the SWF, GAM and SSA models is given by

$$n_{\text{iter}} = {{t_{\text{sim}} } \mathord{\left/ {\vphantom {{t_{\text{sim}} } {\Delta t}}} \right. \kern-0pt} {\Delta t}}$$
(19)

while for the reference model, it is fundamentally higher by an additional term \(12t_{\text{sim}} f_{\text{s}}\),

$$n_{\text{iter}} = {{t_{\text{sim}} } \mathord{\left/ {\vphantom {{t_{\text{sim}} } {\Delta t\; + }}} \right. \kern-0pt} {\Delta t\; + }}\;12t_{\text{sim}} f_{\text{s}}$$
(20)

This difference is explained by the fact that, at each switching event, EMTP switches from a fixed time step trapezoidal integration method to backward Euler with two half-step \((\Delta t/2)\) solutions [42]. This feature helps reduce numerical oscillations when ideal switches are used. It is not required to solve the SWF, GAM, and SSA models because the circuit topology is time-invariant. Furthermore, the simultaneous-switching control algorithm [43], which allows a simultaneous solution of network and control system equations at switching events, also explains why, for the same \(\Delta t,\) the computation time is higher with the reference model.

Finally, the results of the short-circuit test cases in Figs. 11, 15, 16 also show that the developed GAM can effectively be used for accelerating grid-level analyses. The GAM reduces the computation time by a factor of 1/30 in the test case of Fig. 11, while still providing accurate results (error of less than 1% on the peak fault current). As seen in Figs. 15 and 16, the GAM also provides accurate results for typical power system transient analyses such as transient fault current and transient recovery voltage during arcing of circuit breakers. As previously mentioned in Section 4.4, it also provides a significant reduction in computation time for the large-scale test case (factor of 1/100).

5 Conclusion

In this paper, large-signal models of the 3p-DAB converter are developed, analyzed, and implemented in EMTP. It is a key step toward reducing the computation time for time-domain system-level simulation of next-generation DC grids based on the 3p-DAB topology.

Three models are investigated in this paper: SWF, GAM and the SSA. Even if SSA significantly reduces the computation time, it has a limited precision for studying phenomena beyond 1/5 of the converter switching frequency. Because they allow higher time steps, the SWF model and GAM both offer significant reduction in the computation time compared to the ideal-model, while still preserving the converter frequency characteristics up to its switching frequency. However, the maximum time step with the SWF model is limited by the inclusion of the switching-effect, which still leads to high computation time for system-level studies. Overall, the proposed GAM offers the best compromise in terms of accuracy, speed and wideband response.