State–Space Modelling and Stability Analysis of Solid-State Transformers for Resilient Distribution Systems

Abstract

Power grids are currently undergoing a significant transition to enhance operational resilience and elevate power quality issues, aiming to achieve universal access to electricity. In the last few decades, the energy sector has witnessed substantial shifts toward modernizing distribution systems by integrating innovative technologies. Among the innovations, the solid-state transformer (SST) is referred to as a promising technology due to its flexible power control (better reliability) and high efficacy (by decreasing losses) compared with traditional transformers. The design of SST has combined three-stage converters, i.e., the input, isolation, and output stages. The key objective of this design is to implement a modern power distribution system to make it a more intelligent and reliable device in practice. As the power converters are used in SST, they exhibit non-linear behavior and can introduce high-frequency components, making stability more challenging for the system. Besides, the stability issue can be even more complicated by integrating the distributed energy resources into the distribution system. Thus, the stability of SST must be measured prior to /during the design. To determine stability, state-space modeling, and its controller design are important, which this paper explains in detail. Indeed, the system’s stability is measured through the controllability and observability test. Further, the stability analysis is performed using frequency and time-domain diagrams: the Bode plot, Nyquist plot, Nichols chart, Root locus, pole-zero plot, and Eigen plot. Finally, the SST Simulink model is tested and validated through real-time digital simulation using the OPALRT simulator to show its effectiveness and applicability. The stability performance of the proposed SST is evaluated and shows the effectiveness of the controller design of each converter circuit.

1. Introduction

Over the past few decades, extreme events such as natural disasters, cyber-attacks, power system equipment failures, and power thefts have increased, eventually impacting the power system infrastructure, including distribution systems [1]. Consequently, a power system blackout may occur, affecting people’s social and economic life [2]. As traditional power system networks are large ring systems, an extreme event may affect many loads and components [3]. However, much research has been carried out in the area of power system migration, where the main aim was to convert from bulk power networks to small-scale networks [4,5]. The small-scale networks have been established through the development of renewable energies (RESs) and their integration into bulk power networks [6], for example, microgrids (MGs) and smart grids (SGs). These are generally connected to the bulk power system, making them a grid-connected mode [7,8]. Further, it can change to the islanded mode while in events to avoid corresponding impacts on the distribution system and supply the local loads [9]. Furthermore, the deployment of MGs and SGs including electric vehicles can enhance system reliability and resiliency of the power distribution system in responding to extreme operating conditions [10,11].

As noted, it has been seen that the integration of RESs has increased during the past few decades, which plays a crucial role in the distribution systems for grid modernization [12]. However, due to the intermittency of the RESs, it has certain challenges, which make the system complex and unstable. The most noticeable challenge is uncertainty, which means it depends on environmental factors such as solar insolation and wind speed for solar and wind systems, respectively [13]. To deal with these challenges, the advanced power electronics in distribution systems play a crucial role in regard to the control of power (both active and reactive power) flow to enable DC-operated technology [14], protection [15], mitigating the power quality issues, and reducing harmonics. To address these benefits, power electronics are reconfigured in a different way to make it possible for the determination of power quality problems [16]. It is critical for the distribution networks that the reliability factor is always a prime concern. With this objective, using a modern power electronics device, a solid-state device is pronounced, named the SST. SST is treated as an evolving technology for providing power supply to the consumer through MGs or SGs [17]. Besides, the SST is a high-frequency transformer with a power electronic converter and is an alternative to the traditional bulky and non-smart transformer.

In the past decades, researchers have shown interest in modeling and the development of SST for MG and SG applications. In [18,19], the design and response of SSTs are addressed through their converter stages and controllers. In [20], average models and their controllers are proposed and explored through descriptive simulation studies. Further, in [21], the stability aspect of SST and validation, which has been done through the distribution system, is presented. In this regard, many motivating test models of transient instabilities have been developed, and their importance to the link topologies of SSTs in various operating regions is demonstrated [22]. The most prominent manufacturers of SST, namely, ABB, ETH Zurich, EPRI, Siemens, and FREEDM Systems Center, have conducted cutting-edge research to upgrade the conventional transformer to an intelligent one. They have designed and tested various topologies to improve the performance and facilities of SST to the connected devices [23,24,25]. In [26], the FREEDM Systems Center is presented with a start-up SST structure named Generation-1-SST, which can be used in medium-voltage-based MG. Further development has been done with a multi-cell-based DC-DC converter for SST, which is proposed in [27]. In [28], the stability of the SST has been studied with different control methods by incorporating renewable energy resources. Middlebrook’s criterion is used to check the stability. In [29], small-signal stability of SST is studied using a dynamic phasor-based model. Darshit et al. [21] analyzed the SST stability on account of harmonic resonance in the distribution system. Extending to the study on [21], the modified source impedance-based stability criterion has been proposed [30]. The stability analysis considers a Z+Z type cascade system, an impedance analysis in SST explained in [31]. However, the research on SST stability is lacking in the literature and must be considered a vital factor. Thus, the stability aspect of each stage (input, isolation, and output) of the SST needs to be emphasized, which this paper addresses.

This study develops a state–space and a transfer-function-based model with a proper SST controller to reduce system complexity and cost. The state–space parameters (A, B, C, and D) of the converter’s input, isolation, and output stages are estimated. Each converter stage has its own controller and filter circuit. The PI controller is implemented in this design due to its simplicity, robustness, and excellent control action. The transfer function model of each converter is derived from the state–space model. By using the transfer function equation, the stability of the proposed design is tested in terms of frequency response, time response, controllability, and observability. Furthermore, the Simulink design of the proposed system is established in MATLAB, and the controller is designed through MATLAB SISOTOOL. The state–space model is designed for the stability study with and without a controller. In the proposed methodology, the user can simply design the controller part from the unstable model through MATLAB SISOTOOL, thereby making the system robust and stable. Thus, researchers can easily measure the system’s stability, sensitivity, controllability, and observability.

This article is organized as follows. Section 1 provides an introduction to SST and a literature review. Section 2 outlines the fundamentals and different types of SST topology. In Section 3, the continuous-time average state–space model and its transfer function model are formulated. This section categorizes the converter stage into the input, isolation, and output stages with the state–space model. Finally, the result and discussion parts of the proposed model are provided in Section 4, followed by the conclusions in Section 5.

The main contribution of the paper is summarized as follows:

• The state-space model of each converter is derived.
• The stability of converters has been analyzed by integrating it with additional control methods.
• Time-domain and frequency-domain performance have been shown to measure the system’s stability.
• The proposed model has been simulated and validated using an OPAL-RT real-time digital simulator.

2. Fundamental Topologies of SST

Various architectures are available for each of the three stages of SST. The topology is determined depending on the application of SST. The different types of topologies are Type-A (A-1 and A-2), Type-B, Type-C, Type-D, and Type-E [32]. This study adopts the Type-D-based SST topology [33] because it has numerous advantages, such as DC connectivity, fault isolation, proper voltage regulation, easy integration with RESs and storage, and volt-ampere reactive (VAR) compensation. In the design, the architecture for each stage should be chosen appropriately, and the problems should be addressed by integrating them into a single system to deliver regulated voltage at each stage.

The fundamental three-stage architecture of SST is presented in Figure 1. The detailed input, output, and other critical parameters can be seen in Appendix A. The input stage, which is also called the AC–DC stage, contains a single-phase AC–DC IGBT-based converter that operates at a high switching frequency. The isolation stage, which is also called the DC-DC stage, comprises two H-bridges with a high-frequency transformer. A DAB converter is also applied in this stage [34]. The output of the input stage is fed to the DAB converter, and the output of the converter is fed to the inverter or is considered the output stage. In the output stage, a filter circuit is added to mitigate the voltage and power quality issues. The output stage is usually supplied to low-voltage user applications.

3. Continuous-Time Average State–Space Model

In this section, the design of the state–space model and the continuous-time transfer function model is formulated. The controller of each stage is derived to enhance system stability and performance. Furthermore, the different parameters are calculated and presented in tables to determine the stability, frequency level range, and DC gain of each stage.

3.1. Input Stage (AC-DC)

Formulation: The input stage of the SST is the conversion of AC to DC signal, shown as architecture and equivalent diagram in Figure 2a and Figure 2b, respectively. Equations (1) and (2) represent the input voltage; however, in Equation (2), the duty cycle is taken as an average over a switching time. Referred to Figure 2b, Equation (3) is formed by applying the KVL equation. Further, considering changes as a minute value in the DC signal, Equation (3) can be modified as Equation (4). Equation (5) is solved by subtracting Equation (3) from Equation (4). Thereafter, by applying Laplace transform to Equation (5), Equation (6) and its updated version in Equation (7) are obtained, which represent the open-loop transfer function of the input stage with the inductor current.

Further, the transfer function of the input stage through DC-link capacitance current is presented in Equation (8). Similarly, Equations (9)–(11) are solved to reach Equation (12), which represents the ratio of the change in output DC link voltage to any change in input current.

$${\mathrm{U}}_{\mathrm{s}}\left(t\right)={\mathrm{U}}_{\mathrm{d}\mathrm{c}}\mathbb{D}={\mathrm{K}}_{\mathrm{m}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathsf{\omega }\mathrm{t}\right){\mathrm{U}}_{\mathrm{d}\mathrm{c}}$$

(1)

$${\mathrm{U}}_{\mathrm{s}}\left(t\right)={\mathrm{U}}_{\mathrm{d}\mathrm{c}}{\mathbb{D}}_1\left(\mathrm{t}\right)={\mathrm{U}}_{\mathrm{m}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathsf{\omega }\mathrm{t}\right)$$

(2)

$${\mathrm{U}}_{\mathrm{m}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathsf{\omega }\mathrm{t}\right)-{\mathrm{U}}_{\mathrm{d}\mathrm{c}}{\mathbb{D}}_1\left(\mathrm{t}\right) ={\mathrm{L}}_{\mathrm{f}}\frac{\mathrm{d}\left({\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{t}\right)\right)}{\mathrm{d}\mathrm{t}}$$

(3)

$${\mathrm{U}}_{\mathrm{m}}\mathrm{s}\mathrm{i}\mathrm{n}(\mathsf{\omega }\mathrm{t})-{\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left({\mathbb{D}}_1+△{\mathbb{D}}_1\right)(\mathrm{t})={\mathrm{L}}_{\mathrm{f}}\frac{\mathrm{d}\left({\mathrm{i}}_{\mathrm{l}\mathrm{s}}+△{\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{t}\right)\right)}{\mathrm{d}\mathrm{t}}$$

(4)

$$-△{\mathbb{D}}_1\left(\mathrm{t}\right){\mathrm{U}}_{\mathrm{d}\mathrm{c}}={\mathrm{L}}_{\mathrm{f}}\frac{\mathrm{d}△{\mathrm{i}}_{\mathrm{l}\mathrm{s}}}{\mathrm{d}\mathrm{t}}$$

(5)

$$-△{\mathbb{D}}_1\left(\mathrm{s}\right){\mathrm{U}}_{\mathrm{d}\mathrm{c}}=\mathrm{s}{\mathrm{L}}_{\mathrm{s}}△{\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{s}\right)$$

(6)

$$\frac{△{\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{s}\right)}{△{\mathrm{D}}_1\left(\mathrm{s}\right)}=\frac{-{\mathrm{U}}_{\mathrm{d}\mathrm{c}}}{\mathrm{s}{\mathrm{L}}_{\mathrm{s}}}={\mathrm{G}}_{\mathrm{i}\mathrm{n}}$$

(7)

$${\mathrm{C}}_{\mathrm{f}}\frac{\mathrm{d}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}={\mathbb{D}}_1\left({\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{t}\right)\right)$$

(8)

$${\mathrm{C}}_{\mathrm{f}}\frac{\mathrm{d}\left({\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left(\mathrm{t}\right)+△{\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left(\mathrm{t}\right)\right)}{\mathrm{d}\mathrm{t}}={\mathbb{D}}_1\left(\left({\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{t}\right)+△{\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{t}\right)\right)\right)$$

(9)

$$\frac{\mathrm{d}\left(△{\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left(\mathrm{t}\right)\right)}{\mathrm{d}\mathrm{t}}=\frac{1}{{\mathrm{C}}_{\mathrm{f}}}{\mathbb{D}}_1\left(△{\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{t}\right)\right)$$

(10)

$$\mathrm{s}{△\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left(\mathrm{s}\right)=\frac{1}{{\mathrm{C}}_{\mathrm{f}}}{\mathbb{D}}_1\left(△{\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{t}\right)\right)$$

(11)

$$\frac{△{\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left(\mathrm{s}\right)}{△{\mathrm{i}}_{\mathrm{l}\mathrm{s}}\left(\mathrm{t}\right)}=\frac{{\mathbb{D}}_1}{s{\mathrm{C}}_{\mathrm{f}}}={\mathrm{G}}_{\mathrm{v}\mathrm{i}}$$

(12)

State–Space Model: Furthermore, the state–space model presented in Equations (13) and (14) denotes the state and output equation of the system where A, B, C, and D are the state, control, output, and transmission matrices, respectively. The overall transfer function of the input stage has been derived, and the state–space parameters are measured in the results section. Here $x$ is the state variable, which is: inductor current (${\mathrm{i}}_{\mathrm{l}\mathrm{s}}$) and capacitor voltage (${\mathrm{U}}_{\mathrm{d}\mathrm{c}})$ and $u$ is the input as ${\mathrm{U}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$.

$$\stackrel{·}{x}=\mathrm{A}x+\mathrm{B}u$$

(13)

$$y=\mathrm{C}x+\mathrm{D}u$$

(14)

Considering the input stage transfer function as derived in (7) and (12), two controllers have been designed using MATLAB SISOTOOL, as per Figure 3. Thus, corresponding to ${\mathrm{G}}_{\mathrm{v}\mathrm{i}}$ and ${\mathrm{G}}_{\mathrm{i}\mathrm{n}}$, the PI controllers are $G_{PI1}$ and $G_{PI2}$, respectively.

$$G_{PI1}=\frac{-3.82\times {10}^{-6}s+6.5\times {10}^{-6}}{s}$$

$$G_{PI2}=\frac{-2.7\times {10}^{-6}s+259.2\times {10}^{-6}}{s}$$

3.2. Isolation Stage (DC-DC)

In this stage, the DAB converter is considered for DC-DC conversion. A high-frequency transformer is placed between DAB, as depicted in Figure 4a, and the equivalent diagram is depicted in Figure 4b. The DAB converter deals with three main parameters, namely, phase shift between two bridges (Ø), duty cycle ratios ($\mathbb{D}$), and switching frequency (fs). The other parameters are as follows: ${\mathrm{R}}_{\mathrm{t}\mathrm{f}}$ represents the equivalent of winding resistance and switching on resistance; ${\mathrm{L}}_{\mathrm{t}\mathrm{f}}$ is the leakage inductance of the transformer. The leakage inductance is calculated using Equation (20). Refer to Equation (15), the variation in transformer output with different frequencies and leakage inductance is shown in Figure 5.

Moreover, the duty cycle ratio of the proposed scheme is fixed to 0.5. Subsequently, the phase shift modulation technique is used here to control the operation of DAB. The transferred power from one stage of DAB to the other stage can be expressed in Equation (16). Further, the state–space model of the DAB can be formulated as Equation (17) by assuming the two-state variables of the proposed systems, namely, the current through the inductor ($i_L$) and the voltage across the capacitor (${\mathrm{U}}_0$). Equations (17) and (18) are time-variant and nonlinear. To ensure that the system is time-invariant and linear, the traditional state–space averaging procedure is applied to characterize a state variable x(t) by utilizing its Fourier series expansion as given in Equation (19). To make a simple analysis, we have considered the simple full bridge converter circuit in the output side of the transformer with a single DC link. When the Fourier analysis is applied in Equations (17) and (18), Equations (20)–(25) are obtained. In this Fourier analysis, only the 0th and 1st-order terms are considered, while other higher-order terms of state variables are neglected.

$$\mathrm{P}=\frac{{\mathrm{U}}^2}{2\times \pi \times \mathrm{f}\times \mathrm{L}}\varnothing \left(1-\frac{\varnothing }{\pi }\right)$$

(15)

$${\mathrm{P}}_{\mathrm{D}\mathrm{A}\mathrm{B}}=\frac{{\mathrm{U}}_{\mathrm{d}\mathrm{c}}{\mathrm{U}}_0}{2\mathrm{N}{\mathrm{f}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}\mathbb{D}\left(1-\mathbb{D}\right)$$

(16)

$$\mathrm{here}, {\mathrm{L}}_{\mathrm{t}\mathrm{f}}=\frac{{\mathrm{L}}_1}{{\mathrm{N}}^2}+{\mathrm{L}}_2$$

$$\frac{\mathrm{d}{\mathrm{i}}_{\mathrm{L}}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}=-\frac{{\mathrm{R}}_{\mathrm{t}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{i}}_{\mathrm{L}}\left(\mathrm{t}\right)+\frac{{\mathrm{T}}_1\left(\mathrm{t}\right)}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}\left(\mathrm{t}\right)-\frac{{\mathrm{T}}_3\left(\mathrm{t}\right)}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_0\left(\mathrm{t}\right)$$

(17)

$$\frac{\mathrm{d}\mathrm{U}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}=-\frac{1}{\mathrm{R}{\mathrm{C}}_0}\mathrm{U}\left(\mathrm{t}\right)+\frac{{\mathrm{T}}_2\left(\mathrm{t}\right)}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}\left(\mathrm{t}\right)-\frac{{\mathrm{I}}_0}{{\mathrm{C}}_0}$$

(18)

$$\mathrm{X}\left(\mathrm{t}\right)={\sum }_{\mathrm{k}=\infty }^{\infty }{\mathrm{X}}_{\mathrm{k}}\left(\mathrm{t}\right){\mathrm{e}}^{\mathrm{j}\mathrm{k}\mathsf{\omega }\mathrm{t}}$$

(19)

where ω =2πf and X_k(t) is the k^th coefficient in the Fourier series analysis.

$$\frac{\mathrm{d} {\mathrm{U}}_{\mathrm{o}}^0}{\mathrm{d}\mathrm{t}}=-\frac{\mathrm{d} {\mathrm{i}}_{\mathrm{o}}^0}{{\mathrm{C}}_0}-\frac{{\mathrm{U}}_{\mathrm{o}}^0}{{\mathrm{R}\mathrm{C}}_0}+\frac{{\mathrm{T}}_2^0}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{\mathrm{o}}+\frac{{2\mathrm{T}}_2^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}+\frac{{2\mathrm{T}}_2^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}$$

(20)

$$\frac{\mathrm{d} {\mathrm{U}}_{\mathrm{o}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}{\mathrm{d}\mathrm{t}}=-\frac{{\mathrm{i}}_{\mathrm{o}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}{{\mathrm{C}}_0}-\frac{{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}{{\mathrm{R}\mathrm{C}}_0}-\mathsf{\omega }{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}+\frac{{\mathrm{T}}_2^0}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}+\frac{{\mathrm{T}}_2^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^0$$

(21)

$$\frac{\mathrm{d} {\mathrm{U}}_{\mathrm{o}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}}{\mathrm{d}\mathrm{t}}=-\frac{{\mathrm{i}}_{\mathrm{o}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}}{{\mathrm{C}}_0}-\frac{{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}}{{\mathrm{R}\mathrm{C}}_0}-\mathsf{\omega }{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}+\frac{{\mathrm{T}}_2^0}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}+\frac{{\mathrm{T}}_2^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^0$$

(22)

$$\begin{gathered} \frac{\mathrm{d} {\mathrm{i}}_{\mathrm{L}}^0}{\mathrm{d}\mathrm{t}}=-\frac{{\mathrm{R}}_{\mathrm{t}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{i}}_{\mathrm{L}}^0+\frac{1}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}^0{\mathrm{T}}_1^0+\frac{{2\mathrm{T}}_1^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}+\frac{{2\mathrm{T}}_1^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}-\frac{1}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_{\mathrm{o}\mathrm{c}}^0{\mathrm{T}}_2^0 \\ +\frac{{2\mathrm{T}}_2^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}+\frac{{2\mathrm{T}}_2^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}} \end{gathered}$$

(23)

$$\frac{\mathrm{d} {\mathrm{i}}_{\mathrm{L}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}{\mathrm{d}\mathrm{t}}=-\frac{{\mathrm{R}}_{\mathrm{t}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{i}}_{\mathrm{L}}^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}+\mathsf{\omega }{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}+\frac{{\mathrm{U}}_{\mathrm{d}\mathrm{c}}^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{T}}_1^0+{\mathrm{T}}_1^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}^0-\frac{{\mathrm{T}}_2^0}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}+{\mathrm{T}}_2^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}{\mathrm{U}}_{\mathrm{o}}^0$$

(24)

$$\frac{\mathrm{d} {\mathrm{i}}_{\mathrm{L}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}}{\mathrm{d}\mathrm{t}}=-\frac{{\mathrm{R}}_{\mathrm{t}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{i}}_{\mathrm{L}}^{\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}+\mathsf{\omega }{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}+\frac{{\mathrm{U}}_{\mathrm{d}\mathrm{c}}^{\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{T}}_1^0+{\mathrm{T}}_1^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}^0-\frac{{\mathrm{T}}_2^0}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}+{\mathrm{T}}_2^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}{\mathrm{U}}_{\mathrm{o}}^0$$

(25)

The transient behavior of the DAB converter in terms of voltage variations is considerably slower in the input than in the output. Therefore, the 0th term of input voltage is $U_{dc}^0$ = ${\mathrm{U}}_{\mathrm{d}\mathrm{c}}$ and $i_o^0$ = ${\mathrm{i}}_0$. The transient behavior of the first term of input voltage and load current is neglected, which are ${\mathrm{U}}_{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ and ${\mathrm{i}}_{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$. In this design, the converters are operating in 50% of the duty cycle. Thus, the first-order terms of ${\mathrm{T}}_1$ (t) and $T_2$ (t) are similar to zero order and are presented in Equations (26)–(29). Moreover, in Equations (17) and (18), the independent state variables are the inductor current and capacitor voltage, which form the state–space model of the DAB converter with the state vector x(t) and the control vector u(t). Further, Equations (30) and (31) represent the state–space model with the input and output signals of the DAB converter. Therefore, the model parameters are input voltage source (${\mathrm{U}}_{\mathrm{d}\mathrm{c}}$), inductor current ($i_L$), and duty cycle ($\mathbb{D}$). When $T_1^{1imag}$ and $T_1^{1real}$ are introduced into Equations (20) and (25), the state–space model becomes Equation (31). Furthermore, the zero initial states are assumed in Equation (32), which are, ${\mathrm{i}}_{\mathrm{L}}^0$, ${\mathrm{U}}_{\mathrm{o}}^{1\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$, ${\mathrm{U}}_{\mathrm{o}}^{1\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}}$ are similar to zero order, and Equation (32) becomes Equation (33).

$$T_1^{1imag}=-\frac{2}{\pi }$$

(26)

$$T_1^{1real}=0$$

(27)

$$T_2^{1imag}=-\frac{2\mathrm{c}\mathrm{o}\mathrm{s}\mathbb{D}\pi }{\pi }$$

(28)

$$T_2^{1real}=-\frac{2\mathrm{s}\mathrm{i}\mathrm{n}\mathbb{D}\pi }{\pi }$$

(29)

$$\frac{\mathrm{X}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}={\mathrm{A}}_1\mathrm{x}\left(\mathrm{t}\right)+{\mathrm{B}}_1\mathrm{u}\left(\mathrm{t}\right)$$

(30)

$$\mathrm{Y}\left(\mathrm{t}\right)={\mathrm{C}}_1\mathrm{x}\left(\mathrm{t}\right)+{\mathrm{D}}_1\mathrm{u}\left(\mathrm{t}\right)$$

(31)

$$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left[\begin{array}{c}△{\mathrm{U}}_{\mathrm{o}}\\ △i_L^{1real}\\ △i_L^{imag}\end{array}\right]=\hfill \\ \left[\begin{array}{cccccc}-\frac{1}{\mathrm{R}{\mathrm{C}}_0}& -\frac{4\mathrm{s}\mathrm{i}\mathrm{n}\pi \left(\mathbb{D}\pi \right)}{{\pi \mathrm{C}}_0}& -\frac{4\mathrm{c}\mathrm{o}\mathrm{s}\pi \left(\mathbb{D}\pi \right)}{{\pi \mathrm{C}}_0}& 0& 0& 0\\ \frac{2\mathrm{s}\mathrm{i}\mathrm{n}\pi \left(\mathbb{D}\pi \right)}{\pi {\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& -\frac{{\mathrm{R}}_{\mathrm{t}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& {\mathsf{\omega }}_{\mathrm{s}}& 0& 0& 0\\ \frac{2\mathrm{c}\mathrm{o}\mathrm{s}\pi \left(\mathbb{D}\pi \right)}{\pi {\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& -{\mathsf{\omega }}_{\mathrm{s}}& -\frac{{\mathrm{R}}_{\mathrm{t}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& 0& 0& 0\\ 0& 0& 0& -\frac{{\mathrm{R}}_{\mathrm{t}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& \frac{4\mathrm{s}\mathrm{i}\mathrm{n}\pi \left(\mathbb{D}\pi \right)}{\pi {\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& \frac{4\mathrm{c}\mathrm{o}\mathrm{s}\pi \left(\mathbb{D}\pi \right)}{\pi {\mathrm{L}}_{\mathrm{t}\mathrm{f}}}\\ 0& 0& 0& -\frac{2\mathrm{s}\mathrm{i}\mathrm{n}\pi \left(\mathbb{D}\pi \right)}{{\pi \mathrm{C}}_0}& -\frac{1}{\mathrm{R}{\mathrm{C}}_0}& {\mathsf{\omega }}_{\mathrm{s}}\\ 0& 0& 0& -\frac{2\mathrm{c}\mathrm{o}\mathrm{s}\pi \left(\mathbb{D}\pi \right)}{{\pi \mathrm{C}}_0}& {\mathsf{\omega }}_{\mathrm{s}}& -\frac{1}{\mathrm{R}{\mathrm{C}}_0}\end{array}\right]\times \hfill \\ \hfill \left[\begin{array}{c}△{\mathrm{U}}_{\mathrm{o}}\\ △i_L^{1real}\\ △i_L^{1imag}\end{array}\right]+\left[\begin{array}{cc}0& \frac{1}{{\mathrm{C}}_0}\\ 0& \\ -\frac{2}{{\pi \mathrm{L}}_{\mathrm{t}\mathrm{f}}}& 0\end{array}\right]\times \left[\begin{array}{c}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}\\ {\mathrm{i}}_{\mathrm{L}}\end{array}\right]\end{array}$$

(32)

$$\begin{gathered} \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left[\begin{array}{c}△{\mathrm{U}}_0\\ △i_L^{1real}\\ △i_L^{1imag}\end{array}\right]=\left[\begin{array}{ccc}-\frac{1}{\mathrm{R}{\mathrm{C}}_0}& -\frac{4\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathbb{D}\pi \right)}{{\pi \mathrm{C}}_0}& -\frac{4\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathbb{D}\pi \right)}{{\pi \mathrm{C}}_0}\\ \frac{2\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathbb{D}\pi \right)}{\pi {\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& -\frac{{\mathrm{R}}_{\mathrm{t}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& {\mathsf{\omega }}_{\mathrm{s}}\\ \frac{2\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathbb{D}\pi \right)}{\pi {\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& -{\mathsf{\omega }}_{\mathrm{s}}& -\frac{{\mathrm{R}}_{\mathrm{t}\mathrm{f}}}{{\mathrm{L}}_{\mathrm{t}\mathrm{f}}}\end{array}\right]\times \left[\begin{array}{c}△{\mathrm{U}}_0\\ △i_L^{1real}\\ △i_L^{1imag}\end{array}\right] \\ +\left[\begin{array}{cc}0& -\frac{1}{{\mathrm{C}}_0}\\ 0& 0\\ -\frac{2}{\pi {\mathrm{L}}_{\mathrm{t}\mathrm{f}}}& 0\end{array}\right]\times \left[\begin{array}{c}{\mathrm{U}}_{\mathrm{d}\mathrm{c}}\\ {\mathrm{i}}_{\mathrm{L}}\end{array}\right] \end{gathered}$$

(33)

A small signal control for the output transfer function is needed to measure the system stability of the DAB converter with the controller structure. Thus, a small deviation in $\mathfrak{D}$ causes a distinction in the values of ${\mathrm{U}}_o^0$ from its steady-state value. Hence, the corresponding small signal model is defined in Equations (34)–(37). In addition, the nonlinearity of the system is introduced when the control and state variables are multiplied. The approximation solution of nonlinear terms is presented in Equation (38). To this end, given the value of parameters defined in Equations (34)–(37) are placed into Equation (33), then the state–space model can be expressed as in Equation (39) while the corresponding values are given in A, B, C, and D matrix form. Further, the transfer function of the DAB converter can be expressed using a conversion formula, which is given in Equation (40), and the values are assigned and given in Equation (41), where ${\mathrm{G}}_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}-\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}$ is the open-loop transfer function of the DAB.

$$△\mathfrak{D}=\mathfrak{D}-\mathrm{d}$$

(34)

$$△{\mathrm{U}}_o^0={\mathrm{U}}_o^0-{\mathrm{U}}_o^0$$

(35)

$$△i_L^{1real}=i_L^{1real}-I_L^{1real}$$

(36)

$$△i_L^{1imag}=i_L^{1imag}-I_L^{1imag}$$

(37)

$$\begin{gathered} \mathrm{i}\mathrm{n}\left(\pi \mathfrak{D}\right){\mathrm{U}}_o^0=\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi \mathfrak{D}\right)△{\mathrm{U}}_o^0+{\mathrm{U}}_o^0\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi \mathfrak{D}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\pi △\mathfrak{D}\right)+{\mathrm{U}}_o^0\mathrm{c}\mathrm{o}\mathrm{s}\left(\pi \mathbb{D}\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi △\mathfrak{D}\right) \\ =\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi \mathfrak{D}\right)△{\mathrm{U}}_o^0+{\mathrm{U}}_o^0\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi \mathfrak{D}\right)+{\mathrm{U}}_o^0\mathrm{c}\mathrm{o}\mathrm{s}\left(\pi \mathfrak{D}\right) \end{gathered}$$

(38)

(39)

$$\begin{gathered} \mathrm{A}={\left[\begin{array}{ccc}-178& -363.7827& 2.227\times {10}^{-14}\\ 1.8346\times {10}^5& -4.611\times {10}^4& 1.256\times {10}^5\\ 1.1234\times {10}^{-11}& -1.256\times {10}^5& -4.611\times {10}^4\end{array}\right]}_{3\times 3} \\ \mathrm{B}={\left[\begin{array}{c}4.5\times {10}^3\\ 7.0585\times {10}^{-9}\\ -1.1527\times {10}^8\end{array}\right]}_{3\times 1} \\ \mathrm{C}={\left[1 0 0\right]}_{1\times 3} \\ \mathrm{D}={\left[0\right]}_{1\times 1} \end{gathered}$$

$${\mathrm{G}}_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}-\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}=\left[\mathrm{C}{\left(\mathrm{s}\mathrm{I}-\mathrm{A}\right)}^{-1}\mathrm{B}+\mathrm{D}\right]$$

(40)

$${\mathrm{G}}_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}-\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}=\frac{4.5\times {10}^4{\mathrm{s}}^2+4.15\times {10}^9\mathrm{s}+6.072\times {10}^{15}}{{\mathrm{s}}^3+92398{\mathrm{s}}^2+1.798\times {10}^{10}\mathrm{s}+6.264\times {10}^{12}}$$

(41)

3.3. Output (DC-AC) Stage

The DC-AC stage architecture and the equivalent diagram are presented in Figure 6a and Figure 6b, respectively. Based on Figure 6b, the KVL is applied and can be expressed as Equation (42), followed by Equations (43) and (44). Further, the voltage across the capacitor and current through the load and capacitor are presented in Equation (45), Equation (46) and Equation (47), respectively. Equation (48) denotes the inverter output. Thereafter, a small signal change was applied to Equation (44), Equations (49), and (50) are obtained. Then, applying the Laplace transform to Equation (50) yields Equation (51), and the ratio is presented in Equation (52). In order to smoothen the output signal, a filter circuit is added, which is formulated in Equation (53), and the final equation is given in Equation (54). Further, using Equations (52) and (54), the obtained equations are Equations (55) and (56). Equation (57) represents the closed-loop transfer function of the output (DC-AC) stage converter. However, this closed-loop transfer function is designed without a controller. Thus, to design a controller with this transfer function, MATLAB SISOTOOL is used, and the final transfer function becomes Equation (58). Subsequently, the state–space parameters are found using Equation (58), and then controllability and observability are tested in Section 4.

$${\mathrm{U}}_{\mathrm{i}\mathrm{n}\mathrm{v}}\left(\mathrm{t}\right)\mathrm{u}\left(\mathrm{t}\right)-{\mathrm{L}}_{\mathrm{a}\mathrm{c}}\frac{\mathrm{d}{\mathrm{i}}_{\mathrm{a}\mathrm{c}}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}-{\mathrm{U}}_{\mathrm{c}}\left(\mathrm{t}\right)=0$$

(42)

$${\mathrm{L}}_{\mathrm{a}\mathrm{c}}\frac{\mathrm{d}{\mathrm{i}}_{\mathrm{a}\mathrm{c}}\left(\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}={\mathrm{U}}_{\mathrm{i}\mathrm{n}\mathrm{v}}\left(\mathrm{t}\right)\mathrm{u}\left(\mathrm{t}\right)-{\mathrm{U}}_{\mathrm{c}}\left(\mathrm{t}\right)$$

(43)

where

$$u\left(t\right)=\left\{\begin{array}{c}1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{U}}_{\mathrm{i}\mathrm{n}\mathrm{v}}={\mathrm{U}}_0 \hfill \\ 0,\hspace{1em}\hspace{1em}\hspace{1em} {\mathrm{U}}_{\mathrm{i}\mathrm{n}\mathrm{v}}=0 \\ -1,\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{U}}_{\mathrm{i}\mathrm{n}\mathrm{v}}={-\mathrm{U}}_0 \end{array}\right.$$

$${\mathrm{L}}_{\mathrm{a}\mathrm{c}}\frac{\mathrm{d}{\mathrm{i}}_{\mathrm{a}\mathrm{c}}}{\mathrm{d}\mathrm{t}}={\mathrm{U}}_0\left(\mathrm{t}\right)-{\mathrm{U}}_{\mathrm{c}}\left(\mathrm{t}\right)$$

(44)

$${\mathrm{U}}_{\mathrm{c}}\left(\mathrm{t}\right)=\frac{1}{{\mathrm{C}}_{\mathrm{a}\mathrm{c}}}\int {\mathrm{i}}_{\mathrm{C}}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}$$

(45)

$${\mathrm{i}}_{\mathrm{C}}\left(\mathrm{t}\right)={\mathrm{i}}_{\mathrm{a}\mathrm{c}}\left(\mathrm{t}\right)-{\mathrm{i}}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)$$

(46)

$${\mathrm{i}}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{t}\right)=\frac{{\mathrm{U}}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}}{\mathrm{Z}}$$

(47)

$${\mathrm{U}}_{\mathrm{i}\mathrm{n}\mathrm{v}}\left(\mathrm{t}\right)\mathrm{u}\left(\mathrm{t}\right)={\mathrm{U}}_0\left(\mathrm{t}\right)={\mathbb{D}}_2\left(\mathrm{t}\right){\mathrm{U}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$$

(48)

$${\mathrm{L}}_{\mathrm{a}\mathrm{c}}\frac{\mathrm{d}\left({\mathrm{i}}_{\mathrm{a}\mathrm{c}}+△{\mathrm{i}}_{\mathrm{a}\mathrm{c}}\right)}{\mathrm{d}\mathrm{t}}=\left({\mathrm{U}}_0\left(\mathrm{t}\right)+△{\mathrm{U}}_0\left(\mathrm{t}\right)\right)-\left({\mathrm{U}}_{\mathrm{C}}\left(\mathrm{t}\right)+△{\mathrm{U}}_{\mathrm{C}}\left(\mathrm{t}\right)\right)$$

(49)

$${\mathrm{L}}_{\mathrm{a}\mathrm{c}}\frac{\mathrm{d}\left(△{\mathrm{i}}_{\mathrm{a}\mathrm{c}}\right)}{\mathrm{d}\mathrm{t}}=\left(△{\mathrm{U}}_0\left(\mathrm{t}\right)\right)-\left(△{\mathrm{U}}_{\mathrm{C}}\left(\mathrm{t}\right)\right)$$

(50)

$$\mathrm{s}{\mathrm{L}}_{\mathrm{a}\mathrm{c}}\left(△{\mathrm{i}}_{\mathrm{a}\mathrm{c}}\right)=△{\mathrm{U}}_0\left(\mathrm{s}\right)-△{\mathrm{U}}_{\mathrm{C}}\left(\mathrm{s}\right)$$

(51)

$$\frac{△{\mathrm{U}}_0\left(\mathrm{s}\right)}{△{\mathrm{U}}_{\mathrm{C}}\left(\mathrm{s}\right)}=1+\frac{{\mathrm{s}}^2{\mathrm{L}}_{\mathrm{a}\mathrm{c}}{\mathrm{C}}_{\mathrm{a}\mathrm{c}}△{\mathrm{i}}_{\mathrm{a}\mathrm{c}}\left(\mathrm{s}\right)}{△{\mathrm{i}}_{\mathrm{C}}\left(\mathrm{s}\right)}$$

(52)

$$△{\mathrm{i}}_{\mathrm{a}\mathrm{c}}\left(\mathrm{s}\right)=△{\mathrm{i}}_{\mathrm{C}}\left(\mathrm{s}\right)+△{\mathrm{i}}_{\mathrm{L}\mathrm{o}\mathrm{a}\mathrm{d}}\left(\mathrm{s}\right)=△{\mathrm{i}}_{\mathrm{C}}\left(\mathrm{s}\right)+\frac{△{\mathrm{U}}_{\mathrm{C}}\left(\mathrm{s}\right)}{\mathrm{Z}}$$

(53)

$$\frac{△{\mathrm{i}}_{\mathrm{a}\mathrm{c}}\left(\mathrm{s}\right)}{△{\mathrm{i}}_{\mathrm{C}}\left(\mathrm{s}\right)}=1+\frac{1}{\mathrm{s}{\mathrm{C}}_{\mathrm{a}\mathrm{c}}\mathrm{Z}}$$

(54)

$$\frac{△{\mathrm{U}}_0\left(\mathrm{s}\right)}{△{\mathrm{U}}_{\mathrm{C}}\left(\mathrm{s}\right)}=1+{\mathrm{s}}^2{\mathrm{L}}_{\mathrm{a}\mathrm{c}}{\mathrm{C}}_{\mathrm{a}\mathrm{c}}+\frac{\mathrm{s}{\mathrm{L}}_{\mathrm{a}\mathrm{c}}}{\mathrm{Z}}$$

(55)

$$\mathrm{Thus}, \frac{△{\mathrm{i}}_{\mathrm{C}}\left(\mathrm{s}\right)}{△{\mathrm{U}}_{\mathrm{C}}\left(\mathrm{s}\right)}=\frac{\mathrm{s}{\mathrm{C}}_{\mathrm{a}\mathrm{c}}}{1+{\mathrm{s}}^2{\mathrm{L}}_{\mathrm{a}\mathrm{c}}{\mathrm{C}}_{\mathrm{a}\mathrm{c}}+\frac{\mathrm{s}{\mathrm{L}}_{\mathrm{a}\mathrm{c}}}{\mathrm{Z}}}$$

(56)

$${\mathrm{G}}_{\mathrm{o}\mathrm{p}}=\frac{0.00312\mathrm{s}}{3.9\times {10}^{-9}{\mathrm{s}}^2+8.113\times {10}^{-4}\mathrm{s}+1}$$

(57)

$${\mathrm{G}}_{\mathrm{o}\mathrm{p}\left(\mathrm{C}\mathrm{o}\mathrm{n}\right)}=\frac{0.1647\mathrm{s}}{3.047\times {10}^{-14}{\mathrm{s}}^3+6.338\times {10}^{-9}{\mathrm{s}}^2+4.195\times {10}^{-4}\mathrm{s}}$$

(58)

The state–space matrix of the above-mentioned equation is given as

$$\mathrm{A}=1\times {10}^{10}\left[\begin{array}{ccc}-0.000& -1.3768& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$

$$\mathrm{B}={\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}_{ }$$

$$\mathrm{C}=1\times {10}^{12} \left[\begin{array}{c}0\\ 0\\ 5.4053\end{array}\right]$$

$$\mathrm{D}={\left[0\right]}_{1\times 1}$$

4. Results and Discussion

The overall model of the system is developed by concatenating the models of individual parts.

4.1. Time Domain and Frequency Domain Analysis

The proposed SST model is designed and analyzed in terms of time response, frequency response, and state–space characteristics. The time response analysis is conducted by the root locus technique, which determines the absolute stability and shows the location of poles or zeros in the s-plane. This technique also determines the physical performance of the model when the gain of the input to the plant is changed from 0 to $\infty$. The technique is advantageous in controller design because it is organized and saves energy and time. This study is limited to the linearized model, and both eigen analysis and frequency domain testing are applied to the linear model. In addition, time-domain analysis is studied.

Input stage numerical study: To ensure the stable operation of the system, a controller is added through MATLAB SISIOTOOL, given in Equations (59) and (60) for ${\mathrm{G}}_{\mathrm{i}\mathrm{n}}$ and ${\mathrm{G}}_{\mathrm{v}\mathrm{i}}$, respectively. The overall closed-loop transfer function of the input stage converter is given by Equation (61).

$${\mathrm{P}\mathrm{I}}_{\mathrm{i}\mathrm{n}}=\frac{-\left(3.825\times {10}^{-6}\mathrm{s}+6.5\times {10}^{-6}\right)}{\mathrm{s}}$$

(59)

$$\mathrm{P}{\mathrm{I}}_{\mathrm{v}\mathrm{i}}=\frac{-2.7\times {10}^{-6}\mathrm{s}+0.0002592}{\mathrm{s}}$$

(60)

$${\mathrm{G}}_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}=\frac{2.9\times {10}^{-6}{\mathrm{s}}^2+1.717\times {10}^{-6}\mathrm{s}+1.76\times {10}^{-8}}{7.65\times {10}^{-6}{\mathrm{s}}^4+6.5\times {10}^{-6}{\mathrm{s}}^3+6.69\times {10}^{-6}{\mathrm{s}}^2+1.717\times {10}^{-6}\mathrm{s}+1.76\times {10}^{-8}}$$

(61)

$$\mathrm{A}=\left[\begin{array}{cccc}-0.8497& -0.8745& 0.2244& -0.0023\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right]$$

$$\mathrm{B}=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right]$$

$$\mathrm{C}=\left[\begin{array}{cccc}0& 0.3791& 0.2244& 0.0023\end{array}\right]$$

$$\mathrm{D}=\left[\begin{array}{c}0\end{array}\right]$$

Input stage controllability and observability test: To test the system controllability and observability, the controllable matrix (M_c) and observable matrix (M₀) can be measured, which are expressed in Equation (62) and Equation (63), respectively, while their corresponding values are given below. Given the matrices $M_c$ and $M_0$, the ranks of both are 4, which signifies the system is controllable and observable because the ranks of matrics (M_c and M₀) are the same as 4.

$${\mathrm{M}}_{\mathrm{c}}=\left[\mathrm{B} \mathrm{A}\mathrm{B} {\mathrm{A}}^2\mathrm{B} {\mathrm{A}}^3\mathrm{B}\right]$$

(62)

$${\mathrm{M}}_0=\left[\begin{array}{c}\mathrm{C}\\ \mathrm{C}\mathrm{A}\\ \mathrm{C}{\mathrm{A}}^2\\ \mathrm{C}{\mathrm{A}}^3\end{array}\right]$$

(63)

$${\mathrm{M}}_{\mathrm{c}}=\left[\begin{array}{cccc}1& -0.8497& -0.1526& 0.6482\\ 0& 1& -0.8497& -0.1526\\ 0& 0& 1& -0.8497\\ 0& 0& 0& 1\end{array}\right]$$

$${\mathrm{M}}_0=\left[\begin{array}{cccc}0& 0.3791& 0.2244& 0.0023\\ 0.3791& 0.2244& 0.0023& 0\\ -0.0977& -0.3292& -0.0851& -0.0009\\ -0.2462& 0.0003& 0.0210& 0.0002\end{array}\right]$$

Isolation stage controllability and observability test: To test the system controllability and observability, the controllable and observable matrices as ${\mathbf{M}}_{\mathbf{c}}$ and ${\mathbf{M}}_{\mathbf{0}}$ respectively, and the values are given below.

$${\mathrm{M}}_0=1\times {10}^7{\left[\begin{array}{ccc}0.0000& 0& 0\\ -0.0000& -0.0000& 0.0000\\ -6.6707& 1.6839& -4.5691\end{array}\right]}_{3\times 3}$$

$${\mathrm{M}}_{\mathrm{c}}=1\times {10}^{18}{\left[\begin{array}{ccc}0.0000& -0.0000& 0.0053\\ 0.0000& -0.0000& 1.3348\\ -0.0000& 0.0000& 1.5723\end{array}\right]}_{3\times 3}$$

From the M_c and M₀, the ranks of both are 3, which signifies the system is controllable and observable. To ensure the stability of the system and its closed-loop function, a feedback controller and a filter circuit must be added, which is expressed in Equation (64). The corresponding state–space parameters $(\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{D})$ followed by the test matrix (controllable and observable matrix) of the isolation stage are shown below, which is also controllable and observable because the rank of the matrix is the same as that of the test matrix. Further, the closed-loop transfer function of the proposed DAB converter is expressed in Equation (65) and after the assigned value, it is presented in Equation (66).

$${\mathrm{G}}_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}=\frac{0.001362\mathrm{s}+1}{4.916\times {10}^6{\mathrm{s}}^2+0.04229\mathrm{s}}$$

(64)

The corresponding A, B, C, and D matrix are:

$$\begin{gathered} \mathrm{A}=1\times {10}^3{\left[\begin{array}{cc}-8.851& 0\\ 0.0010& 0\end{array}\right]}_{2\times 2} \\ \mathrm{B}=\left[\begin{array}{c}1\\ 0\end{array}\right] \\ \mathrm{C}=1\times {10}^5\left[\begin{array}{cc}0.0028& 2.0300\end{array}\right] \\ \mathrm{D}=\left[0\right] \end{gathered}$$

$$\begin{gathered} {\mathrm{M}}_{\mathrm{c}}=1\times {10}^3\left[\begin{array}{cc}0.0010& -8.5851\\ 0& 0.0010\end{array}\right] \\ {\mathrm{M}}_{\mathrm{o}}=1\times {10}^6\left[\begin{array}{cc}0.0003& 0.2030\\ -2.1707& 0\end{array}\right] \end{gathered}$$

$${\mathrm{G}}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}-\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}={\mathrm{G}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{r}}\times {\mathrm{G}}_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}}$$

(65)

$${\mathrm{G}}_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}-\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{p}}=\frac{61.29{\mathrm{s}}^3+5.697\times {10}^6{\mathrm{s}}^2+8.275\times {10}^{12}\mathrm{s}+6.072\times {10}^{15}}{4.926\times {10}^{-6}{\mathrm{s}}^5+0.4974{\mathrm{s}}^4+9.25\times {10}^4{\mathrm{s}}^3+7.914\times {10}^8{\mathrm{s}}^2+2.649\times {10}^{11}\mathrm{s}}$$

(66)

Output stage controllability and observability test: This sub-section is similar to Section 3.1 and Section 3.2. As in Equations (62) and (63), ${\mathbf{M}}_{\mathbf{c}}$ and ${\mathbf{M}}_{\mathbf{0}}$ are controllable and observable matrices, and the values are given below. The ranks of these matrices are 3, which signifies the system is controllable and observable, which subsequently means that the output stage is controllable and observable.

$${\mathbf{M}}_{\mathbf{c}}=\mathbf{1}\times {\mathbf{10}}^{\mathbf{10}}\left[\begin{array}{ccc}\mathbf{0}& -\mathbf{0.000}& \mathbf{2.950}\\ \mathbf{0}& \mathbf{0}& -\mathbf{0.000}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}\end{array}\right]$$

$${\mathbf{M}}_{\mathbf{0}}=\mathbf{1}\times {\mathbf{10}}^{\mathbf{22}}\left[\begin{array}{ccc}\mathbf{0}& \mathbf{0}& \mathbf{0}\\ -\mathbf{0.000}& \mathbf{0}& \mathbf{0}\\ -\mathbf{0.0001}& -\mathbf{7.4418}& \mathbf{0}\end{array}\right]$$

The stability criterion is shown in

Table 1. Frequency response indices criteria.

Parameter	System
GM = +ve, PM = +ve	Stable
GM = −ve, PM = +ve	Unstable
GM = +ve, PM = −ve	Unstable

. Further, the performance index is measured through frequency domain analysis and presented in

Table 2. Frequency response indices of the input stage.

Parameter	Input Stage	Isolation Stage	Output Stage
GM	Inf	Inf	Inf
PM	80.0634	5.1306	5.1306
ωcg	Inf	Inf	Inf
ωcp	0.8153	2.3237 × 106	2.3237 × 106
dbdrop	−3.5	−3.5	−3.5
BW	1.0110	335.0966	−1.152 × 106
KDC	1	0	335.0966
fpeak	0.7069	33.5	0
gpeak	1.0534	335.0966	33.5

for the input, isolation, and output stages.

The frequency and time domain analysis collectively offers a better understanding of the SST behavior. The designer can recognize the critical points and stability margins to ensure reliable operation and robust performance. For instance, in the bode plot, the response of SST, on account of the diverse frequency ranges, identifies the resonance points to ensure the system’s stability. Similarly, the root locus plot is exhibited here, which can visualize the pole’s movement in the system, which explains how a control parameter can change the system’s stability. The Bode diagram reflects the variation in magnitude and phase with respect to frequency level at a log scale. Various parameters, such as GM, PM, ωcg, and ωcp, can be measured from the Bode plot. Thus, system stability margins can be determined. The Nyquist plot reflects the relationship between the imaginary part of the closed-loop transfer function and the radial frequency (ω) in the real axis with the variation in ‘ω’ from zero to infinite in the polar coordinate. This plot determines the absolute and relative stability even if the system has delays in practical assumption. The plot also provides information such as GM, PM, BW, $f_{peak}$, and $g_{peak}$. The Nichols plot is the extension of the polar plot. In this plot, magnitude and phase-angle loci at log scale versus phase diagram are in the imaginary and real axes. The location of poles and zeros of the system is presented in the pole-zero plot, and the eigenvalue of each converter in the form of a graph is depicted in Figure 7, Figure 8 and Figure 9 for the input, isolation, and output stages, respectively. Through the use of these methods, the SST’s design can be optimized for modern power distribution system applications, where stability is the main concern.

4.2. Simulation Results Analysis

In this section, the simulation results of the three-stage SST are shown. The source voltage, source current, load voltage, load current, and DC-link voltage after completion of SST simulation are shown in Figure 10. The behavior of the input remains the same, even if the load side transient is disturbed. Thus, SST provides excellent isolation through the DAB converter and maintains the same phase angle at the input stage.

4.3. Real-Time Digital Simulation

Furthermore, the Simulink model of SST has been built and executed in real-time simulation using the OPAL-RT simulator. The specification of the real-time digital simulator is OP5707XG, Intel^® Xeon^® processing cores with the power of a Xilinx^® Virtex^®-7 FPGA. The RT lab 2020 is installed on our laptop with a specification of 13th Gen Intel(R) Core(TM) i7-1355U 1.70 GHz, 64-bit operating system.

This key action marks a major advancement in the pursuit of accuracy and real-world application. The incorporation of OPAL-RT as real-time simulation is crucial and often employed for system development, analysis, and validation prior to the physical implementation. The proposed SST model has been developed using OPAL-RT software synchronization, creating an environment closely resembling real-world situations.

Figure 11 represents the real-time synchronization setup, where the SST model has been developed using RT Lab (Figure 11a), synchronized through the OPAL-RT simulator (Figure 11b), and the output can be observed on an oscilloscope (Figure 11c). Furthermore, the performance of the real-time digital model of SST is presented in Figure 12. The input voltage and current are displayed in Figure 12a,b, while the output voltage and current are shown in Figure 12c and Figure 12d, respectively. The DAB output can be observed in Figure 12e. Based on the results, it can be concluded that it exhibits the exact performance of the Simulink model, as presented in Figure 10. The implementation of the proposed SST Simulink model with OPAL-RT for real-time simulation has been successfully validated, representing a significant accomplishment in our research and development work. This makes the proposed SST model more credible and ensures that our considerations are not limited to the virtual world but have relevance to the complexities of the actual world.

5. Conclusions

In this paper, a state–space model is established to measure the stability of each converter stage for determining the SST performance with and without a controller. The analysis is conducted in three phases. In an initial attempt, the state–space model of each converter is derived, and the state–space matrixes are defined. The controllability and observability states are measured to check the system’s stability. Further, the frequency and time domain analyses are performed using the Bode plot, Nyquist, Nichols, and root locus plot. On the basis of these plots, the stability, DC gain, peak resonant, and eigenvalue of the system are measured. Furthermore, the proposed SST model is simulated through the Matlab SIMULINK environment, and the results are provided, validating the systems’ characteristics and seen as smooth performances. Finally, the Simulink model has been validated using a real-time digital simulator, such as OPAL-RT, to show the effectiveness of the system.

This study has particularly drawn attention to the transient performance and the stability of the system, which can be analyzed using the proposed approach, such as the state–space and transfer function-based model. The studied SST model can be implemented in distribution system applications that can cope with voltage disturbance. Besides, the SST can also manage voltage fluctuation due to intermittent sources, such as solar and wind, which, in fact, are necessary for small microgrids as they depend on 100% renewables. Moreover, the SST-based distribution system has numerous advantages, such as efficient power distribution, smart-grid realization, improved voltage regulation and stability, reduced environmental impact as it can enable renewable energy sources, and enhanced grid flexibility and adaptability. With the use of SST, energy providers can justify their investment to the end-user by offering resilience improvement of the distribution system by reducing the potential disturbances due to uncertainties.

The research can be extended to the stability analysis of SST by incorporating renewable energy sources as a microgrid-based system from real-time application aspects. In this paper, we considered only constant load conditions; in the future, different load types will be considered with the impact from the stability point of view. In addition, the detailed state–space model of each converter stage will be analysed.