Development of a Multivariable Deadbeat Controller in dq Coordinates for the Current Loop of a Grid-Connected VSI

Abstract

One of the main concerns in designing control structures for Voltage Source Inverters (VSI) is ensuring the generation of sinusoidal currents that comply with international power quality standards. In this context, the present study investigates the evaluation of multivariable control techniques characterized by their Deadbeat nature, within coordinates dq. One of these techniques incorporates a virtual emulation of the current loop filter behavior. The application of these controllers is demonstrated in the context of a three-phase VSI connected to the electrical grid. These controllers not only show their effectiveness in decoupling control loops but also have greater stability without the presence of phase delays. With a specific emphasis on developing and fine-tuning multivariable Deadbeat controllers using dq coordinates, the study leverages three distinct models to analyze the stability and evaluate the performance of the control strategy. The synthesis of system models and consequently of controllers was made possible by the $\alpha \beta$ and dq coordinate conversion method, developed in this study. This tool holds significant potential not only in synthesis but also in system analysis. The simulated solar setup uses a 5.1 kW photovoltaic system with second-generation modules. It includes an internal current loop, an external energy loop, and a multi-level VSI controlled by Space Vector Pulse Width Modulation (SVPWM). This complex combination of elements serves as the context for the tested control strategies, undergoing thorough testing and evaluation.

Appendix A

1.1 Generalization of the Method

In generalizing the method, we aim to establish an abstract form that can be applied to any transfer function, enabling more efficient analysis and control of these systems. By convolving with the matrix $\mathbf {K_s}$, we will demonstrate how to obtain the transfer function in the dq reference frame from the transfer function in the alpha-beta coordinates. This generalization is crucial for understanding and implementing more advanced control strategies of VSI.

In order to perform the developments, consider:

$$\begin{aligned} \begin{aligned} H^{\alpha \beta }(s)&= \frac{Y^{\alpha \beta }(s)}{X^{\alpha \beta }(s)} \textbf{I}\\&=\frac{\sum _{n=0}^N a_n s^n}{\sum _{n=0}^K b_n s^n}\textbf{I}\\&= \frac{\prod _{n=0}^N (s+z_n)}{\prod _{n=0}^K (s+p_n)} \textbf{I}. \end{aligned} \end{aligned}$$

(62)

Applying convolution with the matrix $\mathbf {K_s}$ on both sides yields:

$$\begin{aligned} \begin{aligned} \sum _{n=0}^K b_n \mathbf {K_s}*(s^nY^{\alpha \beta }(s)\textbf{I}) = \\ \sum _{n=0}^N a_n \mathbf {K_s}*(s^n X^{\alpha \beta }(s)\textbf{I}). \end{aligned} \end{aligned}$$

(63)

By replacing (21) into (63), the expression for the transfer function in the dq reference frame is derived:

$$\begin{aligned} \begin{aligned} \sum _{n=0}^K b_n ((s+\omega \textbf{U})^nY^{dq}(s))= \\ \sum _{n=0}^N a_n ((s+\omega \textbf{U})^n X^{dq}(s)). \end{aligned} \end{aligned}$$

(64)

Or, in another way:

$$\begin{aligned} \begin{aligned}&H^{dq}(s) = \frac{Y^{dq}(s)}{X^{dq}(s)} \\&= (\sum _{n=0}^K b_n (s\textbf{I}+\omega \textbf{U})^n)^{-1} (\sum _{n=0}^N a_n (s\textbf{I}+\omega \textbf{U})^n ) \\&= \prod _{n=0}^K ((s+p_n)\textbf{I}+\omega \textbf{U})^{-1} \prod _{n=0}^N ((s+z_n)\textbf{I}+\omega \textbf{U}). \end{aligned} \end{aligned}$$

(65)

1.2 Matrix Exponential

The matrix exponential simplifies solving systems by eliminating the need to deal with various different cases. For better understanding, some relationships are explained below:

$$\begin{aligned} \begin{aligned} e^{-(\omega \textbf{U} +\mathbf {L_t}^{-1}\mathbf {R_t})T_s}=e^{-(\omega \textbf{U})T_s}e^{-(\mathbf {L_t}^{-1}\mathbf {R_t})T_s}, \end{aligned} \end{aligned}$$

(66)

where:

$$\begin{aligned} \begin{aligned} e^{-(\mathbf {L_t}^{-1}\mathbf {R_t})T_s}=\left[ \begin{array}{c c} e^{-\frac{R_t T_s}{L_t}}&{}0\\ 0&{}e^{-\frac{R_t T_s}{L_t}} \end{array}\right] , \end{aligned} \end{aligned}$$

(67)

$$\begin{aligned} \begin{aligned} e^{-\omega T_s\textbf{U}}=\left[ \begin{array}{c c} cos(\omega T_s)&{}sin(\omega T_s)\\ sin(\omega T_s)&{}cos(\omega T_s) \end{array}\right] . \end{aligned} \end{aligned}$$

(68)

Resulting in:

$$\begin{aligned} \begin{aligned} e^{-(\omega \textbf{U} +\mathbf {L_t}^{-1}\mathbf {R_t})T_s}=e^{-\frac{R_t T_s}{L_t}}\left[ \begin{array}{c c} cos(\omega T_s)&{}-sin(\omega T_s)\\ sin(\omega T_s)&{}cos(\omega T_s) \end{array}\right] . \end{aligned} \end{aligned}$$

(69)

1.3 Dynamic Analysis of the System in abc and dq Coordinates

The system under analysis is described by equations that represent the dynamic behavior of a three-phase system in abc coordinates and its transformations to dq coordinates. By performing mathematical manipulations on these equations, we can investigate how the voltages at the system terminals respond to small variations in time ($\delta t$). This process allows for a detailed analysis of the system’s dynamics to understand the effects of these variations under different conditions. The voltage $V_{abc}(t)$ is given by:

$$\begin{aligned} \begin{aligned} V_{abc}(t)&= E(t)\begin{bmatrix} \sin (\omega (t)t + \theta ) \\ \sin (\omega (t)t + \theta - \frac{2\pi }{3}) \\ \sin (\omega (t)t + \theta + \frac{2\pi }{3}) \end{bmatrix}. \end{aligned} \end{aligned}$$

(70)

And at the instant $(t - \delta t)$,

$$\begin{aligned} \begin{aligned} V_{abc}(t - \delta t)&= E(t - \delta t) \begin{bmatrix} \sin (\omega (t - \delta t)(t - \delta t) + \theta ) \\ \sin (\omega (t - \delta t)(t - \delta t) + \theta - \frac{2\pi }{3}) \\ \sin (\omega (t - \delta t)(t - \delta t) + \theta + \frac{2\pi }{3}) \end{bmatrix}, \end{aligned}\nonumber \\ \end{aligned}$$

(71)

with $\omega (t - \delta t) \approx \omega (t) - \delta \omega$. Defining $\delta \theta = \delta \omega t + \omega (t)\delta t - \delta \omega \delta t$ and for small variations, $\delta \theta \approx \delta \omega t + \omega (t)\delta t$, yields:

$$\begin{aligned} \begin{aligned} V_{abc}(t - \delta t)&= E(t - \delta t)\begin{bmatrix}\sin (\omega (t)t + \theta - \delta \theta ) \\ \sin (\omega (t)t + \theta - \delta \theta - \frac{2\pi }{3}) \\ \sin (\omega (t)t + \theta - \delta \theta + \frac{2\pi }{3}) \end{bmatrix}. \end{aligned}\nonumber \\ \end{aligned}$$

(72)

For signals in dq coordinates, assuming that the PLL is working perfectly, the time delay occurs only in magnitude, as:

$$\begin{aligned} \begin{aligned} V_{dq0}(t - \delta t) = v(t - \delta t)\begin{bmatrix}\sin (\theta ) \\ \cos (\theta ) \\ 0\end{bmatrix} \end{aligned}. \end{aligned}$$

(73)

1.4 Calculation of Total Harmonic Distortion of Current and Recursive Approaches

In order to calculate the THD of the current over time, $THD_I(t)$, it is necessary to know the amplitude of the fundamental component and the other harmonics. The $THD_I(t)$ can be computed by:

$$\begin{aligned} THD_I(t) = 100\% \times \sqrt{\frac{\sum ^N_{h=2} I_h(t)^2}{I_1(t)^2}}, \end{aligned}$$

(74)

where $I_h(t)$ is the amplitude of the harmonics, $I_1(t)$ is the fundamental component, and $N$ is the total number of harmonics considered in the analysis. To simplify the computation of $THD_I(t)$, one can use the relationship $\sum ^N_{h=2} I_h(t)^2 \approx I_{rms}(t)^2 - I_1(t)^2$, which allows representing (74) as:

$$\begin{aligned} THD_I(t) = 100\% \times \sqrt{\frac{I_{rms}(t)^2}{I_1(t)^2} - 1}, \end{aligned}$$

(75)

where $I_{rms}(t)$ is the root mean square value of the monitored current, taking into account its harmonics.. The amplitude of the fundamental component, $I_1$, can be defined by the relationship $I_1(t)=\sqrt{I(\omega ,t)\cdot conj\{ I(\omega , t)\}}$, where $I(\omega ,t)$ is obtained using the Short-Time Fourier Transform (STFT),

$$\begin{aligned} \begin{aligned} I(\omega ,t)&= \frac{1}{B}\int ^{t}_{t-B} i(\tau )e^{-j\omega \tau } d\tau , \end{aligned} \end{aligned}$$

(76)

where $B$ is the width of the window in the STFT, and $i(\tau )$ is the current signal at time $\tau$. Representing (76) in discrete form:

$$\begin{aligned} I(\omega ,n)=\frac{1}{Q}\sum ^{n}_{m=n-Q}i(m)e^{-j\omega m}, \end{aligned}$$

(77)

where $Q=\frac{B}{\Delta t}$ is the number of samples in the STFT window, and $i(m)$ is the $m$-th sample of the current signal. Using (77), a recursive approach to reduce computational effort can be defined as follows:

$$\begin{aligned} \begin{aligned} I(\omega ,n)&=\frac{1}{Q}i(n)e^{-j\omega n}+\frac{1}{Q}\sum ^{n-1}_{m=n-1-Q}i(m)e^{-j\omega m} \\ {}&- \frac{1}{Q}i(n-Q)e^{-j\omega (n-Q)} \\&=I(\omega ,n-1)+\frac{e^{-j\omega n}}{Q}(i(n)-i(n-Q)e^{j\omega Q}). \end{aligned} \end{aligned}$$

(78)

Similarly, the root mean square value of the monitored current, i(t), taking into account its harmonics, can be computed by:

$$\begin{aligned} \begin{aligned} I(n)^2&=\frac{1}{Q}\sum ^{n}_{m=n-Q}i(m)^2\\&=I(n-1)^2+\frac{1}{Q}(i(n)^2-i(n-Q)^2). \end{aligned} \end{aligned}$$

(79)