# Backstepping Controller Design with Inherent Current Limitation for a Voltage-Controlled Grid-Side Converter

Lasse Gnärig, Robin Weiß, Steffen Bernet

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Lasse Gnärig, Robin Weiß, Steffen Bernet. Backstepping Controller Design with Inherent Current Limitation for a Voltage-Controlled Grid-Side Converter. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0198. hal-05101065

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# Backstepping Controller Design with Inherent Current Limitation for a Voltage-Controlled Grid-Side Converter

Lasse Gnärig, Robin Weiß, Steffen Bernet
TECHNISCHE UNIVERSITÄT DRESDEN

Helmholtzstr. 9, 01069

Dresden, Germany

Tel.: +49 (0) 351-46335665

E-Mail: lasse.gnaerig@tu-dresden.de

URL: : https://www.tu-dresden.de

*Index Terms*—Integrator Backstepping, Grid Forming Converters, Current Limitation, Modeling, Control.

*Abstract*—The control and optimization of grid-forming converters remain crucial in power electronics research. A key challenge is limiting the converter's output current during severe grid disturbances to protect equipment. Existing current-limiting strategies either indirectly modify the output voltage or directly control the current, the latter requiring an additional control loop that increases complexity and may introduce instability. To address these challenges, this paper proposes a backstepping-based control strategy that ensures asymptotic stability while maintaining current limitation. A novel selection of control Lyapunov functions enables a continuous control law, ensuring seamless operation across normal and fault conditions without switching between control modes. This feature is particularly beneficial for stability analysis of power electronic-based grids under fault conditions, as it allows for a more straightforward and unified stability assessment. The proposed method is implemented on a two-level voltage source converter (2L-VSC) with an LCL filter. Preliminary simulations validate its effectiveness in ensuring stable operation under both nominal and fault conditions.

## I. INTRODUCTION

The control and optimization of grid-forming converters continues to be of great importance in the research field of power electronics. The primary characteristic of grid forming converters is that their behavior resembles a controlled voltage source behind an impedance [1]. As a result, during severe disturbances in the grid, this feature leads to a fast increase in the output current of the converter. To protect the equipment, one

crucial task is to limit the current under fault conditions. This current limitation can either be applied indirectly through the output voltage or act directly on the current itself [1]. The latter approach necessitates an additional current control loop, which, in turn, increases the complexity of the control algorithm, making controller design more challenging. Furthermore, cascading control loops can lead to undesirable interactions between control instances, potentially resulting in instability under the worst-case scenario [2], [3]. Therefore, design methods that ensure stability are of utmost importance. In grid-forming converters with current control, current limitation is often achieved through setpoint constraints [4]. This method typically introduces nonlinearity into the system, with different control laws governing normal operation and operation under current limitation. As a result, we are faced with additional complexity, particularly in stability analysis and modeling for converter fed power grids [5], [7]. The control strategy presented in this paper addresses these mentioned challenges. A backstepping approach is employed to ensure the asymptotic stability of the controlled system within a defined region [6]. Backstepping control is well established in converter control, see e.g. [8]. A novel aspect of the proposed method is the careful selection of control Lyapunov functions to ensure the limitation of the reference current trajectory while maintaining system stability. In [9] a similar approach is proposed for physical motion systems, which has been adapted in the present work for grid-forming converter systems. A key advantage of this approach is that the resulting control law is applicable both in normal operation and during current-limiting conditions. The proposed design is implemented for a two-level voltage source converter (2L-VSC) with
<figure><img src="image_2.png" /><figcaption>Fig. 1. Electrical circuit of an ideal two-level voltage source converter</figcaption></figure>

an LCL filter. Preliminary simulations demonstrate the effectiveness of the proposed algorithm. The structure of the paper is as follows: Section II introduces the mathematical model of the system. Section III focuses on the design of the backstepping controller. Simulation results are presented in Section IV. Concluding remarks and prospects for future works are given in Section V.

## II. SYSTEM MODEL

The electrical circuit of the considered system is depicted in Fig. 1 and represents an ideal two-level voltage source converter with LC-Filter and a load on the AC-side modeled as current source. In Fig. 1 the meaning of the switching function $q_p: \mathbb{R} \to \{-1, 1\}, p \in \{a, b, c\}$ is illustrated for each phase. Since all further considerations are based on averaged models, in the following we will deal with averaged switching functions $s_p: \mathbb{R} \to [-1, 1], p \in \{a, b, c\}$ [8]. The dc-link voltage $v_{dc}$ is assumed to be constant and due to the isolated star points the sum of the currents $i_p$ and $i_{L,p}$ is zero. With these assumptions the electrical system can be described by

$$C \frac{dv_{abc}}{dt} = i_{abc} - i_{L,abc} \quad (1a)$$

$$L \frac{di_{abc}}{dt} = \frac{v_{dc}}{2} s_{abc} - v_{abc} - v_e \mathbb{1} \quad (1b)$$

with the vectors $v_{abc} = (v_a, v_b, v_c)^T$, $i_{abc} = (i_a, i_b, i_c)^T$, $i_{L,abc} = (i_{L,a}, i_{L,b}, i_{L,c})^T$, $s_{abc} = (s_a, s_b, s_c)^T$, and $\mathbb{1} = (1, 1, 1)^T$. The model introduced here can be written in rotating coordinates (dq0) by applying the transformation $i_{dq0} = T_{dq0}i_{abc}$, where the transformation matrix reads

$$T_{dq0} = \frac{2}{3} \begin{pmatrix} \cos(\theta) & \cos(\theta - \frac{2\pi}{3}) & \cos(\theta + \frac{2\pi}{3}) \\ -\sin(\theta) & -\sin(\theta - \frac{2\pi}{3}) & -\sin(\theta + \frac{2\pi}{3}) \end{pmatrix}, \quad (2)$$

with $\theta$ representing the angle of the rotating reference frame whose time derivative $\dot{\theta} = \omega$ will further be

considered as the angular frequency of the converters output voltage. Transforming the system model (1) as described above leads to

$$C \frac{dv_d}{dt} = i_d - i_{L,d} + \omega C v_q \quad (3a)$$

$$C \frac{dv_q}{dt} = i_q - i_{L,q} - \omega C v_d \quad (3b)$$

$$L \frac{di_d}{dt} = \frac{v_{dc}}{2} s_d - v_d + \omega L i_q \quad (3c)$$

$$L \frac{di_q}{dt} = \frac{v_{dc}}{2} s_q - v_q - \omega L i_d \quad (3d)$$

For the sake of completeness, the voltage $v_e$ between the star point of the filter capacitors at the mid point of the DC-link is given by

$$v_e = \frac{v_{dc}}{2} s_0 \quad (4)$$

which indicates that $s_0$ only affects the common mode voltage $v_e$ and therefore has no influence on the voltages $z_d$ and $z_q$.

## III. CONTROLLER DESIGN

For controller design, a backstepping approach is applied that is carried out in the coordinates $(v_d, v_q, i_d, i_q)^T$. The goal of the design is to obtain a controller that ensures that $(v_d, v_q)^T$ tracks a reference trajectory $(v_d^*, v_q^*)^T$ originating e.g. from a superordinated droop controller. Furthermore, a constraint is given by the limitation of the output of the voltage controller $(i_d^*, i_q^*)^T$ that can be expressed by

$$\left\| (i_d^*, i_q^*)^T \right\|_2 \le \hat{B}, \quad \forall t \ge 0. \quad (5)$$

The objective is now to design an asymptotically stable controller for (3) that complies with (5). In a first step the subsystem (3a), (3b) is considered. Since (5) applies, we have to make sure that the propagation of the error $(v_d - v_d^*, v_q - v_q^*)^T$ is bounded [9]. This implies that the growth of the control Lyapunov function (CLF) $V_1$ has to be restricted according to

$$\left\| \frac{dV}{d\epsilon_v} \right\|_2 \le \kappa ||\epsilon_v||_2, \quad \text{as } ||\epsilon_v||_2 \to \infty, \text{ with } \kappa \in \mathbb{R}_+, \quad (6)$$

where

$$\epsilon_v = (\epsilon_{v,d}, \epsilon_{v,q})^T = (v_d - v_d^*, v_q - v_q^*)^T \quad (7)$$

is the error in the first subsystem. A positive definite CLF that is illustrated in Fig. 2 and that meets the requirements is
<figure><img src="image_3.png" /><figcaption>Fig. 2. CLF $V_1$</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Choice of $\dot{V}_1$</figcaption></figure>

$$V_1 = C \left( \epsilon_{v,d} \tanh \left( \frac{k_v \epsilon_{v,d}}{B} \right) + \epsilon_{v,q} \tanh \left( \frac{k_v \epsilon_{v,q}}{B} \right) \right), \quad (8)$$

where $k_v, B > 0$ are design parameters and the scaled time derivative reads

$$\frac{B}{k_v C} \dot{V}_1 = \epsilon_{v,d} \dot{\epsilon}_{v,d} \left( -\tanh^2\left(\frac{k_v \epsilon_{v,d}}{B}\right) + \tanh\left(\frac{k_v \epsilon_{v,d}}{B}\right) - 1 \right) + \epsilon_{v,q} \dot{\epsilon}_{v,q} \left( -\tanh^2\left(\frac{k_v \epsilon_{v,q}}{B}\right) + \tanh\left(\frac{k_v \epsilon_{v,q}}{B}\right) - 1 \right) \quad (9)$$

In a next step, we substitute the time derivatives $\dot{\epsilon}_{v,q}$ and $\dot{\epsilon}_{v,q}$ by (7), (3a) and (3b). Now, as can be seen in Fig. 3, we can make $\dot{V}_1$ negative definite by the choice $\dot{V}_1 = W_1 = W_{1,d} + W_{1,q}$, where

$$W_{1,d} = -\tanh\left(\frac{k_v \epsilon_{v,d}}{B}\right). \\ \cdot \left(k_v \epsilon_{v,d} + B \tanh\left(\frac{k_v \epsilon_{v,d}}{B}\right) - k_v \epsilon_{v,d} \tanh^2\left(\frac{k_v \epsilon_{v,d}}{B}\right)\right) \\ W_{1,q} = -\tanh\left(\frac{k_v \epsilon_{v,q}}{B}\right). \\ \cdot \left(k_v \epsilon_{v,q} + B \tanh\left(\frac{k_v \epsilon_{v,q}}{B}\right) - k_v \epsilon_{v,q} \tanh^2\left(\frac{k_v \epsilon_{v,q}}{B}\right)\right)$$

In order to fulfill this, we have to make sure that the currents $i_d$ and $i_q$ track the reference trajectories

$$i_d^* = -B \tanh \left( \frac{k_v \epsilon_{v,d}}{B} \right) + i_{L,d} - C \omega v_q + C \dot{u}_d^* \quad (10a)$$

$$i_q^* = -B \tanh \left( \frac{k_v \epsilon_{v,q}}{B} \right) + i_{L,q} + C \omega v_d + C \dot{u}_q^* \quad (10b)$$

<figure><img src="image_5.png" /><figcaption>Fig. 4. Limited propagation of the tracking error $\epsilon_{v,dq}$ through to the next subsystem</figcaption></figure>

If we compare these terms with the constraint (5), we find that (5) is violated. This is due to the feed forward of the load current $i_{L,dq}$ and the decoupling terms in (10), where boundedness cannot be ensured. In contrast to this, the time derivative $\dot{u}_{dq}^*$ of the voltage is bounded by means of an adequate choice of the reference trajectory. We can see from Fig. 4, that the term $-B \tanh(k_v \epsilon_{v,dq}/B)$ is bounded and never exceeds the value of the control parameter $B$. Furthermore, one can show that the function for small errors $\epsilon_{v,dq}$ is approximately linear with the slope $k_v$. There a few options to deal with the not necessarily bounded terms in (10).

* Discard the feed forward of the load current and the decoupling. This measure would compromise the tracking performance. In order to cope with this, the time integral of the tracking error can be added to (10) compensating the steady error.

* Saturate the feed forward signal and the decoupling term. This ensures the boundedness of the reference current trajectory but restricts the region in which asymptotic stability is guaranteed.

In a next step we stabilize the subsystem (3c) and (3d). Due to its dynamics, the converter current cannot follow its reference $(i_d^*, i_d^*)^T$ perfectly. Thus, we have to consider its tracking error $\epsilon_i = (\epsilon_{i,d}, \epsilon_{i,q})^T = (i_d - i_d^*, i_q - i_q^*)^T$. Owing this consideration the actual time derivative of the CLF $V_1$ reads

$$\dot{V}_1 = W_1 + \epsilon_{i,d} \tanh \left( \frac{k_v \epsilon_{v,d}}{B} \right) + \epsilon_{i,q} \tanh \left( \frac{k_v \epsilon_{v,q}}{B} \right) + \frac{k_v}{B} \epsilon_{i,d} \epsilon_{v,d} \left( 1 - \tanh^2 \left( \frac{k_v \epsilon_{v,d}}{B} \right) \right) + \frac{k_v}{B} \epsilon_{i,q} \epsilon_{v,q} \left( 1 - \tanh^2 \left( \frac{k_v \epsilon_{v,q}}{B} \right) \right). \quad (11)$$

Note that introducing a hard bound on the error $\epsilon_i$ is not feasible, since such a control law would lead to extremely large actuator signals. Therefore, in contrast
to [9], a CLF $V_2$ is introduced that results in a linear error dynamics. A promising candidate is

$$V_2 = V_1 + \frac{L}{2}\epsilon_{i,d}^2 + \frac{L}{2}\epsilon_{i,q}^2, \qquad (12)$$

whose time derivative is given by

$$\dot{V}_2 = \dot{V}_1 + L\epsilon_{v,d}\dot{\epsilon}_{v,d} + L\epsilon_{v,q}\dot{\epsilon}_{v,q}. \qquad (13)$$

Here, we eliminate the time derivatives in (13) by setting in (11), (3c), and (3d). Finally we chose the switching positions such that

$$\dot{V}_2 = W_1 - k_i\epsilon_{v,d}^2 - k_i\epsilon_{v,q}^2 \qquad (14)$$

applies. This is fulfilled if the averaged switching positions are given by

$$\frac{v_{dc}}{2}s_d = \Delta_d \left( \frac{BLk_v}{C} - 1 \right) + \Delta_d^2 \left( \frac{k_v}{B}\epsilon_{v,d} + \frac{Lk_v}{C}\epsilon_{i,d} \right) + \\ \Delta_d^3 \frac{BLk_v}{C} + \epsilon_{v,d} \left( 1 - \frac{k_v}{B} - LC\omega^2 \right) - 2L\omega\epsilon_{i,q} - \\ \epsilon_{i,d} \left( \frac{Lk_v}{C} - k_i \right) + v_d^* (1 - LC\omega^2) + LC (\ddot{v}_d^* - 2\omega\dot{v}_q^*) + \\ L(i_{L,d} + \omega i_{L,q}) + 2BL\omega\Delta_q \qquad (15)$$

and

$$\frac{v_{dc}}{2}s_q = \Delta_q \left( \frac{BLk_v}{C} - 1 \right) + \Delta_q^2 \left( \frac{k_v}{B}\epsilon_{v,q} + \frac{Lk_v}{C}\epsilon_{i,q} \right) - \\ \Delta_q^3 \frac{BLk_v}{C} + \epsilon_{v,q} \left( 1 - \frac{k_v}{B} - LC\omega^2 \right) + 2L\omega\epsilon_{i,d} - \\ \epsilon_{i,q} \left( \frac{Lk_v}{C} + k_i \right) + v_q^* (1 - LC\omega^2) + LC (\ddot{v}_q^* + 2\omega\dot{v}_d^*) + \\ L(i_{L,q} + \omega i_{L,d}) - 2BL\omega\Delta_d \qquad (16)$$

where $\Delta_d = \tanh(k_v\epsilon_{v,d}/B)$ and $\Delta_q = \tanh(k_v\epsilon_{v,q}/B)$. As one can see from the block diagramm in Fig. 5, the complete control law is given by (10), (15), and (15).

## IV. IMPLEMENTATION AND SIMULATION

Simulations are carried out as a first step towards verifying the results of the controller design. For this a time discrete model of the controller has been implemented in MATLAB/SIMULINK, whereas the electrical circuit depicted in Fig. 1 is implemented using PLECS. The 2L-VSC is captured by a switched model taking into account the nonlinearities of the plant that are not considered

<figure><img src="image_6.png" /><figcaption>Fig. 5. Block diagram of the control system</figcaption></figure>

during the model based control design. For the simulations the load current sources in Fig. 1 are replaced by a grid-side filter inductance and a passive RL-load. To start with a simple example, we will first consider reference voltages with constant frequency, phase angle, and amplitude, resulting in constant values for $v_d^*$ and $v_q^*$. In order to demonstrate the robustness and ability of the control approach to limit the converter currents, a sudden reduction of the load impedance is investigated by short-circuit the RL-load. It remains to tackle the problem of the feed forward of the load current $i_{L,dq}$ in (10) that may lead to a violation of the constraint (5). Like proposed in the previous section, a provisional solution has been implemented, which is to limit the feed forward of the load current. This is done by replacing $i_{L,dq}$ in the voltage controller (10) by

$$\lambda = B_{\lambda} \tanh \left( \frac{i_{L,dq}}{B_{\lambda}} \right), \qquad (17)$$

where $B_{\lambda} > 0$ is the highest possible magnitude of $\lambda$. Again, we have to point out, that this measure shrinks the region in which asymptotic stability is guaranteed by means of the backstepping controller design. All relevant parameters of the simulation model are listed in TABLE I. In Fig. 6 and Fig. 7 simulation results for a LVRT are shown. The voltage $v_{abc}$ closely tracks its reference during normal operation, with Fig. 8 providing a clear assessment of the tracking performance. As one can see, even under normal operation, there is a slight deviation of the voltage from its setpoint, which is due to the fact that the actual system differs from the model (3) used for controller design. This discrepancy is particularly due to the time delay introduced by the time discretization of the control law and can be canceled by introducing an integral part to (10). Since the current is limited, the voltage controller cannot track the reference trajectory during LVRT. Apart from transient overcurrents (detailed in Fig. 9), the currents remain at a constant and limited amplitude throughout LVRT. The transient over currents
<table><caption>TABLE I
PARAMETERS OF THE SIMULATION MODEL</caption><thead><tr><th>component</th><th>parameter</th></tr></thead><tbody><tr><td>LCL-filter</td><td>L<sub>1</sub> = 800 µH<br>C = 55 µF<br>L<sub>2</sub> = 400 µH</td></tr><tr><td>Load</td><td>R<sub>load</sub> = 22 Ω<br>L<sub>load</sub> = 50 mH<br>R<sub>short</sub> = 1 Ω</td></tr><tr><td>Converter</td><td>v<sub>dc</sub> = 750 V<br>f<sub>switch</sub> = f<sub>sample</sub> = 20 kHz<br>T<sub>blank</sub> = 500 ns</td></tr><tr><td>Voltage reference</td><td>v<sub>d</sub><sup>*</sup> = 325 V<br>v<sub>q</sub><sup>*</sup> = 0<br>ω = 2π50 s<sup>-1</sup></td></tr><tr><td>Controller</td><td>f<sub>sample</sub> = 20 kHz<br>k<sub>v</sub> = 0.8 A V<sup>-1</sup><br>k<sub>i</sub> = 5 V A<sup>-1</sup><br>B = 12 A<br>B<sub>λ</sub> = 16 A<br>β = 30 A</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Simulation results for LVRT. The voltages $v_{abc}$ follow their reference trajectories until the fault occurs at 0.095 s. The current is limited while FRT. After fault clearance at 0.15 s, the pre-fault operating regime is restored.</figcaption></figure>

are due to the fact that the reference trajectory cannot be followed infinitely fast. A closer look at the voltages at fault entry reveals that they transition into a steady-state low-voltage regime without excessive disturbances, as shown in Fig. 10. After the fault is cleared, the initial operating regime is restored within finite time. As shown in Fig. 11, this transition occurs smoothly and without issues.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Simulation results for LVRT. The voltages $v_{dq}$, displayed in rotating coordinates, follow their reference trajectories until the fault occurs at 0.095 s. After fault clearance at 0.15 s, the pre-fault operating regime is restored. The reference values are represented by the dashed lines.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Approximately 2 ms after the converter begins pulsing, the voltages $v_{dq}$ reach a steady-state operating condition. In this regime, a slight deviation of 4 V from the reference value persists.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. After the fault occurs and before the currents enter a limited range, the phase currents exhibit an overshoot.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 10. A closer look at the moment when the voltage drop occurs reveals that the voltage settles into a stationary low-voltage regime after 2 ms.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 11. After fault clearance, the voltages return to the pre-fault regime after approximately 2 ms.</figcaption></figure>

## V. CONCLUSION

A backstepping controller, originally designed for physical motion systems, has been successfully adapted for the first time to a power electronic converter system. The proposed controller effectively bounds the current reference for the subordinate current controller while ensuring asymptotic stability. Unlike existing methods, this approach provides a continuous control law applicable in both nominal operation and low-voltage ride-through conditions. Since there is no switching between different control laws, this characteristic may facilitate the simulative investigation of converter-based power systems during fault conditions. Simulation and initial numerical case studies indicate the controllers effectiveness. Future work should focus on more comprehensive simulations and experimental validation. One notable finding is that load current feedforward may compromises the current limitation – an aspect that requires further investigation

and optimization.

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