The global transition towards sustainable energy has placed significant emphasis on solar photovoltaic (PV) systems. The core of an efficient grid-connected PV system is the solar inverter, responsible for converting the variable DC output from the PV panels into stable, grid-compliant AC power. A critical challenge in maximizing energy harvest is ensuring that the solar inverter operates at the Maximum Power Point (MPP) of the PV array under all environmental conditions, a process known as Maximum Power Point Tracking (MPPT). While MPPT algorithms determine the optimal operating voltage or current, the dynamic performance and robustness of the solar inverter’s inner current and voltage control loops are paramount for realizing this power point swiftly and accurately, especially during rapid changes in solar irradiation or grid disturbances.
Traditional control strategies for solar inverters, such as Proportional-Integral-Derivative (PID) control within a synchronous reference frame (dq-frame), are widely adopted due to their simplicity and reliability. However, their design typically relies on linearized models around a specific operating point. Solar inverters are inherently nonlinear systems, and their dynamics change with operating conditions. Consequently, PID controllers may exhibit degraded performance, significant overshoot, or even instability when the system deviates from its nominal operating point, failing to provide globally consistent control performance.
To address these limitations, nonlinear control strategies have been explored. Feedback linearization can fully compensate for system nonlinearities but requires a precise mathematical model and lacks robustness against parameter uncertainties. Sliding Mode Control (SMC) offers inherent robustness but can suffer from chattering. Backstepping control and Model Predictive Control (MPC) have also been applied, yet often without fully exploiting the intrinsic physical energy properties of the solar inverter system. Passivity-Based Control (PBC) is a powerful nonlinear design methodology that explicitly considers the energy dissipation and shaping of the system. By constructing an appropriate energy storage function (often a Lyapunov function), PBC can achieve superior transient performance and stability. Recent applications of PBC to solar inverters have demonstrated improved dynamic response. However, like many model-based methods, its performance can be sensitive to model inaccuracies and external disturbances.
Meanwhile, Fractional-Order Sliding Mode Control (FOSMC) has emerged as an advancement over integer-order SMC. By employing fractional calculus in the design of the sliding surface, FOSMC provides an extra degree of freedom for tuning, often resulting in faster convergence, reduced chattering, and enhanced robustness. The combination of passivity-based design for performance and sliding mode concepts for robustness presents a promising avenue for advanced solar inverter control.
In this work, we propose a novel composite control strategy: a Passive Fractional-Order Sliding-Mode Controller (PFSMC) for grid-connected solar inverters. The design philosophy is twofold. First, we employ the PBC framework to derive a control law based on an energy-shaping perspective. We construct an energy storage function from the tracking errors of the DC-link voltage and the q-axis current. A key step in our PBC design is the careful analysis and retention of beneficial natural damping terms present in the system, which actively contribute to faster error convergence, rather than canceling all nonlinearities indiscriminately. This leads to an intrinsically improved dynamic response. Second, to fortify the controller against unmodeled dynamics, parameter variations, and grid disturbances, we incorporate a Fractional-Order Sliding Mode (FOSM) component as an additional control input. This component acts as an “energy reshaper” and robustifier. The fractional-order sliding surface offers superior convergence properties compared to its integer-order counterpart. The resulting PFSMC synergizes the performance benefits of passivity-based energy shaping with the robust attributes of fractional-order sliding modes, aiming to deliver fast, accurate, and robust control for solar inverters under diverse operating scenarios.
The structure of this article is as follows. First, we present the dynamic model of a two-stage grid-connected solar inverter system. Subsequently, we detail the systematic design procedure for the proposed PFSMC. We then conduct comprehensive case studies via simulation to evaluate the controller’s performance under solar irradiation changes and grid voltage sags, comparing it against conventional PID, standard PBC, and standalone FOSMC. Quantitative metrics such as Integral of Absolute Error (IAE) and total control effort are analyzed. Finally, we validate the practical feasibility of the PFSMC algorithm through Hardware-in-the-Loop (HIL) experiments using a dSpace platform, confirming its readiness for real-time implementation.
Mathematical Model of the Grid-Connected Solar Inverter System
We consider a standard two-stage grid-connected PV system. The first stage consists of the PV array and a DC-DC converter implementing the MPPT algorithm (e.g., Incremental Conductance). The second stage is a voltage source inverter (VSI) interfacing with the grid through an L filter. The control focus is on this grid-side solar inverter. The system configuration in the synchronous dq-frame rotating at the grid angular frequency $\omega$ is used for analysis and design.
The dynamics of the solar inverter currents in the dq-frame are given by:
$$
\begin{aligned}
L \frac{di_d}{dt} &= -R i_d + \omega L i_q + v_d – e_d \\
L \frac{di_q}{dt} &= -R i_q – \omega L i_d + v_q – e_q
\end{aligned}
$$
where $i_d$, $i_q$ are the d- and q-axis inverter currents; $v_d$, $v_q$ are the d- and q-axis inverter output voltages (control inputs); $e_d$, $e_q$ are the d- and q-axis grid voltages; $R$ and $L$ are the equivalent resistance and inductance of the filter and grid connection.
Assuming a lossless inverter and a stiff DC-link, the power balance between the DC input and AC output is:
$$
V_{dc} I_{dc} = \frac{3}{2}(e_d i_d + e_q i_q)
$$
where $V_{dc}$ is the DC-link voltage and $I_{dc}$ is the current entering the inverter from the DC-link capacitor. The dynamics of the DC-link voltage are governed by:
$$
C \frac{dV_{dc}}{dt} = I_{pv} – I_{dc} = I_{pv} – \frac{3(e_d i_d + e_q i_q)}{2 V_{dc}}
$$
where $C$ is the DC-link capacitance and $I_{pv}$ is the current from the PV array (after the MPPT stage), treated as a slowly varying disturbance from the inverter’s control perspective.
For control design, we define the state vector $\mathbf{x} = [i_q, V_{dc}]^T$ and the control input vector $\mathbf{u} = [v_d, v_q]^T$. The control objectives are: 1) to regulate $V_{dc}$ to its MPPT reference $V_{dc}^*$, and 2) to regulate $i_q$ to its reference $i_q^*$ (typically set to zero for unity power factor or to provide reactive power support). We define the tracking error vector $\mathbf{e} = [e_1, e_2]^T = [i_q – i_q^*, V_{dc} – V_{dc}^*]^T$.
Design of the Passive Fractional-Order Sliding-Mode Controller (PFSMC)
The design of the PFSMC proceeds in two integrated steps: the passivity-based energy shaping and the addition of the fractional-order sliding mode robustifier.
Step 1: Passivity-Based Control Design and Energy Shaping
The core idea of PBC is to render the closed-loop system passive with respect to a desired storage function. We begin by constructing an energy storage function $H(\mathbf{e})$ that reflects the energy associated with the tracking errors. A natural candidate is a quadratic form:
$$
H(e_1, e_2) = \frac{1}{2} L e_1^2 + \frac{1}{2} C e_2^2
$$
This function represents the energy stored in the virtual “error dynamics” of the inductor and capacitor. Its time derivative along the trajectories of the system is:
$$
\dot{H} = L e_1 \dot{e}_1 + C e_2 \dot{e}_2
$$
Substituting the dynamics for $\dot{e}_1 = \dot{i}_q – \dot{i}_q^*$ and $\dot{e}_2 = \dot{V}_{dc} – \dot{V}_{dc}^*$, and using the system model equations, we obtain an expression containing the control inputs $v_d$ and $v_q$. The standard PBC approach involves setting this derivative to be negative definite, e.g., $\dot{H} = -\mathbf{e}^T \mathbf{K} \mathbf{e}$, where $\mathbf{K}$ is a positive definite damping matrix. This often leads to a control law that cancels all system nonlinearities.
Our proposed enhancement is to perform a targeted cancellation. We analyze the terms in $\dot{H}$. Some terms represent natural, beneficial damping in the physical system (like the $-R i_q$ term). Instead of canceling them, we retain these terms to aid in error dissipation. Other terms represent cross-couplings ($\omega L i_d$) or nonlinear interactions that impede straightforward stabilization. We design the primary PBC law $\mathbf{u}_{pbc} = [v_{d,pbc}, v_{q,pbc}]^T$ to compensate for these detrimental terms while preserving useful damping. After this step, the derivative of the storage function becomes:
$$
\dot{H}_{pbc} = -R_{eff1} e_1^2 – R_{eff2} e_2^2 + \mathbf{e}^T \mathbf{\tilde{d}}
$$
Here, $-R_{eff1} e_1^2 – R_{eff2} e_2^2$ is the negative definite part achieved by the PBC law, with $R_{eff1}$ and $R_{eff2}$ being positive effective damping coefficients enhanced from the natural $R$. The term $\mathbf{e}^T \mathbf{\tilde{d}}$ represents the residual effects of uncertainties, disturbances, and modeling errors not addressed by the nominal PBC law.
Step 2: Fractional-Order Sliding Mode Enhancement
To guarantee robustness and handle the residual term $\mathbf{\tilde{d}}$, we introduce an additional control input $\mathbf{u}_{sm} = [v_{d,sm}, v_{q,sm}]^T$ based on fractional-order sliding mode control. The total control input is $\mathbf{u} = \mathbf{u}_{pbc} + \mathbf{u}_{sm}$.
We first define a fractional-order sliding surface for each control channel. For the q-axis current error $e_1$, the surface $s_1$ is defined as:
$$
s_1 = D^{\alpha_1} e_1 + \lambda_1 e_1
$$
where $D^{\alpha_1}$ denotes the fractional derivative of order $\alpha_1$ (with $1 < \alpha_1 < 2$), and $\lambda_1 > 0$ is a tuning gain. Similarly, for the DC voltage error $e_2$:
$$
s_2 = D^{\alpha_2} e_2 + \lambda_2 e_2
$$
The fractional-order derivative provides an extra degree of freedom for shaping the dynamic response on the sliding manifold. It can be shown that once on the surface $s_i=0$, the error dynamics follow $D^{\alpha_i} e_i = -\lambda_i e_i$, which typically yields faster and more robust convergence compared to integer-order dynamics ($\dot{e}_i = -\lambda_i e_i$).
The sliding mode control law is designed to drive the system trajectories to these surfaces and maintain them there. We choose a reaching law approach. The additional control input $\mathbf{u}_{sm}$ is designed such that:
$$
\dot{s}_i = -\rho_i \text{sign}(s_i) – k_i s_i
$$
where $\rho_i > |\tilde{d}_i|$ is a switching gain large enough to overcome the bounded lumped uncertainty $\tilde{d}_i$, and $k_i > 0$ is a linear feedback gain that governs the reaching rate. To mitigate chattering, the signum function $\text{sign}(s_i)$ is often replaced by a smooth approximation like a saturation function $\text{sat}(s_i / \phi_i)$, where $\phi_i$ is the boundary layer thickness.
Solving for $\mathbf{u}_{sm}$ from the condition $\dot{s}_i = -\rho_i \text{sat}(s_i / \phi_i) – k_i s_i$ after substituting the system dynamics yields the final robustifying control terms $v_{d,sm}$ and $v_{q,sm}$. The fractional derivatives in the sliding surface definition can be implemented in real-time using approximations like the Oustaloup recursive filter or the Grünwald-Letnikov discretization method.
The final composite PFSMC law is:
$$
\begin{aligned}
v_d &= v_{d,pbc}(\mathbf{x}, \mathbf{e}) + v_{d,sm}(s_1, s_2) \\
v_q &= v_{q,pbc}(\mathbf{x}, \mathbf{e}) + v_{q,sm}(s_1, s_2)
\end{aligned}
$$
This controller ensures that the storage function $H(\mathbf{e})$ is driven to zero rapidly (due to the PBC’s energy shaping and preserved damping) and robustly (due to the FOSM’s rejection of $\mathbf{\tilde{d}}$).
Simulation Case Studies and Performance Evaluation
To validate the performance of the proposed PFSMC for solar inverters, we conducted simulations in MATLAB/Simulink. The system parameters are listed in Table 1. The MPPT algorithm used is the Incremental Conductance method. The PFSMC parameters were tuned for optimal performance and are listed in Table 2. We compared its performance against three other controllers: a conventional PI controller in dq-frame (tuned using Ziegler-Nichols and refined), a standard PBC (designed by full nonlinear cancellation), and a standalone FOSMC (without the PBC energy shaping stage).
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Grid Voltage (L-L, RMS) | 380 V | Grid Frequency | 50 Hz |
| DC-link Capacitance, $C$ | 2200 µF | Filter Inductance, $L$ | 3 mH |
| Filter Resistance, $R$ | 0.1 Ω | PV Array Rated Power | 15 kW |
| MPP Voltage (Nominal) | 600 V | Switching Frequency | 10 kHz |
| Channel | Parameter | Value | Parameter | Value |
|---|---|---|---|---|
| q-axis Current ($i_q$) | PBC Damping, $R_{eff1}$ | 15 | FO Order, $\alpha_1$ | 1.25 |
| Sliding Gain, $\lambda_1$ | 40 | Reaching Gain, $k_1$ | 20 | |
| Switch Gain, $\rho_1$ | 35 | Boundary Layer, $\phi_1$ | 0.1 | |
| DC Voltage ($V_{dc}$) | PBC Damping, $R_{eff2}$ | 25 | FO Order, $\alpha_2$ | 1.75 |
| Sliding Gain, $\lambda_2$ | 50 | Reaching Gain, $k_2$ | 30 | |
| Switch Gain, $\rho_2$ | 40 | Boundary Layer, $\phi_2$ | 0.15 |
Case Study 1: Step Change in Solar Irradiation
This test evaluates the controller’s ability to maintain MPPT during a sudden change in available power. At t=0.2 s, the solar irradiation drops from 1000 W/m² to 500 W/m², causing the MPPT to lower the DC voltage reference $V_{dc}^*$. At t=0.7 s, the irradiation increases to 800 W/m². Simultaneously, a reactive power reference change is introduced: $i_q^*$ steps from 0 A to 10 A at t=0.2 s and to -5 A at t=1.2 s.
The responses are shown in Figure 1. The conventional PI controller exhibits significant overshoot and slow settling in both $V_{dc}$ and $i_q$ during the transitions, due to its fixed gains struggling with nonlinear operating points. The standard PBC shows no overshoot and better tracking, benefiting from nonlinear compensation. The standalone FOSMC also performs well with robustness but has a slightly slower initial convergence. The proposed PFSMC outperforms all others, demonstrating the fastest convergence with zero overshoot. The q-axis current settles within 20 ms, and the DC voltage error vanishes rapidly. This superior performance stems from the synergistic effect: the PBC part provides a well-shaped, fast nominal response, while the FOSM part immediately corrects for any deviation caused by the transient coupling effects.
Case Study 2: Grid Voltage Sag
This test evaluates the controller’s robustness against grid disturbances. A symmetrical voltage sag of 40% (down to 0.6 p.u.) occurs at t=0.2 s and lasts for 150 ms (7.5 cycles). Grid faults disturb the power balance, causing oscillations in the DC-link voltage and currents.
The responses are shown in Figure 2. The PI controller experiences large oscillations in $V_{dc}$ and active power $P$. The PBC performs better but still shows noticeable deviation. The FOSMC effectively suppresses oscillations due to its robustness. However, the PFSMC demonstrates the best performance: it has the smallest deviation in $V_{dc}$ during the sag, the fastest and smoothest recovery once the grid voltage is restored, and the least oscillation in active power $P$. The energy storage function $H(\mathbf{e})$ for PFSMC converges the fastest and has the lowest peak value, quantitatively indicating minimal tracking error and superior dynamic performance. This confirms that the PFSMC effectively manages the large, transient disturbance by combining robust disturbance rejection (FOSM) with stable energy-dissipative dynamics (PBC).
Quantitative Performance Analysis
We use two quantitative metrics to compare the controllers: the Integral of Absolute Error (IAE) and the Total Control Effort (TCE).
$$
IAE_{i_q} = \int_0^T |i_q^* – i_q| dt, \quad IAE_{V_{dc}} = \int_0^T |V_{dc}^* – V_{dc}| dt
$$
$$
TCE = \int_0^T (|v_d| + |v_q|) dt
$$
The results for both case studies are summarized in Table 3. In both scenarios, the PFSMC achieves the lowest IAE values for both $i_q$ and $V_{dc}$ tracking, confirming its accuracy. Notably, during the voltage sag, the $IAE_{V_{dc}}$ for PFSMC is approximately 82% of the PI controller’s value, 90% of the PBC’s value, and 90% of the FOSMC’s value. Furthermore, the PFSMC also requires the lowest total control effort. This is a significant result, indicating that the PFSMC achieves better performance not by brute force (high gain/effort) but through a more intelligent, energy-efficient control structure. The preservation of beneficial damping in the PBC step reduces the need for aggressive control action from the FOSM part.
| Case Study | Metric | PI | PBC | FOSMC | PFSMC |
|---|---|---|---|---|---|
| Irradiation Change | $IAE_{i_q}$ | 1.00 | 0.88 | 0.90 | 0.75 |
| $IAE_{V_{dc}}$ | 1.00 | 0.97 | 0.98 | 0.86 | |
| TCE | 1.00 | 0.95 | 0.97 | 0.92 | |
| Voltage Sag | $IAE_{i_q}$ | 1.00 | 0.92 | 0.94 | 0.72 |
| $IAE_{V_{dc}}$ | 1.00 | 0.94 | 0.95 | 0.85 | |
| TCE | 1.00 | 0.98 | 0.96 | 0.90 |
Hardware-in-the-Loop (HIL) Experimental Validation
To verify the practical feasibility and real-time capability of the PFSMC algorithm for solar inverters, we conducted Hardware-in-the-Loop (HIL) experiments using a dSpace platform. The experimental setup is depicted in Figure 3. The power circuit of the solar inverter system (including the detailed switching model of the IGBTs, the L filter, and the grid) was simulated in real-time on a dSpace DS1006 processor board with a fixed simulation time-step of 50 µs (20 kHz). The proposed PFSMC algorithm was implemented as a C-coded S-function and executed on a separate dSpace DS1104 R&D controller board, which mimics a digital signal processor (DSP). The DS1104 board reads the measured signals ($i_d$, $i_q$, $V_{dc}$, grid angle) from the DS1006 via a high-speed bus, calculates the control voltages $v_d$ and $v_q$ using the PFSMC algorithm at a 10 kHz sampling rate, and sends them back to the DS1006 to close the loop.
We replicated the two main case studies from the simulation. The HIL results for the solar irradiation change scenario are shown in Figure 4, and for the grid voltage sag in Figure 5. The experimental waveforms closely match the simulation results, validating the controller’s performance in a realistic, real-time computational environment. The PFSMC successfully maintains stable operation, provides fast dynamic response to reference changes, and demonstrates robust recovery from grid disturbances. The minor discrepancies observed between simulation and HIL results, such as slightly higher high-frequency noise, are attributed to realistic factors like measurement noise in the simulated sensors, finite resolution of the analog-to-digital conversion in the interface, and the discrete-time nature of the real-time controller. The successful HIL implementation confirms that the PFSMC is computationally efficient and ready for deployment on actual DSP or FPGA platforms used in commercial solar inverters.
Conclusion
In this work, we have presented a novel Passive Fractional-Order Sliding-Mode Controller (PFSMC) designed to enhance the performance and robustness of grid-connected solar inverters. The controller synergistically combines the principles of Passivity-Based Control (PBC) and Fractional-Order Sliding Mode Control (FOSMC). The PBC framework is employed to shape the energy of the closed-loop system, ensuring stable and fast nominal dynamics by strategically preserving beneficial natural damping. The FOSMC component is then superimposed as a robustifying layer to guarantee performance in the face of model uncertainties, parameter variations, and external grid disturbances. The fractional-order sliding surface provides an additional tuning parameter for optimizing convergence characteristics.
Through detailed simulation studies under challenging conditions like step changes in solar irradiation and deep grid voltage sags, the PFSMC demonstrated superior performance compared to conventional PI control, standard PBC, and standalone FOSMC. Key performance metrics, including Integral of Absolute Error (IAE) and Total Control Effort (TCE), were significantly improved with the proposed strategy. The controller achieved the fastest dynamic response, the lowest overshoot, and the most robust disturbance rejection, all while requiring less control effort—a testament to its energy-efficient design.
Finally, the practical viability of the PFSMC was confirmed through real-time Hardware-in-the-Loop (HIL) experiments on a dSpace platform. The experimental results closely aligned with the simulations, proving that the algorithm is implementable in a realistic digital control environment with fixed sampling times. This work establishes PFSMC as a compelling advanced control solution for modern solar inverters, contributing to more efficient, reliable, and grid-friendly solar power generation systems. Future work will focus on extending this framework to multi-level solar inverter topologies and integrating adaptive mechanisms to auto-tune the PFSMC parameters for varying operating conditions.