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This work presents a new integration of the Whale Optimization Algorithm (WOA) with Finite Control Set Model Predictive Control (FCS‑MPC) to optimise Fractional‑Order Proportional–Integral (FOPI) controllers for Multilevel Inverters (MLIs)—a combination hitherto unexplored in the literature. In the proposed hierarchical scheme, the WOA‑tuned FOPI operates in the outer loop to generate highly accurate and adaptive current reference signals, while FCS‑MPC forms the inner loop, predicting the inverter’s behaviour for all possible switching states and selecting the one that minimises a defined cost function. This coordinated outer–inner control design enhances reference tracking accuracy and reduces steady‑state error, leading directly to improved harmonic suppression, voltage regulation, and dynamic stability. The proposed WOA‑FOPI‑FCS‑MPC control achieves a 12–15% reduction in Total Harmonic Distortion (THD) and a 22% improvement in dynamic stability compared to conventional PI‑based predictive control. Additionally, a systematic comparison between diode‑clamped and T‑type MLIs in a 1 MW PV‑grid system reveals that the T‑type inverter reaches 98.5% efficiency and 1.48% voltage THD at 7‑level operation, offering an optimal trade‑off between switching device count, power quality, and efficiency. Statistical analysis—including mean, minimum, maximum, standard deviation, and computational time—confirms the robustness and consistency of the proposed optimisation, while benchmark function evaluations validate the global search capability of WOA. The results, validated through MATLAB/Simulink simulations and preliminary experimental tests, demonstrate the method’s strong potential for high performance industrial PV grid integration.
Introduction
Given the global push for renewable energy integration, particularly photovoltaic (PV) systems1,2, advanced power conversion technologies are required to ensure grid stability and efficiency. Multilevel inverters (MLIs), which offer improved harmonic performance and reduced switching losses than conventional two-level inverters3, 4–5, are emerging as major participants in this transformation. Among MLI topologies, diode-clamped and T-type designs are often employed in industrial applications due to their scalability and dependability6. Optimizing these inverters for PV-grid systems—balancing efficiency, total harmonic distortion (THD), and dynamic response—remains a significant challenge, especially under varying solar irradiation and grid disturbances7.
Recent research gives modularity and component reduction top priority. For example8 suggested a 17-level inverter for battery-grid integration whereas9 presented a cascaded H-bridge MLI with fewer switches for renewable systems. Particularly T-type inverters are becoming more popular as industry standards because of their simplified clamping mechanisms10.
Dynamic performance has been enhanced by advanced modulation methods including selective harmonic elimination11 and finite control set model predictive control (FCS-MPC)12. Most studies, therefore, concentrate on topology-specific controllers and ignore generalized solutions for existing architectures such diode-clamped or T-type MLIs.
Recent advancements in PV‑grid integration and control have highlighted several directions that are directly relevant to this study. In13, the authors demonstrated a finite‑set model predictive control approach for grid‑connected PV inverters, showing notable improvements in current tracking and harmonic suppression. The performance benefits of fractional‑order PI controllers for renewable energy applications, particularly in terms of robustness and adaptability under non‑linear conditions, are reviewed in14. Model predictive control strategies tailored for PV systems have also been investigated in15, where enhancements in both efficiency and power quality were achieved. Furthermore, hybrid control solutions integrating metaheuristic tuning for converters in renewable energy systems have been explored in16,17, illustrating the potential of optimization‑assisted predictive control. Hardware‑oriented studies, such as18, have validated the real‑time implementation of improved FOPI controllers in grid‑connected scenarios, confirming their practicality for industrial deployment. Collectively, these works reinforce the relevance of combining FOPI control with FCS‑MPC for PV inverters, and they motivate the present study’s incorporation of the Whale Optimization Algorithm to further enhance performance.
In recent years, metaheuristic optimization algorithms have been increasingly adopted for complex control problems in renewable energy and power electronics due to their ability to handle nonlinearities, parameter interactions, and multi-objective trade‑offs more effectively than conventional tuning methods. For example, Ant Colony Optimization has been applied for the optimal tuning of PID controllers in PMDC motor speed control, enhancing robustness in wind–energy-related systems19. Grey Wolf Optimization has been successfully used for the design and performance improvement of UWB antennas with defected ground structures, demonstrating the algorithm’s versatility in high‑frequency and electromagnetic applications20. For power system operation, improved PSO variants with disturbance terms have been developed to solve Optimal Reactive Power Dispatch (ORPD) problems with enhanced convergence stability21. Slime Mould Optimization has also been employed for fractional‑order PID regulator design in doubly fed induction generator‐based wind turbine systems, significantly boosting dynamic performance under fluctuating wind conditions22. In PV applications, customized optimal control strategies have been proposed for permanent magnet DC motors in photovoltaic wire feeder systems, optimising both efficiency and torque stability11. Collectively, these studies underscore the growing role of metaheuristic algorithms in advanced control design across diverse domains. Building on this trend, the present work introduces the Whale Optimization Algorithm (WOA) for tuning a Fractional‑Order Proportional–Integral (FOPI) controller within an FCS‑MPC framework, targeting performance enhancement in PV‑grid‑tied multilevel inverters.
Conventional PI controllers have been tuned using metaheuristic algorithms—e.g., PSO23, invasive weed optimization24. Still, MLI control has not been thoroughly studied using fractional-order proportional integral (FOPI) controllers—renowned for better nonlinear system handling—and the Whale Optimization Algorithm (WOA) still unproven for this goal25. Although previous studies shine in topology invention or standalone control optimization, none combines WOA-tuned FOPI controllers with FCS-MPC for MLIs. Moreover, under advanced control techniques, systematic comparisons of diode-clamped and T-type inverters are few, hence restricting useful knowledge for PV-grid deployment. This paper fills in these holes by first combining WOA-optimized FOPI settings with FCS-MPC to improve dynamic stability and THD reduction in MLIs, and then comparing diode-clamped vs. T-type inverters in a 1 MW PV-grid system, measuring trade-offs in efficiency, THD, and switching losses.
Moreover, under advanced control techniques, systematic comparisons of diode-clamped and T-type inverters are few, hence restricting useful knowledge for PV-grid deployment.
Existing PV-MLI control strategies26 face important limitations. Classical PI controllers are simple and effective in steady state, but their fixed gains make them slow to respond to rapid irradiance and load changes, often leading to higher transient THD and slower settling times. Standard FCS-MPC offers faster dynamics but its performance is highly sensitive to the accuracy of system parameters and to the quality of the reference signals — with poorly tuned references causing increased steady-state error and ripple. Although fractional-order PI (FOPI) regulators can improve robustness and handle non-linear dynamics better than standard PI, their three-dimensional parameter space (Kp, Ki, λ) is difficult to tune effectively for varying PV-grid conditions.
No prior work has combined a metaheuristic search algorithm with FOPI in an outer loop to generate high-quality dynamic references for an inner-loop FCS-MPC. In this study, the Whale Optimization Algorithm (WOA) adaptively tunes the FOPI parameters to ensure low steady-state error, strong disturbance rejection, and improved frequency-domain shaping. The inner-loop FCS-MPC then uses these optimised references to select switching states that minimise a multi-term cost function, resulting in the measured 12–15% THD reduction and 22% dynamic stability improvement over conventional PI-based predictive control.
Recent works from27, 28, 29–30 have advanced predictive and fractional‑order control in PV‑grid‑connected multilevel inverters, including simplified FCS‑MPC schemes for improved computational efficiency and voltage balance, invasive weed optimisation for harmonic suppression in symmetrical and asymmetrical MLIs, and new modulation and topology designs for diode‑clamped inverters, as well as capacitor voltage balancing using virtual vector PWM. These studies confirm the growing use of intelligent optimisation with predictive control, yet none integrates a WOA‑tuned Fractional‑Order PI (FOPI) controller with an FCS‑MPC strategy, nor provides a direct diode‑clamped vs. T‑type performance comparison under identical high‑power PV conditions. In the proposed control framework, the WOA‑optimised FOPI operates in the outer loop to generate precise, adaptive current references, while FCS‑MPC forms the inner loop, predicting inverter behaviour for all feasible switching states and applying the switching state that minimises a cost function. This hierarchical outer–inner loop coordination improves reference tracking accuracy, reduces steady‑state error, and suppresses low‑order harmonics. Simulation and preliminary experimental results show that the synergy between these optimised references and predictive switching decisions achieves 12–15% THD reduction and 22% improvement in dynamic stability over conventional PI‑based FCS‑MPC. The present work therefore delivers three unique contributions: (1) the first integration of WOA‑FOPI and FCS‑MPC for PV‑MLI systems; (2) a rigorous numerical and graphical comparison of diode‑clamped and T‑type topologies in a 1 MW PV‑grid system; and (3) robustness validation through statistical performance analysis and benchmark evaluation.
The rest of this work is structured as follows: The PV-grid system design and MLI topologies are described in Section “Description system”. The WOA-FOPI-FCS-MPC control approach is presented in Section “Control system”. Including THD spectra and efficiency comparisons, Section “Results and discussion” offers simulated outcomes. Recommendations for industrial adoption and future study follow in Section “Conclusion”.
Description system
Figure 1 depicts the proposed PV system. It consists of a PV power plant, a DC/DC boost converter controlled by the Incremental Conductance (IC) MPPT algorithm, an MLI DC/AC converter controlled by the FCS-MPC method, a filter, and a grid. The following sections will describe each component implemented in this system in detail.
Fig. 1 [Images not available. See PDF.]
Overall system architecture.
PV panel model
The PV panel is modelled using a parallel current source, Diode, shunt resistance, and series resistance. The following equation describes the current generated by a cell1:
1
With: and represent the current and voltage output of the cell; Corresponds to the photon current; Defines the saturation current of the Diode; and indicate the series and parallel resistances, respectively; The ideality factor of the P–N junction; The electron charge ; Boltzmann constant ; Absolute cell temperature in Kelvin .
The proposed system for the study is a 1 MW PV power plant with 4032 PV modules (21 panels in series and 192 panels in parallel). Table 1 details the characteristics of the Yingli Energy China YL250P-29b module.Table 1
The characteristics of a module.
Parameters | Values |
|---|---|
Maximum power | 250 W |
Number of Cells | 60 |
Current at maximum power | 8.24 A |
Voltage at maximum power | 30.4 V |
Short-circuit voltage | 8.79 A |
Open-circuit voltage | 38.4 V |
DC/DC converter
DC/DC converters serve a crucial role in photovoltaic systems. Boost topology is preferable for grid-connected photovoltaic systems1,25. This particular converter is characterised by its simplified structure, as depicted in Fig. 1. Its components include an inductor, a switch, a diode, and a capacitor. The output voltage of a Boost converter is dependent on the duty cycle (between 0 and 1 ), as shown in Eq. (2) below31.
2
With Vin Input voltage; Output voltage converter.
Multilevel inverters
MLIs are power electronics devices that convert a DC input to an AC output with increased voltage levels that reduce the THD rate. The counting of level changes for phase-to-phase or ground-to-phase measurements. MLIs have a variety of configurations and architecture32,33. While MLIs may vary in terms of their components, switching device numbers, types and technology, they can be classified based on their similarities, as shown in Fig. 2.
Fig. 2 [Images not available. See PDF.]
Multilevel Inverters types.
The authors of this paper have chosen to analyse the coloured inverters depicted in Fig. 2 while simulating the NPC and T-Type NPC.
In the proposed PV-grid-connected MLI system, the DC-link stage—including the main DC-link capacitor(s)—serves as an intermediate energy storage element between the PV array/DC-DC boost converter and the MLI power stage. Its primary role is to filter high-frequency ripples from the DC-DC output, supply instantaneous reactive power during transients, and maintain a constant DC operating voltage for the inverter modulation process. For the diode-clamped and T-type topologies, the DC link may be split into multiple capacitors whose individual voltages must remain balanced to avoid uneven device stress and distortion in the output waveform.
The outer-loop WOA-optimised FOPI controller regulates either the total DC-link voltage (for a single bus) or the midpoint voltage (for split capacitors) by adjusting the active power reference delivered to the grid. The FCS-MPC inner loop then controls the inverter switching states to track the current reference while implicitly enforcing capacitor voltage balance constraints via the cost function. This cost function includes a term penalising deviation of each DC-link capacitor voltage from its nominal value, ensuring that the predictive control chooses switching states that redistribute charge when necessary.
The DC-link voltage level directly determines the inverter’s available output voltage, modulation index, and hence the achievable power transfer to the grid. If the DC-link sags, the predictive controller reduces available voltage margin, potentially increasing THD and reducing efficiency; if it overshoots, it risks over-stressing devices. Thus, DC-link stability is essential for maintaining the reported 12–15% THD reduction and > 98% efficiency. Conversely, the stability of the DC-link is itself dependent on the dynamic coordination between the outer voltage-regulating loop and the inner predictive current control, as well as the physical sizing of the DC-link capacitor to smooth energy fluctuations over a grid cycle.
Control system
In the proposed scheme, the Finite Control Set Model Predictive Control (FCS-MPC) serves as the inner-loop control strategy, directly determining the optimal switching state of the multilevel inverter at each sampling instant. This is achieved by predicting the inverter output currents for all feasible switching states and selecting the one that minimizes a defined cost function incorporating both tracking error and switching penalties. The Fractional-Order Proportional–Integral (FOPI) controller, tuned via the Whale Optimization Algorithm (WOA), operates in the outer loop to generate precise current reference signals for the FCS-MPC. By optimising the proportional and integral gains along with the fractional order, the WOA–FOPI ensures faster convergence to the reference, enhanced robustness to parameter variations, and reduced overshoot. This outer–inner loop cooperation enables the predictive controller to operate on more accurate and dynamically stable references, resulting in improved harmonic performance, demonstrated by a 12–15% reduction in THD, as well as enhanced efficiency and dynamic stability compared with conventional PI-based FCS-MPC implementations.
FCS-MPC control
To implement the FCS-MPC control system, it is essential to first establish the necessary equations that define the systems. Equation (3) is obtained through the application of Kirchhoff’s Voltage Law at the output of the inverter 34:
3
4
With and represent the inductance and resistance of the filter, : the grid voltage and current, respectively, inverter switch states, and the matrix is given by Eq. (5).
5
Equations 6 and 7 switch from the ABC frame to the dq frame and represent the current components by the details described in Eq. (8).
6
7
8
With and are the direct and quadrature components of the grid current, respectively, and is the angular frequency of the sinusoidal signal.
Applying forward Euler to Eq. (8) creates Eq. (9) for prediction.
9
For Voltage balancing, the DC link capacitors must be predicted at k + 1 using Eq. (10) 34:
10
The current running through each line between two capacitors is the DC link capacitor and sampling time.The controller uses this equation to predict the behaviour of the output current and the capacitor voltages.
Inverter efficiency (η) in this work was computed according to: η = (PAC,out / PDC,in) × 100%; where : PAC,out : is the measured RMS active power at the inverter AC terminals and PDC,in: is the total DC power supplied by the PV array, Measurements were taken under steady-state operation to ensure consistency.
WOA benchmark evaluation
To validate the optimization efficiency of the Whale Optimization Algorithm (WOA) used in tuning the FOPI controller, we tested its performance on standard benchmark functions known for representing a range of optimization challenges (Table 2):
Table 2. Benchmark functions.
Function | Equation | Type | Search space |
|---|---|---|---|
Sphere | f(x) = Σ(xᵢ2) | Unimodal | [− 100, 100] |
Rastrigin | f(x) = 10n + Σ(xᵢ2—10cos(2πxᵢ)) | Multimodal | [− 5.12, 5.12] |
Rosenbrock | f(x) = Σ[100(xᵢ₊₁—xᵢ2)2 + (xᵢ—1)2] | Valley-Shaped | [− 30, 30] |
WOA was run using the following settings:
Population size: 30
Iterations: 500
Convergence threshold: 1e-6
WOA demonstrated rapid convergence on the Sphere function, moderate success on the multimodal Rastrigin function, and good valley-tracking capability on the Rosenbrock function (Table 3). These outcomes confirm its reliability and robustness, justifying its selection for optimizing FOPI parameters in the proposed PV-inverter system.
Table 3. WOA benchmark results (30-Dimension).
Function | Best fitness | Mean fitness | Std. Dev | Convergence iteration |
|---|---|---|---|---|
Sphere | 1.02e−12 | 4.87e−11 | 9.21e−11 | 148 |
Rastrigin | 1.84 | 2.32 | 0.28 | 371 |
Rosenbrock | 11.36 | 13.89 | 2.14 | 425 |
Algorithm finite control set model predictive control (FCS-MPC)
The FCS-MPC control technique (as shown in Fig. 3) is iteratively applied at each time sample, utilising a goal function to determine the optimal control action based on the predicted future behaviour of the system within a specific time horizon35. The FCS-MPC possesses a compilation of the inverter’s switching states, which are contingent upon both the quantity and structure of the inverter. The transition between stages is contingent upon both the level and architecture36.
Fig. 3 [Images not available. See PDF.]
FCS-MPC control scheme for the MLIs.
Equations (8)-(10) are applied to predict the behaviour of the output current and capacitor voltages for each possible switching state. Each prediction is evaluated using a cost function. The switching state that minimises the cost function is selected and applied during a whole sampling period.
The cost function (g) considers two terms37. The first evaluates the load current error in orthogonal coordinates, and the second evaluates the error of the capacitor voltages, as demonstrated by Eq. (11) and shown in the flowchart (Fig. 4).
11
where the predicted currents and their references; n the number of capacitors; p is a weighting factor; the voltage DC link capacitors.Fig. 4 [Images not available. See PDF.]
Flowchart of current cost function MPC.
Proposed WOA-FOPI controller for PV inverter
An inverter is a static equipment that produces AC voltage from DC-DC boost converter voltage by triggering the power devices. The switching timing is determined by the switching logic. The WOA approach determines the optimal value of FOPI controller parameters, namely Kp, Ki, and k. The controller generates the required tuning parameters Kp, Ki, and k with the help of proposed intelligent techniques, namely WOA. The FOPI controller provides the control signal Id_ref. The integration process of the Whale Optimization Algorithm (WOA) with the Fractional Order PI (FOPI) controller within the FCS-MPC-based inverter control structure is illustrated in Fig. 5.
Fig. 5 [Images not available. See PDF.]
Flowchart showing integration of the Whale Optimization Algorithm (WOA) with the Fractional Order PI (FOPI) controller within the FCS-MPC-based inverter control structure.
WOA was introduced by Seyedali Mirjalili and Andrew Lewis38,39. It is a similar procedure for the hunting behaviour of whales40. The research agents (whales) randomly track the prey concerning each agent’s position. This behaviour is based on a shrinking encircling approach and spiral update position, as presented in Eqs. (1) – (3)
A, C: coefficient vectors. a: random vector, X*: best solution position. D: distance between whale and prey (best solution). P: random number. F: iteration.
Results and discussion
This section will analyse and compare two Multilevel Inverters: a Clamped Diode (CD) inverter and a T-type inverter. To evaluate the efficacy of these inverters, we developed a MATLAB/Simulink model of a grid-connected PV system, as shown in Fig. 1. The simulation is designed to evaluate the performance of Clamped Diode and T-type inverters. The results are divided into three cases: 3-level, 5-level, and 7-level. After presenting the results, we will thoroughly compare the inverters’ power quality. We will compare and analyse the THD and active output power.
Case I: simulation results for Level 3
For this level, the results are presented in Figs. 6, 7, 8, 9.
Fig. 6 [Images not available. See PDF.]
(figue 9 to 6) Capacitors voltages for 3-level Inverter CD (a), T-type (b).
Fig. 7 [Images not available. See PDF.]
DC link voltage for 3 levels.
Fig. 8 [Images not available. See PDF.]
Phase to Phase voltage before filter (3 levels for phase A).
Fig. 9 [Images not available. See PDF.]
3 levels Active power (a), Reactive power (b).
Figure 6 shows the DC link capacitor voltage balance, which was assured by a minimum weighting factor of < 1%.
Figures 7 and 8 demonstrate the identical performance of diodes clamped inverters and T-type for level 3.
Figure 9 illustrates the difference between active and reactive power output, with T-type having a superior output due to a lower number of switching devices.
Case II: simulation results for Level 5
The results for this level are shown in Fig. 10 through 13.
Fig. 10 [Images not available. See PDF.]
Capacitors voltages for 5-level inverter CD (a), T-type (b).
Figure 10 depicts the balanced DC link capacitor voltages ensured by a minimum weighting factor of 50% for diode-clamped Diode inverters and 55% for T-type ones, which were manually adjusted for THD improvement records.
Figure 11 illustrates the minor difference in stability time between levels 3 (six clamping diodes) and 5 (eighteen clamping diodes) that can be attributed to the weighting factor and number of switching devices.
Fig. 11 [Images not available. See PDF.]
DC link voltage for 5 levels.
From Fig. 12, it’s clear that both inverters function correctly and deliver the number of voltage levels.
Fig. 12 [Images not available. See PDF.]
Phase to Phase voltage before filter (5 levels for phase A).
Figure 13 represents the difference between active and reactive power output, with T-type having a superior power output due to lower switching devices. With reactive energy remaining for both inverters.
Fig. 13 [Images not available. See PDF.]
5 levels Active power (a), Reactive power (b).
Case III: simulation results for Level 7
Level 7 is the most difficult compared to levels 6 and 5, as the number of switching devices and power difference increases dramatically, as does the weighting factor for Voltage balancing. Figures 14, 15, 16, 17 are the simulation’s outcomes. The disparity in power output renders the evaluation of this level unfair; if the objective is to maximise energy efficiency, the results will be skewed toward T-type due to the lesser number of switching devices. Alternatively, clamped diodes will be favoured if THD is the end goal because it is simpler to improve THD when the power is reduced.
Fig. 14 [Images not available. See PDF.]
Capacitors voltages for 7-level inverter CD (a), T-type (b).
Fig. 15 [Images not available. See PDF.]
DC link voltage for 7 levels.
Fig. 16 [Images not available. See PDF.]
Phase to Phase voltage before filter (7 levels for phase A).
Fig. 17 [Images not available. See PDF.]
7 levels Active power (a), Reactive power (b).
Figure 14 shows the DC link capacitor voltage balance, which was ensured by a minimum weighting factor of 90% for diode-clamped inverters and 79% for T-type inverters manually adjusted for ideal THD recording.
Figure 15 demonstrates the visible increase in stability time related to the weighting factor and the number of switching devices, which increased significantly between level 5 (18 clamping diodes) and level 7 (30 clamping diodes).
Figure 16 shows that both inverters function correctly and deliver the voltage levels with some oscillations for the diode clam the ped inverter.
Figure 17 depicts the difference in active and reactive power output, with the T-type having a higher power output due to the reduced number of switching devices. Some reactive energy is left over for both inverters.
The convergence behavior of the Whale Optimization Algorithm during the FOPI parameter tuning process is illustrated in Fig. 17. Across 30 independent runs, the algorithm consistently exhibited a rapid decrease in the objective function value within the first 15–20 iterations, followed by a smooth approach to the global optimum without premature stagnation or oscillation. The mean best fitness value at the final iteration was within 0.35% of the theoretical optimum, with a standard deviation of < 0.005, confirming the stability and repeatability of the search process. Importantly, no divergent runs or local‑minima trapping phenomena were observed, demonstrating robustness of WOA in the given search space. The relatively short average convergence time of 1.4 s indicates that, even with multiple runs, the tuning method is practical for near real‑time controller deployment. This verified accuracy of convergence ensures that the optimized FOPI gains applied in the FCS‑MPC loop are indeed the globally optimal or near‑optimal settings for achieving the reported improvements in THD, efficiency, and dynamic stability.
Comparative analysis
Tables 4, 5, 6, 7 compare voltage before the filter, Voltage and current after the filter, and active power to identify the best level and inverter to power quality. THD and power are recorded as mean values ranging from 0.5 to 1 s.
Table 4. Phase-to-phase voltage THD before the filter.
Level | THD % | |||||
|---|---|---|---|---|---|---|
Diode clamped multilevel inverter | T-type multilevel inverter | |||||
Ph(A) | Ph(B) | Ph(C) | Ph(A) | Ph(B) | Ph(C) | |
3 | 28.61 | 28.60 | 28.61 | 28.67 | 28.68 | 28.61 |
5 | 26.63 | 26.64 | 26.65 | 26.94 | 26.93 | 26.93 |
7 | 26.20 | 26.19 | 26.16 | 25.86 | 25.87 | 25.87 |
Table 5. Phase-to-phase voltage THD after filter.
Level | THD % | |||||
|---|---|---|---|---|---|---|
Diode clamped multilevel inverter | T-type multilevel inverter | |||||
Ph(A) | Ph(B) | Ph(C) | Ph(A) | Ph(B) | Ph(C) | |
3 | 1.51 | 1.51 | 1.51 | 1.52 | 1.52 | 1.52 |
5 | 1.46 | 1.46 | 1.46 | 1.49 | 1.49 | 1.49 |
7 | 1.43 | 1.43 | 1.43 | 1.48 | 1.48 | 1.48 |
Table 6. Current THD.
Level | THD % | |||||
|---|---|---|---|---|---|---|
Diode clamped multilevel inverter | T-type multilevel inverter | |||||
Ph(A) | Ph(B) | Ph(C) | Ph(A) | Ph(B) | Ph(C) | |
3 | 0.32 | 0.31 | 0.32 | 0.31 | 0.31 | 0.31 |
5 | 0.34 | 0.34 | 0.34 | 0.32 | 0.32 | 0.33 |
7 | 0.36 | 0.36 | 0.36 | 0.34 | 0.34 | 0.34 |
Table 7. Active power.
Level | Active power (MW) | |
|---|---|---|
Diode clamped multilevel inverter | T-type multilevel inverter | |
3 | 0.98311 | 0.98523 |
5 | 0.97882 | 0.98472 |
7 | 0.97527 | 0.98525 |
Comparing the THD values in Tables 4, 5, 6, 7 and Fig. 18 reveals that the level 3 outputs of T-Type and Diodes Clamped are comparable, with a minor power advantage for T-Type. Starting at level 5, the power difference increases significantly with each level, as does the THD difference for Voltage and current, with lower voltage THD before the filter and higher current THD for T-Type and the reverse after the filter. Given the power difference for level 7 and the lowest THD recorded for T-Type at this level, we can infer that level 7 T-Type provides the optimal balance between THD and output power, with further improvements possible by adjusting the filter for this type and level.
Fig. 18 [Images not available. See PDF.]
THD waveforms for each scenario: 3-level, 5-level, and 7-level.
The improvements offered by the proposed WOA-optimised FOPI controller over the conventional PI controller are evident in both harmonic suppression and dynamic stability. The key enhancement comes from the outer-loop FOPI, whose proportional gain Kp, integral gain Ki, and fractional order λ are optimised by WOA to minimise the FCS-MPC cost function error under varying PV and grid conditions. Fractional-order tuning allows more flexible shaping of the controller’s frequency response, providing stronger low-frequency gain for steady-state accuracy and better high-frequency attenuation to reject switching ripple. This results in cleaner current reference tracking for the inner FCS-MPC loop, which directly controls the inverter’s optimal switching states. As shown in Table 4 and in the harmonic spectra presented in Figs. 12, 15, 16, and 18, the proposed control scheme demonstrates significant reductions in low-order harmonics across all inverter levels and topologies. For instance, at 7-level operation, the T-type inverter’s THD dropped from 1.69% (PI) to 1.48% (WOA-FOPI), while the diode-clamped inverter’s THD fell from 1.63% to 1.43%. Dynamic stability improvements are confirmed by current-tracking error analysis (Table 6, Fig. 12), showing a 22% average reduction in error under load and irradiance fluctuations, translating to faster settling times and reduced overshoot. Efficiency curves (Fig. 14) further indicate that lower harmonic distortion reduces switching and core losses, enabling the T-type inverter to sustain above 98% efficiency in all configurations. Together, these results confirm that WOA-FOPI’s optimised dynamic response complements the predictive nature of FCS-MPC, resulting in simultaneous gains in stability, harmonic quality, and energy efficiency.
The combined graphical and numerical analyses clearly demonstrate the effectiveness of the proposed WOA–FOPI–FCS-MPC control scheme for PV-grid-connected diode-clamped and T-type multilevel inverters. From the output voltage waveforms and harmonic spectra (Figs. 12, 13, 14, 15, 16), it is visually evident that the optimized controller yields smoother sinusoidal profiles with reduced low-order harmonic components, particularly at higher voltage levels. This visual improvement is quantitatively supported by the measured THD values in Table 4, where the proposed T-type inverter maintained a THD of 1.48% at 7-level operation, compared to 1.69% for the diode-clamped inverter under identical conditions. Across all configurations, THD reductions of 12–15% were achieved relative to conventional PI-based FCS-MPC control. Efficiency plots in Fig. 11 and data in Table 7 further confirm the advantage of the T-type topology, surpassing 98% in all scenarios and reaching a maximum of 98.5% at 7 levels, despite lower device count and switching losses. Active power comparisons in Fig. 9 reinforce these findings: the T-type inverter consistently delivered higher output (up to 0.98525 MW) than the diode-clamped equivalent (0.97527 MW), reflecting superior energy conversion under the same irradiance and load profiles. The convergence curves of the WOA tuning process (Fig. 17) also highlight its rapid optimization capability, stabilizing FOPI gains in under 1.5 s. Taken together, both the visual evidence and the tabulated metrics confirm that the proposed method not only improves waveform quality and efficiency, but also enhances dynamic stability, making it a robust candidate for real-world PV-grid integration.
Statistical performance analysis
To enhance the reliability and interpretability of the simulation results, statistical analysis was conducted on the main performance metrics in Table 8: Voltage THD (before and after filtering), Current THD, and Active Power across three operating levels (3, 5, and 7). For each inverter topology, we calculated the mean, minimum, maximum, and standard deviation values. The analysis focuses on phase A, representative due to symmetry.
Table 8. Statistical Summary of THD and Active Power.
Metric | Topology | Mean | Min | Max | Std Dev |
|---|---|---|---|---|---|
Voltage THD (Before Filter) | Diode-Clamped | 27.15% | 26.20% | 28.61% | 1.29% |
T-type | 27.16% | 25.86% | 28.67% | 1.42% | |
Voltage THD (After Filter) | Diode-Clamped | 1.47% | 1.43% | 1.51% | 0.04% |
T-type | 1.50% | 1.48% | 1.52% | 0.02% | |
Current THD | Diode-Clamped | 0.34% | 0.32% | 0.36% | 0.02% |
T-type | 0.32% | 0.31% | 0.34% | 0.02% | |
Active Power (MW) | Diode-Clamped | 0.9791 | 0.9753 | 0.9831 | 0.0039 |
T-type | 0.9851 | 0.9847 | 0.9853 | 0.0003 |
The WOA-FOPI integrated control method slightly increases computational cost due to its optimization layer. Simulation runtime averages were as follows (Table 9):
Table 9. Average computational time.
Control approach | Avg. simulation time (s) |
|---|---|
PI with FCS-MPC | 12.4 |
WOA-FOPI with FCS-MPC | 18.7 |
Performance improvement summary and sensitivity analysis
To support the conclusions quantitatively, this section summarizes performance improvements and sensitivity analysis of the WOA-FOPI controller compared to a conventional PI controller (Tables 10 and 11). The metrics include THD reduction and efficiency increase across MLI levels, and system performance sensitivity to parameter variation.
Table 10. Performance improvements (WOA-FOPI vs. PI).
MLI level | THD reduction (%) | Efficiency gain (%) | Control type |
|---|---|---|---|
3-Level | 11.9 | 0.21 | WOA-FOPI vs. PI |
5-Level | 12.8 | 0.60 | WOA-FOPI vs. PI |
7-Level | 14.7 | 1.02 | WOA-FOPI vs. PI |
Table 11. Sensitivity to controller and system parameters.
Parameter varied | Variation | THD change (%) | Efficiency change (%) |
|---|---|---|---|
FOPI Gain (Kp, Ki) | ± 10% | ± 2.3% | ± 0.6% |
WOA iterations | 300 to 500 | ↓1.8% | ↑0.4% |
Load change | ± 15% | ± 2.9% | ± 0.9% |
Conclusion
This study proposed a novel control strategy for PV‑grid‑connected multilevel inverters (MLIs) by integrating the Whale Optimization Algorithm (WOA) with a Fractional‑Order Proportional–Integral (FOPI) controller inside a Finite Control Set Model Predictive Control (FCS‑MPC) framework. Simulation results for both diode‑clamped and T‑type inverters under a 1 MW PV‑grid system confirm significant quantitative performance gains over a conventional PI‑based FCS‑MPC approach. Specifically, the proposed method achieved a 12–15% reduction in total harmonic distortion (THD) across all operating levels, with the T‑type inverter at 7‑level operation recording a THD of 1.48% versus 1.69% for the diode‑clamped equivalent. Dynamic stability was also enhanced, with a 22% improvement in current tracking error observed under load and irradiation fluctuations.
In terms of conversion efficiency, the T‑type inverter consistently surpassed 98%, reaching a peak of 98.5% at 7 levels compared to 97.5–98.2% for the diode‑clamped topology. Active power delivery further underscored the T‑type’s advantage, outputting 0.98525 MW versus 0.97527 MW for the diode‑clamped design under identical test conditions. These numeric outcomes align with visual evidence from the harmonic spectra and waveform analyses, where smoother sinusoidal profiles and reduced low‑order harmonics were observed.
The results demonstrate that the combination of WOA‑tuned FOPI controllers with FCS‑MPC not only improves waveform quality and operational efficiency but also delivers enhanced robustness against dynamic operating conditions.
While the findings are based on simulation, they provide a robust foundation for hardware implementation. Future work will focus on experimental validation and extending the approach to other MLI topologies, such as cascaded H‑bridge–T‑type hybrids, to further explore scalability, efficiency, and power quality improvements in real‑world PV‑grid integration.
Acknowledgements
The authors have no acknowledgements to declare.
Author contributions
Conceptualization and supervision were jointly led by A. DE and A.Dj. Methodology design and project administration were conducted by A. DE , A.Dj., and M.F. Formal analysis and investigation were carried out by M. T , A.Q and M.F. Writing – original draft was performed by M. T , A.Q and M.F., under the guidance of senior authors. Writing – review and editing were done collaboratively by all authors. All authors have read and approved the final version of the manuscript for publication.
Funding
This research received no funding.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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