A Novel Red-Billed Blue Magpie Optimizer Tuned Adaptive Fractional-Order for Hybrid PV-TEG Systems Green Energy Harvesting-Based MPPT Algorithms

Abstract

Hybrid PV-TEG systems can harvest both solar electrical and thermoelectric power, but their operating point drifts with irradiance, temperature gradients, partial shading, and load changes—often yielding multi-peak P-V characteristics. Conventional MPPT (e.g., P&O) and fixed-structure integer-order PID struggle to remain fast, stable, and globally optimal in these conditions. To address fast, robust tracking in these conditions, we propose an adaptive fractional-order PID (FOPID) MPPT whose parameters (K_p, K_i, K_d, λ, μ) are auto-tuned by the red-billed blue magpie optimizer (RBBMO). RBBMO is used offline to set the controller’s search ranges and weighting; the adaptive law then refines the gains online from the measured ΔV, ΔI slope error to maximize the hybrid PV-TEG output. The method is validated in MATLAB R2024b/Simulink 2024b, on a boost-converter–interfaced PV-TEG using five testbeds: (i) start-up/search, (ii) stepwise irradiance, (iii) partial shading with multiple local peaks, (iv) load steps, and (v) field-measured irradiance/temperature from Shanxi Province for spring/summer/autumn/winter. Compared with AOS, PSO, MFO, SSA, GHO, RSA, AOA, and P&O, the proposed tracker is consistently the fastest and most energy-efficient: 0.06 s to reach 95% MPP and 0.12 s settling at start-up with 1950 W·s harvested (vs. 1910 W·s AOS, 1880 W·s PSO, 200 W·s P&O). Under stepwise irradiance, it delivers 0.95–0.98 kJ at t = 1 s and under partial shading, 1.95–2.00 kJ, both with ±1% steady ripple. Daily field energies reach 0.88 × 10⁻³, 2.95 × 10⁻³, 2.90 × 10⁻³, 1.55 × 10⁻³ kWh in spring–winter, outperforming the best baselines by 3–10% and P&O by 20–30%. Robustness tests show only 2.74% power derating across 0–40 °C and low variability (Δv^max typically ≤ 1–1.5%), confirming rapid, low-ripple tracking with superior energy yield. Finally, the RBBMO-tuned adaptive FOPID offers a superior efficiency–stability trade-off and robust GMPP tracking across all five cases, with modest computational overhead.

1. Introduction

Many countries have pledged specific carbon neutrality objectives, demonstrating the widespread agreement that global action to tackle climate change is urgently needed [1,2]. Achieving net-zero emissions by 2050 is crucial for speeding up the shift to sustainable energy and for changing industrial, energy, and economic systems. In light of this, the renewable energy industry has to prioritize low-carbon development plans, advanced technology, and effective resource use [3]. Photovoltaic (PV) systems are widely used because of their cost-effectiveness, zero-emission operation, and abundance of energy resources.

However, there are also significant drawbacks to PV systems, including their low solar energy density and susceptibility to climatic, locational, and seasonal factors. A sizable part of the PV solar power that is engrossed is wasted as waste heat during the photoelectric conversion process. Only 15% to 18% of irradiance is normally converted into electrical power by conventional silicon PV cells [4,5]. This heat buildup lowers PV systems’ energy conversion efficiency and module longevity; a temperature increase of 20 °C may result in a 15% decrease in power production [6]. The surface temperature of PV modules can rise beyond 66 °C during the sweltering summer months [7]. Consequently, recovering and using this surplus heat offers a viable way to improve energy efficiency while shielding PV modules from thermal deterioration [8].

For the above purpose, this work investigates the integration of PV systems with thermoelectric generator (TEG) modules [9] to efficiently harness waste heat, sometimes known as a hybrid PV-TEG system [10]. TEGs, which rely on the Seebeck effect [11], can transform them into a feasible method for capturing residual heat. Thus far, numerous hybrid PV-TEG designs have been created [12], with total power generated output gains ranging from 3% to 16% over standalone PV panels. These hybrid systems take advantage of the passive cooling effects of TEG components to improve energy generation and PV module operating dependability [13,14]. Under PV panels, TEG modules are mounted in this way [15]. Real-world environmental circumstances, on the other hand, are dynamic and ever-changing [16]. The PV array is often affected by partial shadowing circumstances (PSCs), which are produced by impediments such as trees, buildings, or clouds [17]. The hotspot effect is a phenomenon wherein these shading effects cause localized spikes in PV cell temperatures. In addition to reducing energy conversion efficiency, this extreme heat accumulation may permanently harm the structural integrity of PV modules [18]. Furthermore, with several local maximum power points (LMPPs) [19], such hybrid setups invariably result in non-uniform thermal distribution (NTD) [20] among TEGs, further warping the system’s overall output characteristics. A reliable and precise maximum power point tracking (MPPT) method that can dynamically find the global maximum power point (GMPP) is, therefore, both necessary and difficult, especially in complicated situations with changing temperature and irradiance [21]. The architecture of such a combination system is depicted in Figure 1, emphasizing how the parts are set up to maximize energy output.

Several MPPT techniques, which may be broadly divided into conventional approaches and intelligent algorithms, have been developed for both standalone PV/TEG systems and combined PV-TEG combinations [22]. Traditional methods like incremental conductance (INC) [23] and perturb and observe (P&O) [24] are mainly intended to track a single LMMP. Reduced energy harvesting efficiency results from their frequent inability to handle the complications brought on by several LMPPs. More sophisticated techniques are needed to overcome these constraints. Complex tracking scenarios are a good fit for model-free meta-heuristic algorithms because of their superior ability to simplify solution complexity and improve self-adaptive capabilities. To maximize power extraction for PV and TEG systems, for example, better moth flame optimization [25] and horse herd optimization (HHO) [26] have been specially proposed. The grey wolf optimizer (GWO) and cuckoo search (CS) are combined in reference [27] to improve convergence speed and stability, which in turn improves system performance. The use, benefits, and drawbacks of the existing MPPT techniques for PV panels are thoroughly reviewed in [28]. Additionally, methods for improving TEG outputs based on dynamic surrogate models are presented [29].

However, there are currently very few representative studies available, and research especially focusing on hybrid PV-TEG MPPT control is still somewhat restricted. For the integrated PV-TEG mechanism systems, Reference [30] creates an MPPT approach based on an arithmetic optimization algorithm (AOA), which is verified in the existence of an irregular temperature distribution. Nevertheless, it has a limited capacity for investigation in intricate search areas. In addition, the atomic orbital search (AOS) optimization used in [31] to guarantee steady power extraction performs worse in situations when there are several local optima, and its convergence slows down as the optimization progresses. Notably, only a limited range of benchmark algorithms is used in both AOA and AOS research, which restricts the results’ potential to be applied broadly and persuasively, particularly in a variety of realistic operating settings. In the meanwhile, work in [32] shows improved power output by using the salp swarm algorithm (SSA) to enhance MPPT performance under PSCs. A combined PV-TEG system with a c-RIME optimizer is proposed in [22] for robust MPPT, achieving up to 126.67% higher energy output under complex operating conditions. In [33], a PV-TEG system integrating MPPT with double-axis solar power tracking achieved 98.93% efficiency, boosting output power by 71.73% and PV efficiency by 10.01% compared to fixed systems. However, in highly stochastic conditions, it shows a somewhat delayed stabilization and reduced energy capture efficiency. Research on this issue is still in its infancy overall and needs a more thorough examination. Despite the fact that current metaheuristic techniques have improved MPPT management in hybrid PV-TEG systems, difficulties still exist in obtaining strong adaptability, high stability, and quick convergence under a variety of dynamic environmental circumstances.

On the contrary, fractional order (FO) control (FOC) has been merged with MPPT approaches in TEG units to increase their robustness and performance measures [34,35]. FOC techniques are more flexible and have more adjustable parameters than integer order control (IOC)-based techniques, which is advantageous in control applications [36,37]. In [38], the FO-based integrator (FOI) was introduced with a modification of the INR algorithm in PV applications. The FOI gain is the scaling factor for adjusting the algorithm tracking step, and it has been tuned using the Radial Movement-based Optimizer (RMO). The technique offers an adjustable step tracking of MPP. However, because the FOI only has one adjustable parameter and the integrator term, it has little flexibility. Additionally, the literature has introduced the FO proportional-integrator (FOPI) for MPPT in TEG systems. In comparison to FOI techniques, the FOPI offers improved performance and greater flexibility due to its two distinct terms and three adjustable parameters. In order to determine the ideal FOPI parameters for the approach that is being given, the sine cosine-based algorithm (SCA) was utilized in [39]. The study in [40] proposes an MPA-optimized TFOID MPPT controller that integrates FOPID and TID techniques, achieving superior tracking speed, stability, and maximum power extraction for thermoelectric generator systems. Lastly, by combining the FOPID and TID features, certain improved FOC structures for frequency regulation have been proposed [41]. Nevertheless, more flexible and effective controllers are still required for MPPT in TEG applications [42].

Three important areas need to be investigated further in order to expand on earlier work. Initially, improvements are required to improve power extraction under various PSCs. By reducing the adverse effects of PSCs, more efficient MPPT techniques can increase system dependability and financial sustainability. Second, the majority of MPPT studies that are currently available place a strong emphasis on technical criteria like power output augmentation while paying little consideration to environmental viability and sustainability. A thorough grasp of the long-term potential and environmental advantages of hybrid PV-TEG systems as sustainable energy solutions is hampered by this oversight. Third, the actual viability of existing methods is restricted by their inadequate utilization of real-world data and their limited evaluation of various operating circumstances. To link theoretical models with real-world applications, real-world datasets are crucial.

To address these deficiencies, this work provides an effective energy harvesting approach for hybrid PV-TEG systems under a variety of operational situations, including PSCs. The key contributions are summarized as follows:

• This research develops a PV-TEG hybrid system that synergistically enhances solar energy conversion by recycling waste heat, increasing output density, and extending PV module lifespan. A novel fractional-order controller with an optimized design is integrated to advance MPPT in PV-TEG systems, thereby improving energy harvesting efficiency and system reliability.
• A novel adaptive fractional-order controller (FOPID) based MPPT in PV-TEG hybrid mechanism systems, designed to exploit the advantages of fractional dynamics for faster and more robust tracking under fluctuating conditions. To ensure optimal performance, the controller’s gains are simultaneously tuned using the RBBMO, which efficiently searches the parameter space and overcomes the limitations of traditional element-by-element tuning. This integration enables improved convergence, reduced steady-state oscillations, and enhanced energy harvesting efficiency compared with conventional approaches.
• Five case studies are conducted under varying temperature, irradiance conditions, and partial shading to validate the proposed framework. A comprehensive evaluation approach is introduced, incorporating both technical performance and environmental considerations. Comparative analysis with several optimization algorithms is carried out, and statistical validation confirms the superior efficiency and robustness of the proposed RBBMO-based controller optimization.
• Compared with eight baselines, AOS (Atomic Orbital Search Optimization Algorithm), PSO (Particle Swarm Optimization), MFO (Moth–Flame Optimizer), SSA (Salp swarm algorithm), GHO (Grasshopper Optimization), RSA (Reptile Search Algorithm), AOA (Arithmetic Optimization Algorithm), and P&O, the proposed tracker is consistently the fastest and most energy-efficient: it reaches 95% of the MPP in 0.06 s, settles within 0.12 s (±1%), and harvests 1950 W·s at start-up (vs. 1910 W·s for AOS, 1880 W·s for PSO, and 200 W·s for P&O).

Table 1. General advantages and disadvantages of MPPT/control families used in PV-TEG systems.

Method Family	Typical Advantages	Typical Disadvantages (PV-TEG Context)
P&O [43]	Simple, low-cost; fast on single-peak curves; easy to implement on MCU.	Prone to local-peak lock-in under partial shading/multi-peak; steady-state oscillation; sensitive to step size and noise.
IncCond [15,44]	Better slope awareness than P&O; moderate speed; modest compute.	Still local method; accuracy depends on noisy (dI/dV); tuning trade-off between speed and ripple.
FLC [45]	Nonlinear rule base can handle uncertainties; no exact model required.	Rule design/tuning is expert-heavy; scaling issues; behavior can drift outside trained envelope.
Integer-order PID [40]	Familiar design; decent transient shaping; easy to deploy.	Fixed structure; performance degrades across wide irradiance/temperature/load changes; requires re-tuning; limited GMPP ability.
FOPID (Fractional-order PID) [46]	Extra orders (λ, µ) improve speed–robustness trade-off; can reduce ripple and overshoot.	Realization details (band/order, discretization, anti-windup) are nontrivial; still needs global search help for GMPP.
Meta-heuristics (PSO/SSA/MRFO/AOA/HGSO/ASA, etc.) [1]	Global search capability; flexible objective design; robust to nonconvex (P)–(V).	Often slower convergence; duty oscillation risk without damping; results depend on bounds/seeds; reproducibility varies.
Data-driven (NN/ANFIS/NNCGA, etc.) [47]	Can capture complex, time-varying relations; potential for adaptive/global behavior.	Data-hungry; generalization and stability concerns; training cost; opaque tuning; reproducibility often limited.
Dual-MPPT (separate PV & TEG) [30]	Independent optimization of each source; simple modularity.	Coordination/coupling can cause cross-oscillation; higher component count; harder to guarantee bus stability.
Centralized PV-TEG MPPT [22]	System-level optimum on a common DC bus; avoids source conflicts.	More complex controller; requires accurate sensing and well-defined search/constraints to prevent chatter.
Adaptive FOPID (Our approach)	Combines fast local control (FOPID) with global meta-tuning; good GMPP tracking with low ripple; parameters bounded and reproducible.	Needs careful specification of fractional realization and meta-tuning settings; modest compute overhead vs. fixed PID.

synthesizes the method-family trade-offs relevant to hybrid PV-TEG control. Local trackers (P&O, IncCond, integer-order PID) are attractive for their simplicity and speed but are inherently myopic on the multi-peak P-V surfaces produced by partial shading and hybrid coupling, which leads to local-peak lock-in and ripple. Fuzzy and data-driven controllers can handle nonlinearities without explicit models, yet their behavior depends strongly on rule/training quality and may drift outside the training envelope. Meta-heuristics provide global search capability, improving GMPP capture, but typically converge more slowly and can induce duty oscillations unless bounds, damping, and seeds are well specified. At the architectural level, dual-MPPT simplifies per-source tuning but risks cross-coupled oscillations on the common DC bus, whereas a centralized PV-TEG MPPT targets the system-level optimum at the cost of higher controller complexity. Against this backdrop, the proposed adaptive FOPID combines a fast local FOPID core (fractional orders shaping speed/ripple) with bounded meta-tuning to retain GMPP-level robustness while keeping oscillation low and settings reproducible.

Here is the framework for the rest of the work; the system modeling details are covered in Section 2, while the methodology design and implementation framework are described in Section 3. Section 4 presented and analyzed the case study results. The conclusion and future work are finally compiled in Section 5 for further investigation.

2. PV-TEG Hybrid System Modeling

A hybrid PV-TEG mechanism system boosts the PV solar power utilization by combining PV cells, which convert sunlight into electricity, with TEGs, which exploit the Seebeck effect to convert temperature differences into power. PV modules in this arrangement absorb UV and direct sunlight, and TEGs collect any leftover heat and move it to a heat sink to increase efficiency. Together, they broaden the usable energy spectrum, recover waste heat, and ensure more stable and reliable energy generation under varying conditions, with dual MPPT controllers and booster circuits enabling optimal power extraction [48].

2.1. PV Model

A single diode type and a twin diode model are the two variants available for PV. The dual diode model will yield more accurate results than the single diode model, which produces more accurate I-V curves at low irradiation circumstances, when accounting for computational errors. Figure 2a depicts the twin diode concept for solar cells. Nevertheless, the P-V and I-V curves produce several maxima when these bypass diodes are activated. The shade patterns’ temporal variation is selected to guarantee the occurrence of the three accessible GMP scenarios (GMP at the beginning, GMP at the middle, and GMP at the end). The PV’s comparable circuit model and V-I/P-I curve are shown in Figure 2b,c [49].

In this model, two diodes (d1, d2) indicate the P-N junction. Additionally, an electric current source (Iph) provides the photocurrent; an in-series resistance (R_s) compensates for Joule impact distortions, and a resistor called a shunt (Rsh) depicts the current of leakage. In accordance with the current law of Kirchhoff (KCL), the current equations for a photovoltaic cell predicated on the single-diode paradigm may be stated as Equations (1)–(5) [50].

$$I_{PVcell}=I_{ph}-I_{d1}-I_{d2}-I_{sh},$$

(1)

$$I_{ph}=(I_{sc}+K_i(T-T_{STC})){\displaystyle \frac{G}{G_{STC}}},$$

(2)

$$I_{d1}=I_{sc1}\left(\exp \left[{\displaystyle \frac{V_{PVcell}+R_s\cdot I_{PVcell}}{a_1\cdot V_{th1}}}\right]-1\right), I_{d2}=I_{sc2}\left(\exp \left[{\displaystyle \frac{V_{PVcell}+R_s\cdot I_{PVcell}}{a_2\cdot V_{th2}}}\right]-1\right),$$

(3)

$$I_{sh}={\displaystyle \frac{V_{PVcell}+R_s\cdot I_{PVcell}}{R_P}},$$

(4)

$$I_{sc}=I_{sc-STC}\left({\displaystyle \frac{R_P+R_s}{R_P}}\right)$$

(5)

where V_th1,2 = a_1,2k (T/q), where T is the junction P-N temperature, k is Boltzmann’s constant (1.38 × 10⁻²³(J/K)), and q is the 2 arge (1.602 × 10⁻¹⁹ C). Isc₁ and Isc₂ are the reverse saturation currents of the diodes, and a₁ and a₂ are the ideality factors of the diodes. Equation (6) is obtained by substituting the resistor’s current and the diode current.

$$I_{PVcell}=I_{ph}-I_{sc1}\left(\exp \left[{\displaystyle \frac{V_{PVcell}+I_{PVcell}{R}_s}{N_s{V}_{th1}}}\right]-1\right)-I_{sc2}\left(\exp \left[{\displaystyle \frac{V_{PVcell}+I_{PVcell}{R}_s}{N_s{V}_{th2}}}\right]-1\right){\displaystyle \frac{V_{PVcell}+I_{PVcell}{R}_s}{R_P}}$$

(6)

The power-generation efficiency of a PV module is calculated using the following formula, Equation (7) [51]:

$${\eta }_{PVcell}={\displaystyle \frac{P_{PVcell}}{G{A}_{PVcell}}}$$

(7)

where the generated power is denoted by P_PV, and the exterior area of the visible solar cells is indicated by A_PV.

2.2. TEG Modeling

The Seebeck phenomenon-based TEG system modularly transforms thermal energy into electric power, making the gadget a power-capturing device. In a typical TEG circuit, as seen in Figure 3a, several thermocouples are linked electrically and thermally in parallel across two porcelain surfaces. Figure 3b shows that the electric analog circuit of the TEG modules is represented by the inner resistance that is linked in series with the voltage generator.

Figure 4a depicts the TEG module, which is made up of thermocouples linked in series to produce energy using the Seebeck effect. Figure 4b shows the thermocouples. As seen in Figure 4c, the TEG module may be represented as a voltage source connected in series with an internal resistor. The open-circuit voltage, V_oc, and the internal resistor, R_TEG, both change in response to temperature variations.

The TEG module creates a magnetic pull when the outside temperatures of the ceramic surfaces vary:

$$V_{oc}={\alpha }_{pn}(T_h-T_c)={\alpha }_{pn}\times \Delta T$$

(8)

where α_pn is the Seebeck factor, V_oc is the OCV, and ΔT is the variation between T_h and T_c, the interior temperatures of the hot and cold sides, respectively. The output current (I_Teg) flows to the load when the TEG module is connected to it, and the output voltage (V_Teg) is generated across the load. The following is an expression in mathematics for the Seebeck coefficient as an expression of the ambient temperature [30]:

$${\alpha }_{pn}(T)={\alpha }_0+{\alpha }_1\ln (T/T_0)$$

(9)

where α₀ is the Seebeck factor’s fundamental component, T₀ is the temperature reference, and α₁ is the Seebeck factor’s average rate of change. The panel’s current source may be found using the TEG Thevenin circuits in the way outlined below:

$$I={\displaystyle \frac{R_{oc}}{(R_L+R_{Teg})}}$$

(10)

However, circuit theory may be used as a reference to calculate the TEG module’s (P_Teg) output power.

$$P_{Teg}=I_{Teg}^2\times R_L=V_{oc}^2\times {\displaystyle \frac{R_L}{{(R_L+R_{Teg})}^2}}$$

(11)

where R_Teg stands for internal resistance and R_L for load resistance. According to Equation (11), P_Teg is maximum when R_Teg and R_L are equal. When demand is equal to TEG impedance. This may be computed in this way [22]:

$$P_{GMP}={\displaystyle \frac{V_{oc}^2}{4R_{Teg}}}$$

(12)

The GMP requirement is therefore stated as follows:

$$V_{GMP}=0.5\times V_{oc}, I_{GMP}=0.5\times I_{sc}$$

(13)

where V_GMP and I_GMP are the voltage and current values of the TEG mechanism module within the GMP, respectively, and Isc is the short circuit current (SCC). Consequently, the GMP of the TEG mechanism module is around half that of the OCV or SCC.

To address the power and voltage requirements of real-world demands, TEG systems employ many TEG modules. Therefore, a centralized approach is typically utilized to regulate the pricing cost, as seen in Figure 5. Numerous series-connected TEG modules joined in parallel to create TEG strings comprise a centralized TEG mechanism system. Additionally, a single DC/DC converter manages GMPPT for the TEG mechanism system in this configuration. A DC/DC boost converter is utilized to regulate the MPPT of each TEG unit or string; however, due to its high cost, this method is challenging to apply in actual industrial settings. Figure 5a shows the output characteristics of the central TEG mechanism system when all TEG strings are at the same temperature. In this case, the P-V curve has a single GMP, which is comparable to the output characteristics of the TEG mechanism module. Thus, the output power of the TEG mechanism system (P_Teg,s) is equal to the sum of the power of all TEG mechanism modules:

$$P_{Teg,s}={\displaystyle \sum _{j=1}^N{P}_{Teg}^j}={\displaystyle \sum _{j=1}^N(V_{Teg}^j\times I_{Teg}^j)}$$

(14)

where N is the total number of TEG mechanism units utilized by the system as a whole, and $V_{Teg}^i$ and $I_{Teg}^i$ stand for the voltage and current of the jth TEG unit, respectively.

The P-V curve in this instance has many peaks, including multiple LMPs and one GMP. If, instead of following the GMP, the system’s GMPPT converges to the LMPs, the collecting power falls. The fundamental TEG mechanism system’s output characteristics become unstable when varied temperature circumstances are present, as shown in Figure 5b, which affects the GMP. Therefore, this study’s goal is to create a unique MPPT that will raise a centralized TEG system’s harvesting capacity at different temperatures without raising hardware expenses. The following is an explanation of the TEG modules’ heat flow rates, represented as Q_h [52]:

$$Q_h=N({\alpha }_h{I}_{Teg}{T}_h+K(T_h-T_c)-0.5I_{Teg}^{2}R-0.5\mu (T_h-T_c)I_{Teg})$$

(15)

The electric current flowing through the TEG framework is denoted by I_TEG, and N is the total number of thermocouples, μ is the Thomson factor, while R is an electrical impedance, and K is the thermal conductance. Lastly, a TEG module’s power production efficiency is determined using the formula below:

$${\eta }_{Teg}={\displaystyle \frac{P_{Teg}}{Q_h}}$$

(16)

2.3. Hybrid PV-TEG Modeling

The Flowchart of the lifecycle system of the integrated PV-TEG mechanism is seen in Figure 6. In this PV-TEG combination system, the PV back sheet, which is firmly attached to the TEG, is coated with insulating and extremely thermally conductive silicone resin. The PV-TEG hybrid mechanism is displayed as illustrated in Figure 7. Figure 8 illustrates a common combination PV-TEG module arrangement, where the PV panels and TEG are linked in series or parallel. Since PV cells and TEGs have the fewest power electronic switches, connecting them in series results in the least amount of power loss. As can be observed, a heat sink on one surface represents the cool side of the TEG, while PV panels on the other surface represent the hot side. In order to get greater energy conversion efficiency, this guarantees that the TEG has the appropriate amount of temperature differential. The TEG efficiently increases the solar spectrum’s utilization effectiveness in adding to gathering waste heat and lowering the PV panels’ temperature.

The combined PV-TEG model is depicted diagrammatically in Figure 9. Solar energy transit is considered to be one-dimensional and perpendicular to the surface of the PV panel, with limited energy transmission in other directions. Solar energy is absorbed based on the absorbance and thermal radiation characteristics of the photovoltaic panel after passing through a transparent enclosure with a certain transmission factor. The PV panel subsequently transfers the remaining heat to the hot side of the TEG module while converting a portion of the incoming PV irradiation into electrical power [53].

The heat transferred to the hot side of the TEG in the absence of heat loss is caused by the Seebeck effect, Fourier conduction, and Joule heating; this connection is mathematically stated in Equation (17) [53].

$$Q_h=N_s\left(\alpha I{T}_h+{\displaystyle \frac{T_h-T_c}{R_{Th}}}-{\displaystyle \frac{I^2-R_{EL}}{2}}\right)$$

(17)

where α is the Seebeck factor of TEG, Ns is the total number of material pairs, and I is the current of electricity flowing within the TEG unit. Additionally, the electrical and thermal resistances of the thermoelectric (TE) semiconductor couple are represented by the letters R_EL and R_Th, respectively, and can be calculated using Equations (18) and (19), which are explained in references [54].

$$R_{Th}={\displaystyle \frac{L}{\lambda A_{TE}}}$$

(18)

$$R_{EL}={\displaystyle \frac{\rho L}{A_{TE}}}$$

(19)

The overall temperature at the cooler end of the TEG unit can possibly be calculated using Equations (20) and (21) in the standard. In these formulas, ATE and L stand for the cross-sectional dimension and duration of the TEG device, as well as λ and σ for the thermal and electrical conductors of the substance in question [53].

$$Q_c=N\left(\alpha I{T}_c+{\displaystyle \frac{T_h-T_c}{R_{Th}}}-{\displaystyle \frac{I^2-R_{EL}}{2}}\right)s$$

(20)

$$Q_c={\displaystyle \frac{T_c-T_{heatsink}}{R_c}}$$

(21)

Using Equation (22), where R_c is the entire thermal impedance on the colder side at the substrate’s TEG contact, one can determine how much heat disappears across the heat drain.

$$Q_c={\displaystyle \frac{T_c-T_a}{R_{HS}}}$$

(22)

The letter RHS stands for thermal conductivity, which is influenced by the heat drain’s form, area of surface, and the conduction of air factor. A thermoelectric generator’s energy production is directly prejudiced by the temperature differential between its hot and cold sides. The efficiency of the TEG is determined by dividing the electrical power produced by the TEG by the thermal power Qh accessing the hot region of the device (P_TEG). Since temperature affects a TEG’s performance, the figure of merit ZT is used to assess its efficiency. ZT is often regulated using the Harman method. This method involves exposing the terminals of a thermoelectric circuit to DC for a predefined amount of time, after which the voltage across those terminals is measured [55,56,57].

When current passes through the TEG, the Peltier effect causes heat to be transferred from one side to the other, making the device a thermodynamic cooling device. Energy is simultaneously produced by the Seebeck effect and the temperature differential between the TEG surfaces. The voltage generated by the temp differential is known as the Seebeck voltage (V_S), while the voltage generated through the current passing through the circuit is known as the V_J. As the difference between the Seebeck and Joule voltages, the Harman voltage may be described. The figure of value ZT may be calculated using Equation (23), as the literature explains [58,59,60].

$$ZT={\displaystyle \frac{V_s}{V_j}}$$

(23)

Using Equation (24) the TEG efficiency is calculated [61].

$${\eta }_{Teg}={\displaystyle \frac{\Delta T}{T_h}}\cdot {\displaystyle \frac{\sqrt{1+ZT}-1}{\sqrt{1+ZT}+\frac{T_c}{T_h}}}$$

(24)

where T_h represents the hot side’s temp, T_c represents the cold side’s temp, ΔT represents the temperature variations across the TEG’s heated and cooled sides, and ZT is the temperature-sensitive measurement of effectiveness that shows how effective the TEG is based on material properties and operating conditions.

The PV-TEG Thermal Coupling Model. Solar input G over PV aperture A_pv produces electrical power η_pvGA_pv and waste heat (1 − η_pv)GA_pv. A fraction η_c of this waste heat is routed to a secondary absorber → heat-pipe plate → TEG hot shoe (PV is thermally isolated), so the thermal input to the TEG path is as follows:

$$q=(1-{\eta }_{PV})G{A}_{PV}{\eta }_c$$

(25)

Along the absorber → TEG → sink path, the effective series thermal resistance is as follows:

$$R_{th}=R_{abs-hp}+R_{hp}+R_{TEG}+R_{sink}$$

(26)

giving the TEG temperature lift and node temperatures:

$$\Delta T_{TEG}=q{R}_{th}, T_h=T_c+\Delta T_{TEG}$$

(27)

The PV backsheet temperature is bounded by a separate convective–radiative balance and explicitly constrained to T_PV,back ≤ 85 °C, in all scenarios (PV and TEG are not in direct thermal contact). TEG electro-thermal relations. With Seebeck coefficient SSS, internal resistance R_int, and thermal conductance K, the open-circuit voltage and hot-side heat are:

$$V_{oc}=S\Delta T_{TEG}, Q_h=S\overline{T}I+0.5I^2{R}_{\mathrm{int}}+K\Delta T_{TEG}$$

(28)

and the electrical output are P = VI with efficiency η = P/Qh. At matched load (R_L = R_int), the maximum electrical power is:

$$P_{\max }={\displaystyle \frac{V_{oc}^2}{4R_{\mathrm{int}}}}={\displaystyle \frac{S^2\Delta T_{TEG}^2}{4R_{\mathrm{int}}}},$$

(29)

All parameters (η_pv, η_c, R, S, R_int, K) and their ranges are listed in Appendix A (

Table A1. Thermal–electrical coupling parameters.

Symbol	Parameter	Value	Unit	Source/Note
α	Seebeck coefficient (module)	215	µV/K	Datasheet, 25 °C
Rint	Internal resistance	4.2	Ω	Measured (4-wire), 25 °C
K	Thermal conductance (hot–cold)	1.8	W/K	Datasheet
ZT	Figure of merit (avg)	0.9	–	Manufacturer typical
ATEG	TEG contact area	40 × 40	mm2	Geometry
tTIM	TIM thickness	100	µm	Specified adhesive
Rth,c	Contact thermal resistance	0.15	K/W	Vendor app note
(kPV	PV backsheet conductivity	0.2	W·m−1·K−1	Literature
tPV	PV backsheet thickness	0.3	mm	Literature
Rth,hs	Heat-sink thermal resistance	3.0	K/W	Heatsink datasheet
hback	Back-side convection coeff.	10	W·m−2·K−1	No-wind assumption
G	Irradiance	1000	W·m−2	Case study
T	Ambient temperature	25	°C	Case study
Δt	Simulation time step	0.1	s	Transient run

); nominal non-concentrating operation yields ΔT_TEG ≈ 40–80 K (e.g., T_h ≈ 70–110 °C, T_c ≈ 25–35 °C), consistent with the field profiles used in this study.

Figure 8a,b shows the comparable circuit arrangement that takes the power outputs from the PV and TEG units and mixes them. The circuits might have parallel or series connections. The overall efficiency of the PV-TEG combination is determined by the total power generated by the photovoltaic cells and the TEG system’s efficiency. Equation (30) represents the overall energy generation of the system under the assumption that electrical power losses are minimal. Moreover, the electrical efficiency of a PV module is fixed using Equation (31), which is based on the ratio of the panel’s electric power production to the amount of solar radiation received per unit area [62].

$$P{V}_{PV-TEG}=P{V}_{PV}+P{V}_{TEG},$$

(30)

$${\eta }_{PV-TEG}={\displaystyle \frac{P{V}_{PV-TEG}}{AG}}$$

(31)

where P_TEG and PPV stand for the final result of power generated by the solar panel and the TEG, respectively. A is the exterior surface of the PV panel, in which G is the PV illumination energy obtained per unit area on the surface of the solar panel’s module [63].

The total energy conversion efficiency of the combination PV-TEG system may be expressed as follows:

$${\eta }_{PV-Teg}={\eta }_{PV}+{\eta }_{Teg}={\displaystyle \frac{P_{PV}+P_{Teg}}{G{A}_{PV}}}$$

(32)

2.3.1. Thermal Decoupling and Safe Operating Limits

To ensure physical plausibility and PV safety, the TEG is not bonded to the PV backsheet. Instead, solar heat is collected by a secondary black-coated aluminum absorber and delivered to the TEG hot shoe through a plate heat-pipe (or heat-spreader with embedded heat pipes). The PV module is mounted on an independent frame and thermally isolated from the absorber using a low-conductivity standoff/air gap (e.g., PTFE/PEEK posts), so conductive back-heating is negligible. Throughout all scenarios, we constrain the PV backsheet temperature to ≤85 °C. Under non-concentrating irradiance and ambient 25–35 °C, this decoupled layout provides TEG hot-side temperatures of approximately 70–110 °C with ΔT ≈ 40–80 K. A high-ΔT auxiliary stress case (ΔT ≥ 120 K) is executed only on the absorber–TEG loop and does not imply PV backsheet heating.

2.3.2. Compact Thermal Model and Implications

The absorber → heat-pipe plate → TEG → sink path is represented by a series thermal resistance $R_{th}=R_{abs-hp}+R_{hp}+R_{TEG}+R_{sink}$, The thermal input to this path is $q=(1-{\eta }_{PV})G{A}_{PV}{\eta }_c$, where G is the plane-of-array irradiance, A_pv the PV aperture, η_pv the PV electrical efficiency, and η_c a coupling factor accounting for optical/geometry losses and plate spreading. A representative non-concentrating case (G = 1000 W/m², A_pv = 0.20 m², η_pv = 0.19, η_c ≈ 0.6) gives q ≈ 97 W. With typical resistances (R_abs–hp ≈ 0.08 K/W, R_hp ≈ 0.15 K/W, R_TEG ≈ 0.18 K/W, R_sink ≈ 0.17 K/W), R_th ≈ 0.58 K/W and ΔT_TEG ≈ q·R_th ≈ 56 K (e.g., T_h ≈ 86 °C for T_c ≈ 30 °C). Recomputing all scenarios within ΔT_TEG = 40–80 K preserves the core MPPT findings—95%-to-GMPP time of 0.06 s, 0.12 s settling, ±1% ripple, and the ranking versus baselines—with scenario energies shifting by <2%, leaving the conclusions unchanged.

2.3.3. Energy Accounting and Units

All powers are measured at the DC bus and logged at 1 kHz; energies are computed with the trapezoidal rule over the stated interval. Unless noted, results are per PV-TEG test module with aperture A_pv = 0.20 m²; linear scalings to 1.00 m² and 1.0 kWp are provided where relevant. We report energy in J (kJ) and Wh (kWh) using 1 Wh = 3600 J. “Short-window energy” denotes integration over a brief horizon (e.g., the 1-s start-up window: 1.95 kJ = 0.542 Wh for one module), used to characterize transient MPPT behavior. “Daily energy” denotes the diurnal integral under measured irradiance/temperature profiles (Shanxi, non-concentrating) and includes converter losses, yielding 0.88–2.95 Wh per module. All figures/tables now state the time base (window vs. daily) and normalization (per module/per m²/per kWp) to avoid ambiguity.

3. Development of Proposed FOPID Optimized by RBBMO-Based MPPT Method

In the solar power system, the MPPT monitoring procedure is an essential link. It generates a suitable variable duty cycle that uses a pulse width modulation (PWM) generator to pulse the DC/DC converter’s switch. Because of this MPPT regulation, the PV system runs at its peak power. The following step shows the dp/dv feedback-based FOPID and PID control systems.

3.1. Control System Based on the MPPT dp/dv Evaluation Mechanism

The derivative dp/dv indicates the slope of the P_V curve. Otherwise, it is evident that the MPP is at the top of the P-V curve when the slope of dp/dv is zero. In the suggested approach, the MPPT controller is referred to as dp/dv = 0. Figure 10 displays the block diagram for the dp/dv MPPT control, which calculates the slope of dp/dv and compares it to a zero-set reference point. The control approach uses the generated error value to create duty cycle sequences that feed the DC/DC converter switch gate.

Figure 10 shows that the error value, which is the controller input and represents the dp/dv slope, is activated by the PV system’s feedback mechanism. Consequently, the controller uses the PWM generator to supply an output voltage as a control signal (duty cycle variation) to the DC/DC converter switch. This study’s following paragraph compares the proposed fractional order PID (FOPID) controller to a traditional PID controller.

3.2. Fractional Calculus

Classical calculus treats only integer-order operators (e.g., d/dx, d²/dx²). Building on ideas traced to Leibniz and L’Hôpital, fractional calculus was developed by many mathematicians and has become widely used in control engineering. Fractional-order integration and differentiation can be embedded in regulators, with distributed- and variable-order operators providing a compact generalization of the integer-order framework. Owing to their robustness to gain variations and plant uncertainties, fractional controllers have been adopted extensively in recent years. They enable smoother adjustment of gain and phase margins than integer-order designs, albeit with more tuning parameters, and can be configured to meet standard design specifications.

3.3. Classical PID Controller Description

A conventional PID controller combines proportional, integral, and derivative actions in parallel. It runs in a feedback loop that continuously computes the tracking error and updates the control input—even as irradiance and temperature vary. For MPPT, the error is taken as the power–voltage slope e(t) = dP/dV; the MPP is obtained when this slope is zero. The controller therefore regulates the duty ratio to drive e(t) → 0, so the operating point settles with small oscillations about the MPP. The continuous-time PID law:

$$u_i(t)=K_{Pi}\cdot e_i(t)+K_{Ii}{\displaystyle \int e_i(t)+}{K}_{Di}{\displaystyle \frac{d{e}_i(t)}{dt}}$$

(33)

where the derivative, integral, and proportional gains are denoted by K_p, K_i, and K_d, respectively. A traditional PID controller, as shown in Figure 11, is dependable, has a simple structure, and has a good cost-effectiveness ratio.

3.4. Classical Fractional-Order PID Controller Description

In order to increase the efficiency of the PID controller strategy, Podlubny proposed an extension to PID controllers called FOPID, which is described in [46] and contains the differentiator and integrator of µ and λ. Compared to their integer-order counterparts, FOPID controls exhibit non-local dynamics, necessitating the use of the system’s error signal history to determine control action. Because of its outstanding performance and resilience, FOPID is becoming more and more popular. By adding non-integer ordering for the proportional, integral, and derivative components, FOPID controllers are a potent expansion of the conventional PID control architecture. The controller can now handle complicated systems with nonlinear dynamics, time delays, or non-integer order dynamics thanks to this extension, which raises the controller’s degrees of freedom. This FOPID controller uses a differentiator and integrator for λ and µ. The fractional differ-integral is designed using the Riemann-Liouville (R_L) function definition, as given below:

$$\alpha D_t^{\alpha }F(t)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\left({\displaystyle \frac{d}{dt}}\right)}^n{\displaystyle \underset{\alpha }{\overset{t}{\int }}\frac{f(\tau )}{{(t-\tau )}^{1-(n-\alpha )}}d\tau }$$

(34)

where Γ indicates the operator to derive the value of a non-integer and gives Euler’s gamma function, which determines the factorial. Based on the notion of fractional differentiation, the Grunwald—Letnikov function may be mathematically described as follows:

$$\alpha D_t^{\alpha }F(t)=\underset{g\to 0}{lim}{\displaystyle \frac{\mathrm{\Gamma }(\alpha +d)}{\mathrm{\Gamma }(g+1)}}f(t-dg)$$

(35)

It should be noted that the design of a fractional-order operator can be used to condense the integrator. The Laplace transform is used to obtain the transfer function of FOPID, which is expressed as follows:

$$G_c(s)=K_P+K_I{s}^{-\lambda }+K_D{s}^{\mu }$$

(36)

The differential equation of a FOPID controller is shown in the following equation:

$$u(t)=K_{P}e(t)+K_I{D}_t^{-\lambda }e(t)+K_D{D}_t^{\mu }e(t)$$

(37)

The structure of the K_p, K_i, K_d, µ, and λ parameters forms the basis of the FOPID controller idea. The integral and derivative factors are λ and µ, respectively. When it comes to control scheme tuning, the AFOPID controller performs better (see Figure 12). The controller block structure is depicted in Figure 13.

3.5. Adaptive FOPID Tracker for PV-TEG System

A PV array electrically paralleled with a TEG source feeds a DC–DC converter whose duty ratio D ∈ [0,1] is the control input. The measured bus voltage/current are V(t) and I(t), and the electrical power is P(t) = V(t)I(t). Maximum-power operation satisfies dP/dV = 0. We cast MPPT as a slope-nulling regulation task with error:

$$e(t)={\displaystyle \frac{dP}{dV}}(t)\approx {\displaystyle \frac{P(t)-P(t-h)}{V(t)-V(t-h)+\epsilon }}, r(t)=0$$

(38)

where h is the control sampling period and ε > 0, prevents division by zero. The controller outputs a duty increment ΔD; the modulator applies: with standard anti-windup on saturation.

$$D_{k+1}=sat(D_k+u_k,D_{min},D_{max})$$

(39)

In this work, the fractional-order PID controller is shown in Figure 13. As shown in Equations (36) and (37) and with integral order λ ∈ (0, 1) and derivative order μ ∈ (0, 1]. The fractional operators are implemented by an Oustaloup rational approximation over [ωℓ, ωh]:

$$s^{\alpha }\approx k_{\alpha }{\displaystyle \prod _{i=-N}^N\frac{s/{\omega }_{z,i}+1}{s/{\omega }_{p,i}+1}}, {\omega }_{z,i},{\omega }_{p,i}log-spacedin [{\omega }_{\mathcal{l}},{\omega }_h]$$

(40)

or, alternatively, by a finite-memory Grünwald–Letnikov (GL) recursion at sample time h. Let θ = [K_p, K_i, K_d]^⊤ and the regressor:

$$\varphi (t)={[e(t),D^{\lambda }e(t),D^{\mu }e(t)]}^{\mathrm{T}}$$

(41)

Online tuning uses a normalized gradient law with projection and leakage (robust to scale changes and drift):

$$\dot{\theta }=p_s\left(-\mathrm{\Gamma }{\displaystyle \frac{\varphi (t)e(t)}{1+{‖\varphi (t)‖}^2}}-(\theta -{\theta }_0)\right),\mathrm{\Gamma }=\mathrm{diag}({\gamma }_p,{\gamma }_i,{\gamma }_d)\succ 0,\rho >0,$$

(42)

Typical choices are to fix λ and μ and adapt K_p, K_i, and K_d online. A light first-order low-pass is applied to D^μe to attenuate sensor noise.

3.6. RBBMO

An optimizer called the RBBMO mimics the RBBM swarm’s intelligence. In 2024, the RBBMO was released [64,65]. When used in a variety of applications, including TEG procedure [66], vehicle trajectory planning and tracing control [67,68], and fuel cell modeling [69], the RBBMO has demonstrated remarkable performance. The RBBMs’ chase performance, which includes target finding, assault, and food retention, serves as inspiration for the RBBMO. The woods of China, India, and Southeast Asia are the RBBM’s main habitats. The RBBM consumes tiny animals, fruits, and insects. The RBBMs find food more easily and collaborate to boost production when they roam in small groups. The RBBMs also get their sustenance from the earth and trees and store food for later consumption.

The natural behavior of RBBMs is used to expand several important aspects of the RBBMO algorithm. These birds are known for their social intelligence, vocal communication, and cooperative foraging style. They often forage in small, well-organized groups that depend on behaviors like sharing food, alarm calls, and leader-following. In RBBMO, where agents exchange knowledge regarding food sources (solutions) and modify their movements as necessary, these procedures fostered a balance among exploration and exploitation. The method includes biologically grounded ideas like exchange of data, which is analogous to how the magpies use language and visual signals for interaction in abundant in resources areas; adaptive patterns of motion, which are based on the abrupt changes in direction and non-linear migration paths; and learning and retention abilities, which are represented in the RBBMO via adaptable components of memory and fitness-based probability weighting. The conceptual model of the RBMO incorporates the phases of exploration and exploitation.

3.6.1. Initialization

The following is how the RBMO generates a random starting solution X_j,k:

$$X_{j,k}=L{L}_k+r_1(H{L}_k-L{L}_k), \forall j\in Popuandk\in D_{min}$$

(43)

where r₁ is a random number between 0 and 1, Popu is the population’s amount, Dim is the issue dimension, and LLk and HLk are the lower and upper bounds of solution k.

3.6.2. Seeking Food

To increase search effectiveness, RBMs naturally look for small groups of two to five birds or big groups of ten or more birds. Equations (44) and (45) are used to update the positions of seeking agents for small and big sets.

$$X_j(n+1)=X_j(n)+r_2\left({\displaystyle \frac{1}{S}}{\displaystyle \sum _{p=1}^S{X}_p(n)-X_{rc}(n)}\right)$$

(44)

$$X_j(n+1)=X_j(n)+r_3\left({\displaystyle \frac{1}{L}}{\displaystyle \sum _{p=1}^L{X}_p(n)-X_{rc}(n)}\right)$$

(45)

where n is the current iteration, X_j(n + 1), represents the position of the news-seeking agent jth, and r₂ and r₃ are random values between 0 and 1. X_j represents the jth bird, X_rc represents the seeking agent randomly selected in the current iteration, L symbolizes the quantity of big sets of RBBMs randomly selected from the population, and S symbolizes the quantity of tiny sets of RBBMs randomly selected from the population.

3.6.3. Attacking Prey

The RBBMs exhibit excellent expertise and teamwork while attacking their prey. As shown mathematically in Equation (46), little prey is targeted by a small set, and huge prey is targeted by a large set in Equation (47).

$$X_j(n+1)=X_F(n)+CF\cdot r{m}_1\left({\displaystyle \frac{1}{S}}{\displaystyle \sum _{n=1}^S{X}_n(n)-X_{rs}(n)}\right)$$

(46)

$$X_j(n+1)=X_F(n)+CF\cdot r{m}_2\left({\displaystyle \frac{1}{L}}{\displaystyle \sum _{n=1}^L{X}_n(n)-X_{rs}(n)}\right)$$

(47)

where XF(n) represents the food location, $CF={(1-n/N)}^{2n/N}$, and r₁ and r₂ are random values used to generate a typical normal distribution (standard deviation1, mean0).

3.6.4. Storing Food

As follows, the RBBMs store extra food for later consumption:

$$X_j(n+1)=\left\{\begin{array}{l}{X}_j(n)\hspace{1em}\hspace{1em}\forall Fog{j}_j^{old}>Fog{j}_j^{new}\\ X_j(n+1)\hspace{1em}\hspace{1em}\hspace{1em}otherwise\end{array}\right.$$

(48)

The variables $Fog{j}_j^{old}and Fog{j}_j^{new}$ indicate the values of the goal function prior to and during the update of the ith RBBM location, respectively. The RBBMO has a great research plan and a very solid implementation. All of the RBBMO’s information, along with a comparison of its unique characteristics with those of other meta-heuristic optimization algorithms and its pseudocode, can be found in [65]. Figure 14 shows the RBBMO’s flowchart.

3.7. RBBMO for Adaptive FOPID MPPT

RBBMO optimizes the adaptive FOPID by searching over a bounded design vector comprising the initial gains, fractional orders, adaptation gains, leakage, and the Oustaloup frequency band. A population is initialized uniformly (Equation (38)) and evaluated by a multi-objective MPPT cost that aggregates tracking efficiency, settling time, ripple, effort, and overshoot across realistic PV-TEG scenarios. Each iteration alternates exploration via small/large seeking groups (Equations (39) and (40)) and exploitation toward the current best “food” position with a decaying convergence factor (Equations (41) and (42)). After re-evaluation, an elitist storing rule (Equation (43)) keeps improvements and preserves the best. The final best vector Θ^⋆ seeds the online adaptive FOPID, whose gains are then adjusted in real time by a normalized-gradient law with projection, ensuring robust MPPT under irradiance and temperature variations. The Pseudo of the RBBMO for Adaptive FOPID MPPT is shown in Figure 15.

The proposed MPPT scheme uses an adaptive FOPID controller with six design parameters, which are tuned using the RBBMO to improve the energy harvested from the hybrid PV-TEG source. Tuning is performed offline: for each candidate parameter vector, branch voltages and currents (V_PV, I_PV, V_TEG, I_TEG) are used to compute P_PV = V_PVI_PV, P_TEG = V_TEGI_TEG, and the combined output P_PV-TEG; the objective function is the total extracted energy over representative irradiance/temperature and ΔT profiles (optionally with penalties on ripple and control effort). RBBMO iteratively updates the population and retains the best solution at each iteration until convergence, yielding the optimal FOPID parameter set (see Figure 16 for the workflow). The obtained parameters are then implemented online in the real-time MPPT loop, where the FOPID regulates the converter duty to drive the slope dP/dV toward zero and keep the hybrid source operating in the neighborhood of the global MPP under changing conditions.

3.8. Controller and Optimizer Reproducibility

The MPPT control signal is generated by a discrete-time FOPID acting on the slope error e[k] = dVdP[k], with output duty-cycle increment Δ_d[k] saturated to [d_min,d_max] and anti-windup back-calculation. Fractional differintegrators are implemented by a 5th-order Oustaloup approximation over 0.1–100 Hz and discretized at Ts = 1 ms (Tustin). The derivative term uses a first-order roll-off. The slope dP/dV is obtained from a Savitzky–Golay differentiator (window 15 samples, polynomial order 2) applied to V[k], P[k], followed by exponential smoothing; optional Hampel outlier rejection is enabled. The RBBMO meta-tuner runs 25 agents for 60 iterations with fixed coefficients and three random seeds; it searches within explicit bounds for (K_p, K_i, K_d, λ, μ) under box constraints and a ripple/saturation penalty. Full settings, bounds, and seeds are listed in

Table 2. Control, differentiation, and RBBMO settings.

Block	Parameter	Value/Setting	Notes
Sampling and Discretization	Sampling time (Ts)	1 ms (1 kHz)	Common to control and differentiation
	Discretization	Tustin (bilinear)	Pre-warp at 2π·10 Hz for FOPID band center
FOPID Structure	Control law	(Δd[k] = Kp e[k] + Ki(-λI){e}[k] + KdµD{e}[k])	(-λI) and µD are fractional ops
	Fractional orders	λ, µ ϵ [0.8, 1.2])	See bounds below
	Derivative roll-off	(Gd(s) = N/(1 + N/s) with N = 20 · 2π rad/s	Limits noise amplification
	Output limits	d ϵ {dmin,dmax} = [0.05, 0.90]	Duty cycle clamp
	Anti-windup	Back-calculation with (Kaw = 1/Ti); tracking limiter	(Ti = 1/Ki (effective)
Oustaloup Approx.	Order	5 (per frac. operator)	Balanced accuracy/complexity
	Freq. band	0.1–100 Hz (0.628–628 rad/s)	Matches MPPT dynamics band
(dP/dV) Estimation	Raw power	(P[k] = V[k], I[k]) at DC bus	Includes converter losses upstream
	Differentiator	Savitzky–Golay, window 15, poly 2	On (P(V)) stream sampled at 1 kHz
	Regularization	EMA on derivative, (α = 0.2)	(xf[k] = αx[k] + (1 − α)xf[k − 1])
	Outlier filter	Hampel, window 11, 3σ	Optional; default on
	Decimation	200 Hz for controller after filtering	Reduces chatter
RBBMO (Meta-Tuning)	Population size	25	Agents
	Iterations	60	Early stop if no best update in 10 iters
	Coefficients	(w = 0.7) (inertia), (c1 = 1.4), (c2 = 1.2), (ps = 0.2)	Standardized RBBMO params
	Seeds	{2024, 2025, 2026}	Report median over seeds
	Fitness	Maximize (E1s) (1-s energy)—β.ripple—γ.satCount	(β = 0.5,Wh), (γ = 0.01)
	Bounds	(Kp ϵ [0.00, 1.50], Ki ϵ [0, 400], Kd ϵ [0.00, 0.20], λ, µ ϵ [0.80, 1.20])	Box constraints
	Constraint handling	Clamp to bounds; penalty if at bound >10% of time	Prevents edge-seeking
Deployment	Final gains	Best-seed gains (median)	Logged in Supplement YAML
	Runtime adapts.	Small-gain online adaptation (±10%/s cap)	From (Δe) sign and magnitude

; pseudocode and a machine-readable configuration are provided in Figure 15 to enable exact replication. And

Table 3. Matlab/Simulink model specifications.

PV Parameters Value		TEG Parameters Value
Parameters	Value	Parameters	Value
PV output voltage	240 V	Measurement conditions Tc, Th (°C)	30 and 200
PV converter inductance	116.4 mH	Component dimensions (mm × mm × mm)	40 × 40 × 4.4
PV converter capacitance (Cin) and (Co)	48.25 μF	Typical peak power (W)	7.5
Battery capacity	48 Ah	Isc	2.65 A
Initial SoC of BESS	50%	Voc	11 V
Battery inductance	116.4 mH	Number of thermoelectric units	200

provides the hybrid PV-TEG mechanism’s precise characteristic parameters

3.9. Converter Topology, Timing, and Sizing

The PV-TEG front end drives a non-isolated synchronous boost into the DC bus (nominal 240 V as in Table 1). Switching is fsw = 100 kHz with edge-aligned PWM; the MPPT duty command is updated at 200 Hz and latched on the next PWM period. Worst-case design point (from Table 1 sources): Vin = 18–22 V (PV/TEG regulated mix), Vbus = 240 V, Pbus = 50–80 W per test module. With D = 1 − Vin/Vbus ≈ 0.90, we select L = 110 μH to cap the inductor ripple at ΔIL ≤ 30% of Iin at fsw = 100 kHz. The output capacitor bank is sized for ≤1% bus ripple and ≤50 ms recovery to ±1% for a 25% load step; effective Cout ≈ 100 μF (polymer + MLCC) with ESR ≈ 15 mΩ and ESL ≈ 1–2 nH meets both targets. Device models include MOSFET conduction/switching, inductor DCR, and capacitor ESR/ESL; measured/simulated ripple and settling meet specifications across Vin = 16–22 V and 0.8 − 1.2 × load.

4. Results and Discussions

To comprehensively evaluate the proposed RBBMO-tuned adaptive FOPID MPPT in the integrated PV-TEG system, three simulation studies are conducted: (i) start-up/search, (ii) stepwise irradiance, (iii) partial shading with multiple local peaks, (iv) load steps, and (v) field-measured irradiance/temperature from Shanxi Province for spring/summer/autumn/winter. The simulation model is shown in Figure 15. For benchmarking, the proposed method is compared against eight widely used intelligent optimizers, AOS, PSO, MFO, SSA, GHO, RSA, AOA, and P&O, applied under identical environmental profiles and converter constraints. Performance is evaluated along four dimensions: (i) instantaneous output-power response, (ii) cumulative harvested energy, (iii) output-voltage regulation quantified by ripple metrics (Δv^max, Δv^avg), and (iv) switching duty-cycle behavior, including reacquisition time, settling, and jitter.

4.1. Start-Up/Phase Search Assessment

The purpose of this assessment is to evaluate the eight algorithms’ reaction times and convergence stability during the search. Additionally, the hot and cold sides of the TEG were configured to have a substantial temperature differential in order to effectively portray the behavior of the system under dynamic environmental circumstances. Figure 17a displays the power evaluation through the Start-up/search phase. The power trajectories confirm that the proposed controller reaches 95% of the MPP (~2 kW) within 0.06 s and settles within ±1% by 0.12 s, with <0.5% overshoot and steady ripple ±1%. Competing meta-heuristics show slower convergence and larger transient excursions: PSO and AOS reach 95% at 0.18 s and 0.14 s, respectively, and settle only after 0.30–0.35 s; SSA/MFO exhibit pronounced under-/overshoots; AOA stabilizes to a markedly lower plateau. The inset over 0.1–0.3 s highlights these differences in early-time dynamics. Overall, the RBBMO-tuned adaptive FOPID offers the best combination of response speed, convergence stability, and steady-state quality, which aligns with its superior energy capture in Figure 17b.

Figure 17b demonstrates the energy evaluation for the Start-up/search phase. Throughout the 1-s search interval, the cumulative harvested energy for the proposed controller increases almost linearly, reflecting rapid convergence to (and steady operation around) the global MPP with minimal oscillation losses. Reading values at t = 1 s from Figure 17b gives 1.95 kJ for the proposed method, compared with 1.91 kJ (AOS, −2.0%), 1.88 kJ (PSO, −3.6%), 1.78 kJ (MFO, −8.7%), 1.65 kJ (SSA, −15.4%), 1.60 kJ (GHO, −18.0%), 1.05 kJ (RSA, −46%), and 0.42 kJ (AOA, −78%). In addition, the average stability enhancement is approximately 40% relative to SSA, based on the reduction of steady ripple/variance over the 0–1 s interval. The persistent slope advantage over the full horizon highlights the combined benefit of high tracking speed and low ripple.

Additionally, as shown by the duty cycle variation curve in Figure 17c, the RBBMO-tuned adaptive FOPID can quickly reduce the duty’s perturbations in the fewest number of repetitions. Notably, the disturbance amplitude after 0.1 s is only 25% of the SSA. The RBBMO-tuned adaptive FOPID shows an early stability in a small band after a rapid, almost monotonic rise of the duty ratio during the 1-s search time. From Figure 17c, the duty reaches its operating plateau by 0.18–0.20 s with a residual jitter of ±1%, which is markedly lower than the chattering and stepwise swings observed for GHO/RSA/SSA/MFO/PSO/AOS/AOA. In particular, P&O displays a slow ramp and overshoot before settling, while RSA and SSA show large-amplitude perturbations persisting throughout the horizon. The smooth transient of the proposed method indicates well-damped control action and an effective trade-off between exploration and regulation supplied by the RBBMO initialization and the adaptive FOPID update.

Table 4. Search-phase comparison of MPPT algorithms on a hybrid PV-TEG system (1-s interval); metrics include time to 95% MPP, settling time (±1%), overshoot, steady ripple, and cumulative energy.

Algorithm	Time to 95% MPP, t95 (s)	Settling Time (±1%) (s)	Overshoot (%)	Steady Ripple (±%)	Energy @ 1 s (J)	Notes/Rank
Proposed	0.06	0.12	<0.5	1	1950	Best energy and fastest; smooth duty
AOS	0.14	0.30	1–2	2–3	1910	2nd; close to proposed
PSO	0.18	0.35	2–3	2–3	1880	3rd
MFO	0.22	0.40	4–6	3–5	1780	4th
SSA	0.24	0.45	6–8	4–6	1650	5th; pronounced transients
GHO	0.20	0.35	3–5	3–4	1600	6th
RSA	0.28	>0.50	5–8	5–7	1050	7th; slow and noisy
AOA	—(low plateau)	—	—	large	420	8th; fails to reach high plateau
P&O	0.22–0.25	0.40–0.45	5	2–3	200	Slow ramp; lowest energy among baselines

compares nine MPPT strategies during the 1-s search phase on the hybrid PV-TEG model: the proposed RBBMO-tuned adaptive FOPID achieves the fastest convergence (t₉₅ ≈ 0.06 s; settling 0.12 s), the smallest overshoot (<0.5%) and ripple (±1%), and the highest 1-s energy (1.95 kJ). AOS and PSO trail by 2–4% in energy, MFO/GHO/SSA converge more slowly with larger transients, and RSA/AOA (and P&O) underperform substantially.

4.2. Stepwise Irradiance with Constant Temperature

To emulate sun–cloud motion at a fixed temperature, the irradiance is driven through five plateaus as in Figure 18a: 1000 → 500 → 0 → 500 → 1000 W/m² (step changes around 0.3 s, 0.9 s, and 1.1 s). The corresponding power responses in Figure 18b show that the RBBMO-tuned adaptive FOPID acquires the new MPP fastest after every step and maintains the smallest ripple around the final plateau. From the traces, the proposed method reaches ≥ 95% of the local MPP within 0.06–0.08 s after each irradiance change and settles within ±1% in 0.12–0.15 s, while PSO/AOS/GHO typically need 0.18–0.30 s and SSA/MFO longer with larger overshoot–undershoot. The deep valley to 0 W/m² is tracked without ringing, and the recovery to the 2 kW plateau is the quickest among all competitors.

The cumulative-energy comparison in Figure 18c (0–1 s window) reflects these transients: the proposed controller attains 0.95–0.98 kJ at t = 1 s, versus 0.90 kJ (AOS, −3–5%), 0.88 kJ (GHO, −7–9%), 0.83 kJ (RSA, −12–15%), 0.78–0.80 kJ (AOA, −15–20%), 0.57 kJ (SSA, −40%), and 0.68–0.70 kJ (P&O); PSO is the weakest in this test (0.40–0.42 kJ). These values confirm a persistent slope advantage for the proposed method due to faster reacquisition and lower oscillation losses. Consistent with the search-phase study, the average stability enhancement is 30% relative to SSA, measured by the reduction in ripple/variance of the accumulated-energy trajectory. Finally, the duty-cycle behavior in Figure 18d highlights the control quality: the proposed controller quickly establishes the appropriate duty level for each irradiance plateau and holds it with minimal chattering (±1%). In contrast, MFO/SSA/PSO and others exhibit high-frequency swings and occasional spikes, and P&O shows a slow ramp with sustained ripple. The smoother duty profile explains the reduced EMI (electromagnetic interference) tendency and the superior steady-state power seen in Figure 18b.

Table 5. Gradually Stepwise Irradiance with constant temperature comparison of MPPT algorithms on a hybrid PV-TEG system; metrics include time to 95% MPP, settling time (±1%), ripple, and cumulative energy.

Algorithm	Step t95 (s)	Step Settle (s)	Ripple (±%)	Energy @1 s (kJ)	Notes/Rank
Proposed	0.06–0.08	0.12–0.15	≈1	0.95–0.98	Fastest reacquisition; smallest ripple
AOS	0.18–0.25	0.30–0.35	2–3	0.90	2nd
GHO	0.18–0.30	0.30–0.35	3–4	0.88	3rd
RSA	0.25–0.35	0.35–0.45	4–5	0.83	4th
AOA	0.25–0.35	0.40–0.50	5–7	0.78–0.80	5th
MFO	0.30–0.40	0.45–0.55	3–5	~0.74	moderate oscillations
SSA	0.30–0.45	0.50–0.60	4–6	0.57	large dips
P&O	slow	slow	2–3	0.68–0.70	sluggish recovery
PSO	0.25–0.35	0.35–0.45	2–3	0.40–0.42	weakest energy

shows that under irradiance steps, the proposed method reacquires each plateau 2–4 times faster than typical meta-heuristics and keeps energy at 1 s 0.96 kJ, the best by 3–20%.

4.3. Partially Shaded Conditions

Under partial shading, the hybrid PV-TEG array produces a multi-peak P-V characteristic due to bypass-diode activation. As illustrated in Figure 19a, two dominant peaks appear: a left local peak (LP) around V = 170−185 V with P = 1.6−1.7 kW and a right global peak (GP) near V = 290 V with P = 2.0 kW (values read from the plot). Avoiding entrapment at the LP is therefore essential for true MPP operation.

The power responses in Figure 19b show that the RBBMO-tuned adaptive FOPID identifies the global peak on the first attempt and settles at the highest plateau with the smallest ripple. Quantitatively, the proposed method reaches 95% of the GP in 0.06–0.08 s and settles within ±1% by = 0.12–0.15 s, while AOS/PSO/GHO typically require 0.15–0.25 s with larger overshoot–undershoot. RSA converges to a lower suboptimal plateau (1.6–1.7 kW), evidencing entrapment near the LP, and P&O shows the slowest recovery with pronounced transient dips. The cumulative-energy curves in Figure 19c corroborate these trends: at t = 1 s, the proposed controller delivers the highest harvested energy (1.95–2.0 kJ), exceeding AOS by 2–4% and MFO/GHO by 6–10%, and clearly outperforming RSA and PSO (differences > 10%). The inset highlights the persistent slope advantage of the proposed approach in the final 0.1 s window, reflecting both rapid acquisition of the GP and low oscillation losses. Consistent with earlier sections, the average stability enhancement is 30% relative to SSA, as inferred from the reduced variance of the accumulated-energy trajectory.

Finally, the duty-cycle behavior in Figure 19d explains the superior power and energy outcomes: the proposed controller quickly sets the appropriate duty level for the GP and maintains it with ±1% jitter (duty varies no more than ±1% of its own steady value), whereas MFO/SSA/PSO exhibit frequent, high-amplitude swings and P&O shows a slow ramp followed by sustained ripple. The smoother duty profile implies a lower modulation index of the switch waveform, which reduces sideband energy around the switching frequency (i.e., a lower EMI propensity in practice), although formal EMI compliance would require dedicated measurements. Overall, the results demonstrate that the RBBMO-tuned adaptive FOPID reliably escapes local maxima and locks onto the global MPP with the fastest convergence and most stable steady-state among the compared methods.

Table 6. Partial shading comparison of MPPT algorithms on a hybrid PV-TEG system; metrics include time to 95% GP, settling time (±1%), ripple, and cumulative energy.

Algorithm	Time to 95% of GP (s)	Settle (±1%) (s)	Ripple (±%)	Energy @1 s (kJ)	Notes/Rank
Proposed	0.06–0.08	0.12–0.15	≈1	1.95–2.00	Locks GP on first attempt
AOS	0.15–0.22	0.25–0.35	2–3	1.90–1.96	2nd
GHO	0.18–0.25	0.30–0.40	3–4	1.78–1.86	3rd–4th
MFO	0.18–0.25	0.30–0.40	3–5	1.80–1.86	3rd–4th
RSA	—(LP trap)	—	4–6	1.60–1.70	converges near LP
PSO	0.20–0.30	0.35–0.45	2–3	~1.70	slower, undershoot
AOA	—	—	large	<1.6	unstable
P&O	slow	slow	2–3	~1.45	pronounced dips

demonstrates global MPP discrimination under partial shading: the proposed controller locks the GP first and sustains the highest 1-s energy with minimal jitter.

4.4. Gradually Load Steps with Constant Conditions

Under fixed temperature (25 °C) and irradiance (1000 W/m²), the load was swept from 50 Ω to 120 Ω in 10 Ω increments. Figure 20 reports power (left) and duty (right) for four controllers arranged by rows: AOS (a, b), PSO (c, d), P&O (e, f), and the proposed RBBMO-tuned adaptive FOPID (g, h). From the power traces, the proposed method reacquires the new MPP fastest at every load step, showing the smallest transient dip and the highest steady-state plateau. Across all steps, the RBBMO-tuned adaptive FOPID reacquired the new operating point fastest, reaching ≥95% of the local MPP in 0.06–0.08 s and settling within ±1% in 0.12–0.15 s, while PSO/AOS typically needed 0.20–0.35 s, and P&O was slower with persistent ripple. The proposed controller also sustained the highest steady-state power and the smallest transient dips/overshoot, most notably at R = 20 Ω (representative of small electronic loads). Duty-cycle traces confirm a brief adjustment followed by a tight band (±1% jitter), whereas the benchmarks show pronounced chattering, explaining their larger power oscillations and lower energy capture.

4.5. Field-Measured Irradiance and Temperature in Shanxi Province, and Seasonal MPPT Performance

To test real-world robustness, we drove the hybrid PV-TEG model with field data recorded in Shanxi Province (2024) sampled every 10 min. Figure 21a reports the seasonal irradiance profiles, which exhibit the expected bell-shaped envelope with intermittent cloud passages; Figure 21b shows the corresponding ambient temperatures, ranging from 5–18 °C in winter to 24–30 °C in summer. These data introduce natural variability absent in synthetic tests and therefore stress the adaptability of the controllers.

Figure 22a–d compares the instantaneous power over representative days in spring, summer, autumn, and winter. In all seasons, the RBBMO-tuned adaptive FOPID forms the upper envelope of the power trajectories: it ramps earlier in the morning, tracks mid-day peaks more tightly, and suppresses small cloud-induced dips more effectively than the baselines (AOS, AOA, SSA, MFO, GHO, PSO, RSA, P&O). The insets highlight the consistent margin near the noon plateau, where the proposed method maintains a slightly higher and smoother output; this advantage is most visible in summer (high irradiance) and winter (low-temperature, rapidly varying irradiance) conditions.

Figure 23a–d shows daily energy accumulation for spring, summer, autumn, and winter when the hybrid PV-TEG system is driven by the field irradiance/temperature of Figure 21. In every season, the RBBMO-tuned adaptive FOPID forms the upper envelope of all curves: it rises earlier in the morning, keeps a higher mid-day plateau, and preserves its lead to sunset. The insets highlight that the advantage persists even in short windows around the noon peak.

End-of-day energies indicate consistent gains for the proposed method:

• Spring (Figure 23a): proposed 0.88 × 10⁻³ kWh vs. AOS 0.85, AOA 0.83, PSO 0.82, RSA 0.80, P&O 0.72 → 3–8% better than the strongest baseline and 20% over P&O.
• Summer (Figure 23b): proposed 2.95 × 10⁻³ kWh vs. AOS 2.85, AOA 2.75, GHO/MFO/SSA 2.70–2.80, P&O 2.10 → 3–7% over the best competitor and 30% over P&O.
• Autumn (Figure 23c): proposed 2.90 × 10⁻³ kWh vs. AOS 2.80 and others 2.60–2.75 → 3–10% higher.
• Winter (Figure 23d): proposed 1.55 × 10⁻³ kWh vs. AOS/RSA 1.48–1.50 and P&O 1.20 → 3–5% over the best baseline and 25–30% over P&O.

These results confirm, with field variability, that the RBBMO-tuned adaptive FOPID consistently harvests more daily energy than all benchmarks by pairing fast reacquisition with low oscillation losses.

4.6. Robustness Evaluation Under Environmental Uncertainty

To probe robustness and generalizability under realistic operating dispersion, we introduce controlled environmental uncertainties and evaluate MPPT performance using three metrics: average power output P^avg_out, maximum variability Δv^max (peak-to-peak ripple normalized to the mean in a steady window), and average variability Δv^avg (mean absolute deviation normalized to the mean).

4.6.1. Temperature Variation Test

To isolate temperature effects, irradiance is fixed at 600 W/m² while the ambient temperature sweeps 0–40 °C in 5 °C increments (the abscissa in Figure 24a is expressed as a temperature-intensity ratio, 0–100%).

• Average power (Figure 24a). All methods show the expected downward trend with rising temperature, but the RBBMO-tuned adaptive FOPID sustains the highest power at every operating point and degrades the least: it drops only 2.74% (from 75.55 W to 73.48 W). In contrast, RSA falls from 74.17 W to 63.55 W (14.32%), and the deterministic P&O suffers the sharpest decline, from 68.89 W to 41.49 W (40%). Several meta-heuristics also exhibit temperature-sensitivity spikes (e.g., abrupt dips near 80–90% intensity), indicating susceptibility to thermally induced local optima.
• Maximum variability (Figure 24a). The proposed method maintains low and flat Δv^max across the sweep and levels off beyond 50% intensity, evidencing strong convergence stability at higher temperatures. Some competitors (e.g., AOA, GHO) show isolated spikes up to 2% around 70–80%. Notably, P&O appears near-zero in variability, but this reflects early stagnation near a suboptimal LMPP, not robustness, consistent with its low P^avg_out.
• Average variability (Figure 24a). While AOA attains a relatively small Δv^avg in mid-range temperatures, it does so with lower power levels than the proposed method. The RBBMO-FOPID achieves a similarly low variability band (0.01–0.03%) and maintains the highest average power, yielding the best overall stability, performance trade-off.

Overall, under temperature uncertainty, the RBBMO-tuned adaptive FOPID is the most temperature-resilient: it combines minimal power derating, low variability, and the absence of instability spikes, whereas classic P&O underperforms due to fixed-step stagnation and several meta-heuristics exhibit temperature-sensitive dips.

4.6.2. Irradiance Variation Test

To evaluate adaptability to changing solar input, the irradiance intensity ratio was swept from 0–100% (0–1000 W/m²) in 5% steps while holding the ambient temperature at 25 °C.

• Figure 24b shows the expected monotonic rise of the average power output with irradiance for all methods; however, the RBBMO-tuned adaptive FOPID remains highest across the range and becomes the clear leader for irradiance ≥ 80%. At 100% irradiance, the proposed method delivers 230–235 W, exceeding AOS by 2–4%, GHO/MFO by 5–8%, PSO by 8–10%, and P&O by 15%. A pronounced dip around 70% is visible for AOA, suggesting a convergence slip toward a suboptimal operating point.
• The maximum variability results in Figure 24b indicate that the proposed controller maintains a low and smooth profile (generally ≤ 1–1.5%), whereas most meta-heuristics (e.g., AOA, RSA, AOS, SSA) exhibit spikes, especially in the low-irradiance zone (5–30%) where the MPP signal is weak and search becomes noise-sensitive. For example, RSA shows a sharp rise in variability from 0% to 10% irradiance (95% relative increase). Although P&O appears to have very low variability, this stems from early stagnation near a local optimum, consistent with its lower average power.
• Figure 24b (average variability) shows a similar story: the proposed method tracks with consistently low values (0.01–0.02%) and no abrupt excursions, while competitors exhibit multiple peaks up to 0.03–0.05%. AOA attains low variability in some intervals, but at the cost of lower power.

Overall, the RBBMO-tuned adaptive FOPID offers the best performance, stability trade-off under irradiance uncertainty: higher harvested power with lower variability across the full irradiance spectrum.

4.7. Techno-Environmental Assessment

We assess MPPT performance primarily by the harvested energy and, secondarily, by power-stability indices that reflect steady-state quality. The fluctuation metrics are shown in Equation (49):

$$\Delta v^{max}=\underset{t=2,\dots ,T}{max}{\displaystyle \frac{\left|P_{out}(t)-P_{out}(t-1)\right|}{P_{out}^{avg}}},\Delta v^{avg}={\displaystyle \frac{1}{T-1}}{\displaystyle \sum _{t=2}^T\frac{\left|P_{out}(t)-P_{out}(t-1)\right|}{P_{out}^{avg}},}$$

(49)

where T is the total operation time, t is the discrete time index, P_out(t) and P_out(t−1) are PV-TEG power at the current and previous steps, and P_out^avg is the mean output power over the window. (Both Δv^max and Δv^avg are reported in %.)

Table 7. Statistical outcomes for several approaches across four case studies.

Testing Scenarios	Indicators	Proposed	AOS	PSO	MFO	SSA	GHO	RSA	AOA	P&O
Start-up test (search phase)	Energy (J)	1950	1910	1880	1780	1650	1600	1050	420	200
	Δvmax (%)	13.0	26.0	32.0	36.0	39.0	33.0	45.0	58.0	8.0
	Δvavg (%)	0.15	0.30	0.41	0.45	0.49	0.42	0.58	0.60	0.08
Stepwise variations in irradiance	Energy (J)	965	900	410	740	570	880	830	790	690
	Δvmax (%)	20.3	33.1	33.5	26.7	26.4	28.9	23.5	30.0	12.0
	Δvavg (%)	0.32	0.59	0.62	0.48	0.49	0.46	0.34	0.56	0.15
Partial shading	Energy (J)	1950–2000	1900–1960	~1700	1800–1860	1600–1700	1780–1860	1600–1700	<1600	1450
	Δvmax (%)	18.0	25.0	30.0	28.0	32.0	27.0	34.0	38.0	15.0
	Δvavg (%)	0.28	0.40	0.50	0.46	0.55	0.44	0.58	0.60	0.12
Field—Spring	Energy (kWh)	0.00088	0.00085	0.00082	0.00084	0.00081	0.00084	0.00080	0.00083	0.00072
	Δvmax (%)	8.432	6.654	6.1209	5.6754	10.768	8.1127	6.2709	10.5627	10.877
	Δvavg (%)	0.0254	0.0265	0.0676	0.0678	0.0427	0.0397	0.0214	0.0735	0.0635
Field—Summer	Energy (kWh)	0.00295	0.00285	0.00276	0.00280	0.00278	0.00278	0.00270	0.00275	0.00210
	Δvmax (%)	9.43	6.6742	4.4675	5.6708	6.7083	6.435	4.8425	9.3071	10.709
	Δvavg (%)	0.0342	0.0435	0.0345	0.0451	0.054	0.0421	0.0370	0.078	0.0236
Field—Autumn	Energy (kWh)	0.00290	0.00280	0.00268	0.00274	0.00272	0.00272	0.00265	0.00270	0.002235
	Δvmax (%)	7.9982	9.8092	6.4321	7.123	8.6540	9.109	5.0981	7.6190	9.5098
	Δvavg (%)	0.0269	0.0367	0.0717	0.0673	0.0542	0.0324	0.0443	0.0459	0.0397
Field—Winter	Energy (kWh)	0.00155	0.00149–0.00150	0.00147	0.00147	0.00146	0.00147	0.00148–0.00150	0.00147	0.00120
	Δvmax (%)	8.1496	11.546	6.7650	8.2109	10.299	8.2309	6.4509	12.709	12.345
	Δvavg (%)	0.0240	0.0412	0.0809	0.0670	0.0570	0.0389	0.0345	0.0972	0.0908

compiles energy and fluctuation statistics for eight meta-heuristic MPPT methods together with our proposed method (RBBMO-tuned adaptive FOPID). The classical P&O algorithm is omitted from the fluctuation rows because it often stagnates prematurely near a local MPP; this produces deceptively small variability despite low energy harvest.

• Start-up/search phase (1 s window). The RBBMO-tuned adaptive FOPID (Proposed) obtains the highest energy (1950 W·s), beating AOS (+2.1%), PSO (+3.7%), MFO (+9.6%), SSA (+18.2%), GHO (+21.9%), RSA (+85.7%), AOA (+364%), and P&O (+875%). It also exhibits the lowest fluctuations among meta-heuristics (Δv^max = 13.0%, Δv^avg = 0.15%), roughly 50–75% lower than typical baselines (AOS 26%, PSO 32%, … AOA 58%). This indicates fast settling with minimal overshoot.
• Stepwise variations in irradiance. Energy after 1 s is 965 W·s for the proposed controller—higher than AOS (+7.2%), GHO (+9.7%), RSA (+16.3%), AOA (+22.2%), MFO (+30.4%), P&O (+39.9%), and far above PSO (+135%) and SSA (+69%). Stability is again superior (Δv^max = 20.3%, Δv^avg = 0.32%), the smallest among meta-heuristics (others range Δv^max = 23.5–39% and Δv^avg = 0.34–0.62%). The controller reacquires each new MPP quickly and limits oscillation losses.
• Partial shading. The proposed method consistently locks onto the global peak, delivering 1950–2000 W·s, exceeding AOS by ~1–3%, MFO/GHO by ~8–12%, and RSA/PSO by ≥15% (AOA and P&O trail further). Fluctuations remain the lowest class (Δv^max = 18.0%, Δv^avg = 0.28%; others typically 25–38% and 0.40–0.60%). This confirms robust global MPP tracking and a stable, steady state.
• Field-measured irradiance and temperature (Shanxi Province, four representative days). Daily energy is the largest for the proposed in each season: Spring: 0.00088 kWh (3–10% above most baselines). Δv^avg = 0.0254% (top-tier; only GHO is slightly lower at 0.0214%). Summer: 0.00295 kWh (3–9% above strong baselines). Δv^avg = 0.0342% (within the low group). Autumn: 0.00290 kWh (4–9% advantage). Δv^avg = 0.0269% (among the lowest). Winter: 0.00155 kWh (3–8% advantage). Δv^avg = 0.0240% (lowest or co-lowest). Across seasons, the proposed controller pairs higher energy yield with small average fluctuations (≈0.024–0.034%), maintaining a favorable energy–stability trade-off under real weather variability.

A shown in Figure 25, the RBBMO-tuned adaptive FOPID is consistently best in energy harvesting across laboratory and field scenarios while keeping fluctuations low. Its advantage comes from fast reacquisition of the MPP after disturbances and a smooth duty profile that avoids ripple-induced losses. Although we report P&O’s numbers, its deceptively small Δv stems from premature convergence and aligns with its lowest energy across scenarios; it is therefore excluded from stability comparisons.

5. Conclusions

This paper presented an RBBMO-tuned adaptive FOPID MPPT for hybrid PV-TEG conversion. The design couples (i) an offline RBBMO stage that learns effective bounds/weights for K_p, K_i, K_d, λ, μ, (ii) an online adaptive update driven by the dP/dV error so that the fractional orders (λ,μ) and gains track operating changes in real time. Five comprehensive case studies and two uncertainty sweeps demonstrate that the method consistently delivers faster convergence, lower ripple, and higher energy than AOS, PSO, MFO, SSA, GHO, RSA, AOA, and P&O:

• Start-up/search: t₉₅ = 0.06 s and settling = 0.12 s with 1950 W·s energy (closest baseline AOS 1910 W·s).
• Stepwise irradiance: ≥95% reacquisition in 0.06–0.08 s, settling 0.12–0.15 s, and 0.95–0.98 kJ energy at 1 s; ripple ≈ ±1%, about 30% average stability improvement vs. SSA.
• Partial shading: reliable escape from local peaks and lock-in to the global peak with 1.95–2.00 kJ at 1 s (AOS 1.90–1.96 kJ; PSO ≈ 1.70 kJ); duty jitter ≈ ±1%.
• Load steps: fastest reacquisition at every step (0.06–0.08 s to 95%), smallest transient dips (≈5–8%), and tight duty control.
• Field data (Shanxi Province): daily energies of 0.88 × 10⁻³, 2.95 × 10⁻³, 2.90 × 10⁻³, 1.55 × 10⁻³ kWh in spring/summer/autumn/winter, exceeding the strongest baseline by 3–10% and P&O by 20–30%. Seasonal variability is low (e.g., Δv^max = 8.43–9.43% and Δv^avg = 0.024–0.034% for the proposed tracker), whereas P&O’s apparently small variability stems from premature stagnation and low energy.

Robustness analyses confirm resilience to environmental dispersion: with irradiance fixed at 600 W m⁻² and 0–40 °C, the proposed method derates by only 2.74% (RSA 14.32%, P&O 40%); across 0–100% irradiance at 25 °C it preserves the highest average power (≈230–235 W at full sun) while keeping Δv^max generally ≤1–1.5%.

Overall, the RBBMO-adaptive FOPID offers a strong performance–stability trade-off—fast (sub-0.1 s) tracking, ≈±1% ripple, and the best energy yield in all tests—making it a practical MPPT for hybrid PV-TEG systems under real-world variability. The added computational cost of the adaptive fractional structure is modest and acceptable for DSP/MCU deployment.

Validation, Limitations, and Future Work

This study evaluates the proposed controller in simulation using MATLAB/Simulink with field-measured climatic inputs (irradiance/ambient temperature). We do not claim hardware validation here. Model coherence was checked by (i) parameterizing the PV single-diode and TEG electro-thermal models from datasheet ranges (Table 2 and Table 3), (ii) enforcing consistent energy accounting and unit conversions, and (iii) repeating runs across random seeds for the meta-tuning stage to verify ranking robustness. All practice-level remarks (e.g., EMI, lifetime) are framed as qualitative implications only. Future work will (i) extend-loop and grid-tied operation, (ii) add EMI/EMC testing and junction/capacitor thermal-stress cycling, and (iii) provide a screening-level techno-economic sensitivity linking tracking efficiency and ripple to annual energy and component lifetime. All controller/optimizer configs (bounds, seeds, sampling, filters) are released in the supplement to facilitate replication.