Optimal power extraction of PV-TEG hybrid system via fitness-distance-balance-based beluga whale optimization☆

0 Introduction

The energy industry is undergoing a major evolution,transitioning towards a new paradigm that emphasizes distributed and modular systems alongside environmentally sustainable practices.This shift aligns with the principles of the Third Industrial Revolution, which advocates for a sustainable and interconnected energy framework[1-4].The increased exploration and adoption of alternative energy sources, including wind, marine, photovoltaic(PV) systems, and thermal energy, represent a crucial step toward reducing our dependence on conventional fossil fuels.These sustainable alternatives offer a promising route to a more eco-friendly energy landscape.Simultaneously,modern power system infrastructures are increasingly focusing on hybrid power generation techniques, which aim to improve various aspects such as power density,control precision, and efficiency in real-time operation [5-9].

The rapid expansion of PV power generation is largely driven by its numerous inherent advantages,including low operational costs, simple maintenance requirements, and minimal carbon emission [10,11].However, the practical deployment of PV technology faces several challenges,such as high initial installation costs,sensitivity to weather variations,and reduced power output in high-temperature conditions[12].To effectively utilize solar energy,large PV arrays are typically designed for their seamless integration with the electrical grid and are often installed in areas with abundant sunlight[13].It is crucial to recognize that environmental conditions in these installations can vary widely, especially natural and man-made obstacles that can cause partial shading conditions (PSC) on PV arrays.PSCs not only reduce the efficiency of energy production but can also lead to increased temperatures in the PV cells.In severe cases,elevated temperature can cause permanent damage to the cells,compromising the long-term reliability and performance of the entire PV system.This dynamic interplay between the environmental factors and PV performance highlights the complexity of maximizing the benefits of solar energy in real-world applications [14].Moreover, PV modules primarily convert the visible spectrum of sunlight into electricity, thus leaving a considerable portion of solar radiation unused.This unutilized heat not only diminishes the operational efficiency of the modules but also poses a risk to the structural integrity of the PV modules over time due to their gradual wear and tear [15,16].Integrating a thermoelectric generator(TEG) with existing PV systems to create a hybrid setup offers a practical solution for reclaiming the otherwise wasted heat.This hybrid setup is designed to optimally utilize thermal energy, i.e., the heat generated by PV cells as well as the residual heat from solar radiation [17,18].The synergistic effect within this hybrid structure not only boosts the efficiency of power generation but also enhances the operational dependability of PV modules, largely due to the cooling effect provided by the TEGs.Research has shown that employing a PV-TEG hybrid system can improve the power generation efficiency of the hybrid system by approximately 10 % compared to a conventional standalone PV system [19,20].

In the field of PV-TEG hybrid system research,scholars have primarily focused on two key strategies to enhance and fine-tune the performance of these systems.The first strategy involves placing the PV panels directly on the TEG modules, as demonstrated in [21], which has been further refined by employing a well-established heat conduction equation to accurately assess the thermal variations across the TEG modules, as discussed in [22].The second approach utilizes a spectral light separator,described in [23], to redirect photons not absorbed by the PV cells toward the TEG system, thereby boosting the power generation capabilities of the PV-TEG hybrid system.

A wide range of strategies for merging PV and TEG systems has been documented in prior studies, all aiming to optimize the simultaneous harnessing of solar and thermal energy.Aligning with this technological trajectory,tests and analysis of two examples of single irradiance and single temperature have been done in [24] without comparing their results to those obtained from methods such as incremental conductance (INC) and perturb and observe (P&O).An improved fractional order proportional integral derivative control method has been proposed in the literature [25], but it has been compared with only P&O.In another study [26], a maximum power point tracking (MPPT) technique based on the arithmetic optimization algorithm (AOA) has been designed for the hybrid PV-TEG system, where the algorithm operator has been used to update the position.However, AOA is a local optimization algorithm, and often struggles with multiple local maximum power points (LMPPs), making it prone to falling into local optima.In [27], an evaluative exploration has been performed by assessing the performance of the PV-TEG hybrid system through an intricate physical integration of the PV and TEG systems and its optimization done using the atomic orbital search (AOS)method.However, only the traditional P&O algorithm has been used for comparison, and AOS does not consistently maintain efficient energy output across all cases.

By strategically placing the TEG modules under the PV installations, this issue of PSCs in PV systems inevitably results in a diverse thermal energy distribution across the TEG modules,known as the nonuniform thermal distribution (NTD).Similar to PSC that causes multiple peaks in the power-voltage (P-V) curve, NTD leads to the emergence of various LMPPs on the power-temperature (P-T)characteristic curve, indicating varying operational effi-ciencies across the system.However, the output curve of the system typically has a single ideal global maximum power point (GMPP), which is crucial for identifying the optimal operating point [28,29].In this context, a robust and reliable MPPT technology is essential to optimize the system output,especially under dynamic environments with fluctuating irradiation and temperature.

Thus far, traditional analytical methods, such as the INC approach and the P&O technique, have been applied for MPPT in PV as well as TEG systems[30,31].However,these methods are designed to operate based on a single LMPP, making them less effective when multiple LMPPs are present, resulting in reduced energy harvesting, and thus more reliable and powerful techniques are required.Metaheuristic algorithms (MhAs) show strong global search capabilities, rapid convergence rates, and notable reduction in computational costs, which are ideal tools for efficiently managing the dynamic and complex operational characteristics of hybrid PV-TEG systems [32].However, research on optimizing power generation by hybrid PV-TEG systems is still in its early stages,with limited studies on their MPPT performance.Three key areas require further improvement: (1) current research often lacks diverse operational scenarios and real-world data.(2) Though existing methods have enhanced the power output of PV-TEG systems to some extent, there is still room for further improvements with more advanced MPPT strategies.(3) Prior studies mainly focus on the power output and efficiency as the performance indicators of the system and overlook other critical dimensions such as uniformity and stability.

Considering the abovementioned aspects, this work introduces a novel fitness distance balance (FDB)enhanced beluga whale optimization (BWO) algorithm,known as FDBBWO,which aims at optimizing the MPPT of the PV-TEG hybrid system in multiple aspects.The main contributions of this work can be outlined as follows:

1) When the PV system operates independently, the unused heat will not only reduce its efficiency but also gradually cause wear and tear on its components.By integrating the PV and TEG systems,energy efficiency is improved, power generation is increased, and waste heat is effectively utilized.This hybrid system offers a promising solution for a highly efficient,reliable,and sustainable energy production,leveraging the strengths of both technologies to address the growing global demand for clean energy.

2) Most of the previously proposed MPPT algorithms for hybrid systems suffer from poor stability and local optima.To address this, an advanced FDBBWO-based MPPT strategy has been devised that strategically integrates the FDB method into three distinct aspects of BWO, thus enhancing the solution diversity and significantly reducing early convergence and power oscillations during the MPPT process.

3) Many existing studies have set up simple testing environments, thus limiting their practical relevance.To rigorously validate the effectiveness of the proposed FDBBWO in managing the MPPT for the PV-TEG hybrid system across different operating conditions,three case studies have been carefully designed and conducted.Real-world datasets collected from four seasons in Hong Kong have been employed to verify the practical applicability of the proposed MPPT approach.

4) A broader range of criteria has been adopted for a more comprehensive evaluation, focusing not only on the improvement of the overall power output but also in convergence fluctuations (denoted by ΔVavg and ΔVmax).

1 Mathematical modelling of the PV-TEG hybrid system

Considering its proven effectiveness and frequent application in similar studies, the current research mainly adopts the strategy from[20]and[21],viewing it as a foundational model for future investigative endeavors and development initiatives.The structure diagram of the PV-TEG hybrid system used in this study is shown in Fig.1.

Thermal connections primarily arise from their inherent physical relationships.A thorough investigation in [33]methodically analyzes the complex interplay between the operating temperature of the cells and variables such as the ambient environmental conditions, wind dynamics,and solar exposure levels.This study offers correlation techniques ranging from straightforward to complex analyses.Literature[20]highlights a linear equation developed through extensive experimental research, focusing on the detailed relationship between the temperature variations in the PV modules, the surrounding atmospheric conditions, and the intensity of sunlight received.Similarly, in[34], a model aimed at precisely determining the operational temperature range of PV modules has been carefully developed, taking into account critical factors such as the ambient temperature, solar radiation intensity, and wind speed.

1.1 PSCs

In the context of real-world engineering works,the swift movement of clouds and the potential obstructions from neighboring buildings might cause PSCs across large PV installations.The PV-TEG hybrid system proposed in this study capitalizes on waste heat,adhering to the fundamental principles of physical heat transfer.Consequently,TEG arrays within the hybrid system are inevitability subjected to NTD because of the cascading effects caused by PSCs.In other words,in the hybrid system,the influence of PSCs is crucial in its effect on the power characteristic curve,which can cause uneven power distribution across the complex design of the hybrid module.This can, in turn,compromise the coherence of the electrical profile related to the hybrid system while also heightening the challenges tied to MPPT.

Fig.1 Schematic of the PV-TEG hybrid system.

1.2 Modeling of the PV array and mathematical model of the TEG system under PSCs

To optimize the power and voltage output, the cells of the PV modules are often arranged in series and parallel configurations, known as the total-cross-tied (TCT) connections [35].Exposure to PSCs leads to the emergence of distinctive short-circuit currents within each PV module in a specified array [36].If the current in a module strays from the expected string currents,its voltage shifts,turning it into an electrical load that draws energy from the neighboring modules and transforms it into heat.A widely recognized strategy to mitigate this unfavorable phenomenon involves integrating bypass diodes into each PV module,a tactic graphically depicted in Fig.2(a).These diodes, in their functional capacity, prevent reverse current flow,especially under varying irradiance levels.In instances where every PV cell is subjected to PSCs,the diodes retain their inactivity, as illustrated in Fig.2(b).More importantly, when a single PV cell engages the bypass diodes,the subsequent PV array is capable of exhibiting numerous LMPPs along its P-V curve.The incidence of LMPPs directly correlates with the quantity of partially shaded units,as shown in Fig.2(c)and 2(d).The voltage emanating from a shaded PV cell operates within a set of predefined boundaries, which can be expressed as follows:

where V oc represents the open circuit voltage,n represents the number of PV cells that are exposed without shielding,and V Bdiode is a variable that indicates the voltage drop across the diode.Giving due consideration to these parameters is fundamental for a comprehensively understanding the operational efficacy and overall performance of the PV cells.

1.3 Model of the TEG system under PSC

Interconnecting multiple TEG modules in an array of varying configurations is a common practice for ensuring optimal power output.When used as voltage sources,these TEG modules enhance their output power through the structure of parallel and series connections[37].Similar to PV arrays, as the size of the array increases, each module might be exposed to varying operations due to PSC,leading to potential power losses.This disparity in the temperature might initiate power losses, and to mitigate this,bypass diodes were strategically employed to minimize the detrimental impacts of damage to an individual TEG module, as shown in Fig.3.Concurrently, series diodes were incorporated to inhibit the circulation of current among the columns of the array.

Employing a unified approach to arrange N-TEG modules in a methodical series and parallel configuration,mathematically expressed by(2), not only enables efficient electricity production but also its subsequent refinement.

where i=1,2,...,N, and all of the aforementioned variables are specified in the nomenclature section.

The output obtained from the i th module can be expressed as follows:

Fig.2 Schematic of PV cells when they are(a)shadowed and(b)not shadowed.Characteristic curve of the PV cells(c)under PSC and(d)without PSC.

Fig.3 Schematic of the TEG structure with parallel and series connections.

where PTEGi represents the power output from the i th TEG module.

The cumulative output of the entire TEG, essentially equaling the sum of the output derived from the constituent modules, can be written as

1.4 The hybrid system of PV and TEG

In a detailed examination of engineering practices, the employment of series-parallel (SP) connections, widely recognized and implemented in PV power generation systems, have exhibited discernible limitations especially when subjected to PSC, due to the fundamental dependence on uniform solar irradiance across the array.In contrast, arrays composed of TEGs conspicuously stand resilient under analogous conditions, thereby affirming their adaptability and suitability in environments characterized by NTD.It is important to highlight that the TCT connection for the PV array and the SP connection for the TEG array has been deliberately chosen in this study to strategically design their configurations[38].Furthermore, a pair of MPPT controllers has been carefully incorporated in the hybrid system in addition to the boost circuits, to achieve a synergetic integration of power,which is harvested from both types of arrays,as illustrated comprehensively in Fig.4.It is worth noting that the internal parameters of the hybrid system have little impact on the optimization results of the proposed method and have strong applicability.The output of a hybrid system is mainly related to the number of PV and TEG panels contained in it,as well as external environmental factors such as the change in the solar irradiation, temperature, and PSCs.

Fig.4 Schematic of the working mechanism of the PV-TEG hybrid system with an MPPT controller.

By leveraging the integration of the PV and TEG systems, the unused heat generated by the PV system can be effectively reused, whereas the TEG system offers a means to promote the operational efficiency of the PV modules through a cooling mechanism.To manage the thermal interaction between two modules, a previous study [33] explains that the humidity on the hot side of the TEG is influenced by a combination of factors,including the ambient temperature (Tam), wind speed (W s), and intensity of solar irradiance (GT).This multi-factorial influence can be mathematically represented as follows:

Fig.1 illustrates that the TEG module is strategically configured with its hot side situated beneath the PV module, consequently establishing the cold side temperature equivalent to Tam.

The aggregate power output comprises the cumulative energy produced by the PV and TEG units, and can be expressed as

In addition, a distinctive correlation exists between the power conversion efficiency of the hybrid system and the efficiencies of the two isolated systems, which is expressed as follows:

where APV is the area of the PV board.

2 FDBBWO

The global exploration ability of BWO can be ensured by the diversity and randomness of exploration in the exploration phase.In the exploitation phase, optimizing the found solutions and further increasing the likelihood of approaching the GMPP by adopting the FDB improvement strategy are the main goals.The ability to prevent the solution from falling into a local optimum under PSC depends on the effective management in the whale phase.Due to the superiority of FDBBWO in finding a GMPP,maximum number of iterations has been adopted in this study as its termination condition.

2.1 Mathematical model of BWO

Introduced in 2022 through the work illustrated in[39],the BWO algorithm draws its inspiration from the distinct behaviors exhibited by beluga whales in the ocean, in particular, their swimming, predation,and instances of whale fall.Fig.5 visually represents the three characteristic behaviors of beluga whales as integrated into the BWO algorithm, where the stages of swimming, foraging, and experiencing a whale fall are correspondingly related to exploration, exploitation, and whale fall phases.The termination criteria of BWO is usually set in two cases: the fitness value between multiple iterations is less than a certain value, or the maximum number of iterations is reached.

The BWO algorithm comprises three key stages: global exploration, local exploitation, and whale fall.To judiciously regulate the equilibrium between overarching global exploration and nuanced local exploitation within the algorithmic process, a variable, Bf, defined as the balance factor, has been introduced:

Fig.5 Behavioral characteristics of beluga whales,namely,(a)swimming,(b) foraging, and (c) whale fall, incorporated in the BWO algorithm.

where T represents the present number of iterations, Tmax represents the maximum number of iterations, and B0 denotes a random numerical value residing within the(0,1) interval, using which the functionality dynamics of the BWO algorithm can be investigated.When the value of Bf is above the threshold of 0.5, the BWO algorithm performs a macroscopic global search strategy.Conversely,if it falls beneath this threshold,the algorithm pivots toward executing a local, more surgically precise exploitation.

(1) Exploration phase

The position update for each beluga whale during swimming is defined by two equations.Eq.(9) is utilized when the parameters to be optimized are less than onefifth of the total parameters, whereas (10) is employed to find new solution points when this condition is not met.The update process is as follows:

The definitions of all parameters in the abovementioned equations are the same as in [39].

(2) Exploitation phase

The phase of exploitation within the BWO algorithm takes its foundational principles from the insightful observation of predatory strategies implemented by beluga whales in their natural habitats.To enhance convergence,the algorithm incorporates the Le´vy flight strategy, which is modeled as follows:

where the definition of each variable is the same as in[39].

(3) Whale fall

During the other two behavioral characteristics, beluga whales might unfortunately perish and sink to the seabed,forming a whale fall that supports a new ecological system.This process of one beluga whale falling and being replaced by a new one joining the population can be mathematically represented as follows:

where the definitions of all the variables are the same as in[39].

2.2 Mathematical model of FDB

In a previous study [40], the FDB has been used as a strategy during the refinement phase of the MhAs, with the aim of guiding a candidate solution toward its optimum state.Within the FDB approach, two key metrics are used:the fitness value,which is obtained from evaluating the distinct fitness function of each candidate solution,and the spatial Euclidean distance, which measures the separation between the given solutions.As with the BWO termination conditions, two cases are widely used for FDB termination conditions.

For successfully combining the FDB within the MhAs,a systematic approach was adopted.First,the precise position and the corresponding fitness value of each entity within the exploration domain were determined.Subsequently, by arranging the fitness metrics in order, the entity possessing the most favorable fitness emerged.In the concluding phase, an assessment was made of the relative distance separating each entity from this identified optimum counterpart.

where Pbest is the optimum individual,Pi is the position of the individual to calculate the distance, and d is the number of unknown parameters.

Before calculating the FDB score for each solution point, it was necessary to normalize the vector Dp as follows:

Similarly,the fitness function value,F,corresponding to the solution point, was also normalized as follows:

The FDB score was determined using the influence coefficient, w.When conducting research tests on FDB,w was usually taken as 0.5.The vector SP of the FDB score can be expressed as follows:

After calculating the vector SP,the roulette wheel selection technique was employed in this study.This choice deviates from the traditional method of picking the solution point with the highest FDB score.

3 Principle of the FDBBWO-based MPPT for a PV-TEG hybrid system

3.1 PV-TEG model with MPPT controller under PSC

Under particular meteorological conditions,the voltage yields obtained from the PV and TEG systems emerge as variables capable of optimization.The control mechanism for MPPT is instrumental in pinpointing an optimal duty ratio, a parameter intrinsically linked to the peak value of the voltage output [41].Subsequently, this identified duty ratio is then directed toward an insulated gate bipolar transistor (IGBT), setting it up correctly for the next control cycle.The ultimate fitness at every control point can be accurately defined by combining real-world voltage and current data, as follows:

where Pout represents the active power collectively generated by all the PV units. and represent the minimal and maximal constraints, respectively, of the output voltage of a PV unit, defining the boundary conditions within which the output voltage operates.

The calculation procedure of the fitness function for the TEG systems is similar to that used for PV systems,which can be expressed as follows:

Here,Pout is the active power,derived from the comprehensive TEG unit.The parameters and represent the minimal and maximal constraints, respectively, of the output voltage of the TEG system, and establish the permissible operational range for the electrical output.

3.2 Boost converter device

The boost circuit, often characterized as a non-isolated converter designed specifically to amplify the input voltage holds significant importance in the realm of MPPT technology [42], which is particularly noticeable within the dual-phase frameworks of the PV and TEG systems.The reasons behind its recognition include its uncomplicated structural configuration and unparalleled conversion effi-ciency, as mentioned in [43].To further elucidate these properties,Fig.6 presents a schematic of the MPPT architecture rooted in FDBBWO approach, emphasizing its operation within a PV-TEG setup exposed to PSC, utilizing the capabilities of the boost converter.

Fig.6 Schematic showing the working principle of the PV-TEG hybrid system incorporating FDBBWO with an MPPT controller under PSC.

A visual representation showing the working principle of the PV-TEG hybrid system is shown in Fig.6.In this figure, V PV/TEG represents the voltage resulting from the PV/TEG configuration.V out represents the voltage linked to the enhancement of the circuit.The parameters f and T denote the switching frequency corresponding to the IGBT and its subsequent control cycle, respectively.Ip and Ip max represent the standard and maximum currents flowing through the inductor L.The ensuing approach,employed to determine the values for V out, the inductor,L, and the filter capacitance, C1, expressed by Eqs.(25)-(27), is elaborated upon in the subsequent sections:

It should be highlighted that the design of the filter capacitor is primarily aimed towards reducing the influence of the ripple current, a byproduct of the inductor operation,on the performance of the PV unit[44].A comprehensive breakdown of parameters linked to the boost circuit, specifically aligned for the dual subsystems inherent in the PV-TEG hybrid setup,can be seen from Table 1.Broadly speaking, one cannot overlook the fact that DCDC converters inherently face power dissipation.The effi-ciency tied to MPPT can be expressed as

At a given time, t, the hybrid system achieves a specific power output, PPV - TEG（t）.The highest power attainable by the hybrid system at this exact moment in time is represented by a different parameter.

Table 1 Parameter setting of the boost circuit.

ParameterPV systemTEG system CapacitorC1 = C2 = 1 μFC1 = 66 μF, C2 = 200 μF Inductor (L)500 mH250 mH Resistive load (R)200 Ω10 Ω Switching frequency100 kHzfs = 20 kHz

3.3 Overall procedure

Within the context of the PV-TEG hybrid system, the MPPT operation seamlessly integrates the distinct techniques inherent to each subsystem.This approach highlights the importance of a non-model-based MPPT strategy, which relies on accurately measuring two critical parameters,namely,voltage and current,necessitating separate yet compatible MPPT controllers for the PV and TEG components.It is worth mentioning that after testing,the third improvement strategy showed the most desirable optimization performance, and thus all case studies were performed under case 3 in FDBBWO.

4 Case studies

This evaluation section presents four distinct tests: (i)assessing the effect of the step changes in the solar irradiance at constant temperature, (ii) exploring the effects of gradual changes in the solar irradiance and temperature,(iii)analyzing the performance of the hybrid system under random fluctuations in the solar irradiance, and (iv) using real-world data reflecting typical solar irradiation and temperature conditions in Hong Kong.In addition,for a thorough comparative analysis, conventional methodologies,namely,INC,P&O,moth-flame optimization-incremental conductance (MFO-INC), and four other MhAs, including exponential distribution optimizer (EDO), multiple ring algorithum (MRA), reptile search algorithm (RSA),and BWO,were employed.In the experimental simulation of the combination of model and algorithms, SimuNPS was used,which was implemented in the form of a Python script.Ensuring that the optimum parameters for all the methods compared have undergone meticulous scrutiny and validation, a series of comprehensive experimental procedures were executed.This measure safeguarded the integrity of the solution and the computational speed.

Furthermore, to render a more perceptible assessment of the optimization outcomes stemming from the various methodologies within the PV-TEG hybrid system, two indices,as outlined below and designed to compute power fluctuations, were introduced [45]:where the definitions of the parameters are the same as those in [46].In practical applications, the MPPT method based on FDBBWO does not require high-demand hardware investment or a complex software environment, and the development difficulty is moderate.The FDBBWO and the MPPT controller were developed in SimuNPS,with the designing done by Shanghai KeLiang, and the embedded MPPT algorithm was executed using Arduino as the MPPT controller.

Fig.7 Variation in the solar irradiation under PSC.

4.1 Variation in the solar irradiation at constant temperature

Fig.8 Performance on the start-up test employing eight methods based on the hybrid PV-TEG system:(a)photovoltaic and(b)thermoelectric generator system current curves,(c)photovoltaic and(d)thermoelectric generator system voltage curves,and(e)power and(f)energy generated by the hybrid PVTEG system.

This sector describes the results of the simulation studies investigating the effect of changes in the cloud cover on the power generation performance of the PV-TEG array at a constant environmental temperature of .As illustrated in Fig.7, each PV unit experiences a distinct step change in solar irradiance, in accordance with (5).The wind speed (W s) was fixed at 1.5 m/s throughout these observations.The optimization outcomes, observed under conditions of stepped illumination,are presented in Fig.8.A careful review of Fig.8(a)to 8(d)shows that each specific graph supports the observation that FDBBWO demonstrates remarkable convergence speed and stability,surpassing the other evaluated methods.Furthermore,Fig.8(e) demonstrates that FDBBWO manages to attain the optimal power output amidst diverse input conditions across five stages.In addition, Fig.8(f) clearly shows that among all the evaluated methods, FDBBWO achieves the highest energy output, recording a peak energy of 542.12 W·s.This surpasses the outputs of BWO as well as MRA by 3.6%and 11.41%,respectively.Concurrently,as shown in Table 2, the power fluctuations (ΔVavg and ΔVmax) exhibited by FDBBWO are the least pronounced among all the methods tested in this study, with ΔVavg being just 14.37%of that of RSA.In addition,FDBBWO possesses the smallest power fluctuations, ΔVavg and ΔVmax are 0.0144 % and 0.0057 % respectively, and no fluctuation or power oscillation problem near the GMPP appeared after convergence.This superior performance in energy generation, along with reduced power fluctuations attributed to FDBBWO, collectively highlight the efficiency of the FDBBWO approach in MPPT applications.

4.2 Ramp changes in solar irradiation and temperature

In the context of practical engineering, scenarios do occasionally arise where the movement of cloud layers is gradual,thereby causing the light and environmental temperature absorbed by the PV-TEG hybrid system to undergo a sluggish alteration.Aiming to authentically emulate this PSC effect, the irradiance intensity and environmental temperature variations were considered to be gradual, ascending shifts to simulate the response of FDBBWO to such a circumstance.In this test, the wind speed, W s,was fixed at 1.5 m/s.Furthermore, the hot side temperature of the TEG also experienced a ramping shift,as computed using (5), whereas its cold side temperature was fixed at .Fig.9 shows the configuration of the light intensity and temperature for the hybrid system and Fig.10 shows the real-time optimization outcomes obtained from the seven distinctive methods.The local square diagrams in Fig.10(a) to 10(d) indicate that FDBBWO has a higher convergence speed and Fig.10(f)shows that FDBBWO achieves the maximum output energy of 1,440.09 W·s, which is 105.05 % of that of BWO and 106.63%of that of EDO,respectively.In addition,the results in Table 2 indicate that FDBBWO has the smallest power fluctuations (ΔVavg and ΔVmax), indicating that this method has a stronger global searching ability and higher stability compared to the other methods.

4.3 Effect of the stochastic solar irradiation

To imitate scenarios typical of summer situations over a 12-hour daylight period, models that incorporated continuous and seemingly random irradiance were tested and the results thus obtained are shown in Fig.11.The hot side of the TEG module experienced consistent and random thermal fluctuations across 12 h, driven by the parameters of(5), while ensuring a steady temperature of on its cold side.Illustrated in Fig.11 is a random variation curve of five groups of solar irradiance input in the hybrid PVTEG system.On closer inspection, Fig.12(b) distinctly highlights that the power and energy generated by the FDBBWO method always outperforms the other competitive methods.Moreover,the FDBBWO method yields the pinnacle of energy output, eclipsing that achieved by INC and MRA by 10.90%and 12.21%,respectively.The variance in the output indicates that FDBBWO is capable of achieving a significantly higher power output, even amid the minor power fluctuations caused by ongoing,randomized, and continuous changes in irradiance.It should be noted that the power fluctuations, i.e., and ,in the FDBBWO do not always exhibit the lowest values under these conditions.The consistent, gradual changes in irradiance allow other methods to converge more smoothly to the GMPP, whereas FDBBWO continuously seeks the peak power point, driven by its FDB-based enhancement mechanism.

Table 2 Data statistics of the operation results in four case studies obtained using the eight different methods compared in this study.

ScenesIndicesEDOINCMFOINC MRAP&ORSABWOFDBBWO Variations in solar irradiation at constant temperature Energy /(W·s)526.37516.99527.00493.95512.04527.57514.41542.12 ΔVmax /%0.02040.04020.02350.07440.02560.10020.02130.0144 ΔVavg /%0.00650.00590.00660.00680.00590.00690.00630.0057 Energy /(W·s)1,369.42 1,360.221,248.541,229.741,349.651,133.551,426.931,429.32 ΔVmax /%44.2450.1149.3249.6348.9943.5742.3940.23 ΔVavg %%0.00990.01020.01200.01490.01350.01210.00970.0081 Stochastic change in solar irradiation Climbing changes in irradiation and temperature Energy /(10-6 kW·h) 150.29147.03148.35145.31133.84138.47160.51163.05 ΔVmax /%82.3178.4270.9380.4085.1496.2177.2075.74 ΔVavg /%17.9119.6718.9720.2219.6028.7420.1918.95 Measured temperature and solar radiation information for representative days in Hong Kong SpringEnergy /(10-6 kW·h) 35.6932.8434.5525.5823.9635.5835.9850.28 ΔVmax /%191.44379.43194.57253.00251.98182.83185.21194.13 ΔVavg /%17.0626.1317.1720.2119.5216.2916.9918.21 Summer Energy /(10-6 kW·h) 98.7395.2699.2687.4193.3497.73101.17209.32 ΔVmax /%165.65215.72168.05203.78236.39163.03196.12168.51 ΔVavg /%22.6227.2522.9423.5328.6422.9127.4122.36 Autumn Energy /(10-6 kW·h) 101.4697.23105.1691.4096.6395.52107.19220.57 ΔVmax /%166.26144.05125.58149.17194.18344.44263.43113.10 ΔVavg /%17.2314.0912.8012.6214.8728.1819.9911.73 Winter Energy /(kW·h)74.0369.27103.5462.5459.8074.1075.44143.55 ΔVmax /%217.72302.35240.49272.59354.68210.27216.72201.50 ΔVavg /%19.4724.2920.7427.2623.9721.3320.2018.95

Fig.9 Step change in the (a) solar irradiation and (b) ambient temperature caused by the PSC.

Fig.10 Comparative outcomes of the PV-TEG hybrid system subjected to a step change under a stable temperature using seven different methods: (a)photovoltaic and (b) thermoelectric generator system current curves, (c) photovoltaic and (d) thermoelectric generator system voltage curves, (e) power harvested, and (f) energy secured by the PV-TEG hybrid system.

Fig.11 Stochastic irradiance on the PV-TEG hybrid system.

Fig.12 The response of the PV-TEG hybrid system obtained for a stochastic irradiance via eight methods: (a) power harvested and (b)energy obtained by the PV-TEG hybrid system.

4.4 Comprehensive field-obtained temperature and solar irradiation data related to representative days within the Hong Kong region

To verify the applicability of the PV-TEG hybrid system in real-world operational conditions, a comparative analysis of eight different methods was done, using the solar information data from Hong Kong,which is located on the eastern side of the Pearl River Delta of China.The climate of Hong Kong is characterized by a subtropical monsoon pattern, featuring rainy summers with temperatures ranging from 27 to , and relatively dry winters with an average temperature of approximately .The area is particularly affected by typhoons from July to September, often originating from tropical cyclones.For this analysis,data from four distinct days corresponding to the different seasons were collected,with a data sampling interval of 10 min.Fig.13(a)shows the geographical location of the data collection point at N latitude.Fig.13(b)shows the measurement instruments used in this research.To simulate the PSC effect, irradiance levels for five PV panels were set at 100,70,50,40,and 30%relative to the measured data from Hong Kong, reflecting longterm, sequentially decreasing changes.Further, the TEG modules were exposed to NTD as outlined in (5), with wind speed, W s, considered within the range of 1-10 m/s.

Fig.13 (a) Geographical location from which data was collected for the measurement and (b) photograph of the measurement device.

Fig.14(a) presents the collected illuminance data,recorded over four characteristic days within the Hong Kong region and Fig.14(b) shows the surrounding temperature conditions, which are crucial for understanding the environmental conditions.

Figs.15 and 16 clearly show the optimal power output and energy generation achieved by the eight different methods, based on real-world data from Hong Kong.From the figures,it can be seen that the power and energy produced by FDBBWO are the highest, highlighting the significant advantages of this method.As can be seen from Table 2, MhAs demonstrate the capability of achieving superior output energy, coupled with reduced power fluctuations (both ΔVavg and ΔVmax), compared to both INC and P&O in the majority of scenarios.In particular,FDBBWO stands out by delivering the highest energy output across all four distinct seasonal days and also exhibits the smallest fluctuations in power during typical autumn and winter days.

Fig.14 Measured data of the (a) solar irradiation and (b) temperature on typical days in different seasons in the Hong Kong region.

Fig.15 Power harvested by the PV-TEG hybrid system using the eight methods on typical days in (a) spring (b) summer, (c) autumn, and (d) winter seasons.

Fig.16 Energy harvested by the PV-TEG hybrid system using the eight methods on typical days in (a) spring, (b) summer, (c) autumn, and (d) winter seasons.

The boxplots of energy obtained by the different methods on typical days in the four seasons are shown in Fig.17, among which FDBBWO exhibits an efficient and stable optimization effect.Although P&O is more stable than FDBBWO on typical summer days,the energy it obtains is much smaller than FDBBWO, indicating that it is prone to falling into LMPP.Further, MFO-INC performs better than INC in terms of robustness as well as optimization, but it is still inferior to FDBBWO.In scenarios involving long-term, actual data inputs, MhAs demonstrate superior performance over INC and P&O.However, the results that every variant of MhA produce can diverge significantly.Among them, FDBBWO consistently shows the most commendable optimization indicators throughout the four characteristic days spanning all seasons, thereby affirming its notable stability and practicality within tangible engineering applications.

4.5 Sensitivity analysis

Sensitivity analysis is mainly applied to test the effects of changes in the operating temperature and solar irradiance on the MPPT output of the hybrid system obtained using different methods.For the same, three indicators,namely, the average output power, maximum variability,and average variability were introduced in this study to evaluate the performance of each algorithm.

The sensitivity analysis between temperature and power was achieved by setting a temperature ratio (0-100 %),with the interval of 5%between the temperature intensity ratios.Fig.18 presents the sensitivity performance of the eight approaches as a function of the changes in the temperature intensity ratio,from which it is seen that the output power increases as the temperature ratio increases.Furthermore, at different temperature ratios, the average and maximum variability of FDBBWO show relatively small changes, indicating its high stability.

The sensitivity analysis between the solar irradiance and power was achieved by setting a range from 0-1000 W/m2 with a 5 % irradiation intensity interval.The sensitivity performance of the eight methods as a function of the solar irradiance is shown in Fig.19, in which the power output obtained by FDBBWO always exhibits the best performance under different irradiation intensity ratios.In addition, the average and maximum variability acquired by FDBBWO outperforms other algorithms.In general, FDBBWO possesses a convincing performance in terms of power output and power fluctuations.

As shown in Fig.18(a)and Fig.19(a),the power output of MFO-INC is superior to INC in both intensity ratios.However, in contrast to the power output increase of FDBBWO with increasing ratio, MFO-INC experiences a sudden drop in the power output at high radiation and temperature intensity ratios (70-100 %), indicating poor adaptability of MFO-INC in reality.Although the improved INC has a relatively obvious power output improvement compared to INC, it is still easy to fall into a local optimum.Further, in Figs.18 and 19, the simple methods such as INC and P&O, have low power output and small fluctuations in the average and maximum variability, implying solutions trapped in local optima with poor quality in search of optimal solutions.

Fig.17 Boxplots of energy obtained using the eight methods in (a) spring, (b) summer, (c) autumn, and (d) winter seasons.

4.6 Statistical results

Table 2 presents the statistical results obtained by applying MPPT to the PV-TEG hybrid system incorporating eight different approaches across different testing environments, with the best outcomes emphasized in bold for clarity.The data reveals that the FDBBWO algorithm uniformly delivers the highest energy generation across all test conditions, marked by the least power variability (both ΔVavg and ΔVmax)during step changes in irradiance,gradual changes in irradiance and temperature, and on typical days in autumn and winter in Hong Kong.For instance,on a spring day in Hong Kong,the energy output achieved by FDBBWO is 139.74, 141.32, 209.85, 196.56, 145.53,153.11, and 140.88 % higher than that produced by BWO, RSA, P&O, MRA, MFOINC, INC, and EDO,respectively.The optimization efficiency of the algorithm is further proven under extended durations and varying input scenarios, particularly noticeable during fluctuating irradiance levels and when analyzing real-world data from Hong Kong.INC, P&O, and four additional MhA methods tend to prematurely converge to LMPPs, resulting in increased power oscillations (ΔVavg and ΔVmax).In contrast, FDBBWO can track the GMPP efficiently with the smallest power fluctuations in different conditions.In addition,FDBBWO not only outperforms BWO in energy production in each case but also shows reduced power fluctuations in all tested scenarios.For example, using actual data from Hong Kong, the energy production in FDBBWO significantly exceeds that in BWO, doubling it during typical summer and autumn days and nearly doubling it during standard winter days.This highlights the effectiveness of integrating the FDB scheme with BWO,demonstrating that the superior global optimization capability of FDBBWO is well-suited for the complex MPPT demands of PV-TEG hybrid systems.

Fig.18 Sensitivity performance of the eight approaches as a function of the temperature intensity ratio presented in terms of the (a) average output power, (b) maximum variability, and (c) average variability.

5 Conclusions

A hybrid PV-TEG configuration was introduced in this study to address the limitations of the standalone PV and TEG systems.In addition, a novel FDBBWO-based MPPT strategy was designed to alleviate the adverse effect caused by PSC and extract the optimal power output from the PV-TEG hybrid system under dynamic operation conditions.The key contributions of this work can be summarized as follows:

· To realize a proper reutilization of the waste heat generated during PV power generation,the integration of PV and TEG technologies effectively enhances the overall energy output and promotes a more sustainable and effective energy production mode.

· An enhanced FDBBWO-based dynamic MPPT control strategy was devised, in which the FDB mechanism was incorporated into the BWO algorithm for achieving a superior optimization performance.This reduced the risk of the algorithm becoming trapped in LMPPs and improved its convergence stability.

Fig.19 Sensitivity performance of the eight approaches as a function of the irradiation intensity ratio presented in terms of the(a)average output power, (b) maximum variability, and (c) average variability.

· For a thorough evaluation,four experimental scenarios,including diverse temperature and irradiance variation conditions, were employed.Real atmospheric datasets collected from Hong Kong were used to validate the practicability and effectiveness of the devised MPPT control scheme under real-life conditions.For instance, based on the dataset from a spring day in Hong Kong, the energy output facilitated by FDBBWO reached 139.74, 141.32, 209.85, 196.56,153.11, and 140.88 % of those produced by BWO,RSA, P&O, MRA, INC, and EDO, respectively.In general, FDBBWO not only outperformed its competitors in power output but also consistently achieved efficient and stable MPPT under dynamic operating conditions.

· To provide a quantitative multidimensional performance analysis, two additional indicators, ΔVavg and ΔVmax, were introduced to assess the power fluctuations for stability analysis.FDBBWO achieved the smallest ΔVavg and ΔVmax values among the eight approaches.Further,FDBBWO obtained the smallest ΔVavg values of 22.36,11.73,and 18.95%in summer,autumn, and winter, respectively.In conclusion,FDBBWO demonstrated excellent resistance to external disturbances and significantly reduced power oscillations.

Future research directions should focus on the following four aspects:

a) Achieving seamless electrical integration of PV and TEG units to create a unified hybrid system, resulting in cost saving by eliminating the need for separate DC-DC converters.

b) Implementing a dual-input high-gain boost circuit within the hybrid system to increase the power production efficiency and reduce costs related to the boosting process.

c) Expanding the application of FDBBWO-based MPPT controllers for large-scale PV-TEG systems,aiming to enhance the development of extensive grid-connected energy solutions optimized for PVTEG hybrids.

d) Exploring the impact of mixed system models of different sizes, composed of various PV cell types and TEG materials, on their MPPT performance.Further investigation is required in practical applications to achieve optimal performance.

CRediT authorship contribution statement

Bo Yang: Writing - original draft.Boxiao Liang:Methodology. Shaocong Wu: Validation, Software.Hongbiao Li:Data curation.Dengke Gao:Resources.Lin Jiang:Visualization.Jingbo Wang: Writing - review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China(62263014),Yunnan Provincial Basic Research Project (202401AT070344, 202301AT070443).