A Novel Hermite Interpolation-Based MPPT Technique for Photovoltaic Systems Under Partial Shading Conditions

Solar energy is a sustainable and highly promising renewable energy source. The commonly employed Perturbation and Observation (P&O) and Incremental Conductance (INC) methods exhibit advantages such as ease of implementation. However, achieving maximum power through Maximum Power Point Tracking (MPPT) proves challenging under partial shading conditions (PSCs). This paper proposes a novel MPPT based on segmented cubic Hermite interpolation (HPO) to efficiently track the maximum power under all weather conditions. The proposed MPPT is applied to a photovoltaic system comprised of a photovoltaic array and a boost chopper. The feasibility and effectiveness of the proposed HPO algorithm are validated through a comparison with INC and Particle Swarm Optimization (PSO) methods. A solar photovoltaic system based on the Hermite interpolation Maximum Power Point Tracking (HPO-MPPT) algorithm was constructed using MATLAB/SIMULINK software. The system underwent testing under four different lighting conditions, revealing an average tracking efficiency and speed of 99.84% and 0.28 s, respectively, under these conditions. Notably, the proposed method achieved the highest tracking efficiency of 99.99% and the fastest tracking speed of 0.23 s under PSCs.

is influenced by external environmental factors such as solar irradiance, ambient temperature, and load size [3].Furthermore, under partial shading conditions, uncontrollable elements like tree shading, moving clouds, birds, obstructive buildings, dust accumulation, and others affect different modules within the photovoltaic array, resulting in a more complex P-V curve.In such scenarios, the P-V curve may exhibit multiple peaks, with only the highest peak referred to as the Global Maximum Power Point (GMPP).Therefore, tracking the GMPP is crucial for extracting maximum power and enhancing the efficiency of photovoltaic systems.
Classic Maximum Power Point Tracking (MPPT) methods, such as P&O, INC, and Hill Climbing (HC), often encounter challenges in getting trapped in local optima, making it difficult to locate the GMPP.Additionally, a slow tracking speed is another drawback of these methods.Therefore, to locate the GMPP, classical methods typically need to be combined with other approaches.For instance, in [4], the proposal involves partitioning the P-V curve into four regions and using a variable step-size P&O to search the operating region until the GMPP is found.In [5], authors employ a similar approach, combining Golden Section Search (GSS) with INC.Meanwhile, in [6], an improved variable step-size tracking strategy is introduced to reduce power losses caused by step-size oscillations, though tracking inefficiencies persist under partial shading conditions.In [7], the authors adopt Iterative Learning Control (ILC) to periodically regulate perturbation changes, overcoming oscillations and errors near the MPP.However, in certain scenarios, these approaches may inadvertently neglect the identification of the GMPP, resulting in convergence at the LMPP.Furthermore, classical MPPT methods are unable to respond immediately to environmental changes.To address these issues, several emerging intelligent algorithms have been proposed.For instance, Introducing the regulation of human psychological states into MPPT control algorithms, a Human Psychology Optimization control algorithm is proposed, which adjusts the system externally to maintain it in the optimal operating state [8].In [9], Particle Swarm Optimization (PSO) is introduced into the control of MPPT under partial shading conditions, successfully achieving MPP tracking and thereby improving output efficiency.In addressing the steady-state power oscillation issue within conventional PSO algorithms, [10] introduces a solution.The proposed enhancement involves an improved PSO algorithm, which mitigates steady-state oscillations.However, it does not entirely alleviate the prolonged convergence time associated with traditional PSO algorithms.In [11], the authors combine the concepts of circles and their centers with the P&O algo rithm to address the problem of excessive iteration counts.In [12], the authors introduce a Radial Basis Function Network (RBFN) model for Maximum Power Point Tracking (MPPT) control, utilizing a PSO approach and integrating it with Artificial Neural Networks (ANN) [13].This methodology significantly improves the tracking speed of the MPPT algorithm.Nevertheless, it necessitates a considerable volume of experimental data for effective training.
In scientific research, data fitting and approximation methods are commonly employed to define functions for various phenomena.Curve fitting and interpolation are two prevalent approximation techniques, occasionally used in papers to identify the maximum power point of photovoltaic systems.In [14] and [15], the authors utilize curve fitting methods to locate the MPP.The approach proposed in [16] incorporates the Newton interpolation method to assist conventional PSO.Although it accurately identifies the GMPP, the high number of iterations results in low efficiency.In [17] and [18], parabolic and Lagrange interpolation methods are employed, respectively, for data prediction, with the objective of estimating the MPP through collaboration with other MPPT.However, both approaches exhibit a deficiency in precision, failing to accurately approximate the P-V curve.Furthermore, the system experiences pronounced oscillations during steady-state operation.
In this study, the segmented cubic Hermite interpolation method [19] is employed for photovoltaic MPPT, complemented by INC and P&O algorithms, consequently creating an innovative hybrid MPPT algorithm.The primary features and benefits of the proposed algorithm are outlined below: 1) In comparison to MPPT algorithms employing various optimization methods, which exhibit high tracking speeds but complexity, the proposed method offers a simpler structure.It efficiently and rapidly identifies the GMPP using classical MPPT techniques and mathematical computations.
2) The presented approach requires only voltage and current data from the system, eliminating the need for additional parameters.This method, in contrast to other hybrid approaches, boasts a more straightforward structure.For example, compared to the RBFNN method, HPO only requires sampling of six initial voltage and current parameters, while RBFNN requires a dataset that fully covers the input space.Moreover, the faster the environmental changes, the more sample data is needed.3) Within the proposed approach, the INC promptly identifies the vicinity of the MPP.Simultaneously, cubic Hermite interpolation adeptly manages the MPP region, restricting the analysis to this specific area.This proficiency empowers the proposed technique to extract maximum power both effectively and efficiently from the photovoltaic array.The subsequent sections of this paper are organized as follows: Section II furnishes an overview of photovoltaic systems and partial shading conditions.Section III elucidates the fundamental concepts of Hermite interpolation.Section IV

II. PV SYSTEM MODELING AND PARTIAL SHADING
The Single Diode Model (SDM), owing to its exceptional accuracy and structural simplicity, is extensively employed in photovoltaic cells.Comprising a solitary diode, an enduring current source, and a pair of resistors, this model embodies a multitude of advantages.The equivalent model is depicted in Fig. 1.
In the given equation, I 0 represents the reverse saturation current of the diode, U is the solar cell output voltage, α stands for the ideality factor, R h signifies the parallel resistance, R s denotes the series resistance and U th represents the thermal voltage.The terms I pv,n is the short-circuit current measured under standard test conditions ( Variable K I is the short circuit current coefficient, which is usually provided by the manufacturer.Furthermore, T and G respectively denote the operating temperature and irradiance level. A single solar cell can generate only about 0.5 V of voltage.To meet the requirements of higher power and voltage levels, it is necessary to connect several individual cells in parallel or in series to form a photovoltaic array [22].Equation (1) can be modified as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I CHARACTERISTICS OF CRM60S125S
Here, N m and N p represent the quantities of cells connected in series and in parallel.( 3) is a less frequent application in practical scenarios.Instead, the analysis of operational characteristics in photovoltaic arrays commonly depends on the engineering mathematical model for photovoltaic cells [23].This can be modified to (4).In the equation, I mpp , U mpp , U oc , and I sc represent the technical parameters under standard conditions, specifically denoting the current and voltage at the maximum power point, open-circuit voltage, and short-circuit current.
For simulation purposes, the CRM60S125S photovoltaic module was selected.Table I presents the specific parameters of the CRM60S125S photovoltaic panels used in the photovoltaic system.Under uniform shading conditions, each photovoltaic cell within the photovoltaic array receives the same level of irradiance.The output characteristics of the photovoltaic array, represented by the P-V and I-V (current-voltage) curves, exhibit a single-peak curve as shown in Fig. 2(a).
In practical applications and daily life, photovoltaic arrays frequently experience partial or uneven shading caused by factors like cloud cover or other obstructions.When encountering partial shading in the photovoltaic array, the parallel bypass diodes transition between blocking and conducting states.In such instances, (4) becomes inapplicable.For illustrative purposes, we construct the photovoltaic array depicted in Fig. 3. Due to the proportional relationship between the short-circuit current of the photovoltaic array and the illumination intensity, the short-circuit currents of PV1, PV2 and PV3 adhere to I sc1 > I sc2 > I sc3 .When the output current of the photovoltaic array satisfies I sc2 < I ≤ I sc1 , both parallel bypass diodes of PV2 and PV3 conduct, resulting in the output characteristics of the photovoltaic array being identical to that of PV1.When the output current of the photovoltaic array satisfies I sc3 < I ≤ I sc2 , the parallel diode of PV2 undergoes reverse voltage and enters cutoff, allowing simultaneous operation of PV1 and PV2.Similarly, when 0 < I ≤ I sc3 , the parallel diodes of PV2 and PV3 bear reverse voltage and go into cutoff, enabling simultaneous operation of PV1, PV2 and PV3.Consequently, the output characteristics of the photovoltaic array under local shading conditions can be expressed as: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where, C 1 , C 2 are structural coefficients of the photovoltaic cells.Under these conditions, the photovoltaic array's output characteristics are presented in Fig. 2(b), revealing a multi-peak characteristic in its P-V curve under partial shading.The array experiences three different illumination intensities, resulting in three peaks; however, only one GMPP is present.Further generalization suggests that for a photovoltaic array exposed to N intensities of illumination, there will be N-1 LMPP along the P-V curve, with a singular GMPP.The simulation model in Fig. 3 aligns with theoretical expectations and is well-suited for verification.Based on this model, the study conducts simulations and validations for various MPPT algorithms and the proposed hybrid algorithm under partial shading conditions.

III. BASIC CONCEPTS OF HERMITE INTERPOLATION
Interpolation is a fundamental and practical technique in numerical analysis.In practical scenarios, discrete data is frequently encountered.Interpolation facilitates the transformation of such data into functional expressions exhibiting specific relationships [24].The curve produced through segmented cubic Hermite interpolation remains continuous and smooth at every data point.This smooth characteristic enhances the naturalness of the interpolation outcome, thereby preventing abrupt fluctuations or spikes.Furthermore, segmented cubic Hermite interpolation is versatile in accommodating various quantities and distributions of data points.This method is suitable for diverse scenarios, regardless of whether the datasets are dense or sparse.
Given a set of n+1 observation points, the function f (x) can be expressed as an interpolation polynomial of degree at most n, as represented by (6).
By considering the interpolation conditiony i = P (x i )(i = 0, 1, . . ., n), it is possible to derive a system of linear equations with respect to the coefficients a n , a n−1 , . . ., a 1 , a 0 : ) By utilizing (7), the coefficient matrix of the equation system Ax = b can be obtained, where det(A) represents the Vandermonde determinant.When dealing with distinct nodes, the uniqueness of the coefficients for this interpolation polynomial can be established using Cramer's rule.
Given the interpolation nodes x i (i = 0, 1, . . ., n) on the interval [x 1 , x n ], where y = f (x), the function values and first derivative values at x i are as follows: Furthermore, the polynomial H(x) satisfies the following conditions: (9) and ( 10) enable the construction of Hermite polynomials using a maximum of 2n+1 terms: (11) satisfies the interpolation condition: The decomposition of (13) reveals that x i is a first-order zero of β i , whereas x k (k = i) is a second-order zero of β i (x i ): The Lagrange basis functions are defined as follows: Given a polynomial of degree at most 2n+1, denoted as: Combine ( 15) and ( 17) to obtain: According to (18), a can be defined as (19): where a, b are specific constants.Substituting ( 12) into (19) yields: Derivation of ( 16) gives: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.( 19) to ( 21) can be combined to form: By utilizing (11) to (22), Hermite interpolation polynomials can be derived: Define a set of segmented cubic interpolating basis functions α j (x) and β j (x) on the interval [x 0 , x 1 ], where j ranges from zero to one.These basis functions will be used to construct the cubic Hermite interpolating polynomials: ) Now, the unknown coefficients of the Hermite function within the interval [x 0 , x n ] can be determined using (23).In this paper, the process is elucidated using six sample data sets: {(x 0 , y 0 ), (x 1 , y 1 ), (x 2 , y 2 ), (x 3 , y 3 ), (x 4 , y 4 ), (x 5 , y 5 )} (25) Typical P-V curves and interpolated polynomial curves are shown in Fig. 4. For this study, six sample points were selected from the P-V curves and approximated using both polynomial interpolation and cubic Hermite interpolation methods.The selected sample points are as follows:  Next, the obtained results are substituted into (6) to derive the fourth-degree polynomial interpolation, given by: f (x) = −1.66×10−3 x 4 + 0.14x 3 −3.77x 2 +39.32x−12.93 The segmented cubic Hermite interpolation method utilizes five cubic functions to interpolate between six sampling points.The interval is divided into , and the resulting cubic Hermite interpolation polynomials are expressed by (11) to (25):  4, polynomial interpolation can pass through all sample points and is computationally straightforward.However, it is unsuitable for P-V curves as it may produce unstable results in certain intervals.Linear interpolation is limited to performing calculations within specific intervals, necessitating a substantial number of sample points.Furthermore, interpolation calculations within the interval result in a non-smooth curve due to the fixed slope.Conversely, segmented cubic Hermite interpolation accurately approximates the actual P-V curve in all intervals without exhibiting any extreme phenomena.

IV. COMPARISON OF MPPT METHODS IN THE CASE OF PARTIAL SHADING
In typical scenarios, the output voltage of photovoltaic components is unstable and challenging to control.To maintain the stability of the direct current (DC) output voltage, it is necessary to introduce a DC-DC converter [25] in the photovoltaic system for voltage boosting and MPPT adjustment [26].Using the structure illustrated in Fig. 5, the segmented cubic Hermite algorithm transfers the output power of the photovoltaic array Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.to the load.This algorithm only requires sampling of the output voltage and current of the photovoltaic array, and then applies the duty cycle to the DC/DC converter.It possesses a simple structure and lower cost.

A. System Design
The performance of MPPT methods varies under different direct current loads, such as resistive loads or battery loads [27], [28].For a boost converter with a fixed output load, the duty cycle can be calculated using (26).
To maximize the output power of photovoltaic components, the system should operate at their MPP.Solar radiation and module temperature are the primary factors influencing MPP, and these parameters vary over time.The MPPT method of a photovoltaic system matches the internal resistance R mpp and output resistance R o at the MPP of the photovoltaic array, allowing it to find the MPP under varying irradiance and temperature conditions.
The internal resistance at the MPP of the photovoltaic array is calculated using (27).
If the appropriate output resistance is not selected, it will be impossible to locate the MPP.However, when implementing the MPPT method using the structure depicted in Fig. 5, it can only meet the system's requirements under specific conditions.Under uniform illumination conditions, the internal resistance of the photovoltaic cell at the maximum power point varies within the range shown in Fig. 6 as the irradiance changes.In Fig. 6, it is evident that a decrease in irradiance (G) from 800 W/m 2 to 400 W/m 2 shifts the MPP from point A to point B, necessitating an adjustment in the output resistance (R o ).However, manually changing the load resistance when solar irradiance fluctuates is impractical.Therefore, the MPPT algorithm is formulated to iteratively adapt the duty cycle of the boost converter.This ensures alignment between the output resistance and the internal resistance of the photovoltaic array at the MPP, facilitating the extraction of maximum power.The duty cycle in this context is determined through the application of (28).
From (28), it is evident that, for tracking the MPP, the output resistance of the boost converter must be greater than or equal to the internal resistance of the photovoltaic array at the MPP (R o ≥ R mpp ).This guarantees that the duty cycle fluctuates between zero and one.

B. Incremental Conductance Method
The main principle of the INC method is that when the photovoltaic array operates at the MPP, the slope of the P-V curve is zero [29], satisfying (29).As shown in Fig. 2(a), the derivative values to the left of the maximum point are greater than zero, while those to the right are less than zero, corresponding to the output.
According to this principle, small variation values are employed to determine the current operating side concerning the MPP.This ensures the stabilization of the solar cell array within the proximity of the MPP, avoiding fluctuations around it.If the operation is on the right side of the MPP, the variation value is negative.To approach the MPP, the reference output current increases, leading to a reduction in the direct current bus voltage.Conversely, if the operation is on the left side of the MPP, the variation value is positive.To approach the MPP, the reference output current decreases, resulting in an increase in the direct current bus voltage and ultimately achieving a stable operating point.

C. Particle Swarm Optimization
The Particle Swarm Optimization (PSO) algorithm is an effective method for globally optimizing multi-extremal functions, utilizing collective intelligence generated through cooperation and competition among particles in the swarm [30].In each iteration, particles update their positions and velocities using two extreme points: one is the best solution found by the particle itself up to the current moment, denoted as P best , and the other is the best solution found by the entire swarm up to the current moment, denoted as G best , until the optimal solution is reached.In the context of MPPT in photovoltaic arrays, the objective function is the measured output power.The particle positions represent the array voltage.Assuming there are N particles, the positions of the particles correspond to the array voltages U 1 , U 2 , . . ., U N .The updating equations for the velocity v k+1 i and position x k+1 i of the i − th particle in the k + 1 iteration are given by ( 30) and (31).
where, i = 1, 2, . . ., N, N is the total number of particles in the population; k represents the iteration count; w signifies the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

D. Hermite Interpolation
Under uniform irradiance conditions, the segmented Hermite interpolation process employed in the MPPT algorithm is illustrated in Fig. 7. Initially, the algorithm rapidly navigates the vicinity of the maximum power point utilizing the INC method.To facilitate seamless integration with the P&O, the region T, encompassing the MPP subsequent to the application of the INC method, is designated as the interpolation region.Once within this interpolation region T where both voltage and current closely approach the maximum power point on the P-V curve, five sample points (U 1 , I 1 ), (U 2 , I 2 ), (U 3 , I 3 ), (U 4 , I 4 ), (U 5 , I 5 ) are gathered for the corresponding voltage and current.Subsequently, their powers are computed, adhering to the equation: After collecting the required samples, the segmented cubic Hermite interpolation is employed to fit the P-V curve, aiming to determine V mpp and P mpp .The duty cycle D mpp at the maximum power point is calculated using V mpp and P mpp .Simultaneously, a variable step-size P&O method is applied to iteratively refine the computed duty cycle D mpp .The algorithm halts when the optimal duty cycle is detected.If external conditions remain unchanged, the same duty cycle is maintained.However, in the event of external changes, causing a shift in the system's maximum power point, as indicated in the last  For the MPPT of a photovoltaic array under partial shading conditions, a more refined MPPT is required to locate the GMPP accurately.The flowchart in Fig. 8 illustrates the MPPT algorithm for the system under partial shading conditions, as depicted in Fig. 3.In the initial step, the algorithm swiftly tracks the interpolation region using the INC method.Subsequently, the converter is applied with three sampled duty cycles, ensuring D 1 < D int < D 5 .Then, the voltage and current are measured for each data set, and the data set with the maximum power is selected as [D max , P max ].Additionally, three adjacent samples ⎧ ⎨ ⎩ Next, duty cycles D 1 to D 5 are applied in segments using (25) to compute the Hermite interpolation function.Subsequently, an interval [D mpp , P mpp ] can be extracted from the resulting function.Furthermore, a refined P&O with a small step size is employed to fine-tune D mpp , and P mpp corresponding to D mpp is extracted from the actual P-V curve.Then, D mpp is applied to the converter.In the final step, when the sampled power is less than or equal to the set value k, the system reaches the optimal D mpp .At this point, the P&O ceases to prevent steady-state oscillations.
When the operating temperature of the photovoltaic array or the received illumination intensity changes, the position of the MPP also correspondingly shifts.Therefore, a metric is defined, setting the power before and after the environmental change as P s1 and P s2 respectively.When the metric satisfies (34), the algorithm is reinitialized to rediscover the MPP.To achieve this, the algorithm periodically monitors the changes in U pv and I pv to detect variations in external conditions.
V. SIMULATION AND RESULTS Extensive simulation studies have been conducted to assess the performance of the proposed MPPT algorithm.In this section, a comparative analysis of the HPO under various weather and operational conditions is presented through simulation.As depicted in Fig. 5, the photovoltaic array consists of three seriesconnected CRM60S125S photovoltaic modules, and detailed parameters of the DC-DC converter are listed in Table II.Due to the complexity of evaluating all non-uniform climatic conditions, four different shading scenarios, as shown in Table III , were considered.For instance, the illumination time for the first scenario ranges from t = 0 s to t = 2 s, with a duration  of two seconds for each scenario.Additionally, under different conditions, the corresponding temperatures will also undergo changes.These alterations in conditions will significantly modify the GMPP.Simulation experiments were performed on the model shown in Fig. 3 to obtain its output characteristic curves, as illustrated in Fig. 9.
In the comprehensive evaluation of the proposed algorithm's performance, a comparative analysis was undertaken, assessing HPO in comparison with other MPPT algorithms, including INC and PSO.The simulation results of the INC method are shown in Fig. 10(a), and those of PSO are illustrated in Fig. 10(b).For the four illumination scenarios presented in Table III, the simulation results of HPO are depicted in Fig. 11.
In the first scenario, the proposed algorithm initially enters the interpolation region and applies initial duty cycle-sampled voltage and current within the interpolation region.The duty cycle in Fig. 11 decreases from D 1 = 63% to D 5 = 38%, with a reduction magnitude of 5%.At this point, the maximum duty cycle is D 3 = 48%.Subsequently, HPO utilizes these six samples to compute segmented Hermite functions, entering the refinement stage after extracting the maximum value from the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Hermite functions.In the refinement stage, HPO employs a small step P&O method to apply voltage to the boost converter, ultimately yielding P mpp = 74.14W and D mpp = 47.46%.HPO tracks the GMPP within 0.23 seconds, exhibiting a remarkably high efficiency (99.99%).In comparison, INC surpasses the proposed method in tracking speed but lags in tracking efficiency (95.01%).PSO approaches the accuracy of the proposed method in tracking MPP, but requires a significant amount of computation in the initial stages, resulting in a tracking time for GMPP that is five times slower than HPO.During the tracking process, both HPO and INC consistently maintain proximity to GMPP without fluctuations.In contrast, PSO oscillates near GMPP, leading to a substantial loss in efficiency.At t = 2 s and t = 4 s, the system is in the second and third illumination scenarios, where both illumination and temperature undergo simultaneous changes.These methods detect environmental changes and initiate a new search for the GMPP.During this period, the tracking efficiency of INC and PSO significantly decreases, both falling below 90%.In contrast, the proposed algorithm continues to rapidly track the GMPP and maintain high efficiency (above 99%).In the fourth scenario, under uniform irradiance conditions with an intensity of 1000 W/m 2 .At this point, the tracking speed of INC is the fastest among the three algorithms, with an HPO tracking time of around 0.3 seconds and an efficiency of 99.98%.Thus, under uniform illumination conditions, all three MPPT algorithms demonstrated good tracking performance.The tracking results of the three algorithms are summarized in Table IV.

TABLE IV COMPARISON BETWEEN HPO, PSO AND INC BASED MPPT ALGORITHMS UNDER DIFFERENT SCENARIOS
The examination of experimental outcomes reveals that, when subjected to partial shading conditions, both the INC and standard PSO algorithms occasionally encounter challenges in tracing the GMPP.In such instances, they may inadvertently converge into LMPP, impeding their ability to escape.This ultimately leads to incorrect results, resulting in low tracking accuracy and decreased output efficiency of the photovoltaic array.The proposed algorithm, in contrast, adeptly traces the GMPP even in the presence of partial shading conditions.It avoids premature convergence, mitigates the risk of entering LMPP, and demonstrates a comparatively swift tracking speed.

VI. CONCLUSION
Partial shading in a photovoltaic array often causes traditional MPPT control methods to converge to LMPP, leading to a reduced overall system efficiency.This paper proposes a MPPT technique based on segmented cubic Hermite interpolation, applying the concept of Hermite interpolation to MPPT to assist classical INC and P&O algorithms.The algorithm utilizes a fixed-step INC algorithm to enter the interpolation region, applying the initial duty cycle sample set to the boost converter.It calculates a cubic interpolation function using the initial maximum power point to search for the global maximum power point.To enhance system accuracy, a small-step P&O refinement is employed.This algorithm effectively prevents the system from getting trapped in local maximum power points and rapidly tracks the maximum power point under both uniform illumination and partial shading conditions.Finally, by comparing segmented Hermite interpolation with INC and PSO algorithms, the superiority of the HPO algorithm is demonstrated.
In the continuation of this research, the interpolation function will be optimized using different algorithms to enhance the system tracking speed.Furthermore, with the refinement of the algorithms, a control method applicable to grid-connected systems will be developed.

Fig. 2 .
Fig. 2. P-V and I-V output characteristics of CRM60S125S under various environmental conditions.

Fig. 6 .
Fig. 6.Curve depicting the range of internal resistance at the MPP.

TABLE II SPECIFICATIONS
OF THE DC-DC BOOST CONVERTER