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Scientific Reports volume 16, Article number: 9991 (2026) Cite this article
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In this study, an advanced maximum power point tracking (MPPT) control strategy is proposed for a grid-connected photovoltaic (PV) system using the Hippopotamus Optimization Algorithm (HOA). The Incremental Conductance (IC) MPPT technique is integrated with three control approaches: Integral (I), Proportional-Integral (PI), and Fractional-Order Proportional-Integral (FOPI) controllers. The HOA is employed to optimally tune the controller parameters, and its performance is benchmarked against two other nature-inspired algorithms: the Arithmetic Optimization Algorithm (AOA) and the Grey Wolf Optimizer (GWO). A 100 kW grid-tied PV system connected to a medium-voltage distribution network is modeled and simulated in MATLAB/Simulink 2025a. The optimization process aims to minimize four classical performance indices: IAE, ISE, ITAE, and ITSE. Simulation results demonstrate that the HOA-based FOPI-IC-MPPT configuration achieves superior dynamic performance, exhibiting a minimum rise time of 0.0073 s and a maximum extracted power of 100.72 kW. Under the IAE criterion, compared to AOA and GWO, the proposed method reduces the rise time by 9.88% and the settling time by 19.73%. Although the GWO-based controller outperformed in certain metrics (e.g., ISE), the HOA-based approach achieved a better trade-off between dynamic response and maximum power tracking accuracy, making it a promising solution for real-time grid-connected PV applications under variable environmental conditions.
The world is undergoing a rapid and urgent shift toward renewable energy for electricity generation, largely driven by its environmental advantages especially the fact that it doesn’t produce carbon dioxide (CO₂) emissions and the fact that these sources are far more abundant than fossil fuels¹. By 2021, the total installed capacity of renewable energy had reached 3,064GW². That year alone saw an additional 257GW of new renewable energy capacity added². Among all the green technologies, solar photovoltaic (PV)³. power led the way, contributing 133 GW of the new installations, according to data from the International Renewable Energy Agency (IRENA)². A major problem with PV systems is the nonlinear behavior of current-voltage (I-V) and power-voltage (P-V) curves⁴. To ensure maximum power extraction from the PV system, the PV array voltage must be regulated to the pre-determined maximum power point voltage⁵. Also, the changing environmental conditions such as solar irradiance and ambient temperature have significant effects on the power output of these systems. Consequently, correct monitoring of the maximum power point (MPPT) under different weather conditions is essential to guarantee a system captures as much energy as possible. This renders application of effective and accurate MPPT techniques critical. There are four broad groups of MPPT techniques: conventional techniques, artificial intelligence (AI)⁶. based techniques, optimization-based techniques, and hybrid techniques. The most widely used conventional MPPT methods include Incremental Conductance (IC)⁶. Fractional Short-Circuit Current (FSCC), Fractional Open-Circuit Voltage (FOCV), and Perturb and Observe (P&O)⁷. AI-based MPPT methods typically employ Artificial Neural Networks (ANN) and Fuzzy Logic Control (FLC)^8,9. Optimization-based MPPT techniques utilize algorithms such as Harris Hawks Optimization (HHO)¹⁰, Improved Grey Wolf Optimizer (IGWO)¹¹, and Enhanced Squirrel Search Algorithm (ESSA)^12,13. Hybrid MPPT techniques integrate conventional and AI-based approaches to leverage the advantages of both or combine AI techniques with metaheuristic optimization algorithms to enhance tracking precision and convergence speed^9,10. Hence, Fractional-Order Proportional-Integral (FOPI)¹⁴. Controllers are a prominent choice in industrial applications due to their flexible structure and superior performance compared to classical PID variants such as PI or PD^15,16. However, the accurate selection of the optimal parameters, namely the proportional gain (:{K}_{P})​, integral gain(::{K}_{I})​, and the fractional order λ is essential to ensure system stability and dynamic efficiency. In recent years, numerous optimization algorithms have been employed to optimally tune FOPI controllers, including the Marine Predators Algorithm (MPA), Particle Swarm Optimization (PSO)¹⁷. Grey Wolf Optimizer (GWO), Gas Solubility Optimization (GSO)¹⁸.Cheetah optimizer (CO)¹⁹, Grey Wolf Optimizer-Particle Swarm Optimization (GWO-PSO)²⁰. Grasshopper Optimization Algorithm (GOA)²¹. Ant Lion Optimization (ALO)²², Genetic Algorithm (GA)²³, arithmetic optimization algorithm (AOA)²⁴. and the Improved Artificial Bee Colony (IABC) algorithm²⁵, which have been successfully applied across various engineering domains. This study addresses the limitations of existing MPPT tuning methods by introducing the Hippopotamus Optimization Algorithm (HOA) for optimal tuning of IC-based I, PI, and FOPI controllers in a grid-connected photovoltaic system, achieving improved convergence speed, tracking accuracy, and robustness under dynamic operating conditions.
Table 1. provides a comparative analysis of the performance of the Incremental Conductance (IC) maximum power point tracking (MPPT) method when integrated with Integral (I), Proportional-Integral (PI), and fractional-order PI (FOPI) controllers. Accordingly, this study introduces a novel Hippopotamus Optimization Algorithm (HOA). to optimize the parameters of the IC-MPPT method under three distinct control strategies: I, PI, and FOPI. These strategies are implemented within a 100 kW grid-connected photovoltaic system. The proposed method aims to enhance tracking performance, reduce steady-state oscillations, and improve the dynamic response under varying solar irradiance and temperature conditions. By optimally tuning each controller configuration, the HOA-enhanced IC MPPT method demonstrates faster convergence, higher tracking efficiency, and improved stability compared to conventional MPPT techniques.
The key strength of this optimization framework is its capacity to explore a broad range of possibilities when tuning controller parameters. In this study, four widely used performance metrics were applied to identify the most effective control setup²⁶: the Integral of Time-weighted Absolute Error (ITAE), Integral of Absolute Error (IAE), Integral of Time-weighted Squared Error (ITSE), and Integral of Squared Error (ISE).
The core contributions of this research can be outlined as follows:
To the best of the authors’ knowledge, this is the first reported implementation of the HOA for the optimal design of a FOPI controller within an IC MPPT framework for a grid-connected PV power system.
To identify the most effective configuration, all four standard performance metrics IAE, ISE, ITAE, and ITSE were evaluated separately, followed by an overall comparison to determine the optimal control approach.
The proposed HOA-FOPI-IC control strategy was tested on a 100 kW grid-connected PV system, demonstrating its practical suitability for real-world solar energy applications.
Comprehensive testing under various weather conditions, including both minor and major fluctuations, was conducted to assess its tracking accuracy and dynamic stability.
In addition, a comparative study against other optimization techniques, including the Grey Wolf Optimizer (GWO) and the Arithmetic Optimization Algorithm (AOA), highlighted the HOA-based controller’s advantages in convergence speed, tracking performance, and overall robustness.
Photovoltaic (PV) modeling plays a crucial role in analyzing and simulating the performance of solar energy systems under various operating conditions. Several electrical equivalent circuit models have been proposed in the literature, such as the single-diode model (SDM), double-diode model (DDM), and triple-diode model (TDM)^36,37. Among these, the SDM is the most widely adopted owing to its balance between simplicity and accuracy, as it requires only a limited number of equivalent circuit parameters to be determined³⁸ Figure 1. illustrates the equivalent circuit of the PV cell based on the SDM.
The current-voltage characteristics of a PV module consisting of Nₛ series-connected cells can be described mathematically by the SDM as given in Eqs. (1)-(6)^39,40,41.
One-diode equivalent circuit of theoretical and practical PV cells.
where (:{I}_{ph})denotes the photogenerated current, (:{I}_{d})represents the Shockley diode current, and (:{I}_{sh})is the current flowing through the shunt resistance. (:{I}_{o})refers to the diode saturation current, (:V)is the terminal voltage, and (:{R}_{s})and (:{R}_{sh})denote the series and shunt resistances, respectively. The diode ideality factor is given as (:a=0.94504), while (:{N}_{s})represents the number of cells connected in series. The thermal voltage is denoted by (:{V}_{t}). (:K)is the Boltzmann constant (:left(1.38times{10}^{-23}text{J/K}right)), (:{T}_{c})is the PV cell temperature in Kelvin, and (:q)is the electron charge (:left(1.6times{10}^{-19}text {C}right)). (:G)denotes the solar irradiance. (:{I}_{scn})is the short-circuit current under standard test conditions (STC), denoted by the subscript (:n), corresponding to (:{G}_{n}=1000{text{W/m}}^{2})and (:{T}_{cn}={25C}^{^circ:}). (:{K}_{T})represents the temperature coefficient of the short-circuit current. (:{E}_{g})is the bandgap energy of polycrystalline silicon, equal to 1.12 eV at (:{25C}^{^circ:}), and (:{V}_{ocn})denotes the open-circuit voltage under STC.
The capacity of the PV system under study is 100 kW. The PV array is composed of five parallel-connected strings, each consisting of sixty-six series-connected modules of type SunPower SPR-305E-WHT-D. To ensure maximum power extraction, a FOPI regulator based on the IC MPPT technique is employed. The FOPI controller parameters are optimally tuned using the HOA to minimize the conductance error and accurately determine the MPP. Consequently, the FOPI regulator adjusts the duty cycle of a 500 V boost converter, which is cascaded with a grid-tied inverter connected to the medium-voltage utility grid. Fig. 2 illustrates the schematic block diagram of the proposed framework. Furthermore, the I-V and P-V characteristics under different climatic conditions are depicted in Fig. 3 and 4 respectively. demonstrating that the PV output is highly dependent on both irradiance and temperature⁴².
The entire system is under MATLAB simulation.
Characteristics of the used array under different irradiance levels.
Characteristics of the used array under different irradiance levels.
The IC technique is one of the most widely used traditional algorithms for MPPT. This method identifies the MPP by comparing the instantaneous conductance (:(I/V)) with the incremental conductance (:(dI/dV)) ^43,44. According to this principle, the PV array reaches its MPP when (:(dI/dV)=-(I/V)), as expressed in (7). If (:(dI/dV)>-(I/V)), the operating point lies to the left of the MPP. Conversely, when (:(dI/dV)<-(I/V)), the operating point is on the right side of the MPP, as illustrated in Fig. 5.
The process of the INC algorithm.
FOPI controllers have gained widespread popularity in power electronic and industrial applications due to better flexibility and higher robustness compared to classical I and PI controllers in Eq. (8). Unlike classical controllers, FOPI has fast convergence around the reference point and higher steady-state accuracy and is highly successful in nonlinear and time-varying systems. An optimally tuned FOPI controller ensures optimal values of proportional gain (K_P) and integral gain (K_I) and the fractional order (λ) that provide an extra degree of freedom to maximize dynamic response and stability. Various optimization algorithms are employed by researchers to tune FOPI controllers, such as the whale optimization algorithm, genetic algorithm, cuckoo search, and Artificial Bee Colony, in a trial to reduce errors in PV MPPT methods. The overall process of optimal controller design using biological optimization algorithms is illustrated in Fig. 6.
Optimal cost function minimizes error signal (:eleft(tright)) resulting from the IC approach such that optimal MPPT performance utilizing the four standard indices IAE, ISE, ITAE, and ITSE is realized and applies mathematical model in Eq. (9) ⁴⁵ to graphically demonstrate superior effectiveness of resultant developed FOPI-based control method.
where(::{t}_{ss}:)is the steady state response and
.
Use of biological algorithms for the optimal design of MPPT controller.
Launched in 2024⁴⁶. As a novel and original approach to metaheuristic optimization, the HOA sees its first usage. Relying on what hippopotamuses do when navigating groups, defending themselves against threats, and when fleeing away super quickly, the algorithm inquires these actions to guide the process of optimization. As a population base method, the use of HOA is effective in regard to exploration and exploitation, making it a useful method of solving complex and multi variable optimization problems.
The behavioral model of HOA is based on three key phases observed in hippopotamuses:
Exploration phase: Hippopotamuses update their position in rivers or ponds while interacting with the dominant male and the rest of the herd.
Defense phase: In response to predator threats, a hippopotamus may turn aggressively and attempt to repel the attacker.
Exploitation (Escape) phase: If defense fails, the hippopotamus retreats rapidly toward a safer location (typically water).
These natural instincts are mathematically encoded to perform global and local research effectively during the optimization process. This sequence of behavioral strategies is visually illustrated in Fig. 7.
(a–d) Samples of strategies that a hippopotamus utilizes against a predator.
Let the search space be bounded between a lower bound(:left(text{l}text{b}right)) and upper bound(:left(text{u}text{b}right)). The position of each hippopotamus (candidate solution) is initialized as.
Initialization.
Let the search space be bounded between a (:lb) and (:ub). The position of each hippopotamus (candidate solution) is initialized as:
The population matrix is:
Phase 1: Position Update (Exploration).
Male hippopotamuses update their positions relative to the dominant individual as:
Female or immature hippopotamuses may drift from the group as
Phase 2: Defensive Behavior (Enhanced Exploration).
If the predator is too close, the hippopotamus attempts to defend, as shown in Fig. 8:
Graphic representation of phase2.
Phase 3: Escaping from Predators (Exploitation)
To escape, a hippopotamus moves locally around its current position in [0,1] respectively; ϑ is a constant (ϑ = 1.5), (:{Gamma:}) is an abbreviation for Gamma function and (:{sigma:}_{w}) can be obtained by Eq. (19), as shown in Fig. 9.
Depiction of an incremental hippopotamus escaping away from its menacing predator.
The steps of HOA can be summarized as in Fig. 10.
Step-by-step flowchart of the HOA algorithm for metaheuristic optimization.
The Grey Wolf Optimization (GWO) algorithm, introduced by Mir Jalili et al. in 2014⁴⁷. is a nature-inspired metaheuristic that draws on the social behavior and hunting tactics of grey wolves. It replicates how wolves work together to hunt prey and follow a structured leadership system. In a typical wolf pack, roles are divided into four hierarchical levels: the alpha (α) serves as the leader, followed by the beta (β), then the delta (δ), with the omega (ω) occupying the lowest rank. The step-by-step process of the GWO algorithm is shown in Fig. 11.
To ensure optimal performance, the maximum number of iterations ((:text{M}_text{I}text{t}text{e}text{r})) and the number of search agents for each algorithm were selected based on multiple trial runs. Generally, increasing these values leads to more accurate outcomes but at the cost of longer computation time. The lower and upper bounds were initially set using a broad range and then fine-tuned through experimentation to strike a balance between precision and processing efficiency.
Flowchart of the GWO Algorithm.
starting with a wide boundary and changing it if the results were not the best until reaching the suitable boundary. For fair judgment, the same number of iterations, search agent and lower and upper boundaries are selected for each algorithm as demonstrated in Table 2. All optimization algorithms were executed offline to obtain the optimal controller parameters, which were then implemented in the MPPT control loop.
The simulation process follows a structured approach consisting of the following steps:
Choose a suitable number of search agents and iteration counts for each of the selected metaheuristic algorithms HOA, AOA and GWO.
Set the upper and lower limits for the FOPI controller parameters (K_P, K_I, λ) to strike a balance between precision and fast system response.
Use performance metrics such as IAE, ISE, ITAE and ITSE retrieved from the Simulink model as the cost functions for optimization.
Apply the HOA, AOA and GWO algorithms to the PV simulation model, ensuring all defined constraints are respected.
Integrate the optimized parameters produced by the algorithms into the simulation environment.
Evaluate and select the set of controller gains that achieves the most effective overall performance.
The entire simulation framework is visually represented in Fig. 12.
Simulation framework.
The system was assessed under four climate situations, which are shown in every case study. The HOA as well as AOA were applied to the GWO method to select the optimal settings for both I, PI and FOPI controllers based on IAE, ISE, ITAE and ITSE, as presented in Table 3. The approach includes identifying the best index for each algorithm based on how it performs when steady and variable. The best configuration for each algorithm was selected for fair comparison and is presented in Table 4. Among the dynamic response indices, the ISE criterion proved to be the most effective under the HOA-based MPPT technique, achieving the fastest settling time of 0.00069800s, followed by IAE at 0.0006998s, ITAE at 0.0006998s, and ITSE with the slowest response at 0.0009997s. In terms of extracted power, the ISE index yielded the highest value at 100.65 kW, followed by ITSE 100.60 kW, IAE 100.55kWand ITAE 100.52 kW as shown in Fig. 13. Despite these differences in power output, no notable variations were observed among the indices regarding rise time or overshoot. Furthermore, the photovoltaic output voltage demonstrated the highest level of smoothness under the ISE criterion, reaching a minimum of 273.9 V, compared to 273.7 V for ITSE, 273.5 V for IAE and 273.3 V for ITAE, as shown in Fig. 14. Accordingly, the ISE index can be considered the most appropriate criterion for evaluating the performance of the HOA-based MPPT technique. Transitioning to the AOA-based MPPT, the ISE index again demonstrated superior effectiveness by achieving the fastest settling time of 0.0005997s, followed by ITAE at 0.0006995s, ITSE at 0.0059966s and finally IAE with the slowest response at 0.0090909s. The ISE index also yielded the highest extracted power of 100.6 kW, outperforming ITSE 100.4 kW, ITAE 100.3 kW, and IAE 98.5 kW, as shown in Fig. 15. All error-based indices exhibited very similar rise times with only minor differences in overshoot. Regarding the photovoltaic output voltage, the smoothest performance was achieved using the ITSA index, which reached a minimum of 274.5 V, compared to 274.3 V in ITAE, 273.5 V in ISE, and 273 V in IAE, as shown in Fig. 16. Accordingly, the ISE index can be considered the most appropriate criterion for evaluating the performance of the AOA-based MPPT technique. Finally, in the case of the GWO-based MPPT, ITAE demonstrated the lowest settling time, reaching steady-state power at 0.0005993s, followed by ISE at 0.0005995s, ITSE at 0.0005997s, and IAE at 0.0005999s. In terms of extracted power, ITAE yielded the highest value of 99.97 kW, followed by ISE with 99.95 kW, IAE with 96.9 kW, and ITSE with 94.95 kW, as shown in Fig. 17. Although ITAE achieved a slightly shorter rise time than ISE, ITSE reached a steady state faster with a slightly lower overshoot. Furthermore, the PV output voltage was smoothest under the ITAT index, where the minimum voltage was 273 V, followed by ITSA at 272.9 V, ISE at 272.75 V and IAE at 272.65 V, as shown in Fig. 18. Accordingly, the ITAE index can be considered the most appropriate criterion for evaluating the performance of the GWO-based MPPT technique.
As summarized in Table 5, the proposed HOA-based MPPT achieves the highest maximum power output and efficiency, along with the lowest power loss and tracking error, while maintaining a lower implementation cost compared to recent reference algorithms.
PV power-based HOA.
Voltage of the PV-based HOA.
PV power-based AOA.
Voltage of the PV-based AOA.
PV power-based GWO.
Voltage of the PV-based GWO.
Ramp pattern for both increasing temperature and solar irradiance.
The first scenario considers a constant temperature condition with a step change in solar irradiance, as illustrated in Figs. 19 and 20. Figures 21 and 22 present the PV power and voltage responses obtained using the optimal parameter settings of the HOA, AOA and GWO algorithms. Among these, the PV system employing the AOA algorithm demonstrates the fastest settling time, followed by AOA, whereas the MIC algorithm exhibits the slowest response, characterized by pronounced oscillations and the minimum power output 75 kW, despite achieving the smallest overshoot.
Constant temperature.
Step Irradiance.
PV Output Power.
PV Output Voltage.
Furthermore, the PV voltage obtained using the HOA algorithm exhibits the smoothest profile, followed by AOA whereas the GWO algorithm yields the least stable response. Figure 24 illustrates the utility’s three-phase voltage, which has a peak phase voltage of 20 kV. Figure 23 presents the DC-link voltage, regulated at the reference level of 500 V, which remains nearly constant except for a slight increase at t = 0.9s, corresponding to the step change in irradiance from 0 to 1000 W/m². Figure 25 depicts the HOA-based grid current, which closely follows the irradiance pattern under constant temperature conditions.
DC link voltage.
Grid voltage-HOA.
Grid current-HOA.
Constant temperature with a stepwise varying solar irradiance pattern.
bn In the second scenario, both solar irradiance and temperature vary gradually in a ramp-like manner as shown in Figs. 26 and 27. The corresponding dynamic responses of PV output power and voltage obtained using the HOA, AOA and GWO algorithms are presented in Figs. 28 and 29. The results show that PV output power increases proportionally with irradiance and decreases inversely with temperature. At t = 0:0.5s, the minimum PV power achieved by the HOA algorithm is 99.9 kW followed by AOA with 99.5 kW and GWO with 93 kW. Moreover, under low-irradiance conditions, the HOA algorithm demonstrates superior performance compared to AOA and GWO. Therefore, HOA consistently outperforms the other algorithms in both steady-state and transient responses.
Ramp temperature.
Ramp Irradiance.
PV Output Power.
PV Output voltage.
Figures 30, 31 and 32 the DC-link voltage, which is consistently held at the reference level of 500 V; the HOA-based grid current, which closely tracks the irradiance pattern at t = 0.16s under stable temperature conditions and exhibits a slight drop between t = 2–2.1 s as the temperature increases; and the HOA grid voltage, which corresponds to a peak phase voltage of 20KV.
DC link voltage.
Grid current-based HOA.
Grid voltage-based HOA.
Constant temperature with different levels of solar irradiance.
In the third scenario, under constant temperature conditions with varying solar irradiance, as illustrated in Figs. 33 and 34. Figure 35 illustrates the dynamic response of the PV system power under the application of HOA, AOA and GWO algorithms. Since the temperature remains constant throughout the test the PV power output directly follows the variations in solar irradiance. All algorithms successfully tracked the power under different irradiance levels; however, the HOA achieved the best performance during the time interval t = 1.2:1.3s, followed by AOA and finally GWO respectively.
Constant temperature.
Different Irradiance.
PV Output Power.
Figure 36 illustrates the PV output voltage corresponding to the algorithms. The PV voltage obtained using the HOA algorithm demonstrates the highest level of smoothness and stability, achieving a minimum voltage of 270.3 V followed by AOA at 268.8 V and GWO at 267.2 V respectively. Figures 37, 38 and 39 present the DC link voltage with a reference value of 500 V the utility current derived from the HOA algorithm that follows the irradiance pattern under constant temperature conditions, and the utility voltage exhibiting a peak phase voltage of 20 kV respectively.
PV Output voltage.
DC link voltage.
Grid voltage-based HOA.
Grid current-based HOA.
Varying temperature combined with varying solar irradiance.
In this final scenario, which involves variable temperature and variable solar irradiance as illustrated in Figs. 40 and 41 Figs. 42 and 43 present the dynamic responses of the PV system’s output power and voltage under the HOA, AOA and GWO algorithms. All algorithms successfully tracked the PV power in accordance with variations in irradiance and temperature. The results indicate that the HOA algorithm achieved the fastest power tracking performance, followed by AOA and finally GWO. Moreover, the PV voltage decreases with increasing temperature, demonstrating an inverse relationship between voltage and temperature.
Variable irradiance level pattern.
Variable irradiance level temperature.
PV Output Power.
PV Output voltage.
Figures 44, 45 and 46 respectively illustrate the DC-link voltage at the reference value of 500 V the utility current obtained using the AOA algorithm that corresponds to irradiance and temperature variations, and the utility voltage waveform.
DC link voltage.
Grid voltage-HOA.
Grid current-HOA.
In this study, the HOA was used to fine-tune a FOPI-IC-MPPT controller for a 100 kW grid-connected PV system. The effectiveness of this approach was compared with GWO and AOA across four different climate scenarios: constant temperature with step changes in irradiance ramping irradiance with gradually increasing temperature fluctuating irradiance at a constant temperature, and both irradiance and temperature varying simultaneously. Simulation results showed that in the first scenario HOA significantly improved system performance by reducing rise time by 61%, 3% and 4.5% and settling time by 94%, 84.7% and 86.6% compared to MIC, GWO and AOA respectively. This led to a noticeable boost in maximum power output. In the second scenario, HOA demonstrated quicker dynamic responses than both GWO and AOA. In the third scenario, all three algorithms successfully tracked the MPP. In the fourth HOA delivered the fastest tracking of both PV voltage and extracted power, outperforming the other two methods.
Overall, the results confirm that using HOA to optimize the FOPI-IC-MPPT controller offers a robust and efficient solution for grid-connected PV systems. It provides fast response times, lower error rates, and improved energy harvesting, even under challenging and changing environmental conditions. In future work, will focus on evaluating the proposed HOA-based IC-MPPT under realistic irradiance profiles and hybridizing it with artificial intelligence techniques to enable global maximum power point extraction under partial shading conditions in grid-connected photovoltaic systems.
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Addition arithmetic operator
Ideality factor for diode
Position of best-attained solution till now C_Iter
Division arithmetic operator
Incremental conductance term
Error
Bandgap energy of polycrystalline silicon
Solar irradiance
Instantaneous incremental conductance
Diode current
Reverse saturation current of the diode
Photogenerated current
Short circuit current at STC
Current through parallel resistance
Boltzmann constant (1.38 × 10–23 J/K)
Integral gain
Proportional gain
Lambda
Temperature coefficient
Lower boundary of the j^th position
Multiplication arithmetic operator
Maximum number of iterations
Dimension
Function
Math Optimizer Accelerated
Math Optimizer probability
The number of series cells
Electron charge (1.6 × 10–19 C)
Random numbers
Series resistance of PV cell
Shunt resistance
Subtraction arithmetic operator
Temperature of PV cell in Kelvin
Steady-state time response
Upper boundary of the j^th position
Terminal voltage
Open circuit voltage at STC
Thermal voltage
Artificial intelligence
Arithmetic optimization algorithm
Hippopotamus Optimization Algorithm
Current-Voltage relationship
Double diode model of PV cell
Fractional open-circuit voltage
Fractional short circuit current
Grey wolf optimization
Integral absolute error
Incremental conductance
International renewable energy agency ITAE Integral time absolute error
Integral time square error
Integral time absolute error
Maximum PowerPoint tracking
Fractional-Order Proportional-Integral controller
Proportional-Integral controller
Integral controller
Photovoltaic
Power-Voltage relationship
Single diode model of PV cell
Standard test condition
Triple diode model
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The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA, for funding this research (through the project number NBU-FFR-2026-2124-01).
This research was funded by the Deanship of Scientific Research at Northern Border University, Arar, KSA, for funding this research (through the project number NBU-FFR-2026-2124-01).
Electrical Engineering Technical College, Middle Technical University, Baghdad, Iraq
Salah A. Taha & Mohammed Abdulla Abdulsada
Department of Electrical Engineering, Faculty of Engineering at Shoubra, Benha University, Cairo, Egypt
Mohamed Ahmed Ebrahim Mohamed
Department of Electrical Engineering, College of Engineering, Northern Border University, Arar, 91431, Saudi Arabia
Mohammed Alruwaili
Electrical Engineering Department, University of Business and Technology, Jeddah, 23435, Saudi Arabia
Ahmed Emara
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Salah A. Taha: Conceptualization, methodology, supervision, and overall project administration.Mohammed Abdulla Abdulsada: Software implementation, simulation, and data analysis.Mohamed Ahmed Ebrahim Mohamed: Validation, result interpretation, and manuscript drafting.Mohammed Alruwaili: Resources, visualization, and technical review of the manuscript.Ahmed Emara: Formal analysis, manuscript revision, and correspondence with the journal.All authors contributed to manuscript revision, approved the final version, and agreed to be accountable for all aspects of the work.
Correspondence to Salah A. Taha or Ahmed Emara.
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Taha, S.A., Abdulsada, M.A., Mohamed, M.A.E. et al. Enhanced maximum power point tracking using hippopotamus optimization algorithm for grid-connected photovoltaic system. Sci Rep 16, 9991 (2026). https://doi.org/10.1038/s41598-026-40918-4
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Received: 10 November 2025
Accepted: 17 February 2026
Published: 20 February 2026
Version of record: 26 March 2026
DOI: https://doi.org/10.1038/s41598-026-40918-4
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