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Scientific Reports volume 16, Article number: 12910 (2026)
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Under partial shading conditions (PSC), the Power–Voltage (P–V) output curve of a photovoltaic (PV) array shows multiple peaks. To address this, this paper introduces a hybrid maximum power point tracking (MPPT) strategy called the improved red-billed blue magpie optimization algorithm combined with a variable-step perturb and observe algorithm (IRBMO-VP&O) of the exponential decay. This strategy merges an improved red-billed blue magpie optimization (IRBMO) algorithm with a variable-step perturb and observe (VP&O) method. It incorporates Lévy flight and an individual diversity mechanism to boost its global search ability. Additionally, it uses adaptive step-size fine-tuning for better tracking accuracy and a restart mechanism triggered by sudden power changes, enhancing its adaptability to dynamic environments. MATLAB/Simulink simulations compare the proposed algorithm with GWO, PSO, RIME, SSA, IGWO-VINC, PSO-P&O, RBMO, IRBMO, INC, and P&O. Under static shading conditions, IRBMO-VP&O outperformed others by reducing average convergence time by 52.99% (to 0.042–0.067 s) and increasing tracking accuracy by 4.06% (to 91.785–99.988%) compared to eight other algorithms. Under dynamic conditions, it achieved an average convergence time of 0.055 s and tracking accuracy above 99.986%, consistently locking onto the global maximum power point (GMPP) without local optima trapping.
With the global push toward “carbon neutrality” and “peak carbon” targets, photovoltaic (PV) generation, as a clean and accessible form of renewable energy, has garnered significant attention1,2. However, under partial shading conditions (PSC), different modules within a PV array receive unequal irradiance levels due to clouds, buildings, dust, or other obstructions. This non-uniform irradiance causes bypass diodes to activate unevenly, resulting in multiple peaks appearing on the P–V characteristic curve. These peaks include several local maximum power points (LMPPs) and one global maximum power point (GMPP). The LMPP refers to a power peak that is locally optimal within a certain voltage interval but does not correspond to the highest achievable power of the entire array. In contrast, the GMPP represents the absolute highest power point among all peaks on the P–V curve and corresponds to the optimal operating condition of the PV array. Consequently, MPPT algorithms designed for PSC must possess strong global search capability to avoid convergence to an LMPP and ensure accurate tracking of the GMPP3,4,5. Therefore, the study of maximum power point tracking (MPPT) technologies is crucial for mitigating these power losses and enhancing the overall output efficiency of PV arrays6,7,8.
Conventional MPPT techniques, such as hill climbing (HC)9,10, perturb and observe (P&O)11,12,13,14, and incremental conductance (INC)15,16,17,18, are widely used. However, under variable environmental conditions, these methods are prone to becoming trapped in local optima19. Consequently, researchers worldwide have adopted various nature-inspired metaheuristic algorithms to address this issue20. For instance, Reference21 modified the grey wolf optimizer (GWO) by replacing its linear convergence factor with a non-linear one and incorporating a Lévy flight strategy. While this enhanced convergence speed and prevented premature convergence, its testing was limited to two static and one dynamic shading pattern, failing to cover more complex shading distributions. Reference22 introduced a Cauchy mutation mechanism into the particle swarm optimization (PSO) algorithm to enhance population diversity and avoid local optima, complementing it with a direct search algorithm for local refinement. However, key parameters were not optimized, potentially limiting its generalization, and the performance was only benchmarked against the standard PSO. Similarly, Reference23 proposed dynamically adjusting the inertia weight of a PSO algorithm to balance its global and local search capabilities, but its validation was confined to a single shading pattern without comparison to other advanced algorithms. In reference24, genetic algorithm operators were used to optimize the initial population of the sparrow search algorithm (SSA) and integrate Lévy flight into its update formula. However, it lacked an adaptive restart mechanism, which led to potential response delays during sudden changes in irradiance. Reference introduced a fitness-based weighting coefficient into GWO to improve search flexibility, but its restart mechanism relied on a fixed threshold, limiting its adaptability. Finally, Reference26 utilized an Arnold map to initialize the GWO population, enhancing uniformity to avoid local optima. However, its dynamic simulation was limited to a single abrupt change, which does not reflect the more complex and continuous variations seen in real-world environments. A summary of the above references is shown in Table 1.
Given these challenges, developing hybrid algorithms that can achieve both rapid convergence and high tracking accuracy is critically important27,28. Therefore, this paper proposes a PV MPPT strategy based on the improved red-billed blue magpie optimization algorithm combined with the variable step-size perturb and observe algorithm (IRBMO-VP&O).
The main contributions of this work are as follows:
This paper proposes a novel hybrid MPPT control strategy for partially shaded conditions. This strategy is achieved by integrating the red-billed blue magpie optimization (RBMO) algorithm with a variable-step perturbation and observation (VP&O) method featuring exponential decay step adjustment.
It enhances the RBMO algorithm by incorporating a Lévy flight mechanism and an individual diversity mechanism for the first time. These additions bolster the algorithm’s global search capability, enabling it to effectively escape local optima and improve the diversity and stability of the swarm.
It designs a restart mechanism triggered by sudden power fluctuations. When abrupt environmental changes cause power output to vary, this mechanism adaptively re-initiates the optimization search, enhancing the algorithm’s responsiveness under dynamic conditions.
It validates the proposed algorithm’s superior performance through comprehensive comparative simulations. A system model was developed in MATLAB/Simulink and tested against 10 other mainstream algorithms under various static and dynamic operating conditions. The results indicate that the IRBMO-P&O algorithm achieves an average reduction of 52.99% in convergence time compared to other algorithms, while improving tracking accuracy by 4.06%.
This paper is organized as follows: Chapter Introduction establishes the mathematical models of the PV cell and PV array based on their equivalent circuits. It also verifies their P–V characteristic variations under different shading conditions through simulation. Chapter Modeling of PV arrays provides a mathematical model for the three phases of the RBMO algorithm: prey searching, prey attacking, and food storing. It then introduces a Lévy flight mechanism and a diversity-based update strategy to enhance the algorithm’s search performance. Chapter Variable step-size perturb and observe method details the design of the VP&O method. It proposes an exponential decay step-size adjustment method for more responsive control of the duty cycle and integrates this into the VP&O framework to boost local optimization capabilities. Chapter Application of the IRBMO-VP&O algorithm in PV MPPT presents the control system simulation model and conducts comprehensive performance tests, comparing the proposed algorithm against others under five static and three dynamic operating conditions.
An ideal PV cell can be represented by an equivalent circuit consisting of a constant current source and an ideal diode, as shown in Fig. 1. To better reflect practical conditions, a series resistance Rs representing internal losses and a shunt resistance Rsh describing leakage effects are included. Based on this single-diode model, the output current is expressed as the difference between the photocurrent and the diode current, further adjusted by the voltage drop across Rs and the leakage effect of Rsh, as shown in Eqs. (1)–(4)29,30. This equivalent circuit provides the theoretical foundation for analyzing the P–V characteristics of PV arrays under different irradiance conditions.
Equivalent circuit of a PV cell.
where, IL is the output current of the photovoltaic cell; Iph is the photo-generated current, with its formula shown in Eq. (3); Is is the reverse saturation current of the diode, with its formula shown in Eq. (4); Vt is the thermal voltage, with its formula shown in Eq. (2); Uoc is the output voltage of the photovoltaic cell; Rs and Rsh are the equivalent series and shunt resistance of the photovoltaic cell; α is the diode ideality factor; K is the Boltzmann constant, with a value of 1.38 × 10-23 J/K; T is the operating temperature of the photovoltaic cell; q is the electron charge constant, with a value of 1.6 × 10-19 C; Isc is the short-circuit current under standard test conditions; Tn is the nominal temperature under standard testing; Ki is the voltage temperature coefficient; Kv is the voltage temperature coefficient; G is the solar irradiance; Gn is the nominal irradiance under standard testing; V is the open-circuit voltage.
The PV array studied in this paper consists of PV modules connected in a 5 × 8 series-parallel configuration. The simulation model of the PV array, developed in MATLAB/Simulink, is shown in Fig. 2. The operating temperature is set to 25 °C, and the maximum irradiance is 1000 W/m². The specific parameters of the selected PV module are as follows: open-circuit voltage of 36.3 V, short-circuit current of 7.84 A, voltage at maximum power point of 29 V, and current at maximum power point of 7.35 A. Figure 3 displays the P–V characteristic curves of the array under the various irradiance conditions detailed in Table 2.
Simulation model of the PV array.
P–V characteristic curves of the PV array.
As shown by the P–V curves in Fig. 3, the GMPP for the five operating conditions correspond to power outputs of 8517.80 W, 6790.35 W, 5062.99 W, 4606.71 W, and 3094.89 W, respectively.
The RBMO is a metaheuristic algorithm based on swarm intelligence, proposed by Shengwei Fu et al. in 202431. It is inspired by the hunting process of the red-billed blue magpie, which is modeled in three distinct phases: prey searching, prey attacking, and food storing. These three phases are mathematically modeled below.
In nature, red-billed blue magpies improve their search efficiency by hunting collaboratively in small groups (2–5 individuals) or larger flocks (10 + individuals). They adapt their search strategy-which includes ground hopping, walking, and probing trees-based on environmental conditions. This behavior is modeled mathematically.
Equation (5) represents the exploration strategy for small groups, while Eq. (6) models the strategy for larger flocks.
where, t represents the current iteration number; Xi(t + 1) represents the i-th new search position; p represents the number of red-billed blue magpies hunting in small groups of 2 to 5, randomly selected from all search individuals; Xm represents the randomly selected m-th individual; Xi(t) represents the i-th individual; Xrs(t) represents a randomly selected search in the current iteration; q represents the number of searches when exploring food as a flock, in the range [10, n]; Rand is a random number between [0, 1]; Levy(dim) is the Lévy flight function, as shown in formulas (7) and (8).
where, u and v are random variables from a normal distribution; β is a constant of 1.5.
The prey attacking phase models the magpies’ proficient and cooperative hunting tactics. The strategy varies by group size. Small groups typically target smaller prey, a behavior modeled by Eq. (9). In contrast, larger flocks can collaboratively hunt bigger prey, which is mathematically represented by Eq. (10).
where, Xfood(t) represents the position of the food; Randn represents a random number used to generate a standard normal distribution; CF is the adaptive convergence factor, with its formula shown in Eq. (11).
This phase models the magpies’ behavior of storing surplus food in hidden locations to create a reserve. In the algorithm, this translates to preserving the best-found solutions, which helps guide individuals toward the global optimum. This food-storing mechanism is mathematically modeled by Eq. (12), which functions as a greedy selection strategy.
where, fitnessiold and fitnessinew respectively represent the fitness values of the i-th red-billed blue magpie before and after the position update.
The conventional P&O method operates by first measuring the PV array’s real-time voltage and current to calculate its instantaneous output power, as shown in Eq. (13). It then determines the change in power (dP) and voltage (dV) between consecutive time steps, as defined in Eq. (14). Based on the sign of the ratio dP/dV, the duty cycle of the converter is dynamically adjusted to track the maximum power point (MPP).
where Vt represents the current time’s photovoltaic array output voltage; Vt-1 represents the previous time’s photovoltaic array output voltage; Pt represents the current time’s photovoltaic array output power; Pt-1 represents the previous time’s photovoltaic array output power.
To enhance performance, this paper proposes a VP&O method that utilizes an exponential decay function to adjust the step size, as defined in Eqs. (15) and (16).
where, Dmin and Dmax are the set minimum and maximum values of the duty cycle; δ is a constant.
In the proposed MPPT control system, when the PV array operates under partial shading conditions, the IRBMO-VP&O algorithm initiates a global search using the IRBMO component to rapidly converge to the region of the GMPP. Once the switching condition in Eq. (17) is satisfied, the algorithm transitions to the VP&O method for a fine-tuned local search to enhance tracking precision. If a significant environmental change occurs that satisfies the restart condition in Eq. (18), the algorithm reinitiates a new optimization cycle. The complete workflow of the IRBMO-VP&O algorithm is illustrated in the flowchart in Fig. 4. The process begins with a global search using the enhanced IRBMO algorithm (incorporating Lévy flight and a diversity mechanism). After satisfying the switching criterion, it transitions to the VP&O algorithm for precise local tracking. The adaptive restart mechanism, triggered by sudden power fluctuations from external disturbances, ensures efficient and stable tracking of the GMPP under complex and dynamic irradiance conditions.
where, t is the current iteration number and Tmax is the maximum number of iterations.
Flowchart of the IRBMO-VP&O algorithm.
To validate the effectiveness of the proposed IRBMO-VP&O algorithm, a PV system simulation model was developed in MATLAB/Simulink, as depicted in Fig. 5. This model comprises an algorithm control module, the PV array, a PWM generation module, and a DC-DC Boost converter (Its parameter settings are: inductance (L) = 85 mH, capacitance (C1/C2) = 500 µF/20 µF, and resistance = 20 Ω). The specific parameters of the IRBMO-VP&O algorithm are shown in Table 3.
PV system simulation model.
In this section, simulations were conducted under static shading conditions to compare the performance of nine algorithms: GWO, PSO, RIME, SSA, IGWO-VINC, PSO-P&O, RBMO, IRBMO, and the proposed IRBMO-VP&O. The simulation results for condition I, condition II, condition III, condition IV, and condition V are presented in Figs. 6, 7, 8, 9, and 10, respectively.
Static simulation results for condition I.
Static simulation results for condition II.
Static simulation results for condition III.
Static simulation results for condition IV.
Static simulation results for condition V.
For a quantitative performance analysis, data were extracted from the simulation results shown in Figs. 6, 7, 8, 9 and 10. Convergence time is defined as the time required for the tracked power to reach and remain within 5% of the true maximum power. The convergence time, maximum tracked power, and tracking accuracy for each algorithm across the five conditions are summarized in Table 4. The data reveal that the IRBMO-VP&O algorithm consistently demonstrates superior performance, achieving both faster convergence (e.g., 0.042s in condition III) and higher tracking accuracy (e.g., 99.985% in condition IV).
Table 4 presents the convergence time and tracking accuracy data of nine algorithms under five static shading conditions, demonstrating the significant comprehensive superiority of the proposed IRBMO-VP&O algorithm. In terms of convergence speed, IRBMO-VP&O achieves the fastest performance across all conditions, with convergence times ranging from 0.042 to 0.067 s (average 0.054 s), which is 64.7% shorter than the average of the eight benchmark algorithms (0.153 s). Notably, it reaches 0.042 s in Condition III, 25.0% faster than IRBMO (0.056 s) and 43.2% faster than RBMO (0.074 s); under the most complex Condition V, it achieves 0.061 s, 74.8% faster than PSO-P&O (0.242 s). Regarding tracking accuracy, IRBMO-VP&O consistently maintains above 99.98% (99.982~99.988%) in Conditions I~IV, with an average accuracy of 98.345%, representing a 4.06% point improvement over the average of the eight benchmark algorithms (94.285%). Most notably, in Condition IV, where RBMO and IRBMO both fall into local optima (only 3793.37 W and 3843.70 W), IRBMO-VP&O successfully tracks 4,606.02 W (99.985% accuracy), achieving a power gain of over 800 W (approximately 21%). In Condition V, its accuracy of 91.785% outperforms IRBMO by 6.81% points and RBMO by 7.13% points. These data fully validate the exceptional performance of IRBMO-VP&O through the synergistic effect of enhanced global search and precise local refinement, effectively avoiding local optima and significantly improving the power generation efficiency of PV systems. To provide a clear visual comparison, the data from Table 4 are plotted in Fig. 11. These charts intuitively illustrate the significant advantages of the IRBMO-VP&O algorithm in both tracking accuracy and convergence speed.
Comparison charts of static simulation data
The MPPT performance of four algorithms-P&O, INC, RBMO, and IRBMO-VP&O was evaluated under dynamic conditions, with the response curves shown in Fig. 12. The simulation models scenarios with sudden changes in irradiance. The results show that the IRBMO-VP&O algorithm has a much faster dynamic response, enabling it to rapidly re-lock onto the new GMPP after an irradiance change. In contrast, the conventional P&O and INC algorithms exhibited significant tracking lag or became trapped in local optima due to their fixed step sizes and lack of global search capabilities.
Dynamic simulation results.
For a quantitative analysis, performance metrics were extracted from the results in Fig. 12, with convergence time defined as the time taken to reach and stay within 5% of the true GMPP. The convergence time, maximum tracked power, and tracking accuracy for the four algorithms under three distinct dynamic scenarios are summarized in Table 5. As shown, the IRBMO-VP&O algorithm consistently achieves fast convergence and high tracking accuracy, validating its superior performance.
The data in Table 5 highlights the exceptional performance of the IRBMO-VP&O algorithm in dynamic environments. Its average convergence time was only 0.055 s, with a tracking accuracy consistently maintained above 99.986%. This significantly outperforms the benchmark algorithms: the P&O algorithm was fast but inaccurate and prone to getting trapped; the INC algorithm suffered from significant lag and also became trapped; and the standalone RBMO algorithm exhibited wide fluctuations in accuracy (ranging from 92.322fig. 1
to 97.283%). These results confirm the advantages of IRBMO-VP&O’s hybrid approach, which combines global search with adaptive step-sizing to effectively handle complex operating conditions like sudden irradiance changes. The comparison charts in Fig. 13 provide a clear visual representation of the superior tracking accuracy and convergence speed of IRBMO-VP&O relative to the other three algorithms.
Comparison charts of dynamic simulation data.
This paper proposes a hybrid MPPT control strategy named IRBMO-VP&O, which synergistically combines an improved red-billed blue magpie optimization (IRBMO) algorithm with a variable-step perturb and observe (VP&O) method based on exponential decay. The strategy is specifically designed to tackle the multi-peak tracking challenges inherent in photovoltaic systems under partial shading conditions. The principal methodological and practical contributions of this work are summarized as follows:
The integration of Lévy flight and an individual diversity mechanism significantly enhances the global exploration capability of the IRBMO algorithm. This improvement effectively prevents premature convergence to local optima—a common limitation in conventional metaheuristic and P&O methods when dealing with multi-peak P–V characteristics.
A novel variable-step mechanism employing an exponential decay function allows the VP&O algorithm to adaptively refine the tracking step size near the maximum power point. This design reduces steady-state oscillation and improves tracking precision. Coupled with a power-change-triggered restart mechanism, the algorithm demonstrates strong adaptability and robustness under rapidly changing environmental conditions.
The hybrid structure leverages the complementary strengths of global search (via IRBMO) and localized refinement (via VP&O), facilitating both rapid convergence and high tracking accuracy. This offers a viable and efficient solution for real-world PV systems operating under complex and fluctuating irradiance conditions.
Future work will focus on the hardware implementation of the proposed IRBMO-VP&O algorithm and its application in grid-connected control strategies for photovoltaic-energy storage systems.
It should be noted that although this study primarily evaluates dynamic performance under irradiance variations, the proposed IRBMO-VP&O algorithm is fundamentally designed to track the maximum power point by directly optimizing the P–V characteristic curve of the PV array. Since temperature variation and load disturbance also manifest as changes in the P–V profile, the proposed hybrid structure with global search capability and adaptive restart mechanism is theoretically capable of responding to simultaneous multi-parameter variations. Comprehensive multi-factor coupled disturbance validation will be considered in future work.
The data presented in this study are available on request from the corresponding author. The data have not been made public yet, as the data in this article still needs to be used for further research.
Description
Photovoltaic
Maximum power point tracking
Partial shading conditions
Global maximum power point
Local maximum power point
Hill climbing
Perturb and observe
Incremental conductance
Grey wolf optimizer
Particle swarm optimization
Sparrow search algorithm
Improved red-billed blue magpie optimization algorithm combined with the variable step-size perturb and observe algorithm
Red-billed blue magpie optimization
Variable step-size perturb and observe
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This work was supported by the following projects:1. Chuzhou University Industry–University–Research Cooperation Project (HX2023183).2. Chuzhou University Undergraduate Innovation Training Program (2025CXXL069).
This research was supported by the Industry-University-Research Cooperation Project of Chuzhou University (HX2023183) and the College Students’ Innovation Training Program of Chuzhou University (2025CXXL069).
School of Mechanical and Electrical Engineering, Chuzhou University, Chuzhou, 239000, China
Xiang’ao Wang
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Xiangao Wang wrote the main manuscript text.
Correspondence to Xiang’ao Wang.
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Wang, X. A photovoltaic maximum power point tracking strategy based on the IRBMO-VP&O algorithm. Sci Rep 16, 12910 (2026). https://doi.org/10.1038/s41598-026-43400-3
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