Innovative Hybrid War Strategy Optimization with Incremental Conductance for Maximum Power Point Tracking in Partially Shaded Photovoltaic Systems

The motivation behind the hybridization of WSO and IC lies in overcoming the limitations inherent in each individual algorithm when applied to MPPT in PV systems. The IC algorithm, while effective in precisely tracking the MPP by analysing the local gradient of the power curve, can be slow to converge, especially under static PSCs. Moreover, the IC method exhibits remarkable oscillations around the MPP, which can lead to power loss. Conversely, the WSO algorithm excels in global optimization by exploring a broad search space. However, WSO alone may not provide the necessary precision in converging to the exact MPP due to its reliance on population-based heuristics. By integrating WSO with IC, the hybrid algorithm leverages the global search capability of WSO and the local fine-tuning ability of IC, resulting in a more efficient and robust MPPT solution in static PSCs.

The main novelty of our work lies in the development of a hybrid MPPT controller that combines the strengths of the WSO algorithm with the IC method. While classical MPPT techniques, such as P&O and IC, have proven effective under normal conditions, they face challenges due to the complexities introduced by partial shading. Our approach addresses these limitations by leveraging the global optimization capabilities of WSO while benefiting from the local tracking precision of IC, resulting in a more robust and effective solution for maximizing energy production in PV systems.

In particular, our hybrid WSO-IC algorithm improves convergence speed toward the MPP while significantly reducing output power oscillations and energy losses, especially under static PSCs. The results indicate impressive tracking efficiency, ranging from 99.4% to 99.9%, demonstrating its superiority over conventional methods. Additionally, we provide real-time simulations using MATLAB/Simulink, comparing our hybrid algorithm with traditional controllers, thus reinforcing the efficiency and robustness of our method. In summary, our contributions include:

A new MPPT controller that combines WSO and IC to enhance performance in PSCs.

The organization of the article is structured as follows:

Section 2—The PV Conversion Chain: Description and Modelling: This section covers the modelling of solar cells, static boost converters, and the impact of partial shading on PV systems.

A PV cell is described by the single-diode model. This model is generalized to a PV module by considering it as a set of identical cells connected in series and/or in parallel. A model of an elementary cell is constructed based on the most commonly used equivalent electrical circuit, as illustrated in Figure 2 (Drif et al., 2024).

Figure 2.

Equivalent circuit diagram of solar cell.

A PV system is made up of several PV modules that are electrically connected in series (N_s) and parallel (N_p) arrangements to enhance the overall power output. The model for a PV system is articulated through the following equations (Manuel and İnanç, 2022): (1) Ipv=Np⋅Iph−Np⋅IDexpϕ−1−NsVpv+NsNpRsIpvNsNpRsh {I_{pv}} = {N_p} \cdot {I_{ph}} - {N_p} \cdot {I_D}\left[ {{\rm{exp\;}}\left( \phi \right) - 1} \right] - {{{N_s}{V_{pv}} + {{{N_s}} \over {{N_p}}}{R_s}{I_{pv}}} \over {{{{N_s}} \over {{N_p}}}{R_{sh}}}} with (2) ϕ=NsVpv+NsNpRsIpvNsVt \phi = {{{N_s}{V_{pv}} + {{{N_s}} \over {{N_p}}}{R_s}{I_{pv}}} \over {{N_s}{V_t}}} (3) Vt=akbeT {V_t} = {{a{k_b}} \over e}T (4) Iph=Isc+kiT−270G1000 {I_{ph}} = \left( {{I_{sc}} + {k_i}\left( {T - 270} \right)} \right){G \over {1000}} (5) ID=IdrT2983expqEqNskbVt1298−1T {I_D} = {I_{dr}}{\left( {{T \over {298}}} \right)^3}{\rm{exp\;}}\left( {{{q{E_q}} \over {{N_s}{k_b}{V_t}}}\left( {{1 \over {298}} - {1 \over T}} \right)} \right) Several parameters influence the current generated by a solar cell. These include:

I_ph (Photocurrent): The current generated by sunlight hitting the cell.

The DC–DC converter plays a vital role in implementing MPPT within PV systems. MPPT algorithms typically work in tandem with the DC–DC converter to optimize system performance and maximize energy conversion efficiency under fluctuating operating conditions. A boost converter, for instance, is a type of DC–DC converter that increases the output voltage above the input level (Regaya et al., 2020). The power generated by the PV panel is directed to this boost converter, which is regulated by a pulse–width modulation (PWM) signal from the MPPT controller.

The duty cycle (D) of the boost converter is continuously adjusted to optimize the PV system’s power output (MPPT) under varying conditions. As noted by Al-Dhaifallah et al. (2018), this adjustment considers different PV system configurations, fluctuating solar irradiation levels, and ambient temperatures. Additionally, an input capacitor is placed on the PV panel side to filter out high-frequency variations in current.

As detailed in previous research (Chrouta et al., 2023; Kumar and Gupta, 2012), the optimal duty cycle of the DC–DC boost converter, which makes it possible to extract the maximum power generated by the PVG, is given by the following relationship: (6) Dmpp=1−VmppPmpp×Rload {D_{mpp}} = 1 - {{{V_{mpp}}} \over {\sqrt {{P_{mpp}} \times {R_{load}}} }}

P_mpp: Maximum power output of the PV installation when operating at its MPP.

The MPPT block generates a duty cycle signal, which controls the switching device of the boost converter, operating at a frequency of 10 kHz. The choice of a boost converter is driven by the PV system’s need for a higher output voltage, thus requiring a voltage step-up. Additionally, the selection of a boost converter is influenced by its reputation as one of the simplest and most easily controllable types of DC–DC converters, as indicated by Eltamaly et al. (2020).

Considering the continuous conduction mode (CCM) and selecting the current through the inductor and the voltage across the capacitor as state variables, the averaged model of the boost converter, shown in Figure 1, can be defined in Eqs (7) and (8) (Manuel and İnanç, 2022; Menadi et al., 2024). (7) diLtdt=−1−D1Lvot+1Lvpv {{d{i_L}\left( t \right)} \over {dt}} = - \left( {1 - D} \right){1 \over L}{v_o}\left( t \right) + {1 \over L}{v_{pv}} (8) dvotdt=1−D1CoutiLt−1R.Coutvo {{d{v_o}\left( t \right)} \over {dt}} = \left( {1 - D} \right){1 \over {{C_{out}}}}{i_L}\left( t \right) - {1 \over {R.{C_{out}}}}{v_o} where i_L denotes the current through the inductor, v_Cout is the voltage across the capacitor C_out, and D is the control signal (D ∈ [0,1]). In the boost converter circuit, the parameters R, L, and C_out represent the load resistance, the input circuit inductance, and the output filter capacitance, respectively; v_pv is the supply voltage of the boost converter and v_o is the output voltage.

During no uniform irradiance conditions, such as those caused by the shadow of a tree, building, or passing clouds, the PVG system becomes partially shaded. When partial shading occurs, regardless of the cause, as illustrated in Figure 3, the power–voltage (P–V) curves display multiple peaks corresponding to MPPs. Among these peaks, only one is the global maximum power point (GMPP), while the others are local maximum power points (LMPP).

Figure 3.

Configuration of PV modules under different static PSCs. (A) Scenario 1—STC. (B) Scenario 2. (C) Scenario 3. (D) Scenario 4. PSCs, partial shading conditions; PV, photovoltaic; STC, standard test conditions.

PV panels receive varying irradiance levels to simulate PSCs, which can lead to PV curves with multiple peaks. To highlight our contribution, we present a case study of static PSCs featuring four practical test scenarios for simulation:

Scenario 1: A test under standard conditions (1,000 W/m² and 25°C);

Simulations were conducted using MATLAB/Simulink to explore and evaluate the effectiveness of the hybrid WSO-IC algorithm for achieving optimal performance under the proposed static partial shading scenarios. The simulation results were compared with those obtained from the two traditional MPPT methods: P&O and IC.

The IC algorithm is an MPPT technique used in PV systems to optimize energy extraction by calculating and comparing the instantaneous conductance (dI/dV) with the voltage. It continuously adjusts the operating voltage to maintain operation near the MPP with reduced oscillations, offering a balance between simplicity and efficiency. However, its accuracy can be compromised by noise and partial shading.

The IC algorithm adjusts the duty cycle D of the power converter based on the incremental changes in voltage V and current I. The condition for achieving MPP is defined as follows (Al-Dhaifallah et al., 2018): (9) dPdV=0 where P=V.I {{dP} \over {dV}} = 0\;{\rm{where}}\;P = V.I

Expanding this gives: (10) dV.IdV=I+V.dIdV {{d\left( {V.I} \right)} \over {dV}} = I + V.{{dI} \over {dV}}

At the MPP, the output power of the PV system is maximized, which implies that any small change in voltage (V) results in a corresponding change in current (I) that maintains the power at its maximum value. Mathematically, this condition is represented by the relationship given in Eq. (9). When we apply this condition to the derived expression from Eq. (10), this simplifies to: (11) dIdV=−IV {{dI} \over {dV}} = - {I \over V}

The IC algorithm adjusts the duty cycle D based on the relationship between ΔIΔV {{\Delta I} \over {\Delta V}} and IV {I \over V} If ΔIΔV>−IV, decrease D {\rm{If}}\,{{\Delta I} \over {\Delta V}} > - {I \over V},\;{\rm{decrease}}\;D If ΔIΔV<−IV, increase D {\rm{If}}\,{{\Delta I} \over {\Delta V}} < - {I \over V},\;{\rm{increase}}\;D

This ensures a rapid response to changes in the PV system’s operating conditions.

WSO is a metaheuristic algorithm inspired by military strategies. It operates on a population of candidate solutions, each representing a potential duty cycle D, and iteratively improves these solutions based on their fitness (Ayyarao et al., 2022): (12) fitness=−Vpv×Ipv×D fitness = - Vpv \times Ipv \times D

Key steps in WSO include: (1)

Initialization: Initialize a population of candidate solutions.

The operation of this algorithm is described in the flowchart in Figure 4.

Figure 4.

Flowchart of WSO algorithm. WSO, war strategy optimization.

The hybrid WSO-IC algorithm is an advanced technique that combines the advantages of IC and WSO. It is designed to swiftly and efficiently identify the GMPP of a PV system under static PSCs, minimizing response time and enhancing overall system performance. This hybrid approach offers both fast convergence and high accuracy in tracking the MPP, making it a robust solution for PV systems.

Key steps in hybrid WSO-IC include: (1)

IC Phase: Quickly adjusts D based on the local gradient to approach the MPP.

The operation of this algorithm is described in the flowchart given in Figure 5.

Figure 5.

Flowchart of hybrid WSO-IC Algorithm. WSO-IC, war strategy optimization-incremental conductance.

The benefits of the hybrid WSO-IC algorithm can be outlined in the following four main points:

Fast Convergence: The IC method quickly identifies a close approximation of the MPP, and the WSO further refines this by exploring the global search space, resulting in faster convergence to the MPP.

A series of simulation tests were carried out on the proposed MPPT control algorithm. The simulation has been realized under the MATLAB/Simulink environment using the control scheme illustrated in Figure 6.

Figure 6.

Diagram of the 426.3 W peak power PV system simulated in SIMULINK. PV, photovoltaic.

The DC–DC boost converter serves as an interface between the simulated PV installation and a simulated DC load, with the entire system being modelled in the simulation environment. The boost converter is used to control the operating point for current and voltage, thus facilitating the acquisition of the true MPP. In this setup, four different MPPT algorithm blocks are integrated into various sections, with each MPPT method being simulated in succession.

The primary objective of this work is to study and conduct a comparative analysis of four different algorithms for tracking the GMPP of a PV system under the influence of static partial shading. The algorithms considered are IC, P&O, WSO, and a hybrid WSO-IC algorithm. The proposed hybrid WSO-IC algorithm is utilized to further refine the search method in order to accurately track the exact location of the GMPP.

We selected the four algorithms: P&O, IC, WSO, and WSO-IC for comparison due to their complementary approaches in MPPT. P&O is widely used for its simplicity, while IC offers greater precision, especially under changing irradiance conditions. WSO was included as it is an effective global optimization method, particularly useful under PSCs. The hybrid WSO-IC combines the strengths of WSO and IC, aiming to enhance both tracking speed and accuracy. Regarding power losses in the DC–DC converter, these depend to some extent on the control algorithm. More efficient tracking algorithms can reduce oscillations around the MPP, thereby minimizing switching losses and improving overall system efficiency. Algorithms that are slow to converge or prone to oscillations can cause increased switching activity and higher losses in the converter.

The comparative performance analysis of the four algorithms involves observing three parameters to evaluate the tracking time, tracking error, and efficiency of the studied PV system.

Tracking time reflects the convergence time required by the algorithms to achieve maximum power Ppv.

The second parameter, tracking error, measures the deviation between the duty cycle determined by the algorithms and the actual global duty cycle calculated in Table 3, divided by the calculated global duty cycle. This parameter quantifies the accuracy of the algorithms in locating the MPP and is expressed as a percentage.

The third parameter assesses the efficiency of the PV system by calculating the ratio of the generated.

The simulation results are presented in Figures 7–10, corresponding respectively to the power Ppv and duty cycle for the four scenarios 1, 2, 3, and 4 of each of the four algorithms: IC, P&O, WSO, and WSO-IC. The x-axis is consistent across all four cases in each scenario, with a scale ranging from 0 s to 4 s.

Figure 7.

Power Ppv and duty cycle under standard conditions (Scenario 1-STC). STC, standard test conditions.

Figure 8.

Power Ppv and duty cycle under PSCs (Scenario 2). PSCs, partial shading conditions.

Figure 9.

Power Ppv and duty cycle under PSCs (Scenario 3). PSCs, partial shading conditions.

Figure 10.

Power Ppv and duty cycle under PSCs (Scenario 4). PSCs, partial shading conditions.

In the four scenarios 1, 2, 3 and 4, the comparative performance of the hybrid WSO-IC method is demonstrated in Figures 7–10. Remarkably, this method quickly reaches the GMPP with a minimal tracking error in the duty cycle and a high efficiency in extracting the maximum power from the PV installation compared to other methods such as IC, P&O, and WSO. Notably, this objective is achieved without encountering persistent oscillations in the steady state. On the other hand, the WSO method exhibits a faster response time than the other algorithms.

The data collected during the simulation phase allowed for the calculation of the tracking error percentage in the global duty cycle for each method, as indicated in Table 4.

The results clearly show that the WSO-IC outperforms the other suggested approaches in various aspects. Notably, it excels in accurately tracking the global duty cycle (GMPP) and in its ability to operate at its maximum power. The efficiency of the WSO-IC method ranges between 0.995 and 1.000 in the four scenarios studied.

The results shown in Table 4 and Figure 11 underscore the effectiveness of the WSO-IC method in optimizing the MPPT process for PV systems. The superior convergence speed and efficiency percentage of this method highlight its potential as an essential choice for optimizing energy production and overcoming shading challenges in PV applications.

Figure 11.

Quantitative comparison between the performances of IC, P&O, WSO, and WSO-IC methods for different shading patterns. (A) Duty cycle tracking error and (B) power extraction efficiency. IC, incremental conductance; P&O, perturb and observe; WSO, war strategy optimization; WSO-IC, war strategy optimization-incremental conductance.

The hybrid approach, which integrates IC and WSO, is designed to capitalize on the strengths of both methods. The IC algorithm facilitates a quicker initial convergence to a region near the MPP, while WSO fine-tunes the search to ensure the attainment of the global MPP, particularly in partial shading scenarios. Although WSO performs well on its own, incorporating IC aims to decrease the overall duty cycle tracking error and enhance efficiency—key factors for the system’s performance. To comprehensively evaluate the performance of the proposed WSO-IC MPPT controller compared to other methods, a set of key performance indicators (KPIs) was used. These KPIs, detailed in Table 5, include:

The results presented in Table 5 clearly indicate that the WSO-IC algorithm outperforms other methods in several key areas. Based on the simulation obtained data, the percentage efficiency for each method was calculated. It is evident that the WSO-IC method achieves significantly higher efficiency compared to the techniques discussed by Khatib and Muhsen (2020) and Berttahar et al. (2024). The WSO-IC controller also shows a faster response time compared to PSO by Khatib and Muhsen (2020) and GWO by Berttahar et al. (2024) under static PSCs, indicating its ability to quickly achieve the GMPP. This leads to reducing ‘uncaptured’ energy, which our method achieves by reaching the GMPP faster with high efficiency.

These results highlight the effectiveness and the superiority of WSO-IC in optimizing the MPPT process for PV systems. The enhanced convergence speed and efficiency percentage of WSO-IC make it a promising choice for optimizing power generation and addressing shading challenges in PV applications.