Black Widow Optimization-Based PV Array Reconfiguration for Maximized Water Flow in Agricultural Systems Under Partial Shading

Innovation and technology play a crucial role in improving agricultural productivity. One notable area where significant progress is being made is the integration of sensors into agriculture, a trend of the utmost importance. Agriculture is an essential sector of the economy in many countries. At present, a considerable number of farmers rely on traditional sources such as mains electricity or diesel engines for their water pumping needs [1]. However, the expense of diesel pumping has become increasingly onerous. In addition, large tracts of farmland are located in remote areas without electricity [2, 3]. Even in areas with access to electricity, demand for water increases during the summer months, often coinciding with power cuts. Thus, the adoption of PV water pumping appears to be a practical solution to conserve energy, relieve pressure on the electricity grid, and reduce energy costs in the long term.

Partial shading conditions (PSCs) are a major challenge to the efficiency of solar water pumping systems [4, 5]. These conditions are typically caused by obstacles such as trees, tall grass, bushes, telephone poles, birds, hills, and rocks that cast shadows on nearby PV modules [6]. Permanent shading of PV modules in the same area can lead to hot spots and potential damage. To mitigate this, bypass diodes are usually installed on PV modules to avoid this situation. However, this configuration often leads to multiple peaks in the output, significantly reducing the overall power output of the PV array [7]. Various PV array topologies, such as total-cross-tied (TCT) and honeycomb (HC), have been developed to minimize mismatch losses and improve efficiency. Of these topologies, TCT has been shown to generate the highest power output under most shading conditions [7, 8]. However, no single topology consistently outperforms the others in all shading scenarios. Therefore, several static and dynamic reconfiguration methods have been proposed in the literature to improve the PV arrays' output power [9].

Static reconfiguration algorithms use puzzle patterns to disperse shading effects [10]. These algorithms have proven effective for recurring shading scenarios. However, with advances in sensors, solid-state devices, and high-end processors, dynamic reconfiguration is emerging as an enviable solution to counter the impact of partial shading conditions [10-12]. Researchers mainly apply dynamic reconfiguration to TCT configurations, focusing on irradiance equalization to improve output power [13, 14]. This improvement in output power directly increases the efficiency of photovoltaic pumping systems. PV array dynamic reconfiguration technology, which is particularly well suited to water pumping applications due to its ability to scale to smaller PV arrays, cane improve power output under various shading conditions [15]. In addition, the P-V characteristics obtained by dynamic reconfiguration are more consistent, allowing the PV water pumping system to use conventional MPPT techniques for optimal energy extraction.

The P-V curve of a PV module has ae distinct point where maximum power is delivered. MPPT techniques precisely track this point to optimize the PV module's efficiency. There are a variety of MPPT algorithms for extracting maximum power from PV panels [16]. Conventional methods include perturbation and observation (P&O) and incremental conductance (I&C). Hybrid and optimization-based MPPT algorithms offer even more accurate tracking capabilities. Numerous researchers have explored various soft computing and intelligent approaches for MPPT. Notable methods include the Kalman filter (KF) [17], fuzzy logic control (FLC) [18], artificial neural networks (ANN) [19], particle swarm optimisation (PSO) [20], ant colony optimisation (ACO) [21], and the artificial bee colony (ABC) algorithm [22]. These techniques are known for their rapid response when applied to MPPT tasks. Among them, the KF-based MPPT stands out due to its effective use of recursive filtering, leveraging estimations of previous, current, and potentially future states while handling noisy data measurements. In contrast, the Black Widow Optimization (BWO) algorithm—an emerging evolutionary algorithm inspired by the unique mating behavior of black widow spiders—demonstrates superior exploration–exploitation balance, faster convergence, and lower computational complexity [23, 24]. These characteristics make it particularly well-suited for real-time PV array reconfiguration under rapidly varying partial shading scenarios. To the best of our knowledge, this is the first study to apply BWO for dynamic PV array reconfiguration in a TCT topology, offering an original contribution to the field of intelligent PV system control.

Extensive research has been conducted on photovoltaic (PV) water pumping systems, focusing primarily on two main configurations: single-stage and two-stage setups [25-28]. Managing the DC link in single-stage configurations poses significant challenges [28]. Nevertheless, single-stage designs are often preferred for their higher efficiency, stemming from the use of fewer components [26]. Conversely, two-stage power conversion systems provide enhanced flexibility in terms of design, operation, and control. Early studies on solar water pumping predominantly utilized DC motors, which were chosen for their straightforward compatibility with PV arrays [29]. However, these motors typically require frequent maintenance. To address this issue, researchers are investigating the use of advanced magnetic materials in the development of motors tailored to specific applications [30]. These modern motors demonstrate high efficiency and have lower maintenance needs [31], though the substantial cost of magnetic materials contributes to the overall expense of the motor. Other open-wound induction motors (IMs) have also been used for this purpose [27, 31], but their practical availability is problematic. Despite this, IMs remain popular due to their reliability, high availability, robust construction, affordability, and direct compatibility with pump loads [28, 32]. However, the integration of dynamic PV array reconfiguration based on black widow optimization, Kalman filter (KF)-based MPPT controller, and direct torque control (DTC) using a two-level inverter to regulate IMs under PSCs has not previously been explored.

This paper introduces an innovative method for dynamically reconfiguring PV arrays using sensors and switches, aimed at optimizing the output power of TCT PV arrays under partial shading conditions (PSCs). Additionally, a Kalman filter algorithm is employed for MPPT, and direct torque control (DTC) with space vector pulse width modulation (SVPWM) is applied to drive the induction motor (IM) pump. The primary goal of this study is to enhance the performance of PV water pumping systems under PSCs.

The key contributions of this research are outlined as follows:

1. A novel dynamic reconfiguration strategy based on the BWO algorithm is proposed. This algorithm is chosen over other metaheuristics for its superior convergence speed and robustness in handling complex shading conditions. It uses real-time irradiance data to adjust the configuration of PV modules in a TCT array via switches, ensuring maximum power delivery under all PSC scenarios.

2. An MPPT controller based on the KF is adopted to mitigate the occurrence of multiple peaks on the P-V curve, resulting in smoother P-V curves. Typically, under PSCs, conventional MPPT algorithms such as Perturb and Observe (PO) and Incremental Conductance (IC) fail to efficiently track the global MPP. However, with the proposed algorithm, this problem is solved, allowing the MPPT algorithm to perform optimally even under PSCs, thus ensuring efficient system operation.

3. A control scheme including the proposed PV reconfiguration method and KF-based MPPT controller is applied for an IM pumping system with a DTC controller. The results show that an increase of the PV array power under PSCs is achieved which contributes to an enhancement of the water discharge with a gain of 423 l/h.

The rest of this paper is organized as follows: Section 2 presents the design and analysis of the system under study in greater detail. Section 3 describes the adopted control scheme, including the proposed photovoltaic (PV) reconfiguration method and some relevant aspects. Section 4 analyzes the simulation results. Finally, Section 5 summarizes the main conclusions.

In this section, the proposed control for the PV pumping system is described in detail. First, the proposed BWO algorithm for dynamic reconfiguration with the KL-based MPPT technique is presented. Next, the IM modeling and its control scheme are provided.

3.1 Black widow optimization (BWO)-based PV array reconfiguration

In conventional PV systems, series-parallel or TCT configurations are commonly employed to meet specific load demands [37]. Among these, the TCT configuration has proven particularly effective under PSCs due to its superior performance in mitigating mismatch losses. In this work, a dynamic reconfiguration strategy is proposed to further enhance the performance of TCT-connected PV arrays using the Black Widow Optimization (BWO) algorithm. The reconfiguration mechanism relies on real-time irradiance data from each module, collected through sensors. These measurements are used to generate switching control signals that adjust the interconnections between modules. The objective is to dynamically rearrange the modules to optimize the irradiance distribution and extract the maximum possible power from the array under shading.

3.1.1 Overview of the BWO algorithm

The BWO algorithm is a population-based metaheuristic optimization technique that operates using mechanisms analogous to selection, reproduction, and mutation [23, 24]. It does not depend on biological fidelity but instead utilizes abstracted computational procedures to explore the solution space efficiently. Each solution, or individual, represents a candidate configuration for the PV array.

3.1.2 Initial population

The population is initialized with randomly generated candidate solutions:

$W=\left[X^1, X^2, \ldots, X^{N^{\operatorname{Var}}}\right]$            (16)

where, $\mathrm{X}^{\mathrm{N}^{\mathrm{Var}}}$ represents the number of decision variables (i.e., PV module positions), and each $\mathrm{X}^{\mathrm{i}}$,  denotes a floating-point vector encoding a possible interconnection layout of the modules.

3.1.3 Fitness function evaluation

The objective of the optimization process is to minimize power mismatch between array rows, thereby maximizing total power output. This is mathematically formulated through a fitness function that minimizes the difference in total irradiance received by each row:

$\begin{array}{r}\text { fitness }=\operatorname{minimize}(\max (\operatorname{sum} G(n, 1)) \\ -\min (\operatorname{sum} G(n, 1)))\end{array}$            (17)

This formulation ensures that rows receive similar irradiance levels, thereby equalizing current outputs and minimizing bypass diode activation.

3.1.4 Reproduction and cannibalism mechanism

High-quality individuals (configurations with better fitness) are selected for mating through a crossover operator. New offspring are generated by combining portions of parent solutions. To maintain population diversity and improve convergence, a selection pressure mechanism is applied: a portion of the lowest-performing individuals is removed from the population (analogous to the biological concept of cannibalism, though abstracted here as an elimination step).

3.1.5 Mutation and replacement

To avoid premature convergence and local minima, a low-probability mutation operation is applied to certain solutions, introducing new variability into the population. The best-performing individuals are retained for the next generation.

3.1.6 Implementation in PV array reconfiguration

In this study, the BWO algorithm interfaces with a switching matrix responsible for modifying the physical connections between PV modules. At each reconfiguration interval, the algorithm processes the irradiance data and returns the optimal layout that maximizes the array’s power output under the current PSC scenario.

The complete process of the proposed BWO-based PV array reconfiguration is illustrated in Figure 3.

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Figure 3. Flowchart of the BWO Algorithm for dynamic PV array reconfiguration

3.2 Kalman filter-based MPPT control algorithm

Enhancing the energy efficiency of the SPVWPS relies heavily on the performance of the MPPT controller, which is essential for optimizing the power output of PV cells across varying environmental conditions. The voltage-power and voltage-current characteristics of the solar panel change with different levels of irradiance, revealing that each shift in irradiance results in a new maximum power point (V_mpp, I_mpp). To effectively track these changes, the KF is utilized in the MPPT process, with its specific operating principles detailed in the following sections. An iterative KF approach is applied to estimate state variables by refining predictions derived from noisy measurement data. This process involves two key phases that are repeated at each iteration: the prediction phase (temporal updates) and the correction phase (measurement updates). These iterative adjustments continue until an accurate representation of the measured value is achieved. During the prediction phase, the KF projects the future state and estimates the error covariance, resulting in a priori predictions for the next iteration. The underlying mathematical expressions for this prediction process are as follows [38]:

• Forecast the state for the upcoming step

$x_{k-}^{-}=A x_{k-1}+B u_{k-1}$             (18)

• Project the covariance error

$P_k^{-}=A P_{k-1} A^T+Q$           (19)

In the Kalman filter $(\mathrm{KF})$ algorithm, $x_k^{-}$represents the predicted state estimate at iteration $k$, derived from the state estimate $x_{k-1}$ from the previous iteration using the measurement $z_{k-1}$. The term A denotes the transitione state constante modele appliede toe thee prior estate, while Be ise ae constante corresponding to the econtrol-input emodel used for the control process. The control input at iteration $k-1$ is denoted by $u_{k-1}$. The variable $P_k^{-}$indicates the prior error covariance at iteration k, P_{k- 1} is the posterior error covariance from iteration k-1, and Q represents the process noise covariance linking the prior and posterior error covariances.

The correction phase of the KF algorithm involves three main equations for refining the predicted value [17]:

• Kalman Gain Calculation:

$K_k=H_k^{-} C^T\left(C H_k^{-} C^T+R\right)^{-1}$           (20)

• Estimate Update Using Measurement:

$x_k=x_k^{-}+K_k\left(z_k-C x_k^{-}\right)$           (21)

• Covariance Error Update:

$P_k=\left(I-P_k C\right) P_k^{-}$           (22)

4.png

Figure 4. Flowchart of the MPPT algorithm utilizing a kalman filter

In these equations, $x_k$ is the corrected state at iteration k based on the measurement $z_k$, R is the measurement noise covarianc, Kk is the Kalman gain, Ce is a system constant dependent on the KF frame work and the observed data space, and Pk erepresents the posterior error covariance ate iteration k. For Maximum Power Point Tracking (MPPT), the KF algorithm must monitor the operating point's position in real time, adjusting to environmental changes such as variations in irradiance and temperature impacting thee PVe arrays. According to the P-Ve curve, power increases withe a positive slope until reaching the MPP, followed by a decrease with a negative slope beyond that point. The KF uses its predictive and corrective equations toe derive the optimal duty cycle (α) for the DC-DC boost converter, ensuring the PV system operates at the MPP. Thee flowchart of the KF-based MPPT algorith is shown in Figure 4.

3.3 DTC control for IM using a two-level inverter

The DTC strategy controls the two-level inverter switches toe drivee the IM. By selecting one of eight voltage vectors (six active: V1-V6, and two zero: V0, V7), the inverter's state is determined. These vectors are divided into six 60-degree sectors, as depicted in Figure 5. The optimal voltage vector is chosen based on the motor's flux position and the desired changes in flux and torque magnitude, as outlined in Table 3 [39]. Figure 6. Flowchart illustrating the implementation of the direct torque control (DTC) strategy in a two-level inverter-fed induction motor. The flowchart includes flux and torque estimation, hysteresis control, sector identification, and voltage vector selection.

5.png

Figure 5. Space voltages for two-level inverter

6.png

Figure 6. DTC strategy for a two-level inverter driving an induction motor

This paper presented a new approach to improve the performance of a solar PV water pumping system under PSCs. By combining dynamic PV array reconfiguration using the BWO algorithm, KF-based MPPT controller, and DTC for an IM using a two-level inverter, significant progress has been made in optimizing system efficiency and reliability. The proposed BWO algorithm effectively increases the PV output power by dynamically reconfiguring the PV array in response to changing irradiation conditions, resulting in a more consistent and efficient power generation. This increase in the PV output power directly influences motor speed through the DTC mechanism, thereby improving mechanical power. As a result, improved mechanical power translates into higher hydraulic power, leading to substantially improved water flows. Simulation studies have shown that the proposed approach effectively increases water flow. Specifically, the first case showed a gain of 216 l/h, the second a gain of 432 l/h, and the third a gain of 288 l/h. This improvement holds great promise for promoting sustainable agricultural practices, particularly in regions where access to conventional energy sources is limited. By addressing the challenges posed by PSCs and improving the performance of the SPVWPS, this research may contribute to the advancement of renewable energy technologies in agricultural applications, ultimately helping to achieve the goal of food security and environmental sustainability. In terms of practical feasibility, the proposed system is well-aligned with contemporary hardware capabilities. The required irradiance sensors, one per PV module, are commercially available and cost-effective. The reconfigurable switch matrix—necessary for implementing dynamic topologies—can be realized using microcontroller-controlled relays or MOSFETs, without excessive hardware complexity. Moreover, the BWO algorithm and KF-based MPPT operate efficiently enough to be executed on embedded systems such as FPGAs or DSPs in real time. This ensures the viability of real-time control and makes the implementation of the proposed technique both practical and scalable for deployment in rural, off-grid settings.