Global Maximum Point Research Articles

Configuration optimization of PV array for stratospheric airship under nonuniform radiation condition

The nonuniform solar irradiation on the stratospheric airship surface leads to overlooked mismatch losses in photovoltaic (PV) array. Proper configuration for PV array is crucial but frequently neglected in airship energy system research. In this paper, a mathematical model coupled with the layout of PV array on airship surface, radiation model, electrical model and energy balance model is established. The deployment of PV pack based on the distribution characteristics of solar radiation on airships surface is analyzed. Three maximum power point tracking configurations for the PV array are investigated in detail. The results indicate that mounting PV pack in axial series on stratospheric airships leads to higher power output compared to parallel-connected deployment, PV packs on stratospheric airships exhibit significant output power differences along the circumferential direction and minor differences along the axial direction, the peak of daily maximum output difference ratio is merely 0.4 % between the PV array under the distributed maximum power point tracking (DMPPT) and the partial distributed global maximum point tracking (PDMPPT) configuration, and PDMPPT configuration needs much fewer module, it emerges as the optimal choice for stratospheric airship PV array. This investigation can provide references for the further PV array configuration optimization under nonuniform radiation condition.

A modified invasive weed optimization for MPPT of PV based water pumping system driven by induction motor

A novel approach called Modified Invasive Weed Optimization (MIWO) technique has been developed and combined with the Perturb and Observes (P&O) algorithm to enhance the extraction of maximum power from photovoltaic (PV) panels in the presence of partial shading conditions. The conventional P&O algorithm falls short in extracting the maximum power from PV systems under partial shading conditions due to the existence of multiple maximum points. In such scenarios, optimization techniques can be employed to search for the global maximum point. The proposed MIWO-based P&O algorithm updates the reference voltage to ensure that the PV system operates at the Maximum Power Point (MPP) based on the prevailing weather conditions. This MIWO based PV system is further fed to water pumping system. A PV-based water pumping system is utilized for both irrigation and domestic purposes. Additionally, a sensorless vector control-based induction motor is employed in this study to drive the pump. The objective of this research is to demonstrate the achievement of an efficient PV-based water pumping system without the need for battery storage. Various results based on MIWO are compared with PSO and GWO. The results are presented based on various water pumping applications and the availability of solar irradiance during rapid climate changes. MATLAB/Simulink simulations, along with hardware-based experiments, are provided to validate the effectiveness of the proposed method under both transient and steady-state conditions.

Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials

We are concerned with the existence and multiplicity of normalized solutions to the fractional Schrödinger equation (−Δ)su+V(εx)u=λu+h(εx)f(u)inRN,∫RN|u|2dx=a,, where (−Δ)s is the fractional Laplacian, s∈(0,1), a,ε>0, λ∈R is an unknown parameter that appears as a Lagrange multiplier, h:RN→[0,+∞) are bounded and continuous, and f is L2-subcritical. Under some assumptions on the potential V, we show the existence of normalized solutions depends on the global maximum points of h when ε is small enough.

Multiple normalized solutions for ( 2 , q ) -Laplacian equation problems in whole R N

This paper considers the existence of multiple normalized solutions of the following ( 2 , q ) -Laplacian equation: { − Δ u − Δ q u = λ u + h ( ϵ x ) f ( u ) , i n R N , ∫ R N | u | 2 d x = a 2 , where 2 < q < N , ϵ > 0 , a > 0 and λ ∈ R is a Lagrange multiplier which is unknown, h is a continuous positive function and f is also continuous satisfying L 2 -subcritical growth. When ϵ is small enough, we show that the number of normalized solutions is at least the number of global maximum points of h by Ekeland's variational principle.

On Logarithmic Holder Condition and Local Extrema of Power Takagi Functions

This paper studies one class of real functions, which we call Takagi power functions. Such functions have one positive real parameter; they are continuous, but nowhere differentiable, and are given on a real line using functional series. These series are similar to the series defining the continuous, nowhere differentiable Takagi function described in 1903. For each parameter value, we derive a functional equation for functions related to Takagi power functions. Then, using this equation, we obtain an accurate two-sides estimate for the functions under study. Next, we prove that for parameter values not exceeding 1, Takagi power functions satisfy the Holder logarithmic condition, and find the smallest value of the constant in this condition. As a result, we get the usual Hölder condition, which follows from the logarithmic Hölder condition. Moreover, for parameter values ranging from 0 to 1, we investigate the behavior of Takagi power functions in the neighborhood of their global maximum points. Then we show that the functions under study reach a strict local minimum on the real axis at binary-rational points, and only at them. Finally, we describe the set of points at which our functions reach a strict local maximum. The benefit of our research lies in the development of methods applicable to continuous functions that cannot be differentiated anywhere. This can significantly expand the set of functions being studied.

Multiplicity and stability of normalized solutions to non-autonomous Schrödinger equation with mixed non-linearities

AbstractThis paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2, \end{cases} \end{equation*} where $a, \epsilon, \eta \gt 0$, q is L2-subcritical, p is L2-supercritical, $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and h is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ϵ is small enough. The solutions obtained are local minimizers and probably not ground state solutions for the lack of symmetry of the potential h. Secondly, the stability of several different sets consisting of the local minimizers is analysed. Compared with the results of the corresponding autonomous equation, the appearance of the potential h increases the number of the local minimizers and the number of the stable sets. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$.

Effects of PSO Algorithm Parameters on the MPPT System Under Partial Shading Condition

Maximum Power Point Tracking (MPPT) systems enable photovoltaic (PV) panels to work at their Maximum PowerPoint (MPP). To do this, several algorithms have been developed, including conventional, intelligent, and meta-heuristic. Once a partial shading condition (PSC) occurs, more than one peak emerges in the power-voltage curve of photovoltaic arrays. Under PSCs, conventional algorithms get stuck at the local maximum point and fail to reach the global maximum point. Being an alternative method, Particle Swarm Optimization (PSO) algorithm has been frequently employed for MPPT systems under PSCs. This algorithm has some parameters that affect its performance to reach the global MPP of the PV panel. Therefore, with well-tuned parameters, the effectiveness of the PSO will increase for the different PSCs. In this study, the effects of the cognitive learning and social learning parameters of the PSO algorithm are investigated under different PSCs. To achieve this, an MPPT system, including a boost-type DC-DC converter, is created in MATLAB®/Simulink®. Simulation studies show that the PSO algorithm fails to track global MPP with constant cognitive and social learning parameters under changing partial shading conditions. Furthermore, the results show that these two parameters affect the time to reach the MPP of the PSO algorithm.

Transmissive Metasurfaces Assisted Wireless Communications on Railways: Channel Strength Evaluation and Performance Analysis

We propose a new wireless paradigm for railways – transmissive metasurface (TMS) assisted communications. It compensates for the Doppler shift brought by high mobility and reduces signal degradation due to train carriages. Specifically, the elements of the TMS panel attached to train windows manipulate the links between the base station (BS) and the onboard users. One fundamental problem with it is: does the channel introduced by TMS outperform the traditional direct channel? To answer this, we compare the gains of direct and cascaded channels and introduce the cascaded-outperform-direct probability (CODP), which is the probability that the latter exceeds the former. We then derive two simplified closed-form CODP expressions by approximating the cascaded channel gain as mixed Gaussian and Gamma distributions. Moreover, BS-related parameters impact the CODP; we show that the CODP has (i) global maximum points concerning the azimuth angle of the path from the BS to the TMS, the distance from the BS to the railway, and the BS height, respectively, and (ii) a minimum point in terms of the azimuth angle of the path from the BS to the TMS. Finally, we provide numerical results to verify our analysis and derivations.

PSO based Solar PV system for SVPWM Controlled Induction Motor for Agricultural Applications

Nowadays, renewable energy sources play a major role in agricultural applications. The recommended Particle Swarm Optimization (PSO) algorithm with DC-DC converter supports in refining the output voltage. This is beneficial however since the partial shading condition of PSO technique has high output power with fewer searching steps of MPPT. The objective of this work is to propose a PSO based boost converter using Space Vector Pulse Width Modulation (SVPWM) method. PSO has been anticipated to track the Global Maximum Point (GMP) under the partial shading condition. The welfares of the SVPWM technique are fixed switching frequency and lower harmonic content. The centrifugal pump used in agricultural system is obsessed by SVPWM controlled three-phase induction motor. The extent of the motor as well as the photovoltaic array evaluations are certain as the water could be pumped out even at temperature fluctuations and irradiation levels. The efficacy of the proposed design is observed under the MATLAB/SIMULINK.

Analysis of the relationship between global breast density and maximum points of breast density in a sample of Brazilian women

Maximum points of breast density have been strongly associated with breast cancer masking than global breast density, which is a widely used measure today. The objective of this work was to verify the correlation between two measures of global breast density (VBDGLOBAL and DABGLOBAL) and a measure of maximum point breast density (VBDMP). Mammographic images of 4.020 patients were analyzed using the Volpara software, which calculated or extracted the variables needed for the study from the DICOM header. Two-tailed partial correlation tests were performed between the variable VBDGLOBAL with VBDMP and DABGLOBAL with VBDMP in the following contexts: keeping PA and CBT constant, keeping only CBT constant, and keeping only PA constant. The Pearson test was also used to verify the bivariate correlation between VBDGLOBAL with VBDMP and DABGLOBAL with VBDMP. For the two-tailed partial correlation tests between VBDGLOBAL with VBDMP, keeping the CBT and PA variables constant resulted in r = 0.845 (p < 0.05). When kept constant only the CBT, r = 0.875 (p < 0.05), and keeping only the PA constant r = 0.866 (p < 0.05). Pearson's test showed r = 0.883 (p < 0.05). For the two-tailed partial correlation tests between the DABGLOBAL with VBDMP quantities, the results were r = 0.675 (p < 0.05), r = 0.725 (p < 0.05) and r = 0.701 (p < 0.05) for constant CBT and PA, constant CBT and constant PA, respectively, while the Pearson test resulted in r = 0.738 (p < 0.05). We conclude that a woman who has high global breast density is also highly likely to have maximum points of breast density.

Normalized solutions to lower critical Choquard equation with a local perturbation

In this paper, we study the existence and non-existence of normalized solutions to the lower critical Choquard equation with a local perturbation $ \begin{equation*} \begin{cases} -\Delta u+\lambda u = \gamma (I_{\alpha}\ast|u|^{\frac{N+\alpha}{N}})|u|^{\frac{N+\alpha}{N}-2}u+\mu |u|^{q-2}u, \quad {\rm{in}}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx = c^2, \end{cases} \end{equation*} $ where $ \gamma, \mu, c&gt;0 $, $ 2&lt;q\leq 2+\frac{4}{N} $, and $ \lambda\in \mathbb{R} $ is an unknown parameter that appears as a Lagrange multiplier. The results of this paper about this equation answer some questions proposed by Yao, Chen, Rǎdulescu and Sun [Siam J. Math. Anal., 54(3) (2022), 3696-3723]. Moreover, based on the results obtained, we study the multiplicity of normalized solutions to the non-autonomous Choquard equation$ \begin{equation*} \begin{cases} -\Delta u+\lambda u = (I_\alpha\ast [h(\epsilon x)|u|^{\frac{N+\alpha}{N}}])h(\epsilon x)|u|^{\frac{N+\alpha}{N}-2}u+\mu|u|^{q-2}u, \ x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx = c^2, \end{cases} \end{equation*} $where $ \epsilon&gt;0 $, $ 2&lt;q&lt;2+\frac{4}{N} $, and $ h $ is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of $ h $ when $ \epsilon $ is small enough.

Power reserve control strategy of PV system for active power reserve under dynamic shading patterns

The de-loading technique in photovoltaics is used to reserve some active power for frequency regulation purposes. However, the selection of the de-loading method depends upon the shading conditions of photovoltaics. Many de-loading methods have already been developed for uniform shading conditions. Whereas for partial shading conditions, few methods are available for static shading patterns, while, the area of dynamic shading is untouched. Under dynamic shading, the global maximum power point changes continuously. Therefore, in this paper, in order to develop a power reserve under dynamic shading patterns of partial shading conditions, a trained artificial neural network is used. The information about the local and global maximum points is obtained from this neural network, whereas, the system de-loading is achieved using a de-loading algorithm. The neural network is trained using different irradiance patterns which lead to two or three peaks on the power vs. voltage curve of photovoltaics. The control is easy to implement and can handle up to three peak system. The simulation and experimental results have been used to verify the effectiveness of the control.

A novel hybrid grey wolf optimized fuzzy logic control based photovoltaic water pumping system

AbstractThis paper describes the novel scientific contribution of a modified hybrid grey wolf optimization‐fuzzy logic controller (GWO‐FLC) based on maximum power point tracking (MPPT) for a photovoltaic (PV) system. The modified GWO eliminates the lower rank (δ) and lowest rank (Ψ) wolves, which provides a rapid search method to track global maximum point (GMPP) in a lesser period. The hybrid GWO‐FLC based MPPT technique provides tracking of GMPP with zero oscillations and high PV power tracking under rapid weather fluctuations. The GWO technique acquires GMPP and provides soft tuning of fuzzy parameters to achieve high PV power tracking with zero oscillations around GMPP. The experimental responses confirm changing environmental situations. The dSPACE real‐time platform has been employed to validate the proposed hybrid MPPT controller design, which provides optimal PV power tracking with accurate efficiency, fast convergence velocity, and less computational period. The switched reluctance motor (SRM) driven PV water pump is proposed in this research work. The proposed PV water pumping has low cost and no voltage, and current sensors are employed to generate a gating pattern of an SRM based mid‐point converter, which reduces the losses.

Implementation of the Bio-Inspired Metaheuristic Firefly Algorithm (FA) Applied to Maximum Power Point Tracking of Photovoltaic Systems

In this paper, an algorithm for the maximum extraction of energy generated by photovoltaic (PV) systems was presented. The tracking of the global maximum point of the system is complex due to the non-linearity of the current-voltage (I-V) characteristic curve of the photovoltaic modules, as they vary according to the temperature and solar irradiation in the module. To obtain the best energy efficiency in these systems, it is important that the generation is delivering the maximum power available through the arrangement. In order to solve such problems, in this work an efficient MPPT-FA method was proposed, which showed good traceability when compared to traditional methods. Most traditional MPPT techniques are not able to find the global maximum point to extract the maximum power provided by the PV system. Finally, the Firefly Metaheuristic MPPT method proved to be robust against several partial shading scenarios. Simulations were presented to demonstrate the effectiveness of the proposal when compared to the traditional MPPT-PO method.

A novel sliding mode control and modified PSO-modified P&O algorithms for peak power control of PV

Photovoltaic (PV) energy is a renewable source which provides clean energy in power systems. Because of low efficiency of energy conversion in PV systems, harvesting maximum power from these systems is important. Many papers have been focused on this challenge to obtain the maximum power from PV array. Sliding mode control is an efficient control method to achieve maximum power. Although, due to non-linear curve of P–V and I–V of PV array in one side and uncertain PV generated power due to climate and temperature conditions, different evolutionary algorithms must be applied for accurate maximum power point tracking. Perturbation and observation (P&O) is a well-known method in this field but it has some defects in case of shadow-condition multiple maxima. In this paper, considering analysis of two sliding mode control structures, modified particle swarm optimization (PSO) algorithm has been used in combination with modified P&O method called modified PSO-modified P&O (MPSO-MP&O) to compensate the problem of P&O tracking in shadow conditions. The proposed method selects the global maximum point by searching all maximum points in many iterations. Also, the achievement stability of the maximum point is interesting in different temperature conditions and sudden changes of physical parameters of PV system. Simulation results show the efficiency of the proposed method performance in obtaining the maximum point faster with more PV power compared to other efficient methods in this field.

The subdifferentiability of functions convex with respect to the set of Lipschitz concave functions

A function defined on normed vector spaces X is called convex with respect to the set LĈ := LĈ (X,R ) ofLipschitz continuous classically concave functions (further, for brevity, LĈ -convex), if it is the upper envelope of some subset of functions from LĈ. A function f is LĈ -convex if and only if it is lower semicontinuous and bounded from below by a Lipschitz function. We introduce the notion of LĈ -subdifferentiability of a function at a point, i. e., subdifferentiability with respect to Lipschitz concave functions, which generalizes the notion of subdifferentiability of classically convex functions, and prove that for each LĈ -convex function the set of points at which it is LĈ -subdifferentiable is dense in its effective domain. The last result extends the well-known Brondsted – Rockafellar theorem on the existence of the subdifferential for classically convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using elements of the subset LĈ θ ⊂ LĈ, which consists of Lipschitz continuous functions vanishing at the origin of X we introduce the notions of LĈ θ -subgradient and LĈ θ -subdifferential for a function at a point.The properties of LĈ -subdifferentials and their relations with the classical Fenchel – Rockafellar subdifferential are studied. Considering the set LČ := LČ (X,R ) of Lipschitz continuous classically convex functions as elementary ones we define the notions of LČ -concavity and LČ -superdifferentiability that are symmetric to the LĈ -convexity and LĈ -subdifferentiability of functions. We also derive criteria for global minimum and maximum points of nonsmooth functions formulated in terms of LĈ θ -subdifferentials and LČ θ -superdifferentials.

Centralized Control Topology for PV Farms Shading Detection and GMPP Searching Restarting Condition

The power output of the solar panels follows a power-voltage (P-V) characteristic containing only one Global Maximum Point (GMP) in the normal conditions. However, under Partial Shading Conditions (PSC), the unbalanced irradiance in the panels creates Local Maximum Points (LMP) in the P-V curve. Standard control techniques for Maximum Power Point Tracking (MPPT) can not properly locate the GMP, stagnating in LMPs and generating losses in the energy harvesting. Specific techniques to locate the GMP are presented in the literature. However, the condition to restart the GMP is not widely discussed. The main challenges of global search algorithms are related to the restarting conditions. Avoiding unnecessary searching and providing an assertive GMP restarting condition is crucial for PV systems operation. In every GMP search, the solar inverters oscillate the power exchanged with the grid, causing frequency and voltage variations depending on the size of the PV plant. This paper proposes a novel technique that uses a centralized controller to identify the shaded inverters, creating flags that locally start the GMP searching. The solution minimizes the number of times the search is performed by providing an assertive GMP restarting condition, saving energy, and avoiding unnecessary output power oscillation. The proposed control technique was evaluated using the data of a real 150-kW solar farm containing five inverters with two MPPT trackers each.

Photovoltaic-Based Electric Vehicle Charging Station using GA Algorithm

Detailed modeling of rapid charging Electric Vehicle (EV) stations connected to a hybrid grid-Renewable Energy Systems (RES), such as solar, mini-hydro, and wind, has been proposed. This assists in maximizing profit and lowering grid energy consumption. Since Photovoltaic (PV) systems exhibit several peaks under partial shading conditions; it is very difficult to track the global maximum point. Therefore, a broad search is needed to solve this nonlinear problem. Due to the variety of solutions, GA is preferred for solar MPPT. In comparison to the Genetic Algorithm (GA) and Maximum Power Point Tracking (MPPT) technique, the economic considerations obtained optimizes profit. It is also obvious that the proposed strategy lowers the grid's influence on the system network by capping the amount of electricity that may be exchanged between the system network and the grid. The voltage and current of PV array with GA method and GA MPPT technique results on the utility grid respectively.

Strict convexity of the objective function and uniqueness of the maximum point in a model with three arbitrary random priorities

The main result of this paper is the proof of the strict concavity of some function of integral form depending on three random variables, which we call priorities. This function is an objective function in the so-called model with priorities, in which the arbiter, following expert opinions, distributes funds among the enterprises and institutions under his jurisdiction. This result implies an important corollary about the existence and uniqueness of a local maximum point (which is also a global maximum point) of the objective function. This is a significant generalization of the corresponding result of N.V. Neumezhitskaia, S.I. Uglich and T.A. Volosatova, published in December 2020.

The location of hot spots and other extremal points

In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we consider the torsional rigidity function of a bar, the first mode of a vibrating membrane, and the temperature of a heat conductor grounded to zero at the boundary. Our main results are presented for domains with mean convex boundary and compare that distance to the inradius of the relevant domain. For the torsional rigidity function, the obtained bound only depends on the space dimension. The more general case of a boundary which is not mean convex is also considered. However, in this case the estimates also depend on some geometrical quantities such as the diameter and the radius of the largest exterior osculating ball to the relevant domain, or the minimum of the mean curvature of the boundary. For the first mode, the relevant bound only depends on the space dimension, as well. Moreover, it largely improves on an earlier estimate obtained for convex domains by the first author and co-authors. The bound related to the temperature depends on time and the initial distribution of temperature. Such a bound is substantially consistent with what one obtains in the stationary situation. The methods employed are based on elementary arguments and existing literature, and can be extended to other situations that entail quasilinear equations, isotropic and anisotropic, and also certain classes of semilinear equations.