Search Results (25)

The increasing penetration of distributed energy resources (DERs) and power electronic loads challenges the modeling and control of modern distribution networks (DNs). The traditional models often fail to capture the complex aggregate dynamics required for advanced control strategies. This paper proposes a novel framework for DN power regulation based on Neural Ordinary Differential Equations (NODEs) and Model Predictive Control (MPC). NODEs are employed to develop a data-driven, continuous-time dynamic model capturing the aggregate relationship between the voltage at the point of common coupling (PCC) and the network’s power consumption, using only PCC measurements. Building upon this NODE model, an MPC strategy is designed to regulate the DN’s active power by manipulating the PCC voltage. To ensure computational tractability for real-time applications, a local linearization technique is applied to the NODE dynamics within the MPC, transforming the optimization problem into a standard Quadratic Programming (QP) problem that can be solved efficiently. The framework’s efficacy is comprehensively validated through simulations. The NODE model demonstrates high accuracy in predicting the dynamic behavior in a DN against a detailed simulator, with maximum relative errors below 0.35% for active power. The linearized NODE-MPC controller shows effective tracking performance, constraint handling, and computational efficiency, with typical QP solve times below 0.1 s within a 0.1 s control interval. The validation includes offline tests using the NODE model and online co-simulation studies using CloudPSS and Python via Redis. Application scenarios, including Conservation Voltage Reduction (CVR) and supply–demand balancing, further illustrate the practical potential of the proposed approach for enhancing the operation and efficiency of modern distribution networks. Full article

In this study, we present a methodology for predicting timber board foot volume using a forest landscape model, incorporating allometric equations and forest inventory data. The research focuses on the Ozark Plateau, a 48,000-square-mile region characterized by productive soils and varied precipitation. To simulate timber volume, we used the LANDIS PRO forest landscape model, initialized with forest composition data derived from the USDA Forest Service’s Forest Inventory and Analysis (FIA) plots. The model accounted for species-specific growth rates and was run from the year 2000 to 2100 at five-year intervals. Timber volume estimates were calculated using both quadratic mean diameter (QMD) and tree diameter in the Hahn and Hansen board foot volume equation. These estimates were compared across different forest types—deciduous, coniferous, and mixed stands—and verified against FIA plot data using a paired permutation test. Results showed high correlations between QMD and tree diameter methods, with a slightly lower volume estimate from the QMD approach. Projections indicate significant increases in board foot volume for key species groups such as red oak and white oak while showing declines toward the end of the model period in groups like shortleaf pine due to age-related mortality and regeneration challenges. The model’s estimates closely align with state-level FIA data, underscoring the effectiveness of the integrated approach. The study highlights the utility of integrating landscape models and forest inventory data to predict timber volume over time, offering valuable insights for forest management and policy planning. Full article

The finite-time turnpike property describes the situation in an optimal control problem where an optimal trajectory reaches the desired state before the end of the time interval and remains there. We consider a machine learning problem with a neural ordinary differential equation that can be seen as a homogenization of a deep ResNet. We show that with the appropriate scaling of the quadratic control cost and the non-smooth tracking term, the optimal control problem has the finite-time turnpike property; that is, the desired state is reached within the time interval and the optimal state remains there until the terminal time T. The time $t_0$ where the optimal trajectories reach the desired state can serve as an additional design parameter. Since ResNets can be viewed as discretizations of neural odes, the choice of $t_0$ corresponds to the choice of the number of layers; that is, the depth of the neural network. The choice of $t_0$ allows us to achieve a compromise between the depth of the network and the size of the optimal system parameters, which we hope will be useful to determine the optimal depths for neural network architectures in the future. Full article

To address the problem of large tractive resistance in traditional subsoiling methods, this paper designed a pulse-type gas explosion subsoiler, as well as an air-blown double-ended chisel type subsoiling shovel and a conduit. The mathematical equation of the influence of the structural parameters of the subsoiler on the groove profile is established. The EDEM 2022 software was used to simulate the subsoiling operation process. The soil disturbance law of the chisel subsoiler was analyzed by the change of soil particle velocity. The optimum value interval of quadratic regression orthogonal rotation combination test factors was determined by using the steepest climb test, with specific tillage resistance and filling power as evaluation indicators. Based on the Box–Behnken design test, a second-order regression model of response value and significance parameter was obtained, and an optimal combination was found by optimizing the significance parameter. The effects of subsoiling air pressure, pulse width and pulse interval on evaluation indicators were analyzed by the response surface method; the test results show that when the air pressure was 0.8 MPa, the pulse width was 0.17 s and the pulse interval was 0.12 s, and the specific tillage resistance was 0.4421 N/mm² and the filling power was 18.5%; a comparative test between the pulse-type gas explosion subsoiler and a continuous gas explosion subsoiler was carried out, and the specific tillage resistance was reduced by 12.2% and the filling power was reduced by 10.5%; the comparative test shows that the pulse-type gas explosion subsoiler has smaller tractive resistance per unit area and smaller disturbance to soil. The research results provide a theoretical basis and reference for the optimization and improvement of gas explosion subsoilers. Full article

Computation of the solution of the nonlinear Caputo fractional differential equation is essential for using $q,$ which is the order of the derivative, as a parameter. The value of q can be determined to enhance the mathematical model in question using the data. The numerical methods available in the literature provide only the local existence of the solution. However, the interval of existence is known and guaranteed by the natural upper and lower solutions of the nonlinear differential equations. In this work, we develop monotone iterates, together with lower and upper solutions that converge uniformly, monotonically, and quadratically to the unique solution of the Caputo nonlinear fractional differential equation over its entire interval of existence. The nonlinear function is assumed to be the sum of convex and concave functions. The method is referred to as the generalized quasilinearization method. We provide a Caputo fractional logistic equation as an example whose interval of existence is $[0,\infty ).$ Full article

This paper aims to design a quadratic optimization model of an investment portfolio based on value-at-risk (VaR) by entering risk-free assets and company liabilities. The designed model develops Markowitz’s investment portfolio optimization model with risk aversion. Model development was carried out using vector and matrix equations. The entry of risk-free assets and liabilities is essential. Risk-free assets reduce the loss risk, while liabilities accommodate a fundamental analysis of the company’s condition. The model can be applied in various sectors of capital markets worldwide. This study applied the model to Indonesia’s mining and energy sector. The application results show that risk aversion negatively correlates with the mean and VaR of the return of investment portfolios. Assuming that risk aversion is in the 5.1% to 8.2% interval, the maximum mean and VaR obtained for the next month are 0.0103316 and 0.0138270, respectively, while the minimum mean and VaR are 0.0102964 and 0.0137975, respectively. The finding of this study is that the vector equation for investment portfolio weights is obtained, which can facilitate calculating investment portfolio weight optimization. This study is expected to help investors control the quality of appropriate investment, especially in some stocks in Indonesia’s mining and energy sector. Full article

In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an interval variable. The four methods reviewed here in order to solve this problem are: (i) the method of classic interval analysis used by Elishakoff and Daphnis, (ii) the direct method based on minimizations and maximizations also used by the same authors, (iii) the method of quantifier elimination used by Ioakimidis, and (iv) the interval parametrization method suggested by Elishakoff and Miglis and again based on minimizations and maximizations. We will also compare the results yielded by all these methods by using the computer algebra system Mathematica for computer evaluations (including quantifier eliminations) in order to conclude which method would be the most efficient way to solve problems relevant to interval quadratic equations. Full article

Situated at middle-to-high latitudes with limited thermal resources, Northeast China is the primary maize-producing region in China. It is also one of the regions most significantly impacted by climate change. Given the persistent impact of climate change, it is crucial to elucidate the effects of the varying thermal conditions and low temperatures for different sowing dates on the growth, development, and grain maturity of spring maize. To ensure secure maize production and disaster prevention, choosing the optimal sowing time for spring maize holds significant implications for the judicious utilization of climatic resources, risk mitigation, and the provision of meteorological guidance. Moreover, it can serve as a technical reference for relevant departments to conduct climate evaluation, disaster monitoring, prediction, and assessment, as well as impact analysis of corn production safety. Additionally, it can provide meteorological evidence to ensure food security and promote the sustainable development of modern agriculture. An interval sowing experiment of spring maize was conducted in Harbin in the north of Northeast China. Two varieties were used in the experiment. Four sowing dates were set, and the interval between adjacent sowing dates was 10 days. The local perennial sowing time, 5 May, was set as the second sowing date, with one date set later and two dates set earlier. During the experiment, the growth process, grain dry matter, seed moisture content, yield components, and temperature of spring maize were observed. The impact of temperature conditions on maize growth and yield formation was analyzed in this paper through mathematical statistics, which further led to the establishment of a monitoring and evaluation model for assessing the effect of thermal conditions and temperature on maize. The results showed that the growth rate of spring maize was closely related to temperature. When the average temperature, minimum temperature, and maximum temperature increased by 1 °C, the average emergence rate increased by 1.05%, 0.99%, and 1.07%, respectively, and the average vegetative growth rate increased by 0.16%, 0.16%, and 0.09%, respectively. The change rate of ≥10 °C active accumulated temperature was significantly correlated with the change rate of the dry weight of the grain kernel, which conformed to the quadratic equation of one variable. The temperature influence coefficients of different sowing dates varied from 1.0% to 1.7%. The relationship between the accumulated values of 10 ℃ active accumulated temperature and the grain moisture content of spring maize was a logarithmic function. From 10 to 50 days after anthesis, the effect of temperature can explain about 95% of the change in grain moisture content. After physiological maturity, the effect of thermal conditions can only explain 56–83%. The temperature influence coefficient ranges from 1.3% to 13.8%. Comparatively speaking, the second sowing date is the most suitable sowing date. Early sowing is prone to encounter low temperatures, resulting in underutilization of the early heat, while late sowing is prone to less heat. Both conditions are not conducive to better improve the yield of spring maize. Full article

This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval $L_1[0,T]$ and, secondly, the existence of nonincreasing solutions in unbounded interval $L_1\left(R_{+}\right)$. Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution $\psi \in L_1[0,T]$ and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution $\psi \in L_1\left(R_{+}\right)$ on some parameters. Finally, some examples are provided to verify our investigation. Full article

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained. Full article

Aiming at the problem that the non-probabilistic reliability analysis method of slope engineering, which is based on an interval model, cannot consider the cross-correlation of geotechnical parameters, a non-probabilistic reliability analysis method of slopes based on a multidimensional parallelepiped model is proposed. This method can effectively alleviate the problem of difficult data survey in the field of geotechnical engineering. Using the limited sample data of soil parameters, the multidimensional parallelepiped model is constructed. The performance function of the slope is constructed based on Latin hypercube sampling and the quadratic response surface method. Then, the limit state equation of the slope can be standardized using the multidimensional parallelepiped model. The non-probabilistic reliability indexes of the slope are calculated based on the global optimal solution to judge the stability state of the slope. The example analysis verifies the feasibility of the proposed method. The results show that the correlation of shear strength parameters of soil has a great influence on the non-probabilistic reliability indexes of the slope. When the correlation coefficients of the shear strength parameters are between −1.0 and 0, the smaller the correlation coefficient is, the greater the non-probabilistic reliability index of the slope is; when the correlation coefficients of the shear strength parameters are between 0 and 1.0, the non-probabilistic reliability index of the slope does not change with the correlation coefficient. The non-probabilistic reliability indexes of the slope obtained using the multidimensional parallelepiped model are between the results obtained using an ellipsoid model and those obtained using an interval model, which are validated by Monte Carlo method and relatively more reasonable. In the absence of a large number of geotechnical sample data, this method provides a new way for slope stability analysis and expands the application field of calculation methods based on non-probabilistic theory. Full article

Differential equations are useful mathematical tools for solving complex problems. Differential equations include ordinary and partial differential equations. Nonlinear equations can express the nonlinear relationship between dependent and independent variables. The nonlinear second-order neutral differential equations studied in this paper are a class of quadratic differentiable equations that include delay terms. According to the t-value interval in the differential equation function, a basis is needed for selecting the initial values of the differential equations. The initial value of the differential equation is calculated with the initial value calculation formula, and the existence of the solution of the nonlinear second-order neutral differential equation is determined using the condensation mapping fixed-point theorem. Thus, the oscillation analysis of nonlinear differential equations is realized. The experimental results indicate that the nonlinear neutral differential equation can analyze the oscillation behavior of the circuit in the Colpitts oscillator by constructing a solution equation for the oscillation frequency and optimizing the circuit design. It provides a more accurate control for practical applications. Full article

Numerical simulation and inversion imaging are essential in geophysics exploration. Fourier transform plays a vital role in geophysical numerical simulation and inversion imaging, especially in solving partial differential equations. This paper proposes an arbitrary sampling Fourier transform algorithm (AS-FT) based on quadratic interpolation of shape function. Its core idea is to discretize the Fourier transform integral into the sum of finite element integrals. The quadratic shape function represents the function change in each element, and then all element integrals are calculated and accumulated. In this way, the semi-analytical solution of the Fourier oscillation operator in each integral interval can be obtained, and the Fourier transform coefficient can be calculated in advance, so the algorithm has high calculation accuracy and efficiency. Based on the one-dimensional (1D) transform, the two-dimensional (2D) transform is realized by integrating the 1D Fourier transform twice, and the three-dimensional (3D) transform is realized by integrating the 1D Fourier transform three times. The algorithm can sample flexibly according to the distribution of integrated values. The correctness and efficiency of the algorithm are verified by Fourier transform pairs. The AS-FT algorithm is applied to the numerical simulation of magnetic anomalies. The accuracy and efficiency are compared with the standard Fast Fourier transform (standard-FFT) and Gauss Fast Fourier transform (Gauss-FFT). It shows that the AS-FT algorithm has no edge effects and has a higher computational speed. The AS-FT algorithm has good adaptability to continuous medium, weak magnetic catastrophe medium, and strong magnetic catastrophe medium. It can achieve the same as or even higher accuracy than Gauss-FFT through fewer sampling points. The AS-FT algorithm provides a new means for partial differential equation solution in geophysics. It successfully solves the boundary problems, which makes it an efficient and high-precision Fourier transform approach with promising applications. Therefore, the AS-FT algorithm has excellent advantages in solving partial differential equations, providing a new means for solving geophysical forward and inverse problems. Full article

In scientific journals, it is increasingly common to find articles presenting methods for solving problems not based on idealistic mathematical models containing perfectly accurate coefficient values that cannot be obtained in practice, but on models in which coefficient values are affected by uncertainty and are expressed in the form of intervals, fuzzy numbers, etc. However, solving tasks with interval coefficients is not fully mastered, and a number of such problems cannot be solved by currently known methods. There is undeniably a research gap here. The article presents a method for solving problems governed by the quadratic interval equation and shows how to find the tolerant optimal control value of such a system. This makes it possible to solve problems that could not be solved before. The paper introduces a new concept of the degree of robustness of the control to the set of all possible multidimensional states of the system resulting from its uncertainties. The method presented in the article was applied to an example of determining the optimal value of nitrogen fertilization of a sugar beet plantation, the vegetation of which is under uncertainty. It would be unrealistic to assume precise knowledge of crop characteristics here. The proposed method allows to determine the value of fertilization, which gives a chance to obtain the desired yield for the maximum number of field conditions that can occur during the growing season. Full article

The hybrid and the dressed metric formalisms for the study of primordial perturbations in Loop Quantum Cosmology lead to dynamical equations for the modes of these perturbations that are of a generalized harmonic-oscillator type, with a mass that depends on the background but is the same for all modes. For quantum background states that are peaked on trajectories of the effective description of Loop Quantum Cosmology, the main difference between the two considered formalisms is found in the expression of this mass. The value of the mass at the bounce is especially important, since it is only in a short interval around this event that the quantum geometry effects on the perturbations are relevant. In a previous article, the properties of this mass were discussed for an inflaton potential of quadratic form, or with similar characteristics. In the present work, we extend this study to other interesting potentials in cosmology, namely the Starobinsky and the exponential potentials. We prove that there exists a finite interval of values of the potential (which includes the zero but typically goes beyond the sector of kinetically dominated inflaton energy density) for which the hybrid mass is positive at the bounce whereas the dressed metric mass is negative. Full article