Search Results (48)

With the increasing penetration of wind power, enhancing the renewable energy accommodation rate and reducing the carbon footprint of the IES, this study proposes a comprehensive evaluation method to assess the impact of a novel dynamic Green Certificate Trading (GCT) and Green Hydrogen Certificate Trading (GHCT) joint mechanism. First, considering the integration of the IES into the carbon trading market, a coupled dynamic GCT-GHCT framework is established. This framework links dynamic green electricity certificate revenues with green hydrogen certificate revenues, leveraging cross-subsidization to incentivize renewable energy consumption. Subsequently, an optimal operation model for the IES is formulated with the objective of minimizing comprehensive costs, which encompass energy procurement, green certificates, carbon trading, and wind curtailment penalties. A piecewise linearization approach is applied to transform the optimization model into a Mixed-Integer Linear Programming problem for efficient solving. Furthermore, based on the dispatch results, a multidimensional evaluation index system is constructed, extracting key indicators from economic, technical, and environmental perspectives. To ensure the rationality of the evaluation, a dynamic reward–penalty asymmetric cloud matter-element (ACME) comprehensive evaluation method based on game theory combinatorial weighting is introduced to calculate the index weights and the final comprehensive evaluation value. Finally, multi-scenario simulations are conducted to verify the superiority of the integrated GCT-GHCT trading framework. The results reveal that the proposed approach not only maximizes renewable energy integration but also provides a robust decision-making tool for the low-carbon transition of multi-energy systems. Full article

Greenhouse gas emissions reductions are urgently necessary to mitigate the effects of climate change. Several protocols and agreements are in place to reduce emissions, but global emissions continue to rise nonetheless. This is in part due to emissions offshoring: the shift of manufacturing from countries with developed economies to countries with developing economies. While many countries have achieved reductions through technological advancements, offshoring remains an issue, demonstrated by a global emissions increase despite developed economies reducing their emissions. Trends of atmospheric emission of nitrogen oxides (NO_X), which can be used as a surrogate for gauging sustainability with respect to fossil fuel combustion, confirm this issue. Canada has a developed economy and purports to have reduced emissions in recent decades. Countries accounting for 90% of the dollar value of manufactured goods imported to Canada from 1990 to 2022 were analyzed. Canada shows a decrease in NO_X emissions attributed to manufacturing alongside an increase in imports of manufactured products. Human Development Index (HDI), a United Nations metric for the development level of a country, was plotted against relative manufacturing NO_X emissions on a country basis. There are distinct trends over the time period among low, medium, high, and very high HDI categories for the top countries importing to Canada. Piecewise linear regressions were run for each country, allowing the number of breaks to be equivalent to the number of HDI category changes spanned over the time period. As the HDI category increased, the number of countries with an inverse relationship between HDI and NO_X emissions grew. Countries with very high HDI almost all showed that the HDI increase corresponded to lower emissions of NO_X, while countries with lower HDI categories showed a reduction in this trend. The results support the theory that Canada has offshored manufacturing emissions rather than decreasing the emissions they are responsible for in terms of their manufactured goods. Full article

Differential equations with fractional order play an important role in modeling some natural phenomena. This paper investigates the dynamics of the fractional-order commensal symbiosis model with the Allee effect. This model describes the relationship between prey and predator populations. The piecewise-constant approximation technique is applied to discretize this model. Equilibrium points are established, and local stability conditions are calculated using fractional-order linearization and eigenvalue-based arguments. Moreover, the bifurcation theory is successfully invoked to discuss the period-doubling bifurcation. In particular, sufficient conditions are effectively determined for the emergence of the period-doubling bifurcation. We utilize the hybrid control approach to control the behavior of the considered system. Then, some numerical examples are presented to demonstrate the accuracy and validity of the theoretical results. The findings indicate that fractional order and Allee effects improve system dynamics and substantially improve stability limits and bifurcation structures, providing new insights into how to handle the dynamics of ecological systems. Full article

This paper investigates the transition behaviour and stochastic resonance phenomenon in a class of bifurcation systems with a symmetric piecewise smooth potential, induced by a control parameter, under the combined influence of multiplicative white noise, additive white noise, and a periodic force. As the control parameter increases, the symmetric piecewise smooth potential of the system evolves from tristability to bistability. To study stochastic resonance in this system, an approximate Fokker–Planck equation is first derived based on Novikov’s theorem and the Fox approximation method. Subsequently, the approximate stationary probability density of the system is obtained from the Fokker–Planck equation, revealing the occurrence of a stochastic P-bifurcation. Then, within the bistable and multistable regimes, the effects of the bifurcation parameter, multiplicative noise intensity, and additive noise intensity on the mean first passage time (MFPT) are analysed. Finally, based on the mean first passage time, the response amplitude for stochastic resonance is derived via linear response theory, and the influences of the bifurcation parameter, noise intensities, correlation time, and signal frequency on the response amplitude are examined. In the bifurcation regime, the correctness of the expressions is verified numerically. It is found that multistability reduces the mean first passage time, and stochastic resonance is further analysed using the Fokker–Planck equation. Full article

Recently, piecewise differential operators have been introduced to capture crossover dynamics in physical systems. In the evolution of corruption, the underlying dynamics can shift across different regimes. These crossovers occur due to policy changes, economic shocks, or shifts in social behavior. To demonstrate the crossover dynamics of a corruption mathematical system, we use a piecewise operator. The piecewise operator is divided into three pieces: a classic or integer order operator, a fractional operator, and a stochastic operator. For the fractional order case, we use the constant proportional Caputo (CPC) operator, which is a straightforward linear combination of the Riemann–Liouville (RL) integral and the Caputo derivative. Theoretical analysis such as existence and uniqueness of solutions for the fractional case under CPC derivative, is elucidated via notions of fixed point theory, specifically the implication of Perov’s fixed point result and for the stochastic model using Ito calculus. Numerical results are presented for the proposed model. Graphical analysis of the corruption model is performed using PW operators across three distinct intervals to portray the crossover dynamics of the considered system. Also, the influence of various parameters on the crossover dynamics of the corruption model is illustrated via numerical simulations. Sensitivity of parameters is demonstrated via some statistical experiments, such as scatter plots and Pearson correlation coefficients, quantifying the relationship between key parameters of the system. The validity of the result is verified by comparing the system dynamics with real data dynamics via 2D graphs. Full article

Classical DEA models typically assume a linear valuation approach in performance assessment. However, in practical applications, many DMU inputs and outputs exhibit nonlinear valuation. A linear valuation may fail to accurately capture the variations in value across different DMUs. One critical challenge in efficiency evaluation is the presence of undesirable outputs, which negatively affects DMU performance. To address this, decision-makers aim to incorporate the impact of undesirable factors into efficiency measurements, enabling them to identify high-performing DMUs under comparable conditions and use them as benchmarks for inefficient ones. In response to this issue, this study introduces a novel approach based on the SBM Network DEA model to enhance airport efficiency within a two-stage framework while accounting for undesirable outputs. By applying piecewise linear theory, the model assigns lower weight to excessive quantities of undesirable outputs, effectively distinguishing DMUs that generate fewer undesirable outputs from those producing higher amounts. Furthermore, this research offers a practical benchmarking strategy for inefficient airports, aiming to improve their efficiency while considering the priority of each stage. Full article

We introduce a smoothed variant of the Smoothly Clipped Absolute Deviation (SCAD) thresholding rule for wavelet denoising by replacing its piecewise linear transition with a raised cosine. The resulting shrinkage function is odd, continuous on $\mathbb{R}$, and continuously differentiable away from the main threshold, yet retains the hallmark SCAD properties of sparsity for small coefficients and near unbiasedness for large ones. This smoothness places the rule within the continuous thresholding class for which Stein’s unbiased risk estimate (SURE) is valid. As a result, unbiased risk computation, stable data-driven threshold selection, and the asymptotic theory of Kudryavtsev and Shestakov apply. A corresponding nonconvex prior is obtained whose posterior mode coincides with the estimator, yielding a transparent Bayesian interpretation. We give an explicit SURE risk expression, discuss the oracle scale of the optimal threshold, and describe both global and level-dependent adaptive versions. The smooth SCAD rule therefore offers a tractable refinement of SCAD, combining low bias, exact sparsity, and analytical convenience in a single wavelet shrinkage procedure. Full article

This study introduces a sensorless control approach for permanent magnet synchronous motors (PMSMs) that employs an Improved Non-Singular Fast Terminal Sliding Mode Controller (IMNFTSMC) and an Improved Non-Singular Fast Terminal Sliding Mode Observer (IMNFTSMO). The IMNFTSMC employs a novel hybrid reaching law and a continuous piecewise square root switching function to achieve faster convergence and effective chattering reduction over the conventional Sliding Mode Controller (SMC). This design successfully replaces two critical components: the discontinuous constant velocity term (a key component of the traditional SMC reaching law that is a primary source of control chattering in PMSM torque regulation) and the high-gain exponential term (which tends to induce overshoot during transient speed adjustments and degrade steady-state control precision). In the IMNFTSMO, a hybrid approach combining linear and non-singular terminal sliding modes eliminates phase lag associated with low-pass filtering in traditional sliding mode observers, improving rotor position and speed estimation accuracy. Stability of both IMNFTSMC and IMNFTSMO is rigorously proven using Lyapunov stability theory.Validation through extensive simulations and hardware experiments, including challenging zero-speed start, speed stepping, and load disturbance tests, confirms the proposed strategy provides improved dynamic response, effective anti-disturbance capability, and high accuracy for rotor position and speed estimation compared to established benchmark methods, demonstrating its feasibility for mid-to-low speed sensorless PMSM drives. Full article

We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input–output pairs of an unknown function defined on a tropical semifield, the problem is to determine an approximating rational function formed by two Puiseux polynomials as numerator and denominator. With specified numbers of monomials in both polynomials, the approximation aims at evaluating the exponent and coefficient for each monomial in the polynomials to fit the rational function to the data in the sense of a tropical distance function. To solve the problem, we transform it into an approximation of a vector equation with unknown vectors on both sides, where one side corresponds to the numerator polynomial and the other side to the denominator. Each side involves a matrix with entries dependent on the unknown exponents, multiplied by the vector of unknown coefficients of monomials. We propose an algorithm that constructs a series of approximate solutions by alternately fixing one side of the equation to an already-found result and leaving the other side intact. Each equation obtained is approximated with respect to the vector of coefficients, which yields this vector and approximation error, both parameterized by exponents. The exponents are found by minimizing the error with an optimization procedure based on an agglomerative clustering technique. To illustrate, we present results for an approximation problem in terms of max-plus algebra (a real semifield with addition defined as maximum and multiplication as arithmetic addition), which corresponds to an ordinary problem of piecewise linear approximation of real functions. As our numerical experience shows, the proposed algorithm converges in a finite number of steps and provides a reasonably accurate solution to the problems considered. Full article

This paper investigates the nonlinear dynamics of the wheel-side planetary reducer, considering the tooth wear effect. The tooth wear model based on the Archard adhesion wear theory is established, and the impact of tooth wear on meshing stiffness and piecewise-linear backlash of the planetary gear system is discussed. Then, the torsional vibration model and dimensionless differential equations considering tooth wear for the wheel-side planetary reducer are established, in which meshing excitations include time-varying mesh stiffness (TVMS), piecewise-linear backlash, and transmission error. The dynamic responses are numerically solved using the fourth-order Runge–Kutta method. On this basis, the nonlinear dynamics, such as the bifurcation and chaos properties of the wheel-side planetary reducer with tooth wear, are analyzed. Simulation results demonstrate that the existence of tooth wear reduces meshing stiffness and increases backlash. The reduction in the meshing stiffness changes the bifurcation path and chaotic amplitude of the system, inducing chaotic phenomena more easily. The increase in the gear backlash causes a higher amplitude of the relative displacement and more severe vibration. Full article

This article introduces two numerical methods based on the Theory of Functional Connections (TFC) for solving linear ordinary differential equations that involve step discontinuities in the forcing term. The novelty of the first proposed approach lies in the direct incorporation of discontinuities into the free function of the TFC framework, while the second proposed method resolves discontinuities through piecewise constrained expressions comprising particular weighted support functions systematically chosen to enforce continuity conditions. The accuracy of the proposed methods is validated for both a second-order initial value and boundary value problem. As a final demonstration, the methods are applied to a third-order differential equation with non-constant coefficients and multiple discontinuities, for which an analytical solution is known. The methods achieve error levels approaching machine precision, even in the case of equations involving functions whose Laplace transforms are not available. Full article

Bilinear systems can be developed from the point of view of time-varying linear differential equations or from the symmetry of Lie theory, in particular Lie algebras, Lie groups, and Lie semigroups. For bilinear control systems with bounded control range, we analyze when a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists, for the induced system on projective space. We use the system semigroup by considering piecewise constant controls and use spectral properties to extend the result to bilinear systems in ${\mathbb{R}}^d$. The contribution of this paper highlights the relationship between all the existent control sets. We show that the controllability property of a bilinear system is equivalent to the existence and uniqueness of a control set of the projective system. Full article

In this study, we examine the bifurcations and dynamics of a piecewise linear van der Pol equation—a model that captures self-sustained oscillations and is applied in various scientific disciplines, including electronics, neuroscience, biology, and economics. The van der Pol equation is transformed into a piecewise linear system to simplify the analysis of stability and controllability, which is particularly beneficial in engineering applications. This work explores the impact of increasing the number of linear segments on the system’s dynamics, focusing on the stability of the equilibria, phase portraits, and bifurcations. The findings reveal that while the bifurcation structure at critical values of the bifurcation parameter is complex, the topology of the piecewise linear model remains unaffected by an increase in the number of linear segments from three to four. This research contributes to our understanding of the dynamics of nonlinear systems with piecewise linear characteristics and has implications for the analysis and design of real-world systems exhibiting such behavior. Full article

Rock materials failures are accompanied by the co-existence of various failure mechanisms, including rock fracturing, shearing, and compaction yield. These mechanisms manifest macroscopically as multiple failure modes and nonlinear strength characteristics related to stress levels. Considering the limitations of current rock mechanics strength theories, which are primarily derived from single failure mechanisms, this study evaluates the applicability of alternative strength theories. Based on the extensional-strain criterion and the PMC (Paul-Mohr-Coulomb) model, a piecewise linear strength model was proposed that is suitable for analyzing multiple failure mechanisms in rocks, revealing the intrinsic mechanisms of multi-mechanism rock material failure. A multiple failure mechanism strength model in the form of inequalities was proposed, using the generalized shear stress, mean stress, and stress Lode angle as parameters. Strength tests conducted on sandstone and granite rock material samples under different stress conditions revealed distinct piecewise linear strength characteristics for both rock types, validating the rationality and applicability of the multiple failure mechanism model. The findings construct a multi-mechanism failure model for rocks, providing enhanced predictive capabilities and aiding in the prevention of rock structural failures. Full article

We consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order $\alpha \in (0,1)$, and the non-linear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of $O(\Delta t^{\alpha })$ in the mean square norm, where $\Delta t$ denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory. Full article