Piecewise Linear Approximation Research Articles

Real-Time Power Market Clearing Model With Improved Network Constraints Considering PTDF Correction and Fast-Calculated Dynamic Line Rating

The increasing penetration of renewables in power systems reconfigures the generation mix and geography distribution, resulting in more frequent power flow congestion. Conventionally, a relatively conservative dispatch margin for transmission congestion in network constraints was reserved to ensure safe operation due to potential deviations introduced by DC power flow modeling and conservative line rating. This article proposes a real-time power market clearing model with improved network constraints considering power transfer distribution factor (PTDF) correction and fast-calculated dynamic line rating. The clearing model based on PTDF correction is proposed first employing the Taylor expansion with the voltage and phase angle approximation. An improved dynamic line rating model is further applied into the clearing model based on the piecewise linear approximation method, which calculates real-time conductor temperature faster with sufficient precision. We test and verify the validity of the proposed model based on three independent systems including the IEEE 39-bus system, the IEEE 2383-bus system, and the real power grid system in Yunnan province of China, respectively. The results show that the improved clearing model reduces the calculated power flow deviation between the clearing results and real conditions compared with traditional security constraints, where the efficiency of power grid operation is also increased.

Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source

The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.

Time-Reduced Model for Multilayer Spiking Neural Networks

Spiking neural networks (SNNs) is a type of biological neural network model, which is more biologically plausible and computationally powerful than traditional artificial neural networks (ANNs). SNNs can achieve the same goals as ANNs, and can build a large-scale network structure (i.e. deep spiking neural network) to accomplish complex tasks. However, training deep spiking neural network is difficult due to the non-differentiable nature of spike events, and it requires much computation time during the training period. In this paper, a time-reduced model adopting two methods is presented for reducing the computation time of a deep spiking neural network (i.e. approximating the spike response function by the piecewise linear method, and choosing the suitable number of sub-synapses). The experimental results show that the methods of piecewise linear approximation and choosing the suitable number of sub-synapses is effective. This method can not only reduce the training time but also simplify the network structure. With the piecewise linear approximation method, the half of computation time of the original model can be reduced by at least. With the rule of choosing the number of sub-synapses, the computation time of less than one-tenth of the original model can be reduced for XOR and Iris tasks.

Branch-and-price algorithms for large-scale mission-oriented maintenance planning problems

This paper presents a novel column-generation-based approach for solving large-scale instances of the joint selective maintenance and repairperson assignment problem (JSM-RAP) for mission-oriented systems in industrial settings. Such systems perform consecutive missions separated by scheduled finite-duration breaks during which some of their components are imperfectly maintained by repairpersons, aiming to maximize system reliability in subsequent missions. The resulting mathematical model is computationally expensive, even for problems of moderate size. The proposed approach decomposes the JSM-RAP into a master problem and multiple subproblems that are solved to generate maintenance patterns, i.e., columns. Two methods are developed to handle the mixed-integer nonlinear subproblems: a piecewise-linear approximation and an exact reformulation into mixed-integer exponential conic programs. Branch-and-price algorithms are developed by embedding the column-generation method into a branch-and-bound tree to restore solution integrality and guarantee its optimality. Furthermore, we use a stabilization scheme to accelerate convergence. Numerical experiments validate the proposed approach and demonstrate its added value in terms of computation time and solution quality. Problem instances of very-large size, similar to real-life industrial production plants, are solved efficiently. Results also show that increasing the number of maintenance levels grants more flexibility to the optimizer to find combinations of components and maintenance actions that better use the limited resources.

An immersed Crouzeix–Raviart finite element method in 2D and 3D based on discrete level set functions

This paper is devoted to the construction and analysis of immersed finite element (IFE) methods in three dimensions. Different from the 2D case, the points of intersection of the interface and the edges of a tetrahedron are usually not coplanar, which makes the extension of the original 2D IFE methods based on a piecewise linear approximation of the interface to the 3D case not straightforward. We address this coplanarity issue by an approach where the interface is approximated via discrete level set functions. This approach is very convenient from a computational point of view since in many practical applications the exact interface is often unknown, and only a discrete level set function is available. As this approach has also not be considered in the 2D IFE methods, in this paper we present a unified framework for both 2D and 3D cases. We consider an IFE method based on the traditional Crouzeix–Raviart element using integral values on faces as degrees of freedom. The novelty of the proposed IFE is the unisolvence of basis functions on arbitrary triangles/tetrahedrons without any angle restrictions even for anisotropic interface problems, which is advantageous over the IFE using nodal values as degrees of freedom. The optimal bounds for the IFE interpolation errors are proved on shape-regular triangulations. For the IFE method, optimal a priori error and condition number estimates are derived with constants independent of the location of the interface with respect to the unfitted mesh. The extension to anisotropic interface problems with tensor coefficients is also discussed. Numerical examples supporting the theoretical results are provided.

Linear and piecewise linear formulations for a hierarchical facility location and sizing problem

This paper presents a hierarchical location and sizing problem in the presence of joint partial coverage and unreliable facilities. We consider a 3-level hierarchical production-distribution system of a supply chain network in which the lowest level facilities act as the first points of contact for customers and the upper level facilities supply the lower level facilities. For the problem, we first develop an integer nonlinear program which determines the number, location, and size of two types of facilities as well as their primary and backup assignments within the network such that the weighted total demand coverage is maximized under budget constraints. Adopting a special network mapping technique, we then develop an equivalent mixed integer linear programming formulation. Next, we propose two competing piecewise linear approximations, one based on a separable programming approach and the other on a tangent line approximation method. We finally assess the performance of the three proposed formulations via numerical experiments carried out for a variety of problem instances with different sizes under three optimality gap settings and two linearization approximation error level alternatives. Our results show that both approximations are promising and outperform the exact formulation both in terms of computation time and solution quality.

Temperature Drift Compensation of a MEMS Accelerometer Based on DLSTM and ISSA.

In order to improve the performance of a micro-electro-mechanical system (MEMS) accelerometer, three algorithms for compensating its temperature drift are proposed in this paper, including deep long short-term memory recurrent neural network (DLSTM-RNN, short DLSTM), DLSTM based on sparrow search algorithm (SSA), and DLSTM based on improved SSA (ISSA). Moreover, the piecewise linear approximation (PLA) method is employed in this paper as a comparison to evaluate the impact of the proposed algorithm. First, a temperature experiment is performed to obtain the MEMS accelerometer's temperature drift output (TDO). Then, we propose a real-time compensation model and a linear approximation model for neural network methods compensation and PLA method compensation, respectively. The real-time compensation model is a recursive method based on the TDO at the last moment. The linear approximation model considers the MEMS accelerometer's temperature and TDO as input and output, respectively. Next, the TDO is analyzed and optimized by the real-time compensation model and the three algorithms mentioned before. Moreover, the TDO is also compensated by the linear approximation model and PLA method as a comparison. The compensation results show that the three neural network methods and the PLA method effectively compensate for the temperature drift of the MEMS accelerometer, and the DLSTM + ISSA method achieves the best compensation effect. After compensation by DLSTM + ISSA, the three Allen variance coefficients of the MEMS accelerometer that bias instability, rate random walk, and rate ramp are improved from 5.43×10-4mg, 4.33×10-5mg/s12, 1.18×10-6mg/s to 2.77×10-5mg, 1.14×10-6mg/s12, 2.63×10-8mg/s, respectively, with an increase of 96.68% on average.

An efficient spatial branch-and-bound algorithm using an adaptive branching rule for linear multiplicative programming

In this paper, a spatial branch-and-bound algorithm with an adaptive branching rule is proposed for solving linear multiplicative programming (LMP) problem. In the solution algorithm, LMP problem is first transformed into an equivalent problem, and a novel reformulation is then introduced to convert the nonconvex constraints into differences of square terms form, subsequently using a piecewise linear approximation for the concave part. By using the proposed adaptive branching rule for dividing rectangles and iteratively refining the piecewise linear approximations, the process of solving LMP problem can be translated into solving a series of second order cone relaxations (SOCR). Also, we discuss the bound on the optimality gap as a function of the approximation errors at the iterate, and estimate the computational complexity in the order of O(ɛ) to attain an ɛ-optimal solution. Finally, preliminary numerical results demonstrate that the proposed algorithm can efficiently find the global optimal solutions for test instances.

Hydroacoustic communication channel capacity

Objectives . Capacity, describing the maximum rate of information transmission, is an important characteristic of any communication channel. The main purpose of this work isto determine the capacity of a hydroacoustic communication channel with constrained average intensity of the transmitted signal. An additional aim consists in finding the optimal spectrum of a transmitted signal and calculate its boundary frequencies. A model of a single-path channel was considered, which is characteristic of the deep sea with the receiver or transmitter placed at a sufficient depth. Methods . Concepts of applied hydroacoustics, the theory of random processes, and information theory were used. Results . An expression for gain in a hydroacoustic communication channel has been obtained. A novel expression derived for the spectral level of sea noise caused by sea surface waves is based on piecewise linear approximation of the curves of the spectral levels of noise obtained from four sources: turbulence, shipping, sea waves, and the thermal noise of the sea. Dependencies of the hydroacoustic channel capacity on communication distance, intensity of the transmitted signal, and sea state, are characterized. The definition of the optimal spectrum itself is determined along with the lower and upper boundary frequencies of the optimal spectrum of the transmitted signal. The dependence of the bandwidth usage on the intensity of the input signal at various communication distances has been investigated. Conclusions . On the basis of the Francois–Garrison attenuation coefficient, channel capacity was correlated with the parameters of the marine environment: temperature, salinity, and pH in the study area. At a given intensity of the input signal, channel capacity was shown to decrease significantly with increasing distance and sea wave intensity. It is also shown that the width of the optimal spectrum decreases with increasing distance. Sea wave noise was noted to affect significantly the shape of the optimal spectrum and its boundary frequencies. The possibility of cases where bandwidth usage increases with increasing distance at a given input signal intensity cannot be ruled out.

Full-Wave 3-D Transient Analysis With Method of Moments and Numerical Laplace Transform Including Resistive Non-Linear Elements

Methods based on both time ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> )-domain and frequency ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> )-domain have been used for electromagnetic transient (EMT) analysis using both circuit-theory and field-theory approaches. Methods based on circuit theory are fast and convenient but may become inapplicable to many engineering problems due to simplifications such as the transverse electromagnetic assumptions. Moreover, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> -domain-based techniques rely on approximate methods to account for frequency dependence, which is an essential characteristic of electric conductors, equipment and soil environment. In this paper, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</i> -domain solutions obtained from the method of moments (MoM) discretization of the electric field integral equation (EFIE) are converted to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-</i> domain using the numerical Laplace transform (NLT) by means of post-processing. This circumvents the need to formulate the EFIE and MoM based on the complex frequency which would be required for their application to conventional NLT. Hence, existing MoM implementations can be used to perform a full-wave 3-D EMT analysis for a wide range of power system applications. Examples include energizations of 3-D power system networks with fast and non-vanishing excitations such as step functions, as well as modeling non-linear elements using the piecewise linear approximation. Results from experimental measurements, finite-difference <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> -domain method, and EMT-type software confirm the validity of the proposed method for power system transients in the range of microseconds down to nanoseconds.

Real‐time emission and cost estimation based on unit‐level dynamic carbon emission factor

AbstractThe real‐time carbon emission estimation of generators helps quantify the carbon emission costs and reduce power system emissions of power generation. Accurate estimation relies on the accuracy of the carbon emission factors (EFs) and power generation measurements. The dynamic carbon emission factor (DEF) of generators was proposed recently as a linear function of the output power. However, there is a significant deviation between the modelled DEFs and the actual measurement EFs, especially when the output power is low. This paper first presents the general definition of the DEF to characterize the emissions and focuses on the unit‐level DEF (UDEF). The piecewise non‐linear UDEF (P‐UDEF) model is then proposed, which can better represent the unit emission characteristics. Then an accurate piecewise linear cost approximation method is proposed considering the segment points and extreme points of both P‐UDEF and generation costs function. Last, the system carbon emissions and costs estimation are estimated by combined economic emission dispatch (CEED), and the reduction potential is evaluated. Case studies on an IEEE 30‐bus system with piecewise linear cost functions show that the proposed P‐UDEF can realize real‐time emission and cost estimation as well as reduce the total system emissions by considering the incomplete combustion cost of generating units.

Resonant mode coupling approximation for calculation of optical spectra of stacked photonic crystal slabs. Part II

We propose the further development of the resonant mode coupling approximation for the calculation of optical spectra of stacked periodic nanostructures in terms of the scattering matrix. We previously showed that given the resonant input and output vectors as well as background scattering matrices of two subsystems, one can easily calculate those for the combined system comprising two subsystems. It allows us to write a resonant approximation for the combined system and speed up the calculation significantly for typical wave scattering problems. This approach is best suitable for isolated resonances when the approximation of a constant background scattering matrix is reasonable. In this article, we aim at expanding the applicability of the resonant mode coupling approximation by utilizing more complicated approximations for the background matrices. In particular, we show that consideration of energy-dependent correction terms for the background matrices remarkably reduces the calculation error of the resonant parameters. Here we first consider a linear approximation of the background scattering matrices which is subsequently used as a base for a piecewise-linear approximation. We show that the latter allows one to keep the approximation error negligibly small with only a few sample points and to apply the resonant mode coupling approximation within almost arbitrary large energy ranges. Finally, we consider the approximation of background matrices by arbitrary matrix functions and propose a technique to derive resonant poles in this case. The methods described here could be considered an alternative approach for calculating the optical spectra of stacked systems.

A Method for Calculating the Derivative of Activation Functions Based on Piecewise Linear Approximation

Nonlinear functions are widely used as activation functions in artificial neural networks, which have a great impact on the fitting ability of artificial neural networks. Due to the complexity of the activation function, the computation of the activation function and its derivative requires a lot of computing resources and time during training. In order to improve the computational efficiency of the derivatives of the activation function in the back-propagation of artificial neural networks, this paper proposes a method based on piecewise linear approximation method to calculate the derivative of the activation function. This method is hardware-friendly and universal, it can efficiently compute various nonlinear activation functions in the field of neural network hardware accelerators. In this paper, we use least squares to improve a piecewise linear approximation calculation method that can control the absolute error and get less number of segments or smaller average error, which means fewer hardware resources are required. We use this method to perform a segmented linear approximation to the original or derivative function of the activation function. Both types of activation functions are substituted into a multilayer perceptron for binary classification experiments to verify the effectiveness of the proposed method. Experimental results show that the same or even slightly higher classification accuracy can be achieved by using this method, and the computation time of the back-propagation is reduced by 4–6% compared to the direct calculation of the derivative directly from the function expression using the operator encapsulated in PyTorch. This shows that the proposed method provides an efficient solution of nonlinear activation functions for hardware acceleration of neural networks.

A Variational Formulation for Fins with Nonzero Contact Thermal Resistance at the Base

This paper considers the steady-state heat transfer process in a fin with a Robin boundary condition at the base (instead of the usual Dirichlet boundary condition at the base). Robin boundary condition models the effect of the thermal resistance between the base of the fin and the surface on which the fin is placed. This work presents an equivalent minimum principle, represented by a convex and coercive functional, ensuring the solution’s existence and uniqueness. In order to illustrate the use of the proposed functional for reaching approximations, the heat-transfer process in a trapezoidal fin considering a piecewise linear approximation is simulated. The Appendix presents a case in which the exact solution in a closed form has been achieved.

Application of spectral approximation to model the dynamic properties of wireless communication channels

This work is devoted to the application of approximation methods for modeling wireless radio communication channels. The rationale and prospects of the use of unmanned aerial vehicles as part of flying self-organizing networks for the transmission of high-speed information in the conditions of «smart cities» are demonstrated. The problem of using these networks is also referred, associated with the technical complexity of ensuring acceptable reliability and quality of wireless communication, associated with the multipath of signal propagation and a number of other factors. It is shown that in this aspect, the development of mathematical modeling methods of analyzing signals at the inputs of UAV radio receivers in order to evaluate their amplitude- phase transformations by a communication channel is of special significance. It has been established that the connection between the signals on the transmitting and receiving sides of an arbitrary wireless radio communication channel, assuming its linearity, can be uniquely determined by a complex transfer function in the frequency domain, which in practice is very complex and difficult to describe analytically. In this regard, an approach to its approximation by an equivalent model, described by fractional- rational functions of a complex variable, physically realized by mixed connections of various linear inertial and inertial links, is proposed, and for modeling dynamic characteristics, a numerical-a nalytical modeling technique based on the spectral method and piecewise linear approximation. A results of applying the proposed solutions are shown.

Stability test for neutral type delay systems: a piecewise linear approximation scheme*

A necessary and sufficient test for the stability of neutral type linear time-delay systems is presented. Our approach combines the Lyapunov functional expressed in terms of the delay Lyapunov matrix for this class of systems with a piecewise linear approximation of the functional argument. As a result, a stability criterion is given in terms of the non-negative definiteness test of a matrix. An example is presented to validate the obtained result.

Differentiated approach to the diagnostics of economic security and resilience of Russian regions (case of the Volga federal district)

Relevance. Given the volatile global economy, unfriendly policy towards Russia and other external challenges, it is necessary to improve tools for predicting threats and risks to regional economic security and resilience. To this end, individual projections and indicators, as well as complex models should be examined. Research objective. The study aims to develop a new differentiated approach to diagnostics of indicators (projections) of economic security and resilience at the regional level, which can help visualize and evaluate threats to economic meso-systems. Data and Methods. Comparative and indicative approaches, ranking, piecewise linear approximation (scaling) and correlation analysis were used in the study. Results. As a result, the study presents the author's system of indicators of economic security and risks to the regional economy based on a differentiated approach. 36 indicators were grouped into 4 projections – general economic, social, technological and financial – and divided into sub-projections. In order to provide adequate and comparable estimates in the regional and temporal context, various methodological principles were used: application of relative indicators; assessment of cost indicators of regional development using the number of fixed market baskets (FMB) of the region (ratio of the regional cost indicator and the cost of a fixed market basket). Such approach allowed us to evaluate regional economic security in dynamics. The diagnostics of resilience of regional economic systems was performed in the context of individual projections by comparing crisis and relatively stable periods. Conclusions. The methodology was tested using data from regions of the Volga Federal District. The study revealed specific projections, sub-projections and indicators affected by threats, as well as demonstrated the ability of regions of the Volga Federal District to face the crisis and, in particular, resist sanctions.

A Bayesian Filtering Method for Wiener State-Space Systems Utilizing a Piece-wise Linear Approximation

In this paper, we develop a filtering algorithm for Wiener systems written in state-space form which considers correlated noise sources. The output non-linearity is approximated by using a piece-wise linear function. The probability function of the output signal conditioned to the system state is written as a Gaussian mixture distribution. A Gaussian sum filter algorithm to obtain the a posteriori probability density function of the state given the current and past output is developed. The associated statistics of the system state are obtained. The benefits of our proposal are illustrated via numerical simulations.

Teaching assistant selection in Thailand by using an extended VIKOR based on piecewise linear approximation of fuzzy numbers

Because of the COVID-19 situation, selection for a teaching assistant position to get a TA scholarship in a university in Thailand needs to be performed online by the formed committee. Due to the online process and the limited number of scholarships offered by the university, beyond the face-to-face interview, multiple-criteria decision analysis can help to select a proper student. In this study, we use the extended VIKOR method with fuzzy numbers to help committees to select the students from the applicants. The criteria and the weights of the criteria are provided with the help of committees. Both trapezoidal and triangular linguistic variables are used to find the solution and to observe the range of the possible result. The different weights supporting the strategy of maximum group utility are varied to detect the potential alternatives. The ranking results are also compared with the one obtained from the TODIM approach to illustrate the appropriate alternative.

Homoclinic–Heteroclinic Bifurcations and Chaos in a Coupled SD Oscillator Subjected to Gaussian Colored Noise

The so-called coupled smooth and discontinuous (SD) oscillator whose stiffness term leads to a transcendental function is a simple mass-spring system constrained to a straight line by two parameters, which are the dimensionless distances to the fixed point. This paper studies the homoclinic–heteroclinic chaos in a coupled SD oscillator subjected to Gaussian colored noise. In order to investigate the chaos thresholds analytically, the piecewise linearization approximation is used to fit the transcendental function. Stochastic nonsmooth Melnikov method with homoclinic–heteroclinic orbits is developed to study chaos thresholds of oscillators with tri-stable potential. Based on stochastic Melnikov process, the mean square criterion and the rate of phase-space flux function theory are used to study the chaotic motions of a coupled SD oscillator under weak noise and strong noise, respectively. The obtained results show that it is effective to use the piecewise linear approximation to analyze chaos in the coupled SD oscillator subjected to Gaussian colored noise. It also lays the foundation for chaos research of other nonsmooth mechanical vibration systems under random excitation.