Piecewise Linear Research Articles

QPA: A Quantization-Aware Piecewise Polynomial Approximation Methodology for Hardware-Efficient Implementations

Piecewise polynomial approximation (PPA) on nonlinear functions plays an important role in high-precision computing. In this article, we proposed QPA, an integration of error-flattened quantization-aware PPA methods, to generate the optimized coefficients for efficient hardware implementations targeting any polynomial order. QPA incorporated four key features to minimize the fitting error and the hardware cost, including using the Remez algorithm to compute the minimax fitting polynomial, combining the fitting and quantization operations to get an error-flattened characteristic, assigning specific coefficient bit width to each multiplier to reduce the hardware cost, and fine-tuning the truncated coefficients to further reduce the fitting error. Experimental results showed that our methods consistently achieved the lowest fitting error compared with the state-of-the-art error-flattened piecewise approximation methods. We synthesized the proposed designs with 28-nm TSMC CMOS technology. The results showed that the proposed designs achieved up to 37.0% area reduction and 50.5% power consumption reduction compared to the state-of-the-art error-flattened piecewise linear (PWL) method, and up to 27.0% area reduction, 21.4% delay reduction, and 20.8% power consumption reduction compared to the state-of-the-art error-flattened piecewise quadratic (PWQ) method.

Heterogeneity in resilience patterns and its prediction of 1-year quality of life outcomes among patients with newly diagnosed cancer: An exploratory piecewise growth mixture model analysis

PurposeThis study was designed to explore the impact of a new cancer diagnosis on resilience of patients and whether the resilience patterns could predict Quality of Life (QoL) in the first year. MethodsAn exploratory linear piecewise growth mixture modeling (PGMM) with one hypothetical dot (3 months since diagnosis, T1) was employed to identify different resilience patterns and growth in 289 patients with different cancer diagnoses at five assessment occasions (T0-T4). Logistic regression analysis was performed to select potential predictors and receiver operating characteristic (ROC) curve analysis was utilized to test PGMM’s discriminative ability against 1-year QoL. ResultsFive discrete resilience trajectories with two growing trends were identified, including “Transcendence” (7.3%), “Resilient” (47.4%), “Recovery” (18.7%), “Damaged” (14.9%) and “Maladaption” (11.8%). Advanced stage, colorectal cancer, and receiving surgery therapy were significant predictors of negative resilience trajectories (“Damaged” or “Maladaption”). Discriminative ability was good for PGMM (AUC = 0.81, 95%CI, 0.76–0.85, P < 0.0001). ConclusionHeterogeneity is identified in resilience growth before and after 3 months since diagnosis. 26.7% newly diagnosed patients need additional attention especially for those with advanced colorectal cancer and receiving surgery therapy.

LEO Satellite Clock Modeling and Its Benefits for LEO Kinematic POD

High-accuracy Low Earth Orbit (LEO) satellite clock and orbital products are preconditions to realize LEO augmentation for high-accuracy GNSS-based positioning on the ground. There is a high correlation between the orbit and clock parameters in the kinematic Precise Orbit Determination (POD) process. While future LEO satellites are planned to be equipped with better clocks, the benefits of modeling high-stability LEO satellite clocks are not yet thoroughly investigated, particularly when mid- to long-term systematic effects induced by the complex LEO relativistic effects and the external environment remain in the clocks. Through clock modeling, this study attempts to reduce not only the short-term noise of radial kinematic orbits, but also mis-modeled effects caused by, e.g., real-time GNSS orbital and clock errors. To explore the benefits of clock modeling, the clocks need to be first detrended by the mid- to long-term systematic effects. While over-detrending limits the orbital improvements, weak detrending would also hamper strong clock modeling and easily lead to performance degradations. A balance between the strengths of the detrending and the model thus needs to be investigated for different clock types. In this study, the Piece-Wise Linear (PWL) model of different time lengths and a 2.5-state filter with different strengths (h values) are tested using real data from GRACE FO-1 with an Ultra-Stable Oscillator (USO) on board. Using the CNES real-time GPS products, it was found that when detrending the clocks with a smoothing window of 300 to 500 s, one could generally expect an improvement larger than 10% in the estimation of radial orbits when applying a PWL model with a length from 300 to 1200 s. Improvements of this size can also be expected when using the 2.5-state model with h−1 (for Flicker Frequency Noise) from 10−28 to 10−30.

Approximate Solution Using Elementary Functions of Mixed Problem with Boundary Conditions of the Second Kind for One-Dimensional Wave Equation

The paper considers a mixed problem with boundary conditions of the second kind for a one-dimensional wave equation. The solution to this problem is written in integral form using the Green’s function. For practical use, this solution is of little use, since, firstly, the Green’s function is a trigonometric series and, therefore, its calculation presents certain difficulties, secondly, it is necessary to calculate approximately the five integrals with the Green’s function included in the solution of the problem, and, thirdly, it is extremely difficult to estimate the error of the approximate calculation of the solution. In this work, these difficulties are overcome, namely, simple expression for the Green’s function is found in terms of a periodic piecewise linear function, the integrals included in the approximate solution are calculated using periodic piecewise linear, piecewise quadratic and piecewise cubic functions, and, finally, a simple and efficient estimate of the approximation error is obtained. The error estimate is linear in the grid steps of the problem and uniform in the spatial variable at any fixed point in time. Thus, an approximate solution of the problem with an arbitrarily small error is effectively expressed in terms of elementary functions. An example of solving the problem by the proposed method is given, and graphs of the exact and approximate solutions are plotted.

A method for power flow calculation and optimal dispatch of gas–thermal–electricity multi energy system considering unit commitment

The optimal scheduling of integrated energy system considering unit commitment is a mixed integer nonlinear programming (MINLP) problem with 0-1 binary variables, and the efficient solution of MINLP has always been a research hotspot. In this paper, a power flow calculation method of gas thermal power multi energy coupling system is proposed. Considering the output of coal-fired, gas-fired and new energy units, a simplified unit commitment calculation model is established, which can greatly reduce the number of binary variables and constraints. At the same time, an optimal scheduling model with minimum economic cost and minimum carbon emission is established. For the nonlinear and non convex problem of air pressure equations, the incremental piecewise linearization method based on the idea of feasible region segmentation is used to transform it into MILP model, and the effective optimal solution is obtained. Finally, an example is given to verify the effectiveness and accuracy of the proposed algorithm.

Modeling of FGM contacts using piecewise linear variation in Young’s modulus

Functionally graded materials (FGM) are used as coatings at contact interfaces to tailor the desired surface properties while avoiding sudden changes in material properties. This work investigates the full-sliding frictional contact between a rigid indenter and FGM-coated half-space under plane-strain conditions. The FGM-coating is divided into a number of sublayers, with Young’s modulus varying linearly in each sublayer, effectively making the overall variation piece-wise linear. This approach overcomes the disadvantage of discontinuity of material properties at the interfaces of sublayers that exists in multi-layered piecewise constant modeling. Series solution approach based on the solution to the singular integral equation that characterizes the 2D contact is considered. In this approach, pressure distribution at the interface is expressed as a series expansion. This approach can be used to analyze any arbitrarily shaped contacts. The results show that the piecewise linear FGM modeling approach with tractions expressed as modified Fourier series is a more accurate and efficient way of representing FGM material property as compared to piecewise constant material properties, which are used in most of the existing literature. The number of layers needed to converge in continuous variation is also very less compared to the piece-wise constant and hence the computational time is less.

Optimal intersection design and signal setting in a transportation network with mixed HVs and CAVs

It is widely recognized that human-driven vehicles (HVs) and connected and autonomous vehicles (CAVs) are expected to coexist and share the urban traffic infrastructure in the transportation network for a long time. To fully utilizes CAVs’ potential to reduce congestion in the transitional period, this study proposes and addresses the intersection design and signal setting problem in the transportation network with mixed HVs and CAVs. Due to the difference in terms of communication technology and autonomous driving technology for HVs and CAVs, three types of intersections have been proposed to amplify the efficiency-improvement benefit from CAVs by separating CAVs from HVs in a temporal or local-spatial dimension: the conventional signalized intersection, the novel signalized intersection with a dedicated CAV phase and dedicated CAV approaches, and the intelligent signal-free intersection. The problem is to determine the spatial layout of different types of intersections in the transportation network, the cycle time, and green time duration for each phase of signalized intersections that minimize the total travel cost, in which the route choice behavior of heterogeneous travelers has been respected based on the user equilibrium principle. A mixed-integer nonlinear programming model is developed to formulate the proposed intersection design and signal setting problem based on the link-node modeling method, in which the path enumeration is avoided. Then, by employing various linearization techniques (e.g., disjunctive constraints, logarithmic transformation, piecewise linearization with logarithmic-sized binary variables and constraints, outer-approximation technique), the proposed model can be further transformed into a relaxed sub-problem in the form of mixed-integer linear programming. A globally optimal solution algorithm embedding with solving a sequence of relaxed sub-problems and nonlinear mixed complementarity problems is proposed to converge to a global optimum. The results of numerical experiments illustrate that the proposed methodology can significantly improve the performance of the whole network. Moreover, it consistently outperforms the optimization model considering only conventional signalized intersections under various CAV market penetration rates.

Adaptive Piecewise Equivalent Circuit Model With SOC/SOH Estimation Based on Extended Kalman Filter

Battery modelling plays a critical role in battery management tasks. A model that provides accurate estimations of state of charge and state of heath in varying operating conditions could significantly improve the performance of battery management systems. Departing from existing literature, this paper presents a self-adaptive Piecewise Equivalent Circuit Model (PECM) based on Extended Kalman Filter (EKF). While traditional Equivalent Circuit Models (ECM) are typically parameterized and validated for a specific range of working conditions (temperature, current and etc.), PECM is able to adapt itself to any working condition in real time. Established in the form of a combination of linear and nonlinear piecewise functions, the model parameters are continuously adjusted based on the measurement of voltage, current, and temperature. Another advantage of PECM is it does not require any prior tests in the lab, for example the Open Circuit Voltage (OCV) test which is time consuming and needs to be calibrated when aged. PECM is accurate, flexible and efficient. It has been validated for different battery chemistries, duty cycles, and temperatures. Furthermore, PECM comes with the State of Charge (SOC) and State of Health (SOH) estimation, which is shown in the model validation process and the degradation study. The results demonstrate that the piecewise parameter adaptation proposed in this paper can be applied to a range of different battery chemistries and at different aged states.

Gaussian kernelized feature selection and improved multilayer perceptive deep learning classifier for software fault prediction

Software fault prediction is the significant process of identifying the errors or defects or faults in a software product. But, accurate and timely detection is the major challenging issue in different existing approaches to predicting software defects. A novel Gaussian linear feature embedding-based statistical test piecewise multilayer perceptive deep learning classifier (GLFE-STPMPDLC) is introduced to improve software fault prediction accuracy and minimize time consumption. First, the input data are collected from the dataset. Next, the software metrics selection is carried out to select the significant metrics using Gaussian kernelized locally linear embedding with lesser software fault prediction. Then classification is carried out by Kaiser Meyer piecewise multilayer perceptive deep learning classifier for software fault prediction. The novelty of Kaiser–Meyer–Olkin (KMO) correlation test analyzes testing and training instances. The innovation of the Heaviside step activation function is applied for analyzing the KMO correlation test results and providing the final software fault prediction results. Finally, accurate fault prediction outcomes are achieved at the output layer with lesser error. Simulation of proposed GLFE-STPMPDLC technique achieves better 5%, 3%, 3% and 3% enhancement of fault prediction accuracy, precision, recall, and f-measure and 13% faster prediction time compared to conventional methods.

A Two-Degree-of-Freedom Vibro-impact Triboelectric Energy Harvester for larger bandwidth

The efficiency of the energy harvesters can be improved by increasing the harvester bandwidth. Toward this, we presented a Two-Degree of Freedom (2-DOF) Vibro-impact Triboelectric Energy Harvester by combining multi-modality and piecewise linearity of two close resonant frequencies. The harvester structure consists of a primary cantilever beam attached to a secondary cantilever beam through a tip mass. The secondary beam is attached in the opposite direction to the primary beam. The secondary beam’s bottom surface acts as a triboelectric generator’s upper electrode. A lower electrode with bonded Polydimethylsiloxane (PDMS) insulator is attached at some gap separation distance underneath the upper electrode to create an impact structure. When the system vibrates, an impact between the triboelectric layers generates an alternating electrical signal. A 2-DOF system with lumped parameter theoretical model was developed to extract the governing equations. In addition, the experimental results were extracted to validate the theoretical model. Afterward, parametric analysis based on the variation of gaps, resistance, and surface charge density was theoretically investigated. Finally, the proposed harvester demonstrated an increase in the maximum output voltage by more than 300%, and a 250% increase in the bandwidth, by changing the excitation level from 0.1 g to 0.7 g. The result of this study can pave the way for an efficient energy harvester that can scavenge ambient vibrations over a wide range of excitation frequencies.

A Hierarchical Motion Planning System for Driving in Changing Environments: Framework, Algorithms, and Verifications

In this article, a hierarchical real-time motion planning system is proposed to solve complex navigation problem in realistic dynamic traffic environments. First, a longitudinal safety spacing model is established to describe the possible collision risk in the lane-changing (LC) process, and a piecewise linearization method is proposed to convert the nonlinear constraints into linear constraints. Second, a decision-making strategy is proposed to decide whether the trajectory should be replanned according to the real-time prediction of the surrounding environments, and select the optimal lane. Third, the quintic B-spline curve method and the quadratic programming method are integrated to obtain the optimal LC trajectory, considering factors of safety, comfort, and efficiency. The terminal position objective function based on the artificial potential field is designed to soften the hard security constraints and guide the LC trajectory toward the center of safety zone. Finally, a model predictive control method based on the kinematics vehicle model is utilized to track the planned trajectory. The experimental results of the miniature intelligent vehicle group verify the real-time performance and effectiveness of the proposed motion planning system.

Short-term operation of cascade hydropower system sharing flexibility via high voltage direct current lines for multiple grids peak shaving

The escalating pressure for peak shaving presents a significant challenge for the power grid's operation, particularly with the rapid growth of renewable energies. Huge cascade hydropower stations (CHS) send power to the local power grid and multiple power grids via long-distance high voltage direct current (HVDC) transmission lines, how to share hydropower flexibility for multiple power grids in peak shaving while considering constraints of hydropower and HVDC lines challenges power dispatching management. This study proposed a multiple power grids peak shaving model of cascade hydropower stations with HVDC sharing flexibility for receiving power grids. Firstly, the objective of minimizing the peak-valley difference of multiple power grids is adopted. The model then takes into account the operation constraints of CHSs and HVDC tie-lines to share hydropower flexibility. Finally, the model is transformed into a Mixed-Integer Linear Programming problem through piecewise linearization. The case studies of practical examples in southwest China show that the proposed model reduces peak-valley differences by 23.3%, 46.9%, 99.1% and 19.6%, 34.3%, 99.5% for three receiving power grids in two typical days by sharing hydropower flexibility. Moreover, adjusting the HVDC daily power regulation times and electricity deviation tolerance helps coordinate power grids to improve their peak shaving performance.

Expected value model of microgrid economic dispatching considering wind power uncertainty

New energy microgrid is one of the development trends of the future power grid. As there are wind turbines and other new energy equipment in the microgrid, its output is easily affected by unstable weather factors. Moreover, the complexity of dispatching caused by the storage battery in the microgrid and the non-linear factors of the diesel generator have brought great challenges to the economic dispatching of the microgrid. To tackle these problems, this paper establishes the expected value model of microgrid economic dispatching considering the uncertainty of wind power generation and the factors such as battery and diesel generator and proves that this expected value model is convex. On this basis, the piecewise linearization method is proposed to find the global optimal solution of the economic dispatch. Finally, a representative microgrid is optimized to verify the effectiveness of expected value model.

Self-Scheduling of a Generation Company with Carbon Emission Trading

A carbon emission trading self-scheduling (CETSS) model was proposed. The proposed model considered not only carbon emission allowance constraints but also carbon emission trading. A new method was presented for solving CETSS problems based on piece-wise linearisation and second-order cone linearisation. The effectiveness and validity of the proposed model and method were illustrated by 10-100 unit systems over 24 hours.

Polaron hopping through piecewise-linear functionals

Polarons have a large impact on transport properties of materials and on related applications in photovoltaics. This work demonstrates that enforcing the piecewise linearity condition leads to robust polaron properties upon variation of the functional. Considered properties include electron densities, lattice bonds, formation energies, hyperfine and superhyperfine parameters, hopping barriers, and transfer rates. This lends justification to the use of efficient semilocal functionals, thus paving the way to accurate and systematic studies of polaronic transport properties from first principles.

Tapping into the potential CO2 emission reduction of a crude oil transportation system from carbon footprint perspective

With more countries adopting a net-zero carbon emission goal, low carbon has become the focus of the transformation of the petroleum industry. Improper operations of the crude oil transportation system, one of the primary energy consumers of oilfields, usually cause severe environmental problems. Following the life cycle assessment method and considering the special process characteristics of the crude oil transportation system, a detailed carbon footprint analysis was conducted from the design to the recycling phase to identify the key phase for carbon emission control. Combined with process modifications, a multi-objective operation optimization model constrained by thermal constraints and insertion oil was developed to reach economic and environmental goals. Then, the multi-objective mixed integer nonlinear programming model was turned into a mixed integer linear programming model by the piecewise linearization method and augmented ε-constraint method. The models were validated using a typical case. The carbon footprint analysis showed that the average CO2e emissions were 2.68 t CO2e/(104 km m3•a) over the lifecycle, and the CO2e emissions during the operation phase were the highest (74.04%) among all phases. Moreover, the optimization model reduced the total cost and CO2e emissions by up to 18.72% and 16.97%. Additionally, sensitivity analysis revealed that the optimal operation scheme was highly sensitive to the crude oil price and transportation volume. The total CO2e emissions per unit volume exhibited a concave trend as the transportation volume increased, indicating the existence of an optimal transportation volume that achieved the lowest emissions. The study captured the weak points of carbon emissions over the lifecycle and tapped into potential CO2 emission reduction, providing a reference for the design and operation of the future pipeline system.

Discrete integration for measuring aerodynamic loads on trains in crosswinds − realizable strategies of discretization and discrete integration

Discrete integration occupies an important place in train aerodynamic tests in crosswinds. Improved delayed detached eddy simulations based on shear stress transport k-ω turbulence models were carried out to calculate the side force, lift, rolling moment around the lee rail, and surface pressure on bluff and streamlined vehicles. The changepoints and piecewise linearities of the pressure coefficients were evaluated, and the maximum coefficient of determination in the elements was 0.9973. A realizable strategy of the discretization based on the Lagrange rectangular elements was suggested, including the largest lengths and numbers of the elements. From this, a strategy of the discrete integration was presented to measure the aerodynamic loads, considering the real orientation of the elements. The maximum errors of the mean aerodynamic load coefficients of the bluff and streamlined vehicles were 4.1% and 2.2% (except the mean lift coefficient of the bluff vehicle), respectively. The errors were less than those in the previous studies, especially for the streamlined vehicle, which reduced by up to 8.7%. The unsteady aerodynamic loads with no delay obtained by the strategies were near to natural ones in the frequency range that people would be concerned about in crosswinds (at the Strouhal number of less than 0.4). Some suggestions were made for using the strategies in the full-scale tests and model tests, which provided a foundation for further studies of the running safety in natural crosswinds.

Threshold Optimized CSWPL Behavioral Model for RF Power Transistors Based on Particle Swarm Algorithm

In this letter, the particle swarm optimization (PSO) is employed to optimize the threshold values of the canonical section-wise piecewise linear (CSWPL) function-based behavioral model of power transistors. The effectiveness of the proposed method is validated through measurements carried out on a 10-W GaN power transistor produced by Wolfspeed. Compared with the existing simultaneous perturbation stochastic approximation (SPSA) method, the developed modeling technique not only provides superior prediction performance across different input power levels but also finds the optimal thresholds much more efficiently.

Uncertainty-Informed Operation Coordination in a Water-Energy Nexus

The widespread deployment of smart heterogeneous technologies and the growing complexity in our modern society calls for effective coordination of the interdependent lifeline networks. In particular, operation coordination of electric power and water infrastructures is urgently needed as the water system is one of the most energy-intensive networks, an interruption in which may quickly evolve into a dramatic societal concern. The closely-intertwined ecosystem of water and power infrastructures is commonly known as water-energy nexus. This paper develops a novel analytic for uncertainty-aware day-ahead operation optimization of the interconnected power and water systems (PaWS). Joint probabilistic constraint (JPC) programming is employed to capture the uncertainties in wind resources and water demand forecasts. The proposed integrated stochastic model is presented as a non-linear non-convex optimization problem, where the non-linear hydraulic constraints in the water network are linearized using piece-wise linearization technique, and the non-convexity is efficiently tackled with a Boolean solution methodology to convert the proposed model with JPCs to a tractable mixed-integer linear programming (MILP) formulation that can be quickly solved to optimality. The suggested framework is applied to a 15-node commercial-scale water network jointly operated with a power transmission system using a modified IEEE 57-bus test system. The numerical results demonstrate the of the proposed stochastic framework, resulting in cost reduction (13&#x0025; on average when compared to the traditional setting) and energy saving of the integrated model under different realizations of uncertain renewable energy sources (RESs) and water demand scenarios. Additionally, the scalability of the proposed model is tested on a modified IEEE 118-bus test system connected to five water networks.

Implementation of Pipelined Multi_Precision (1, 2 and 4) Floating-point Arithmetic Operations

Abstract: The implementation of pipelined multi-precision-based arithmetic operations are carried out. In the existing system, the floating-point operation is based on single precision and is implemented on a divider. The proposed design has been implemented using single, double and quad precision using the universal piece-wise linear (PWL) approximation method and a modified Goldschmidt algorithm. The proposed design performs addition, subtraction, multiplication, and division using the universal PWL method to reduce maximum error. Small multipliers are used in the modified Goldschmidt algorithm. The pipelining process has been used in order to improve the speed of execution and accuracy. This pipelined architecture is described in Verilog and timing performance is measured with Xilinx timing analyser.