Piecewise Linear Programming Research Articles

Coupling Control of Traffic Signal and Entry Lane at Isolated Intersections Under the Mixed-Autonomy Traffic Environment

There is a growing number of studies on the traffic control strategies of signal timings and vehicle trajectories at signalized intersections, while lane assignments are widely pre-specified and fixed. Meanwhile, existing strategies generally require a fully connected and automated vehicles (CAVs) environment. To fill up the gaps, this study contributes to a two-dimensional (spatiotemporal) control strategy by jointly optimizing traffic signals, lane settings, and vehicle trajectories at isolated signalized intersections under the mixed traffic of connected automated and human-driven vehicles. Specifically, based on the pseudo-platoons, signal timing plans and settings of approach lanes are jointly optimized by a piece-wise linear programming model. Then, vehicle trajectory control is integrated into the collaborative control framework to smooth vehicle trajectories. Three groups of numerical experiments are conducted to verify the effectiveness and efficiency of the proposed control method. Results show that the proposed algorithm outperforms the actuated control in terms of vehicle travel time under both under-saturated and over-saturated traffic conditions.

Current fluctuations reduction strategy based on optimal three‐space‐vector pulse‐width modulation for service‐life improvement of DC‐link electrolytic capacitors in three‐phase voltage‐source inverter system

AbstractThe authors propose a current fluctuation reduction strategy based on optimal three‐space‐vector pulse‐width modulation (TSVPWM), which is used to improve the service‐life of the dc‐link electrolytic capacitors in two‐level three‐phase voltage‐source inverter (VSI) system. Differing from the typical SVPWM‐based method that employs two adjacent active voltage vectors and zero voltage vectors to synthesise the reference voltage, the proposed strategy selects the required voltage vectors based on the minimum dc‐link electrolytic capacitor current fluctuation. With the proposed strategy, an objective function for the root‐mean‐square value of dc‐link electrolytic capacitor current is first constructed under the constraint of eight permissible voltage vectors. And then, by solving the minimum of the objective function through the piecewise linear programming, the optimal three voltage vectors under the different modulation ratios and load power factors are obtained accordingly, so as to achieve the minimum dc‐link electrolytic capacitor current fluctuations during each switching period. Meanwhile, a PWM implementation scheme based on positive–negative double carrier signals is presented, which not only simplifies the implementation process but also effectively reduces the switching losses of power devices. Finally, the experimental results demonstrate that the proposed strategy can minimise the dc‐link electrolytic capacitor current fluctuation under different working conditions.

Development of Upper Euphrates Basin hydro-economic model and hydropower generation optimization

Abstract Hydro-economic optimization models are common in hydropower reservoir modeling to aid system operators and planners. In these models, operations are driven by the economic value and constrained by the availability of water. The objective is to either minimize total costs or maximize total benefits. In this study, a hydro-economic optimization model for the integrated reservoir system of the Upper Euphrates Basin, with major tributaries providing water flow to the Euphrates River, is introduced. These model the 10 large-scale reservoirs of the basin with a total installed capacity of 3,255 MW. Water management and hydropower decision-making operations are evaluated with a piecewise linear programming algorithm in monthly time steps using a 45-year historical hydrology between 1971 and 2016. The model aims to maximize hydropower revenue over a long-term time horizon with energy prices varying by month. Reservoir storage and turbine release decisions are optimized for multiple hydropower plants connected in serial or parallel. Hydropower generation, revenue, reservoir storage, capacity ratios and generation reliability results are analyzed. Results show that these hydropower plants generate about 9,481 Gigawatt hour (GWh) of energy with an average turbine capacity use of 36% and obtain a revenue of 620 million $ per year.

Convexifying Market Clearing of SoC-Dependent Bids From Merchant Storage Participants

State-of-charge (SoC) dependent bidding allows merchant storage participants to incorporate SoC-dependent operation and opportunity costs in a bid-based market clearing process. However, such a bid results in a non-convex cost function in the multi-interval economic dispatch and market clearing, limiting its implementation in practice. We show that a simple restriction on the bidding format removes the non-convexity, making the multi-interval dispatch of SoC-dependent bids a standard convex piece-wise linear program.

Portfolio optimization under a minimax rule revisited

In this paper, we revisit the bi-criteria portfolio optimization model where the short selling is permitted, and a trade-off is sought between the expected return rate of a portfolio and the maximum of the uncertainty measured by a general deviation measure for all the investments comprising a portfolio. We solve this bi-criteria model by first converting it into a collection of weighted sum piecewise linear convex programs, and then analysing their optimality conditions. We not only provide explicit analytical formulas for all the efficient portfolios, but also explore as a whole the set of all the efficient portfolios and its structure such as dimensionality and distribution. We generalize the classical Two-fund Theorem by providing some collections of finitely many efficient portfolios to generate or estimate the set of all the efficient portfolios. We also notice that our efficient portfolios are almost the risk parity ones in the sense that the risks are allocated equally across the investments. Moreover, we illustrate the reliability of our model by carrying out Monte Carlo simulations to test the performance of some efficient portfolios versus inefficient ones.

Closed-Loop Elastic Demand Control under Dynamic Pricing Program in Smart Microgrid Using Super Twisting Sliding Mode Controller.

Electricity demand is rising due to industrialisation, population growth and economic development. To meet this rising electricity demand, towns are renovated by smart cities, where the internet of things enabled devices, communication technologies, dynamic pricing servers and renewable energy sources are integrated. Internet of things (IoT) refers to scenarios where network connectivity and computing capability is extended to objects, sensors and other items not normally considered computers. IoT allows these devices to generate, exchange and consume data without or with minimum human intervention. This integrated environment of smart cities maintains a balance between demand and supply. In this work, we proposed a closed-loop super twisting sliding mode controller (STSMC) to handle the uncertain and fluctuating load to maintain the balance between demand and supply persistently. Demand-side load management (DSLM) consists of agents-based demand response (DR) programs that are designed to control, change and shift the load usage pattern according to the price of the energy of a smart grid community. In smart grids, evolved DR programs are implemented which facilitate controlling of consumer demand by effective regulation services. The DSLM under price-based DR programs perform load shifting, peak clipping and valley filling to maintain the balance between demand and supply. We demonstrate a theoretical control approach for persistent demand control by dynamic price-based closed-loop STSMC. A renewable energy integrated microgrid scenario is discussed numerically to show that the demand of consumers can be controlled through STSMC, which regulates the electricity price to the DSLM agents of the smart grid community. The overall demand elasticity of the current study is represented by a first-order dynamic price generation model having a piece-wise linear price-based DR program. The simulation environment for this whole scenario is developed in MATLAB/Simulink. The simulations validate that the closed-loop price-based elastic demand control technique can trace down the generation of a renewable energy integrated microgrid.

Jerk-limited feedrate scheduling and optimization for five-axis machining using new piecewise linear programming approach

In this paper, a new computation method and an optimization algorithm are presented for feedrate scheduling of five-axis machining in compliance with both machine drive limits and process limits. Five-axis machine tool with its ability of controlling tool orientation to follow the sculptured surface contour has been widely used in modern manufacturing industry. Feedrate scheduling serving as a kernel of CNC control system plays a critical role to ensure the required machining accuracy and reliability for five-axis machining. Due to the nonlinear coupling effects of all involved drive axes and the saturation limit of servo motors, the feedrate scheduling for multi-axis machining has long been recognized and remains as a critical challenge for achieving five-axis machine tools’ full capacity and advantage. To solve the nonlinearity nature of the five-axis feedrate scheduling problems, a relaxation mathematical process is presented for relaxing both the drive motors’ physical limitations and the kinematic constraints of five-axis tool motions. Based on the primary optimization variable of feedrate, the presented method analytically linearizes the machining-related constraints, in terms of the machines’ axis velocities, axis accelerations and axis jerks. The nonlinear multi-constrained feedrate scheduling problem is transformed into a manageable linear programming problem. An optimization algorithm is presented to find the optimal feedrate scheduling solution for the five-axis machining problems. Both computer implementation and laboratorial experiment testing by actual machine cutting were conducted and presented in this paper. The experiment results demonstrate that the proposed method can effectively generate efficient feedrate scheduling for five-axis machining with constraints of the machine tool physical constraints and limits. Compared with other existing numerical methods, the proposed method is able to find an accurate analytical solution for the nonlinear constrained five-axis feedrate scheduling problems without compromising the efficiency of the machining processes.

Method of hill tunneling via weighted simplex centroid for continuous piecewise linear programming

This paper works on a heuristic algorithm with determinacy for the global optimization of continuous piecewise linear (CPWL) programming. CPWL is widely applied since it can be equivalently transformed into D.C. programming, and further, concave optimization over a polyhedron. Considering that the super-level sets of concave piecewise linear functions are polyhedra, we propose the hill tunneling via simplex centroid (HTSC) algorithm, which is able to escape a local optimum to the other side of its contour surface by cutting across the super-level set. The searching path for the hill tunneling is established by using the centroid of a constructed simplex. In the numerical experiments, the proposed HTSC algorithm is compared with CPLEX and the hill detouring (HD) method, which shows its superior performance on the numerical efficiency and the global search capability.

Piecewise linear programming approach to solve multi-objective matrix games with I-fuzzy goals

ABSTRACT The intuitionistic fuzzy set (I-fuzzy set) plays an effective role in game theory when players face ‘neither this nor that’ situation to set their goals. This study presents a maxmin–minmax solution to multi-objective two person zero-sum matrix games with I-fuzzy goals. In this article, a class of piecewise linear membership and non-membership functions for I-fuzzy goals is constructed. These functions are more effective in real games because marginal rate of increase (decrease) of such membership functions (non-membership functions) is different in different intervals of tolerance errors. Finally, one numerical example is given to examine the effectiveness and advantages of the proposed results.

On the approximate reanalysis technique in topology optimization

A classical problem in topology optimization concerns the minimization of the compliance of a static structure, subject to a volume constraint upon the available material. Assuming that the structure is under small displacements and it is composed of a linear elastic material, the evaluation of the objective function demands the solution of a linear system. Hence, within the computational optimization process of addressing topology optimization problems, the cost of evaluating the objective function may be an issue, especially as the discretized mesh is refined. This work pursues the approximate reanalysis technique in combination with the Sequential Piecewise Linear Programming method for obtaining optimized structures. Numerical evidences are presented to corroborate the usage of this blend in a study composed by three distinct strategies in three benchmark test problems. A further analysis has been performed concerning the impact of the computation of the gradient vector of the objective function, pointing out room for additional savings.

Hill tunneling method via peak subpoints for continuous piecewise linear programming

Hill tunneling method via peak subpoints for continuous piecewise linear programming

Risk-Limiting Unit Commitment in Smart Grid With Intelligent Periphery

This paper proposes the risk-limiting unit commitment (RLUC) as the operational method to address the uncertainties in the smart grid with intelligent periphery (GRIP). Three key requirements are identified for the RLUC in GRIP. The first one requires the RLUC to be modeled as a multistage multiperiod unit commitment problem considering power trades, operational constraints, and operational risks. The second one requires the RLUC considering the conditional prediction to achieve a globally optimal solution. It is addressed by using conditional probability in a scenario-based form. The last one requires the risk index in the RLUC to be both valid and computationally friendly, and it is tackled by the utilization of a coherent risk index and the mathematical proof of a risk chain theorem. Finally, the comprehensive RLUC in GRIP satisfying all the three requirements is solved by an equivalent transformation into a mixed integer piecewise linear programming problem. Case studies on a nine-bus system, a realistic provincial power system, and a regional power grid in China demonstrate the advantages of the proposed RLUC in GRIP.

A comparison of piecewise linear programming formulations for stochastic disassembly line balancing

Recently, several mathematical programming formulations and solution approaches have been developed for the stochastic disassembly line balancing problem (DLBP). This paper aims at finding optimal solutions for the stochastic DLBP. Two second-order cone programming (SOCP1 and SOCP2) models and five piecewise linear mixed integer programming (PwLP) models are presented. The PwLP formulations involve two specially ordered sets of type 2 (S1 and S2) models and three convex combination (CC1, CC2 and CC3) models. In each modelling category, the latter models strengthen the initial S1 and CC1 models. Our computational analysis of a total 240 instances of ten problems demonstrates that all the seven models can be used to solve practical-sized DLBP problems to optimality using GUROBI. The SOCP2 model and the strengthened S2 and CC2 models lead to lower computation times, compared to SOCP1, S1, CC1 and CC3, respectively. Using the strengthened S2 and CC2 formulations, the CPU times of the CC3 model available in the literature can be reduced by 50 and 40%, respectively. Besides analysing the optimal solutions and the differences of the computation times, we present insights gained from our results.

A piecewise linear programming algorithm for sparse signal reconstruction

In order to recover a signal from its compressive measurements, the compressed sensing theory seeks the sparsest signal that agrees with the measurements, which is actually an l norm minimization problem. In this paper, we equivalently transform the l norm minimization into a concave continuous piecewise linear programming, and propose an optimization algorithm based on a modified interior point method. Numerical experiments demonstrate that our algorithm improves the sufficient number of measurements, relaxes the restrictions of the sensing matrix to some extent, and performs robustly in the noisy scenarios.

Objective variation simplex algorithm for continuous piecewise linear programming

This paper works on a modified simplex algorithm for the local optimization of Continuous PieceWise Linear (CPWL) programming with generalization of hinging hyperplane objective and linear constraints. CPWL programming is popular since it can be equivalently transformed into difference of convex functions programming or concave optimization. Inspired by the concavity of the concave CPWL functions, we propose an Objective Variation Simplex Algorithm (OVSA), which is able to find a local optimum in a reasonable time. Computational results are presented for further insights into the performance of the OVSA compared with two other algorithms on random test problems.

A parametric simplex algorithm for biobjective piecewise linear programming problems

In this paper we attempt to develop a parametric simplex algorithm for solving biobjective convex separable piecewise linear programming problems. The algorithm presented in this paper can be regarded as an extension of the parametric simplex algorithm for solving biobjective linear programming problems to the piecewise linear case. Analogous to the linear case, this parametric simplex algorithm provides a decomposition of parametric space. A numerical example is presented to illustrate the implementation of the algorithm.

Sparse signal reconstruction via concave continuous piecewise linear programming

Compressed sensing (CS) is a new paradigm for acquiring sparse and compressible signals which can be approximated using much less information than their nominal dimension would suggest. In order to recover a signal from its compressive measurements, the conventional CS theory seeks the sparsest signal that agrees with the measurements via a great many algorithms, which usually solve merely an approximation of the l0 norm minimization. In this paper, CS has been considered from a new perspective. We equivalently transform the l0 norm minimization into a concave continuous piecewise linear programming based on the prior knowledge of sparsity, and propose a novel global optimization algorithm for it based on a sophisticated detour strategy and the γ valid cut theory. Numerical experiments demonstrate that our algorithm improves the best known number of measurements in the literature, relaxes the restrictions of the sensing matrix to some extent, and performs robustly in the noisy scenarios.

Supply chain network design and operation: Systematic decision-making for centralized, distributed, and mobile biofuel production using mixed integer linear programming (MILP) under uncertainty

Biomass resources are dispersed and subject to seasonal and geographical uncertainties. Therefore, supply chain network design and management can significantly influence the economic viability of a biofuel technology. Fast pyrolysis offers several advantages for biofuel production. It is a relatively cheap process and can be conducted in centralized, decentralizes, or even mobile configurations. Furthermore, it does not overlap with the human food supply chain, using wastes or lignocellulosic feedstocks. In this article, a mixed integer (piece-wise) linear program (MILP) was developed to determine the optimal supply chain design and operation, under uncertainty. Rigorous process modelling and detailed economic analysis were coupled with exhaustive search of potential production locations and biomass resources in order to enhance the fidelity of the solution. The optimisation results suggest that a combination of geographically centralized pyrolysis and upgrading centres would suffice for supply chain management under deterministic conditions. However, under uncertain scenarios, it is advantageous to deploy mobile pyrolyzers to add extra flexibility to the process operation. Further analysis suggested that as the mobile pyrolyzers are commercialized and their unit price is reduced, this technology has the potential to become a key member of the biofuel supply chain.

Tropical determinant on transportation polytopes

Let Dk,l(m,n) be the set of all the integer points in the transportation polytope of kn×ln matrices with row sums lm and column sums km. In this paper we find the sharp lower bound on the tropical determinant over the set Dk,l(m,n). This integer piecewise linear programming problem in arbitrary dimension turns out to be equivalent to an integer non-linear (in fact, quadratic) optimization problem in dimension two. We also compute the sharp upper bound on a modification of the tropical determinant, where the maximum over all the transversals in a matrix is replaced with the minimum.

Piecewise Linear Quadratic Programming Adaptive Vibration Control of Active Constrained Layered Damping Beam

Piecewise Linear Quadratic Programming Adaptive Vibration Control of Active Constrained Layered Damping Beam