Piecewise Linear Cost Functions Research Articles

Scheduling a set of jobs with convex piecewise linear cost functions on a single-batch-processing machine

We investigate a general single-batch-processing machine scheduling problem of minimizing the total costs of all jobs with general convex piecewise linear cost functions, where the processing time, due dates, and sizes of jobs are non-uniform. Since the studied problem is NP-hard and the objective is irregular, we adopt a practical 3-step method that starts from a specific job sequence, finds the optimal and near-optimal schedules of the given job sequence, and then iteratively improves the schedules. For the restricted problem where the job sequence is given, we propose a dynamic programming algorithm to obtain the optimal schedule. To further reduce the running time, we also devise a span-limit tree search (SLTS) approach to find the near-optimal schedule. We then design a local search method to improve the solutions solved by SLTS. Finally, we design a branch and bound algorithm with a lower bound method for small-sized instances. All proposed methods are based on our analysis of the mathematical properties of the scheduling problem, and extensive numerical experiments are conducted to demonstrate their effectiveness and efficiency.

A tailored adaptive large neighborhood search algorithm for the air cargo partitioning problem with a piecewise linear cost function

A tailored adaptive large neighborhood search algorithm for the air cargo partitioning problem with a piecewise linear cost function

Capacitated lot sizing problem with periodic carbon emission constraints and multiple resources

We study the single item capacitated lot sizing problem with multiple resources and periodic carbon emission constraints that impose an upper bound for the average emission per product produced in any period. Although the uncapacitated version of this problem can be solved in polynomial time, generalisation of the problem including the resource capacities is NP-Hard, in general. We present important structural properties for the optimal solutions of the problem. We consider the special cases with two resources and under non-speculative costs, construct the piecewise linear total production cost function when the resource capacities, and the emission and cost parameters are time-invariant, and develop a polynomial time dynamic programming algorithm (DP) to solve them. Then, we generalise the procedure to construct the total production cost function and the DP for the general setting with fixed number of capacitated resources. We test our algorithm for different problem instances, and compare it with a commercial solver and a DP available in the literature for solving the lot sizing problem with piecewise concave production cost functions. The results reveal that our DP outperforms the other one, and it performs better than the commercial solver when the number of breakpoints of the total production cost function is small.

Passive Attack Detection for a Class of Stealthy Intermittent Integrity Attacks

This paper proposes a passive methodology for detecting a class of stealthy intermittent integrity attacks in cyber-physical systems subject to process disturbances and measurement noise. A stealthy intermittent integrity attack strategy is first proposed by modifying a zero-dynamics attack model. The stealthiness of the generated attacks is rigorously investigated under the condition that the adversary does not know precisely the system state values. In order to help detect such attacks, a backward-in-time detection residual is proposed based on an equivalent quantity of the system state change, due to the attack, at a time prior to the attack occurrence time. A key characteristic of this residual is that its magnitude increases every time a new attack occurs. To estimate this unknown residual, an optimal fixed-point smoother is proposed by minimizing a piece-wise linear quadratic cost function with a set of specifically designed weighting matrices. The smoother design guarantees robustness with respect to process disturbances and measurement noise, and is also able to maintain sensitivity as time progresses to intermittent integrity attack by resetting the covariance matrix based on the weighting matrices. The adaptive threshold is designed based on the estimated backward-in-time residual, and the attack detectability analysis is rigorously investigated to characterize quantitatively the class of attacks that can be detected by the proposed methodology. Finally, a simulation example is used to demonstrate the effectiveness of the developed methodology.

Optimal sizing for microgrids integrating distributed flexibility with the Perth West smart city as a case study

With the decreasing cost of green technologies and the increasing ambitions to reach the net-zero carbon emissions target, more communities are engaged in renewable deployment and energy-intensive technologies such as heat pumps and electric vehicles will be intensively adopted in the near future. The integration of these appliances in lower grid levels will likely require grid reinforcements. However, some of these appliances are flexible and there is an opportunity to explore their flexibility potential to optimise the investment costs further. This paper proposes an optimal design strategy for a grid-connected site that returns the renewable generation and storage’s optimal sizing capacities and the required network reinforcement capacity. The novelty of the work is integrating network upgrade costs and considering flexibility from distributed flexible resources across planning and operation. The problem is formulated as a mixed integer piecewise linear problem, with the capacities of generation, storage and network upgrade as decision variables. The piecewise linear cost function related to the upgrade costs figuring in the objective function is then recast as a mixed-integer problem, and the flexible resources are modelled through an approximation method as a single virtual flexible asset. The application of the strategy on the Perth West smart city project as a case study demonstrates the importance of considering flexibility in the planning phase. The costs related to the storage system can decrease by up to 76%, and the overall costs by up to 35%, with the highest levels of savings, reached for the highest rates of electric vehicle adoption.

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.

Optimizing inland container logistics through dry ports: A two-stage stochastic mixed-integer programming approach considering volume discounts and consolidation in rail transport

It is essential for container shipping companies to plan for the optimum number and locations of dry ports as well as efficient inland container logistics operations to reduce costs and environmental impacts while improving customer service. The plan for optimum inland container transportation network design must account for demand uncertainty to eliminate financial difficulty due to redundant investment. In this regard, this study proposed a two-stage stochastic mixed-integer programming model for optimizing inland container logistics through dry ports. The model contributes to the state of the art in current research by including a piecewise-linear cost function for railway transportation to account for volume discounts, integrating full container scheduling with transport mode selection and empty container relocation for consolidation, and reflecting the fact that the amount of import and export full containers transported between customers (consignees and consignors) and seaports are exogenously decided. The solution results demonstrated the definite performance superiority of the progressive hedging algorithm over the extensive form solution. Additionally, the value of stochastic solution calculation showed that the application of stochastic solution might result in significant cost savings compared to the application of mean value deterministic solution. The proposed model can be applied for practical necessities to design robust optimum inland container logistics operations using intermodal rail transport with the progressive hedging algorithm as an efficient solution approach.

A hybrid electricity pricing mechanism for joint system optimization and social acceptance within energy communities

This article presents a framework for local electricity pricing mechanisms for households designed for social acceptance. The optimization goal of the mechanisms is to flatten the neighborhood electricity profile. The motivation and need for such mechanisms result from the expectation that the energy transition leads to high peaks in the distribution grid, both in electricity consumption and renewable generation, posing a significant challenge to these grids.Following the literature, quadratic cost functions have the potential to achieve the envisioned system optimization. However, their drawback is that consumers find the resulting pricing mechanisms too complicated and are generally not willing to participate in systems offering such prices. In contrast, the simpler pricing mechanisms currently used in practice are socially accepted. However, these mechanisms lack sufficient incentive to reduce electricity peaks in the distribution grids.Our approach is to combine these concepts in a hybrid pricing mechanism for local energy communities, using a piecewise linear cost function, that approximates a quadratic function. The resulting pricing mechanism is evaluated in a field test, and, based on the feedback of participating consumers and other criteria defined in literature, we can conclude that the proposed mechanism is socially accepted.In the implementation of this hybrid pricing mechanism, its performance can be improved such that it obtains results comparable to those of quadratic cost functions. Based on these findings, a detailed numerical evaluation, and the results from the field test, we conclude that the presented pricing mechanism has the potential for being used in practice.

Piecewise Linear Valued CSPs Solvable by Linear Programming Relaxation

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of the complexity of infinite-domain VCSPs with piecewise linear homogeneous cost functions. Such VCSPs can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by reducing the problem to a finite-domain VCSP which can be solved using the basic linear program relaxation. It follows that VCSPs for submodular PLH cost functions can be solved in polynomial time; in fact, we show that submodular PLH functions form a maximally tractable class of PLH cost functions.

Reverse 1-centre problem on trees under convex piecewise-linear cost function

ABSTRACT Given a network as well as a prescribed vertex s in which a facility is located, the aim of solving reverse 1-centre problems is to decrease edge lengths under a budget constraint in a way that the maximum distance between s and the other vertices is minimized. This paper considers the reverse 1-centre problem on general tree networks with weights associated to vertices under a convex piecewise-linear cost function. It is shown that the problem can be transformed into a parametric minimum cost flow problem. A polynomial time approximation scheme (PTAS) is first developed using the bisection method. Then, the problem on unweighted trees is investigated. Using sensitivity analysis, a pseudo-polynomial time algorithm is designed. Finally, a hybrid algorithm is developed to obtain an exact solution in polynomial time.

An exact algorithm for two-dimensional vector packing problem with volumetric weight and general costs

The volumetric weight of a package has become an essential factor in calculating the delivery cost of shipments in the international logistics market. In this work, we extend the two-dimensional vector packing problem by considering a more realistic cost structure, which is a general function of volumetric weight. The problem is to pack a set of different items into some identical bins without violating weight limits and volume capacities so that the total delivery cost is minimized. We develop an exact approach based on a branch-and-price algorithm and subset-row inequalities for the problem. To efficiently solve the pricing problem in column generation, a label-setting algorithm with an effective label dominance rule and a bounding procedure is presented. A stronger label dominance rule is derived for the case where the cost function is convex. The computational results show that the exact method is effective in solving the various test instances of the problem. If the volumetric weight is not considered, the exact method can be adapted to solve the two-dimensional vector packing problem with piecewise linear cost function and outperformed the existing exact algorithm by computing 27 optimal solutions for previously open instances.

The impacts of convex piecewise linear cost formulations on AC optimal power flow

Despite strong connections through shared application areas, research efforts on power market optimization (e.g., unit commitment) and power network optimization (e.g., optimal power flow) remain largely independent. A notable illustration of this is the treatment of power generation cost functions, where nonlinear network optimization has largely used polynomial representations and market optimization has adopted piecewise linear encodings. This work combines state-of-the-art results from both lines of research to understand the best mathematical formulations of the nonlinear AC optimal power flow problem with piecewise linear generation cost functions. An extensive numerical analysis of non-convex models, linear approximations, and convex relaxations across fifty-four realistic test cases illustrates that nonlinear optimization methods are surprisingly sensitive to the mathematical formulation of piecewise linear functions. The results indicate that a poor formulation choice can slow down algorithm performance by a factor of ten, increasing the runtime from seconds to minutes. These results provide valuable insights into the best formulations of nonlinear optimal power flow problems with piecewise linear cost functions, an important step towards building a new generation of energy markets that incorporate the nonlinear AC power flow model.

Estimation of cost efficiency with piecewise linear cost function in integer-valued data envelopment analysis

The cost efficiency is one of the types of the used efficiency concepts in the data envelopment analysis (DEA). In the literature of the cost efficiency, the prices are often applied to the units in a fixed and linear form, while in some cases, as the subjects of the savings and observance of the consumption patterns, the prices are usually considered in terms of stepwise (piecewise linear) and not fixed (linear). This paper proposes the new models to evaluate of the cost efficiency in the fully competitive setting by using the costs in a piecewise linear form. The models are developed for the non-fully competitive and integer-valued inputs cases. Finally, the proposed models are used to determine the cost efficiency of the confectioneries in Islam Abad Gharb city from Iran country in the case study.

Sizing of Energy Storage with Network Constraints Economic Dispatch of Fuel Cell Micro Turbine and Renewable Energy Sources

Micro-Grid is the appropriate solution to various problems in the power system. Different types of energy sources, likewise Fuel cell (FC), Micro-Turbine (MT) and renewable energy sources can be integrated with micro-grids (MG). The battery energy storage has played a crucial role to support the power mismatch of on-grid or off-grid MG. Therefore the optimal size of battery energy storage along with the optimal cost-based calculation has become an essential part for the micro-grid operator. The piecewise linear cost method is used for the cost based analysis. The main contribution of this paper is: (i) the optimal size of battery energy storage has been determined with a Fuel cell (FC) and Micro-Turbine (MT) based distribution generation (DG). (ii) The impact of battery storage with DG and renewable energy sources (RES) has been considered. (iii) The total benefit and market benefit has been maximised. (iv) The unit-commitment cost of FC and MT with spinning reserve, piecewise linear cost function, ramp rate, minimum up and downtime constraints has been considered for the sizing of battery storage. (v) The network constrained has been found to obtain minimum daily energy loss for the optimal size of battery storage. (vi) The state of charge (SOC) of battery, the power output of DG’s and RES, power loss, battery cost per day, operating cost of generation, etc. have been determined. The optimal sizing of battery energy storage determination is helpful for the both Microgrid operators as well as designers. The IEEE-33 bus test system with ZIP load has been carried out for analysis and result validation. The general algebraic modeling system (GAMS) is used to solve the deterministic optimisation problem.

Real-Time Pricing of Smart Grid Based on Piece-Wise Linear Functions

Abstract In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption; cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satisfaction of different users and the supplier. Quadratic utility, logarithmic utility, and quadratic cost functions are widely used in social welfare maximization models of real-time pricing. These functions are not universal; they have to be discussed in detail for individual models. To overcome this problem, a piece-wise linear utility function and a piece-wise linear cost function with general properties are proposed in this paper. By smoothing the piece-wise linear utility and cost functions, a social welfare maximization model can be transformed into a differentiable convex optimization problem. A dual optimization method is used to solve the smoothed model. Through mathematical deduction and numerical simulations, the rationality of the model and the validity of the algorithm are verified as long as the elastic and cost coefficients take appropriate values. Thus, different user types and the supplier can be determined by selecting different elastic and cost coefficients.

Algorithm for Searching an Equilibrium in a Routing Game with Piecewise Constant Cost Functions

We consider nonatomic routing games which are used to model transportation systems with a large number of agents and suggest an algorithm to search for the user equilibrium in such games, which is a generalization of the notion of Nash equilibrium. In general, finding a user equilibrium in routing games is computationally a hard problem. We consider the following subclass of routing games: games with piecewise constant cost functions, and construct an algorithm finding equilibrium in such games. This algorithm is based on the potential function method and the method of piecewise linear (PWL) costs enumeration which finds min-cost flow in a network with PWL cost functions. If each cost function increases, then the complexity of our algorithm is polynomial in the parameters of the network. But if some cost functions have decreasing points, then the complexity is exponential by the number of such points.

Sterilization network design

With the increasing costs of health care, service providers are looking for innovative ways to cut cost without sacrificing high service quality. Resource pooling is a promising trend in different health care areas, and sterilization is no exception. Specialized medical sterilization centers (SCs) that offer reusable medical device (RMD) sterilization services for hospitals have the potential to cut cost and improve efficiency through better utilization of resources, risk-pooling and economies of scale. However, it is unclear whether the resulting cost savings can offset the additional transportation costs and the operational complications associated with a centralized system. We compare three schemes to organize RMD sterilization in a group of hospitals: fully distributed, centralized processing, and centralized processing and stock keeping. The sterilization network design problem is formulated as a mixed-integer concave minimization program (MISOCP) that considers economies of scale, service level requirements and variable demand, with the objective of minimizing capacity, transportation, and inventory holding costs. The mathematical model is reformulated as a mixed-integer second-order cone program with a piecewise-linear cost function so it can be solved efficiently. We also consider the case when the first two moments of the RMD demand distributions are not known with certainty but are rather estimated based on sample data. We show that, with proper selection of the uncertainty structure, the robust nonlinear optimization problem can be tractably reformulated as a MISOCP as well. Testing on a realistic case study under different scenarios reveals that significant cost savings can be achieved by consolidating sterilization services. Compared to the distributed scheme, we found that cost saving in the second scheme is attributed primarily to improved resource utilization and economies of scale. Risk-pooling in the last scheme results in an additional small cost saving that has to be weighed against other operational and legal considerations.

Factoring the Cycle Aging Cost of Batteries Participating in Electricity Markets

When participating in electricity markets, owners of battery energy storage systems must bid in such a way that their revenues will at least cover their true cost of operation. Since cycle aging of battery cells represents a substantial part of this operating cost, the cost of battery degradation must be factored in these bids. However, existing models of battery degradation either do not fit market clearing software or do not reflect the actual battery aging mechanism. In this paper we model battery cycle aging using a piecewise linear cost function, an approach that provides a close approximation of the cycle aging mechanism of electrochemical batteries and can be incorporated easily into existing market dispatch programs. By defining the marginal aging cost of each battery cycle, we can assess the actual operating profitability of batteries. A case study demonstrates the effectiveness of the proposed model in maximizing the operating profit of a battery energy storage system taking part in the ISO New England energy and reserve markets.

Polynomial time algorithms and extended formulations for unit commitment problems

Recently, increasing penetration of renewable energy generation has created challenges for power system operators to perform efficient power generation daily scheduling, due to the intermittent nature of the renewable generation and discrete decisions of each generation unit. Among all aspects to be considered, a unit commitment polytope is fundamental and embedded in the models at different stages of power system planning and operations. In this article, we focus on deriving polynomial-time algorithms for the unit commitment problems with a general convex cost function and piecewise linear cost function, respectively. We refine an time, where T represents the number of time periods, algorithm for the deterministic single-generator unit commitment problem with a general convex cost function and accordingly develop an extended formulation in a higher-dimensional space that can provide an integral solution, in which the physical meanings of the decision variables are described. This means the original problem can be solved as a convex program instead of a mixed-integer convex program. Furthermore, for the case in which the cost function is piecewise linear, by exploring the optimality conditions, we derive more efficient algorithms for both deterministic (i.e., time) and stochastic (i.e., time, where N represents the number of nodes in the stochastic scenario tree) single-generator unit commitment problems. We also develop the corresponding extended formulations for both deterministic and stochastic single-generator unit commitment problems that solve the original mixed-integer linear programs as linear programs. Similarly, physical meanings of the decision variables are explored to show the insights of the new modeling approach.

A branch-and-price algorithm for the two-dimensional vector packing problem with piecewise linear cost function

The two-dimensional vector packing problem with piecewise linear cost function (2DVPP-PLC) models a practical packing problem faced by many companies that use courier services. We propose a branch-and-price algorithm to solve the 2DVPP-PLC exactly. The column generation procedure is a key component that affects the performance of a branch-and-price algorithm. The pricing problem exhibits an interesting structure that allows us to decompose it into subproblems that form a lattice. We explore dominance relations on the lattice and design an efficient algorithm for the pricing problem. Experimental results show that our branch-and-price algorithm is capable of solving 2DVPP-PLC test instances effectively.