Multiple Time Grids Research Articles

Asynchronous switching and sparse discretization with time scaling for control vector parameterization optimal control

This work introduces a novel approach for solving open-loop Optimal Control Problems (OCPs) using multiple time grids, sparse discretization and a time scaling transformation. Control Vector Parameterization (CVP) method parameterizes control variables to create a finite-dimensional problem. The Variable Time Nodes Control Vector Parameterization (VTNCVP) discretization strategy allows each control component an independent time grid, offering enhanced input design flexibility and potentially yielding improved outcomes. A novel time scaling technique for OCPs with multiple time grids is presented that is easier to comprehend than existing methods and can circumvent some numerical difficulties. Additionally, Multiple Grid (MG)–Sparse Variable Time Nodes (MG-SVTN) is presented. This novel discretization strategy employs asynchronous switching with user-specified numbers of subdivisions for input controls, potentially yielding comparable objective function values to VTNCVP while reducing the dimensionality of decision variables. Two example problems with path constraints are solved using the methods.

A novel multi-operation sequencing formulation for the crude oil scheduling problem with restricted overlapping constraints

The scheduling of crude oil operations holds critical significance in the petrochemical industry due to its profound influence on downstream unit operations and, consequently, overall business profitability. This work presents a novel multi-operation sequencing formulation considering operations with restricted overlapping constraints. This operation-centric model also distinguishes itself by incorporating constraints posed by refinery practice such as downstream product management, multiparcel unloading, brine settling, and changeover detection. A new symmetry-breaking strategy is applied to remove symmetric solutions properly and expedite problem solving while ensuring solution quality. Case studies of various examples are conducted to validate the efficacy of the proposed model. Comparative analyses are performed with the unit-specific event-based formulation, which enables multiple time grids tailored to different units and is widely used in continuous-time modeling. Results further demonstrate the great potential of the presented formulation in generating tighter bounds and accelerating convergence.

Establishment of a pMCAO model in SD rats and screening for behavioral indicators suitable for long-term monitoring

ABSTRACT Objective: This study aimed to establish a permanent middle cerebral artery occlusion (pMCAO) model in rats to simulate the pathological process of stroke patients with no reperfusion. And screen highly sensitive items that could be used to detect long-term behavioral abilities in rat of intraluminal suture models. Method: Established the pMCAO model then tested the rats for the bilateral asymmetry, modified neurological severity score, grid-walking, cylinder, rotating, and water maze test from week 1 to week 16. Results: The infarct volume of the model rats was stable (26.72% ±1.86%). The sensorimotor test of bilateral asymmetry, grid-walking, cylinder, and mNSS test showed significant differences from week 1 to week 16 after injury. The water maze test at week 16 showed significant differences in spatial exploration and learning ability between the two groups. We confirmed that there was no significant difference between MRI and TTC staining in detecting the degree of brain injury, which facilitated the diversity of subsequent detection methods. We also confirmed that at multiple time points, grid, cylinder and water maze test were significantly positively correlated with rat brain infarct volume. Conclusion: They are suitable for the long-term observation of behaviors in the sequela stage of stroke in rat.

A multi-objective approach to determine time series aggregation strategies for optimal design of multi-energy systems

The optimal topology and operation of multi-energy systems (MES) can be determined by mathematical programming models, which are commonly difficult to solve due to the complex structures of the models and large-scale time-series data. The time series aggregation (TSA) strategies are usually adopted to reduce the complication of the data input and thus improve the solution efficiency. Nevertheless, for the optimal design of MES, significant differences exist in the structures and performances of the systems resulting from different TSA strategies. It is crucially important to determine appropriate TSA strategies in the optimal design of MES. In this work, a two-step method was proposed to determine TSA strategies based on a multi-objective optimization framework. In the first step, the preliminary design for the multi-energy system was obtained by different TSA strategies for the optimal design and operation of MES. In the second step, the total annual cost and loss of load probability (LSLP) of the system were taken as two objectives, and a multi-objective optimization approach was established to determine the appropriate TSA strategy for the optimal design of the MES, where the structure and performance of the system were considered simultaneously. The implementation and effectiveness of the proposed method were demonstrated by the optimal design of an MES with energy storage. The deviations of the design scheme and system performance obtained by different TSA strategies were comprehensively compared and analyzed. The results indicate that the feasibility of the optimal design schemes of the MES can be verified by using the original full-time series data, and the deviation of the results obtained by different TSA strategies can be quantified. The reduction of LSLP is at the expense of the increase of the costs. In the multiple time grids strategy (TG1) with the number of time steps (TS) 730, the TAC of the system increases from $9.79 × 107 to $1.56 × 108 when the LSLP of the system decreases from 7.90% to zero. The determination of the strategy based on the result of the minimum LSLP is critical to meet the demand in the energy system. The appropriate TSA strategies for the optimal design of MES can be selected and determined in compliance with specific requirements.

Discrete and continuous-time formulations for dealing with break periods: Preemptive and non-preemptive scheduling

This paper presents new mixed-integer linear programming (MILP) approaches for handling preemption both in discrete and continuous-time scheduling formulations. Preemption refers to the capability of interrupting the execution of a task when encountering a pre-defined break period, assuming that the task continues immediately after the end of such time window. We rely on Generalized Disjunctive Programming to derive the constraints for the continuous-time formulations and on a compact convex hull reformulation to make them computationally efficient. We investigate both the general precedence and multiple time grids representation concepts. Generalization of the discrete-time formulation is simpler, involving a change in the model parameters. Validation and comparison of the mathematical formulations is done through the solution of sixteen benchmark problems, involving instances with one to four sets of breaks. The results show that the general precedence formulation is computationally more effective for flexible flowshops, being outperformed by the discrete-time approach when considering common rather than machine-dependent breaks. For single stage plants with parallel units, the continuous multiple time grid formulation prevails.

Multiple time grids in operational optimisation of energy systems with short- and long-term thermal energy storage

As a vital part of future low carbon energy systems, storage technologies need to be included in the overall optimisation of energy systems. However, this comes with a price of increasing complexity and computational cost. The increase in complexity can be limited by using simplified time series formulations in the optimisation process, e.g. typical days or multiple time grids. This in turn will affect the computational cost and quality of the optimisation results. The trade-off between these two aspects has to be quantified in order to appropriately use the simplification method. This paper investigates the implementation of the multiple time grids approach in the optimisation of a solar district heating system with short- and long-term thermal energy storage. The multiple time grids can improve the optimisation computational time by over an order of magnitude. Nevertheless, this is not a general rule since it is shown that there is a possibility for the computational time to increase with time step size. Furthermore, the benefits of multiple time grids become more evident in optimisation with a longer time horizon, reaching almost two order of magnitude improvement in computational time for the case with 6 years time horizon and 5% MIP gap.

A Control Parameterization Approach with Variable Time Nodes for Optimal Control Problems

AbstractThis paper presents a novel computational approach to deal with optimal multivariable control problems using a control vector parameterization approach with multiple time grids, where each of the control variables has its own time grid of parametrization. Both the control parameters and time nodes in the grid partition are treated directly as variables to be optimized. Based on the derived relationship between the gradients of time nodes and the ones of interval lengths, the gradient formulae for parameters are presented. Compared with the existing approaches, for which all the control variables are parameterized on the same time grid, the proposed method is more general and flexible. To illustrate, two numerical cases are tested, and the results demonstrate that fewer parameters are needed to achieve the same level of optimization.

On the solution of large-scale mixed integer programming scheduling models

In this paper, we show how four recently developed modeling and solution methods can be integrated to address mixed integer programs for the scheduling of large-scale chemical production systems. The first method uses multiple discrete time grids. The second adds tightening constraints that lower bound the total production and number of batches for each task and material based on the customer demand, while the third generates upper bounding constraints based on inventory and resource availability. The final method is a reformulation that introduces a new integer variable representing the total number of batches of a task. We apply the aforementioned methods to large-scale problems with a variety of processing features, including variable conversion coefficients, changeovers, various storage policies, continuous processing tasks, setups, and utilities, using a discrete-time model. We illustrate how these methods lead to significant improvements in computational performance.

Theoretical framework for formulating MIP scheduling models with multiple and non-uniform discrete-time grids

We present a framework for the formulation of MIP scheduling models based on multiple and nonuniform discrete time grids. In a previous work we showed that it is possible to use different (possibly non-uniform) time grids for each task, unit, and material. Here, we generalize these ideas to account for general resources, and a range of processing characteristics such as limited intermediate storage and changeovers. Each resource has its own grid based on resource consumption and availability allowing resource constraints to be modeled more accurately without increasing the number of binary variables. We develop algorithms to define the unit-, task-, material-, and resource-specific grids directly from problem data. Importantly, we prove that the multi-grid formulation is able to find a schedule with the same optimal objective as the discrete-time single-grid model with an arbitrarily fine grid. The proposed framework leads to the formulation of models with reduced number of binary variables and constraints, which are able to find good solutions faster than existing models.

Cyclic Scheduling of Pulp Digesters with Integrated Heating Tasks

This paper addresses a multistage batch plant scheduling problem under energy constraints. These reflect the limited availability of a thermal heating utility that is shared among parallel digesters of different capacities for the production of pulp. Depending on the processing sequence, more or less steam will be available for a given digester, which will affect the duration of its heating stage and the overall cycle time. Such integrated heating tasks resemble direct heat integration, which has been addressed through models based on generic frameworks for process representation (e.g., State-Task Network, Resource-Task Network, State-Sequence Network) and relying on a single time grid, either discrete or continuous. A new multiple time grid continuous-time model is now proposed where the complex energy constraints are derived from the higher level modeling framework that is Generalized Disjunctive Programming. The results show a considerable better performance compared to RTN discrete and continuous-time formulations, due to a substantially lower integrality gap and model size.

Multiple and nonuniform time grids in discrete-time MIP models for chemical production scheduling

The modeling of time plays a key role in the formulation of mixed-integer programming (MIP) models for scheduling, production planning, and operational supply chain planning problems. It affects the size of the model, the computational requirements, and the quality of the solution. While the development of smaller continuous-time scheduling models, based on multiple time grids, has received considerable attention, no truly different modeling methods are available for discrete-time models. In this paper, we challenge the long-standing belief that employing a discrete modeling of time requires a common uniform grid. First, we show that multiple grids can actually be employed in discrete-time models. Second, we show that not only unit-specific but also task-specific and material-specific grids can be generated. Third, we present methods to systematically formulate discrete-time multi-grid models that allow different tasks, units, or materials to have their own time grid. We present two different algorithms to find the grid. The first algorithm determines the largest grid spacing that will not eliminate the optimal solution. The second algorithm allows the user to adjust the level of approximation; more approximate grids may have worse solutions, but many fewer binary variables. Importantly, we show that the proposed models have exactly the same types of constraints as models relying on a single uniform grid, which means that the proposed models are tight and that known solution methods can be employed. The proposed methods lead to substantial reductions in the size of the formulations and thus the computational requirements. In addition, they can yield better solutions than formulations that use approximations. We show how to select the different time grids, state the formulation, and present computational results.

Generalized Disjunctive Programming as a Systematic Modeling Framework to Derive Scheduling Formulations

We propose linear generalized disjunctive programming (GDP) models for the short-term scheduling problem of single stage batch plants with parallel units. Three different concepts of continuous-time representation are explored, immediate and general precedence, as well as multiple time grids. The linear GDP models are then reformulated using both big-M and convex hull reformulations, and the resulting mixed-integer linear programming models compared through the solution of a set of example problems. We show that two general precedence models from the literature can be derived using a big-M reformulation for a set of disjunctions and a convex hull reformulation for another. The best performer is, however, a multiple time grid model which can be derived from the convex hull reformulation followed by simple algebraic manipulations to eliminate the disaggregated variables and reduce the sets of constraints, thus leading to a more compact and efficient formulation.

Scheduling inspired models for two-dimensional packing problems

We propose two exact algorithms for two-dimensional orthogonal packing problems whose main components are simple mixed-integer linear programming models. Based on the different forms of time representation in scheduling formulations, we extend the concept of multiple time grids into a second dimension and propose a hybrid discrete/continuous-space formulation. By relying on events to continuously locate the rectangles along the strip height, we aim to reduce the size of the resulting mathematical problem when compared to a pure discrete-space model, with hopes of achieving a better computational performance. Through the solution of a set of 29 test instances from the literature, we show that this was mostly accomplished, primarily because the associated search strategy can quickly find good feasible solutions prior to the optimum, which may be very important in real industrial environments. We also provide a comprehensive comparison to seven other conceptually different approaches that have solved the same strip packing problems.

Short-Term Scheduling of Multistage Batch Plants with Unlimited Intermediate Storage

This paper presents a new multiple time grid, continuous-time, mixed integer linear program model for the short-term scheduling of multistage, multiproduct batch plants. The formulation is based on the resource−task network process representation and explicitly considers a single intermediate storage unit per stage to keep track of material transfer between processing units belonging to consecutive stages of production. It is compared to other existing unit-specific and sequence-based approaches through the solution of a few example problems taken from the literature. Alternative objective functions such as total cost, makespan, and total earliness minimization, as well as revenue maximization, are considered. The results show that the proposed formulation performs similarly to the one of Castro and Grossmann (2005), which is limited to problems involving a single product batch per stage. In addition, it can be up to 3 or 4 orders of magnitude faster than the one of Shaik and Floudas (2008), due to a lower integrality gap, the use of a smaller number of event points, and fewer constraints.

Simultaneous batching and scheduling of single stage batch plants with parallel units

AbstractThis article deals with the optimal short‐term scheduling of single stage batch plants with sequence‐dependent changeovers together with the optimal selection of the number of batches to produce. The novelty of the article is that instead of following the traditional approach of considering one processing task per batch, all batches of the product are now aggregated into a single task. Integer variables that hold the number of batches to produce are used to characterize these aggregated tasks. Two conceptually different continuous‐time models are proposed. They rely on either multiple time grids or global precedence sequencing variables for event representation and generate a mixed integer linear program. The new formulations are compared to a traditional resource‐task network multiple time grid approach as well as to a recent bounding model with immediate precedence sequencing variables. The results for several example problems show the new multiple time grid formulation as the best overall performer. When compared to the traditional approach, one order of magnitude savings in computational effort are achieved due to the need of fewer event points to get to the global optimal solutions. © 2007 American Institute of Chemical Engineers AIChE J, 2008

Optimal Periodic Scheduling of Multistage Continuous Plants with Single and Multiple Time Grid Formulations

This paper presents a new mixed integer nonlinear program (MINLP) model for the periodic scheduling of multistage, multiproduct continuous plants featuring equipment units in parallel that are subject to sequence dependent changeovers. The formulation is based on the resource task network (RTN) process representation, features combined processing and changeover tasks, and assumes that each order is executed only once on each stage. The new multiple time grid formulation is compared to an also RTN-based, single time grid formulation, through the solution of a few example problems taken from the literature. The results show that the proposed formulation is significantly more efficient computationally, essentially because smaller problems are generated and, because the number of tasks to execute can easily be predicted, no iterative procedure over the total number of event points in the time grid is required to find the optimal solution. On the other hand, the single time grid formulation is able to sometimes find better solutions to the problem because of its ability to consider, for each order, the execution of more than a single instance of the processing task per stage.

An efficient MILP model for the short-term scheduling of single stage batch plants

This paper presents a multiple time grid continuous time MILP model for the short-term scheduling of single stage, multiproduct batch plants. It can handle both release and due dates and the objective can be either the minimization of total cost or total earliness. This formulation is compared to other mixed-integer linear programming approaches that have appeared in the literature, to a constraint programming model, and to a hybrid mixed-integer linear/constraint programming algorithm. The results show that the proposed formulation is significantly more efficient than the MILP and CP models and comparable to the hybrid model. For one large instance, both methods exceeded the time limit but the hybrid method failed to find a feasible solution. The results also show that a discrete-time formulation performs very efficiently even when a large number of time intervals are used.