Trajectory Piecewise-linear Model Research Articles

Lattice Trajectory Piecewise Linear Method for the Simulation of Diode Circuits

In this paper, we present an approach to nonlinear system approximation, called the lattice trajectory piecewise linear (LTPWL) model. The approach involves determining a lattice piecewise linear (PWL) approximation to the state trajectory of a nonlinear system. It has been shown in the literature that the lattice PWL expression can represent any PWL function in any dimension. After the LTPWL approximation has been obtained, the order of each model piece is reduced using a Krylov projection technique. Compared to existing trajectory piecewise linear (TPWL) models, which are quasi-PWL in the whole region, LTPWL models are virtually linear in each subregion. Besides, the single output LTPWL model can be seen as a special kind of TPWL model, in which only one weight is 1, and the other weights are 0. In general, for multiple output LTPWL model, the weights set to be 1 for each component are different, which makes the LTPWL model more flexible in approximation of nonlinear function. The proposed strategy is applied to simulate diode circuits, and the experimental results show that the performance of the LTPWL model is better than that of the traditional TPWL model in terms of approximation accuracy and generalization ability.

Economic MPC of Wastewater Treatment Plants Based on Model Reduction

In this paper, we consider the problem of economic model predictive control of wastewater treatment plants based on model reduction. We apply two model approximation methods to a wastewater treatment plant (WWTP) described by a modified Benchmark Simulation Model No.1 to overcome the intensive computation associated with economic model predictive control (MPC). Two computationally efficient models are obtained based on trajectory piecewise linearization (TPWL) and reduced order TPWL. To obtain the reduced order TPWL model, a proper orthogonal decomposition (POD)-based method is utilized. Further, the reduced order model is linearized to obtain a TPWL-POD model. The objective is to design controllers which minimize the overall economic cost. Accordingly, we design economic MPC (EMPC) controllers based on each of the models. The economic control cost can be described as a weighted summation of effluent quality and overall operating cost. We compare the accuracy of the two proposed approximation models with different linearization point numbers. We evaluate the average

Model order reduction of nonlinear models based on decoupled multi-model via trajectory piecewise linearization

In this paper a novel model order reduction method for nonlinear models, based on decoupled multi-model, via trajectory piecewise-linearization is proposed. Through different strategies in trajectory piecewise-linear model reduction, model order reduction of a trajectory piecewise-linear model based on output weighting (TPWLOW), has been developed by authors of current work. The structure of mentioned work was founded based on Krylov subspace method, which is appropriate for high order models. Indeed the size of the Krylov subspaces may increase with the number of inputs of the system. As a result, the use of Krylov subspace method may become deficient the case for multi-input systems of order few decades. This paper aims to generalize the idea of model reduction of TPWLOW model by concentrating on balanced truncation technique which is appropriate for medium size systems. In addition, the proposed method either guarantees or provides guaranteed stability in some mentioned conditions. Finally, applicability of the reduced model, instead of high-order decoupled multi-model in weighting multi-model controllers, is investigated through the control of a nonlinear Alstom gasifier benchmark.

Model order reduction for dynamic simulation of slender beams undergoing large rotations

In this paper a model order reduction technique for the dynamic simulation of beams undergoing large rotations is presented. The finite element model for the motion of such beams is based on the corotational formulation. The trajectory piecewise linear model order reduction (TPWLMOR) method with second order Krylov subspace is used to obtain the reduced order model from the finite element model. Improvements are suggested to improve the accuracy and the computational efficiency of the TPWLMOR model. Several numerical examples which include forced undamped and damped beams are presented to validate the proposed method.

New strategies in model order reduction of trajectory piecewise‐linear models

AbstractModel order reduction based on trajectory piecewise linearization (TPWL) is a beneficial technique for approximating nonlinear models. One efficient method for building projection matrix in TPWL reduction is by aggregation of projection matrices of linearization points (LPs). However, in this method, the size of projection matrix will also grow up by increasing the number of LPs, which yield the increment of the size of reduced model. In other words, the size of reduced model will depend on the number of LPs. In this paper, we will address this issue and propose two new strategies for obviating this problem. Contrarily to former works in TPWL modeling, we established a model via TWPL based on output weighting of parallel linear models. Then, we proposed two reduction strategies for suggested TPWL model. The first algorithm inspires from former works in this field but in a parallel structure that enable segregation of projection matrices whereas the second algorithm remedies the problem by considering the high‐order TPWL model as a unit linear model and reduces this model like a linear model but uses back projection method for constructing different outputs. The efficiency of methods is shown by comparison with former TPWL methods through vast simulations. Copyright © 2015 John Wiley & Sons, Ltd.

Fast Solution of Transient Elastohydrodynamic Line Contact Problems Using the Trajectory Piecewise Linear Approach

In order to obtain a fast solution scheme, the trajectory piecewise linear (TPWL) method is applied to the transient elastohydrodynamic (EHD) line contact problem for the first time. TPWL approximates the nonlinearity of a dynamical system by a weighted superposition of reduced linearized systems along specified trajectories. The method is compared to another reduced order model (ROM), based on Galerkin projection, Newton–Raphson scheme and an approximation of the nonlinear reduced system functions. The TPWL model provides further speed-up compared to the Newton–Raphson based method at a high accuracy.

A Global Maximum Error Controller-Based Method for Linearization Point Selection in Trajectory Piecewise-Linear Model Order Reduction

We propose a new linearization point selection method based on a global maximum error controller for the trajectory piecewise-linear (TPWL) model order reduction (MOR). This method is based on a simple fact that the simulation cost of the TPWL model is very low, and selects the state at which the responses of the current TPWL model and the full nonlinear model have the maximum difference as a new linearization point. Numerical results show that the proposed method can generate the TPWL model of smaller size and higher accuracy, and can easily be extended to generate the TPWL model for multiple training inputs.

Model order reduction for dynamic simulation of beams with forcing and geometric nonlinearities

The objective of the paper is to investigate the applicability of a model order reduction technique for dynamic simulation of beams with forcing and geometric nonlinearities. A cantilever and a doubly clamped beams actuated by an electrostatic force are considered in the paper. The governing partial differential equations for the two cases which account for the nonlinearities are presented. These equations are spatially discretized using the Galerkin finite element method (FEM). The resulting system of nonlinear ordinary differential equations is reduced using the trajectory piecewise linear model order reduction (TPWLMOR) method. Simulation indicates that the use of the original TPWLMOR method leads to the presence of a phase error in the long term dynamic simulation of the models. To improve the accuracy of the dynamic response, a modification to the original TPWLMOR based on minimization of residual at linearization point is proposed. Further, the parameters affecting the accuracy of the modified TPWLMOR are studied.

Subspace Trajectory Piecewise-Linear Model Order Reduction for Nonlinear Circuits

AbstractDespite the efficiency of trajectory piecewise-linear (TPWL) model order reduction (MOR) for nonlinear circuits, it needs large amount of expansion points for large-scale nonlinear circuits. This will inevitably increase the model size as well as the simulation time of the resulting reduced macromodels. In this paper, subspace TPWL-MOR approach is developed for the model order reduction of nonlinear circuits. By breaking the high-dimensional state space into several subspaces with much lower dimensions, the subspace TPWL-MOR has very promising advantages of reducing the number of expansion points as well as increasing the effective region of the reduced-order model in the state space. As a result, the model size and the accuracy of the TWPL model can be greatly improved. The numerical results have shown dramatic reduction in the model size as well as the improvement in accuracy by using the subspace TPWL-MOR compared with the conventional TPWL-MOR approach.

Reduced-order modeling for thermal recovery processes

Thermal recovery can entail considerably higher costs than conventional oil recovery, so the use of computational optimization techniques in designing and operating these processes may be beneficial. Optimization, however, requires many simulations, which results in substantial computational cost. Here, we implement a model-order reduction technique that aims at large reductions in computational requirements. The technique considered, trajectory piecewise linearization (TPWL), entails the representation of new solutions in terms of linearizations around previously simulated (and saved) training solutions. The linearized representation is projected into a low-dimensional space, with the projection matrix constructed through proper orthogonal decomposition of solution “snapshots” generated in the training step. Two idealized problems are considered here: primary production of oil driven by downhole heaters and a simplified model for steam-assisted gravity drainage, where water and steam are treated as a single “effective” phase. The strong temperature dependence of oil viscosity is included in both cases. TPWL results for these systems demonstrate that the method can provide accurate predictions relative to full-order reference solutions. Observed runtime speedups are very substantial, over 2 orders of magnitude for the cases considered. The overhead associated with TPWL model construction is equivalent to the computation time for several full-order simulations (the precise overhead depends on the number of training runs), so the method is only applicable if many simulations are to be performed.

Compact thermoelectric coupled models of micromachined thermal sensors using trajectory piecewise-linear model order reduction

Numerical simulation has become an effective way to design and optimize micromachined thermal sensors. To improve the speed of development process, fast simulation is indispensable. This paper investigates compact models, which are accurate low order representations of high order finite element models, for micromachined thermal sensors. Because the thermal field and the electric field act on and influence each other, the compact models should be established from the thermoelectric coupled full models. Thermoelectric coupling and temperature dependent resistivity make the problems strong nonlinearity. Therefore a powerful nonlinear model order reduction method, named trajectory piecewise-linear (TPWL) method, is employed. Its core idea is approximating the nonlinear model piecewise-linearly along a training trajectory. The performance of the TPWL method and fast TPWL method are compared. And the effects of linearization point number and local reduced basis order on accuracy, efficiency and size of the TPWL compact models are studied. Moreover, the expandability of the TPWL compact models is also discussed. Results show that the TPWL compact models are suitable for the design and optimization of micromachined thermal sensors.

Reduced-order flow modeling and geological parameterization for ensemble-based data assimilation

Reduced-order modeling represents an attractive approach for accelerating computationally expensive reservoir simulation applications. In this paper, we introduce and apply such a methodology for data assimilation problems. The technique applied to provide flow simulation results, trajectory piecewise linearization (TPWL), has been used previously for production optimization problems, where it has provided large computational speedups. The TPWL model developed here represents simulation results for new geological realizations in terms of a linearization around previously simulated (training) cases. The high-dimensional representation of the states is projected into a low-dimensional subspace using proper orthogonal decomposition. The geological models are also represented in reduced terms using a Karhunen–Loève expansion of the log-transmissibility field. Thus, both the reservoir states and geological parameters are described very concisely. The reduced-order representation of flow and geology is appropriate for use with ensemble-based data assimilation procedures, and here it is incorporated into an ensemble Kalman filter (EnKF) framework to enrich the ensemble at a low cost. The method is able to reconstruct full-order states, which are required by EnKF, whenever necessary. The combined technique enables EnKF to be applied using many fewer high-fidelity reservoir simulations than would otherwise be required to avoid ensemble collapse. For two- and three-dimensional example cases, it is demonstrated that EnKF results using 50 high-fidelity simulations along with 150 TPWL simulations are much better than those using only 50 high-fidelity simulations (for which ensemble collapse is observed) and are, in fact, comparable to the results achieved using 200 high-fidelity simulations.

Enhanced linearized reduced-order models for subsurface flow simulation

Trajectory piecewise linearization (TPWL) represents a promising approach for constructing reduced-order models. Using TPWL, new solutions are represented in terms of expansions around previously simulated (and saved) solutions. High degrees of efficiency are achieved when the representation is projected into a low-dimensional space using a basis constructed by proper orthogonal decomposition of snapshots generated in a training run. In recent work, a TPWL procedure applicable for two-phase subsurface flow problems was presented. The method was shown to perform well for many cases, such as those with no density differences between phases, though accuracy and robustness were found to degrade in other cases. In this work, these limitations are shown to be related to model accuracy at key locations and model stability. Enhancements addressing both of these issues are introduced. A new TPWL procedure, referred to as local resolution TPWL, enables key grid blocks (such as those containing injection or production wells) to be represented at full resolution; i.e., these blocks are not projected into the low-dimensional space. This leads to high accuracy at selected locations, and will be shown to improve the accuracy of important simulation quantities such as injection and production rates. Next, two techniques for enhancing the stability of the TPWL model are presented. The first approach involves a basis optimization procedure in which the number of columns in the basis matrix is determined to minimize the spectral radius of an appropriately defined amplification matrix. The second procedure incorporates a basis matrix constructed using snapshots from a simulation with equal phase densities. Both approaches are compatible with the local resolution procedure. Results for a series of test cases demonstrate the accuracy and stability provided by the new treatments. Finally, the TPWL model is used as a surrogate in a direct search optimization algorithm, and comparison with results using the full-order model demonstrates the efficacy of the enhanced TPWL procedures for this application.

Use of Reduced-Order Modeling Procedures for Production Optimization

Summary The determination of optimal well settings is very demanding computationally because the simulation model must be run many times during the course of the optimization. For this reason, reduced-order modeling procedures, which are a family of techniques that enable highly efficient simulations, may be very useful for optimization problems. In this paper, we describe a recently developed reduced-order modeling (ROM) technique that has been used in other application areas, the trajectory piecewise linearization (TPWL) procedure, and incorporate it in production-optimization computations. The TPWL methodology represents solutions encountered during the optimization runs in terms of Taylor-series expansions around previously simulated states. This requires a small number of preprocessing (training) simulations using the full (high-fidelity) model, during which pressure and saturation states and Jacobian matrices are saved. These states and matrices are then projected into a low-dimensional space using proper orthogonal decomposition (POD). Simulations in this reduced space can be performed very efficiently; in this work, we observe runtime speedups of a factor of 450. Overall speedups are, however, less because of the preprocessing overhead. We assess the TPWL representation for simulations of waterflood in a heterogeneous 3D model containing more than 20,000 gridblocks and six wells. The high degree of accuracy of the TPWL model is first demonstrated for several testing simulations in which producer- and injector-well settings differ from those used in the training runs. The TPWL representations are then used in optimizations involving the determination of optimal bottomhole pressures (BHPs) for a reservoir model with four production wells and two injection wells. A gradient-based algorithm is applied for the optimizations. In the first case, the BHPs of the producers and injectors are optimized at six different times (36 control variables) and in the second case at 15 different times (90 control variables). Results for optimized net present value (NPV) using TPWL are shown to be in consistently close agreement with those computed using high-fidelity simulations. Most significantly, when the optimal well settings obtained using the TPWL procedure are applied in high-fidelity models, the resulting NPVs are within approximately 0.5% of the values determined using the high-fidelity simulations. Our overall conclusion is that the TPWL representation may be quite useful in production-optimization problems.

Linearized reduced-order models for subsurface flow simulation

A trajectory piecewise linearization (TPWL) procedure for the reduced-order modeling of two-phase flow in subsurface formations is developed and applied. The method represents new pressure and saturation states using linear expansions around states previously simulated and saved during a series of preprocessing training runs. The linearized representation is projected into a low-dimensional space, with the projection matrix constructed through proper orthogonal decomposition of the states determined during the training runs. The TPWL model is applied to two example problems, containing 24,000 and 79,200 grid blocks, which are characterized by heterogeneous permeability descriptions. Extensive test simulations are performed for both models. It is shown that the TPWL model provides accurate results when the controls (bottom hole pressures of the production wells in this case) applied in test simulations are within the general range of the controls applied in the training runs, even though the well pressure schedules for the test runs can differ significantly from those of the training runs. This indicates that the TPWL model displays a reasonable degree of robustness. Runtime speedups using the procedure are very significant–a factor of 100–2000 (depending on model size and whether or not mass balance error is computed at every time step) for the cases considered. The preprocessing overhead required by the TPWL procedure is the equivalent of about four high-fidelity simulations. Finally, the TPWL procedure is applied to a computationally demanding multiobjective optimization problem, for which the Pareto front is determined. Limited high-fidelity simulations demonstrate the accuracy and applicability of TPWL for this optimization. Future work should focus on error estimation and on stabilizing the method for large models with significant density differences between phases.