Reduced order modeling for smart grids' simulation and optimization

POD reduced-order unstructured mesh modeling applied to 2D and 3D fluid flow

This paper presents some new findings in model order reduction of multi input-multi output (MIMO) systems of modern power grid with wind generation. A widely-used model reduction technique called Balanced Truncation (BT) is compared with a relatively new Interpolatory approach of Iterative Rational Krylov Algorithm (IRKA). It is demonstrated that both BT and IRKA can produce an acceptable accuracy during model reduction for power grids with synchronous generators. Whereas, when modern grid with wind generation is considered, BT fails to produce a stable reduced order model (ROM) while IRKA is still able to derive accurate stable ROM retaining the poorly-damped inter-area modes of the system.

In recent years, autonomously guided parachute systems have been developed with the goal of increasing the landing accuracy of airdrop operations. These precision airdrop systems utilize an onboard global positioning system (GPS) coupled with a guidance, navigation and control (GNC) system to continuously monitor and correct the flight path of the parachute system during descent. A critical component of the GNC system is an algorithm that predicts the response of the system to various aerodynamic forces in real-time that are typically based on a low-order multi-degree of freedom (DOF) dynamics model. Reduced order modeling is the process of representing a complex dynamic system with many DOF with a model having relatively few DOF. The goal of this study is to examine the use of large scale finite element models to construct and calibrate Reduced Order Models (ROM) for parachute systems. The resulting ROM would provide parachute designers with a simple tool to examine parachute behavior under various operating conditions. The use of numerical simulations to calibrate the ROM would help to eliminate the number of physical tests needed to characterize a parachute system’s properties. In this study, a 6 degree of freedom ROM is adopted. The system dynamic properties for the ROM and the resultant forces and moments acting on the ROM are obtained from finite element simulations. The 6 DOF model originally proposed by Cockrell and Doherr and later used by Dobrokhodov is adopted for the ROM. The finite element (FE) code, TENSION, is used to perform the large scale simulations and extract the ROM dynamic properties and resultant forces. Several examples are performed that verify the ROM procedure by comparing ROM predictions to FE simulations for various parachute system trajectories. The performance of the ROM is shown to be excellent for in-plane motion under an oscillating wind field. It is shown, however, that the ROM procedure deteriorates as the FE model is refined by adding more DOF. It is demonstrated that, for highly deformable FE models with many DOF, more sophisticated ROM procedures are needed.

Non-linear analysis is of increasing importance in wind energy engineering as a result of their exposure in extreme conditions and the ever-increasing size and slenderness of wind turbines. Whilst modern computing capabilities facilitate execution of complex analyses, certain applications which require multiple or real-time analyses remain a challenge, motivating adoption of accelerated computing schemes, such as reduced order modelling (ROM) methods. Soil structure interaction (SSI) simulations fall in this class of problems, with the non-linear restoring force significantly affecting the dynamic behaviour of the turbine. In this work, we propose a ROM approach to the SSI problem using a recently developed ROM methodology. We exploit a data-driven non-linear ROM methodology coupling an autoencoder with long short-term memory (LSTM) neural networks. The ROM is trained to emulate a steel monopile foundation constrained by non-linear soil and subject to forces and moments at the top of the foundation, which represent the equivalent loading of an operating turbine under wind and wave forcing. The ROM well approximates the time domain and frequency domain response of the Full Order Model (FOM) over a range of different wind and wave loading regimes, whilst reducing the computational toll by a factor of 300. We further propose an error metric for capturing isolated failure instances of the ROM.

We discuss methods for model order reduction (MOR) of linear systems with many input and output variables, arising in the modeling of linear (sub) circuits with a huge number of nodes and a large number of terminals, like power grids. Our work is based on the approaches SVDMOR and ESVDMOR proposed in recent publications (1; 2; 3; 4; 5). In particular, we discuss efficient numerical algorithms for their implementation. Only by using efficient tools from numerical linear algebra, these methods become applicable for truly large-scale problems.

Vehicle drivability is an important factor which more and more customers have started assessing before buying a vehicle. Customers carry out this assessment based on both vehicle reviews/ratings and based on the test drives. One of common maneuver which a customers perform during the test drive is sudden accelerator pedal tip-in or tip-out to accelerate or coast the vehicle. Clunk and shuffle are the phenomena that usually occur during this scenario causing driver discomfort. The clunk and shuffle are caused by the backlash and compliance physical properties of the driveline. Consequently, control strategy needs to be developed which can provide a fast driveline response without clunk and shuffle. One major input to develop a control strategy is the knowledge of the vehicle states and parameters based on the available measurements, which is the major focus of this work. This work begins with a discussion of various existing estimation strategies that have been used to estimate the states of vehicle along with their merits and demerits. Then a full order model, developed in the previous works, is validated for a locked torque converter case along with its reduced order model which is used for estimator development. The error in the simulated shuffle frequency for the full order model and reduced order model is less than 1%. The reduced order model is then used to develop an observable state space model to estimate the backlash state and size of the model. The estimators developed are validated and the robustness analysis is done for different scenarios of torque inputs, delays and sampling times. It is found that the sampling time of the estimators and measurement inputs significantly effect the estimates of lash traversal time and backlash size with a mean error of 9% in lash traversal time estimate and 2% error in lash size for 10 ms sampling time. Furthermore, the estimators are found to be more robust to the variations in the wheel speed measurements as compared to variations in the engine speed measurements.

With the increasing need for flexible operation (shorter startup time and higher load change rate etc.), clearance monitoring between the rotor and the stationary components in steam turbines is becoming more important. This is because as load change rate increases, minimum radial and axial clearances during operation tend to be smaller due to thermal deformation of steam turbines, and the risk of contact between the rotor and the stationary components becomes higher. This situation has accelerated development of clearance sensors. However, it is still difficult to monitor all possible points of contact only with physical sensors due to limited installation location and short lifetime in high temperature environment. From the above background, we have been developing a virtual clearance monitoring (VCM) technique based on a novel data fusion approach that utilizes both physical and data-driven models. Specifically, a reduced order model (ROM) is used as physical model in order to enable real-time prediction with an accuracy similar to that of finite element analysis (FEA). Then, the prediction error of the physical model is corrected by using a residual model built by machine learning from the past clearance sensor values and the corresponding physical model-based prediction results. As will be explained in this report, this technique has an advantage that the clearances can be predicted in real-time based only on operating data such as steam conditions at inlet and outlet, and some temperatures in the parts not modeled in the ROM. Therefore, the virtual sensor based on this technique can be used as a replacement for the physical sensor after it has failed. Furthermore, this technique can also be used to preliminarily study unsteady clearance behavior for inexperienced operating conditions. This paper describes how to build the ROM from a finite element model for thermal-structural analysis of an entire steam turbine by model order reduction (MOR), and the detail of the VCM technique, and a VCM system installed in a measurement room of a state-of-the-art GTCC power plant manufactured by Mitsubishi Power. In addition, the verification results of the VCM system are presented. In this research, the ROM and the residual model were built using the data obtained from four operations with different start-up modes each other. Then, VCM was performed for 12 operating cases. As a result, this survey revealed the followings: (1) This system is capable of real-time prediction with output intervals of roughly 2 seconds. (2) As for radial clearance prediction error during rotor rotating, the RMSEs and the absolute values of minimum value errors are less than or equal to 7.2 % and 7.0 % respectively relative to an initial radial clearance value during the steam turbine stopping. From the above results, we conclude that this VCM technique based on data fusion approach is effective in terms of computational speed and prediction accuracy. This means that if a physical clearance sensor fails, the radial clearance can be continuously monitored by a virtual clearance sensor with a residual model built using the data when the sensor was working normally. In the future, we plan to further improve the accuracy of this technique through improvement in physical modeling.

A reduced order model (ROM) is proposed to simulate the aeroelasticity of compressor blades in both coupled and uncoupled mode at low computational cost. The ROM describes the structural dynamics of blades using the mode superposition technique, while effective aerodynamic forces are modeled using various system identification methods. The ROM takes advantage of the aerodynamic influence coefficients (AIC) to model the aerodynamic loads of vibrating blades. System identification of unsteady aerodynamic forces is performed using ARX polynomial models, nonlinear ARX polynomials, Hammerstein-Wiener models and recurrent neural networks. Two different training signals are tested, and their performance is compared. The uncoupled results are validated against traditional Fourier method, showing good agreement in estimated aerodynamic damping. The different formulations of the coupled ROM are then compared to the coupled full order model (FOM) of reference: consistent predictions for the damping ratio are achieved, with a substantial reduction of the computational cost with respect to full order models of reference.

<p>Order reduction of linear-time invariant systems employing two methods; one using the advantages of Routh approximation and other by an evolutionary technique is presented in this paper. In Routh approximation method the denominator of the reduced order model is obtained using Routh approximation while the numerator of the reduced order model is determined using the indirect approach of retaining the time moments and/or Markov parameters of original system. By this method the reduced order model guarantees stability if the original high order model is stable. In the second method Particle Swarm Optimization (PSO) is employed to reduce the higher order model. PSO method is based on the minimization of the Integral Squared Error (ISE) between the transient responses of original higher order model and the reduced order model pertaining to a unit step input. Both the methods are illustrated through numerical examples.</p>

A hybrid projection/data-driven reduced order model for the Navier-Stokes equations with nonlinear filtering stabilization

Simple reduced order models (ROMs) for the aerodynamic coefficients - lift, drag, and pitch moment - of a pitching NACA0012 airfoil are presented. The ROMs are designed for quick computation of the transient aerodynamic characteristics of the airfoil and are developed utilizing computational fluid dynamics (CFD) simulation results. The entire aerodynamic system is modeled as a single input, multi output system yielding three independent systems to be characterized. A systematic, two step process is employed to develop the ROMs for each aerodynamic system. First, a CFD simulation is conducted to determine the linearity of each system, and any nonlinear system is restructured as a nonlinear operator followed by a linear system to allow for the use of linear system identification techniques. A second CFD simulation is conducted to determine the frequency response of each linear system, and the coefficients of each ROM are extracted by fitting a second order model to each frequency response function. The ROMs are validated against an independent CFD simulation of a pitching airfoil and are shown to accurately model each aerodynamic coefficient.

An efficient data-driven multiscale stochastic reduced order modeling framework for complex systems

There has been a growing demand for nonlinear model updating procedures in the structural dynamics community as advanced vehicles have started to incorporate nonlinearity into their designs. Finite element (FE) model updating is difficult for a nonlinear structure for several reasons: there may be many unknown parameters in the FE model so that multiple solutions may exist, and it is very expensive to compute the structure’s nonlinear normal modes from a full FE model that are an excellent metric to use for nonlinear updating. A recent work showed that some of these challenges can be overcome by updating a nonlinear reduced order model (ROM) rather than the full FE model. The updated ROM can be used to compute response statistics, stresses and ultimately life of the structure. However, one drawback of this approach is that one does not gain physical insight into which parameters in the FE model were in error, and so it is difficult to transfer the lessons learned to future FE models. This work explores the feasibility of updating a FE model to correlate with a ROM with a set of known parameters. The target ROM parameters consist of the linear stiffness and the nonlinear stiffness coefficients which have been identified previously by updating the ROM to correlate with experimental measurements. In this process an optimization routine is setup in which the free parameters are those of the FE model, such as boundary stiffness springs, imperfections, pre-stress, etc. The optimization procedure is wrapped around a ROM creation algorithm that iteratively tunes the FE parameters, creates a ROM, and evaluates the ROM parameters with respect to the target ROM in order to minimize the objective function. This work will demonstrate the procedure on a numerical example of a curved beam in order to verify its effectiveness on the nonlinear model updating.

This chapter presents a strategy to enable a team of mobile robots to adaptively sample and track a dynamic spatiotemporal process. We propose a distributed strategy where robots collect sparse sensor measurements, create a reduced -order model (ROM) of the spatiotemporal process, and use this model to estimate field values for areas without sensor measurements of the dynamic process. The robots then use these estimates of the field, or inferences about the process, to adapt the model and reconfigure their sensing locations. We use this method to obtain an estimate for the underlying fl ow field and use that to plan optimal energy paths for robots to travel between sensing locations. We show that the errors due to the reduced -order modeling scheme are bounded, and we illustrate the application of the proposed solution in simulation and compare it to centralized and global approaches. We then test our approach with physical marine robots sampling a spatially nonuniform time -varying process in a water tank.

The reduction of the large order models of complex systems into a sufficiently low dimensional order in such a way that important dynamic characteristics of the real system are preserved, is a nontrivial task. The motivations for such a model order reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. In the present article, major techniques in model order reduction are briefly discussed. The characteristic similarities and differences are highlighted, their advantages and disadvantages are underlined, and further research requirements and trends are commented upon. Introduction and Perspective One of the central issues in the active control of complex systems such as large flexible space structures (LFSS) is the derivation of a correct mathematical model for both the controlled and the uncontrolled dynamical systems. Theoretically, there are infinitely many elastic modes or degrees of freedom (DOF) in the distributed parameter (DP) models of LFSS, usually with very low natural damping. Moreover, the flexible modes contribute to the actual deformation of the structure.' The truly infinite dimensional character of LFSS models has to be approximated by some fidelity finite (but usually very large) dimensional model. The normal approach taken by engineers to achieve this end is via modal models with a large number of modes that provide a reasonable representation of the spacecraft dynamic characteristics. However, a difficult problem still remains to be the development of a model of lowenough dimensional order that it can be utilized by the onboard controller, yet high enough dimensional order that it preserves the dynamic characteristics of the real system represented and controlled. The motivations for such a reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. Discretization procedures will not be discussed herein and it will be assumed that a large finite dimensional model has been generated somehow and for implementation and other practical considerations, the model needs to be further reduced. *This work was-supported, in part, by the Air Force Wright Aeronautical, Air Force Systems Command, Flight Dynamics Lab, Structures and Dynamics Division, while the author was employed with HR-Textron, Valencia, California. There are various approaches to model order reduction; some are or pseudo-optimal and others are ad-hoc methods based on practical considerations and engineering experience and judgment. These techniques are known by different names such as reduction, condensation, economization, aggregation, optimal projection, and other combinations thereof. It has often been pointed out that such techniques generally constitute application of Rayleigh-Ritz/Galerkin optimization, and matrix transformation methods to the eigenvalue/eigenvector formulation for structural dynamic problems. The reduction or the condensation methods are based on transformations of the coordinates in the equations of motion that essentially maintain the invariance of the quadratic forms of the potential and kinetic energies. An important feature of these methods is that the reduced order model often loses the basic characteristics of the original system. Considerable progress was made by Likins, Ohkami, and wong5 in this respect. However, they failed to develop a general enough criterion that could reduce the system model in an sense without significantly affecting the eigenvalues of the original model. The optimization or the mathematical reduction procedure, on the other hand, is based on the reduction of the eigenvalue problem to a smaller size based on some optimality criterion (usually quadratic). In these techniques, the reductions are carried out without actually resorting to approximations, i .e. , without truncatiag any coordinates or states of the original system. Balanced model reduction of linear timeinvariant dynamical systems is essentially based on the controllability and observability relations of the states of the system. Subsystem models are obtained by deleting those states that contribute the least to the controllability and observability (or the impulse response) of the original system7 and thus are only in this sense. A more recent approach to model reduction is proposed in Skelton's work,* where each state of the system model is assigned a relative to a given basis, via a quadratic criterion, and the states with the least cost are deleted in a systematic manner. The resulting reduced model is a function of the state-space basis, and thus there is no guarantee for optimality for all choices. The latest development in model order reduction techniques is the work by Hyland and erns stein.^ Herein first order necessary conditions for reduced order modeling of linear time-invariant systems are derived via a pair of modified Lyapunov equations coupled by a nonorthogonal projection. This approach reveals the possibility of multiple extrema forsome of the abovementioned methods. Copyright @ 1988 by the American Institute of Aeronautics and Astronautics. Inc. All Rights Reserved.