In this work, we present a nonlinear model reduction approach for reducing two commonly used nonlinear dynamical models of power grids: the effective network (EN) model and the synchronous motor (SM) model. Such models are essential in real-time security assessments of power grids. However, as power grids are often large-scale, it is necessary to reduce the models in order to utilize them in real-time. We reformulate the nonlinear power grid models as quadratic systems and reduce them using balanced truncation based on approximations of the reachability and observability Gramians. Finally, we present examples involving numerical simulation of reduced EN and SM models of the IEEE 57 bus and IEEE 118 bus systems.
We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees asymptotic stability and passivity of the reduced order model as well as the positive definiteness of the mass and stiffness matrices. Moreover, we receive an a priori gap metric error bound. Finally, we show that our method based on positive real balanced truncation preserves the structure of overdamped second order systems.
The paper presents a model reduction framework geared towards the analysis and design of systems that switch and oscillate. While such phenomena are ubiquitous in nature and engineering, model reduction methods are not well developed for non-equilibrium behaviors. The proposed framework addresses this need by exploiting recent advances on dominance theory. Classical balanced truncation for linear time-invariant systems is used to develop a dominance-preserving model reduction method for Lure systems, i.e. systems that can be decomposed as the feedback interconnection of a linear system and a s…
Balanced Truncation of Linear Time-Invariant Systems over Finite-frequency Ranges
Max Planck Institute for Dynamics of Complex Technical Systems（马普复杂技术系统动力学研究所）；School of Mechatronic Engineering and Automation, Shanghai University（上海大学机电工程与自动化学院）
Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems in a manner that accounts for either open- or closed-loop observability and controllability aspects of the system. Two computational challenges have so far prevented its deployment on large-scale systems: (a) the energy functions required for characterization of controllability and observability are solutions of high-dimensional Hamilton-Jacobi-(Bellman) equations, which have been computationally intractable and (b) the transformations to construct the reduced-order models (ROMs) …
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL), UMR CNRS 9189, CNRS, Ecole Centrale de Lille（里尔信息、信号与自动化研究中心 (CRIStAL)，“CNRS联合研究单位9189”，法国国家科学研究中心，里尔中央理工学院）；Department of Electrical and Computer Engineering, Rice University（莱斯大学电气与计算机工程系）；CNRS CRIStAL UMR 9189, Université de Lille 1, Sciences et Technologies（法国国家科学研究中心 CRIStAL “联合研究单位9189”，里尔第一大学科学与技术学院）
In standard balanced truncation model order reduction, the initial condition is typically ignored in the reduction procedure and is assumed to be zero instead. However, such a reduced-order model may be a bad approximation to the full-order system, if the initial condition is not zero. In the literature there are several attempts for modified reduction methods at the price of having no error bound or only a posteriori error bounds which are often too expensive to evaluate. In this work we propose a new balancing procedure that is based on a shift transformation on the state. We first derive a …
Compression and Distillation of Data Quadruplets in Non-intrusive Reduced-order Modeling
Yangtze University（长江大学）；Hubei Normal University（湖北师范大学）；Wuhan University of Science and Technology（武汉科技大学）；Shanxi University（山西大学）；The University of Western Australia（西澳大学）
In this paper, we explore the role of tensor algebra in balanced truncation (BT) based model reduction/identification for high-dimensional multilinear/linear time invariant systems. In particular, we employ tensor train decomposition (TTD), which provides a good compromise between numerical stability and level of compression, and has an associated algebra that facilitates computations. Using TTD, we propose a new BT approach which we refer to as higher-order balanced truncation, and consider different data-driven variations including higher-order empirical gramians, higher-order balanced prope…
We propose a new framework to design controllers for high-dimensional nonlinear systems. The control is designed through the iterative linear quadratic regulator (ILQR), an algorithm that computes control by iteratively applying the linear quadratic regulator on the local linearization of the system at each time step. Since ILQR is computationally expensive, we propose to first construct reduced-order models (ROMs) of the high-dimensional nonlinear system. We derive nonlinear ROMs via projection, where the basis is computed via balanced truncation (BT) and LQG balanced truncation (LQG-BT). Num…
In this work, the empirical-Gramian-based model reduction methods: Empirical poor man's truncated balanced realization, empirical approximate balancing, empirical dominant subspaces, empirical balanced truncation, and empirical balanced gains are compared in a non-parametric and two parametric variants, via ten error measures: Approximate Lebesgue $L_0$, $L_1$, $L_2$, $L_\infty$, Hardy $H_2$, $H_\infty$, Hankel, Hilbert-Schmidt-Hankel, modified induced primal, and modified induced dual norms, for variants of the thermal block model reduction benchmark. This comparison is conducted via a new me…
In time-limited model order reduction, a reduced-order approximation of the original high-order model is obtained that accurately approximates the original model within the desired limited time interval. Accuracy outside that time interval is not that important. The error incurred when a reduced-order model is used as a surrogate for the original model can be quantified in absolute or relative terms to access the performance of the model reduction algorithm. The relative error is generally more meaningful than an absolute error because if the original and reduced systems' responses are of smal…
We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a system-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical sy…
We present a novel reformulation of balanced truncation, a classical model reduction method. The principal innovation that we introduce comes through the use of system response data that has been either measured or computed, without reference to any prescribed realization of the original model. Data are represented by sampled values of the transfer function {or the impulse response} corresponding to the original model. We discuss parallels that our approach bears with the Loewner framework, another popular data-driven model reduction method. We illustrate our approach numerically in both conti…
This paper presents an interpolatory framework for time-limited $H_2$ optimal model order reduction named Limited Time Iterative Rational Krylov Algorithm (LT-IRKA). The algorithm yields high fidelity reduced order models over limited time intervals of the form, $\begin{bmatrix}0 & τ\end{bmatrix}$ with $τ< \infty$ for linear time invariant (LTI) systems. Using the time limited $H_2$ norm, we derive interpolation based $H_{2,τ}$ optimality conditions. The LT-IRKA yields a near optimal $H_2(τ)$ reduced order system. The nearness to the exact $H_2(τ)$ optimal reduced system is quantized in terms …
Kernel Methods for the Approximation of Nonlinear Systems
Laboratory for Computational and Statistical Learning, Massachusetts Institute of Technology（麻省理工学院计算与统计学习实验室）；Department of Mathematics, Duke University（杜克大学数学系）；Department of Mathematics, Imperial College London（帝国理工学院伦敦数学系）
Nonlinear Model Reduction in Power Systems by Balancing of Empirical Controllability and Observability Covariances
Energy Systems Division, Argonne National Laboratory（阿贡国家实验室能源系统部）；Department of Electrical Engineering, Guangxi University（广西大学电气工程学院）；Energy and Transportation Sciences Division, Oak Ridge National Laboratory（橡树岭国家实验室能源与交通科学部）
