Deep Koopman model predictive control for enhancing transient stability in power grids

Ping, Zuowei; Yin, Zhun; Li, Xiuting; Liu, Yefeng; Yang, Tao

doi:10.1002/rnc.5043

Cited by 31 publications

(18 citation statements)

References 36 publications

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“…( 15). Several works employed a linear Koopman approach in identification and control studies and observed a higher performance over conventional linear identification methods [27][28][29][30][31][32][33][34]. However, when significant nonlinearity is present, Eq.…”

Section: Koopman Theory For Controlled Systemsmentioning

confidence: 99%

“…( 13), which is also known from standard system identification literature [4]. Deep learning for Koopman models with inputs was applied in [30,32,34]. However, these works identified linear models, Eq.…”

Section: Koopman Theory For Controlled Systemsmentioning

confidence: 99%

“…Specifically, we employ Koopman theory to identify low-order latent dynamics describing the nonlinear system evolution. While the existing literature on Koopman modeling for control has focused on linear [27][28][29][30][31][32][33][34] and bilinear [35][36][37][38] forms, we consider a MIMO Wiener structure. As we show, when combined with Koopman theory, the Wiener structure provides strong model reduction capabilities beyond those of linear and bilinear models.…”

Section: Introductionmentioning

confidence: 99%

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Identification of MIMO Wiener-type Koopman Models for Data-Driven Model Reduction using Deep Learning

Schulze,

Doncevic,

Mitsos

2022

Preprint

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We use Koopman theory to develop a data-driven nonlinear model reduction and identification strategy for multiple-input multiple-output (MIMO) input-affine dynamical systems. While the present literature has focused on linear and bilinear Koopman models, we derive and use a Wiener-type Koopman formulation. We discuss that the Wiener structure is particularly suitable for model reduction, and can be naturally derived from Koopman theory. Moreover, the Wiener block-structure unifies the mathematical simplicity of linear dynamical blocks and the accuracy of bilinear dynamics. We present a Koopman deep-learning strategy combining autoencoders and linear dynamics that generates low-order surrogate models of MIMO Wiener type. In three case studies, we apply our framework for identification and reduction of a system with input multiplicity, a chemical reactor and a high-purity distillation column. We compare the prediction performance of the identified Wiener models to linear and bilinear Koopman models. We observe the highest accuracy and strongest model reduction capabilities of low-order Wiener-type Koopman models, making them promising for control.

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Section: Koopman Theory For Controlled Systemsmentioning

confidence: 99%

Section: Koopman Theory For Controlled Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Identification of MIMO Wiener-type Koopman Models for Data-Driven Model Reduction using Deep Learning

Schulze,

Doncevic,

Mitsos

2022

Preprint

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“…Furthermore, the input might also be projected through the nonlinear lifting, which endangers controllability. Despite these theoretical considerations and due to its simplicity, a linear time invariant (LTI) Kooman model with nonlifted input is generally used in practice, especially with control methods such as linear quadratic regulation (LQR) and model predictive control (MPC), [12], [15], [18]. However, there is no discussion on the validity of these models or on the approximation error that is introduced.…”

Section: Introductionmentioning

confidence: 99%

Koopman Form of Nonlinear Systems with Inputs

Iacob¹,

Tóth²,

Schoukens³

2022

Preprint

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The Koopman framework proposes a linear representation of finite dimensional nonlinear systems through a generally infinite dimensional globally linear representation. Originally, the Koopman formalism has been described for autonomous systems and the extension for actuated continuous-time systems with a linear input or a control affine form has only recently been addressed. However, such a derivation for discrete-time systems has not yet been developed. Thus, a particular Koopman form is generally assumed, predominantly a linear time invariant (LTI) model, as it facilitates the use of control techniques such as linear quadratic regulation and model predictive control. However, we show that this assumption is insufficient to capture the dynamics of the underlying nonlinear system. In the present paper, we systematically investigate and analytically derive lifted forms under inputs for a general class of nonlinear systems in both continuous and discrete time, using the fundamental theorem of calculus. We prove that the resulting lifted representations give linear state-space Koopman models where the input matrix becomes state (and input, for the discrete-time case)-dependent, hence it can be seen as a specially structured linear parameter-varying (LPV) description of the underlying system. We also provide error bounds on how much the parameter variation contributes and how well the system behaviour can be approximated by an LTI Koopman representation. The introduced theoretical insight greatly helps for performing proper model structure selection in system identification with Koopman models as well as making a proper choice for LTI or LPV techniques for the control of nonlinear systems through the Koopman approach.

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“…On basis functions designing problem, we can autonomously train a neural network to take the place of basis functions instead of manually trying different kernel functions with different hyper-parameters. From this perspective, notable researches about combining deep learning and the Koopman operator have fueled its application in many fields such as fluid dynamics [15], power grid [16], vehicle dynamics [17], molecular kinetics [18], atomic scale dynamics [19], highway traffic dynamics [20], chaos system [21] et.al. These works usually adopt an anto-encoder (AE) framework, and the encoder is used to approximate the Koopman eigenfunctions.…”

Section: Introductionmentioning

confidence: 99%

CKNet: A Convolutional Neural Network Based on Koopman Operator for Modeling Latent Dynamics from Pixels

Xiao¹,

Xu²,

Lin³

2021

Preprint

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For systems with only known pixels, it is difficult to identify its dynamics, especially with a linear operator. In this work, we present a convolutional neural network (CNN) based on the Koopman operator (CKNet) to identify the latent dynamics from raw pixels. CKNet learned an encoder and decoder to play the role of the Koopman eigenfunctions and modes, respectively. The Koopman eigenvalues can be approximated by the eigenvalues of the learned system matrix. We present the deterministic and variational approaches to realize the encoder separately. Because CKNet is trained under the constraints of the Koopman theory, the identified dynamics is linear, controllable and physically-interpretable. Besides, the system matrix and control matrix are trained as trainable tensors. To improve the performance, we propose the auxiliary weight term for multistep linearity and prediction losses. Experiments select two classic forced dynamical systems with continuous action space, and the results show that identified dynamics with 32-dim can predict validly 120 steps and generate clear images.

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Deep Koopman model predictive control for enhancing transient stability in power grids

Cited by 31 publications

References 36 publications

Identification of MIMO Wiener-type Koopman Models for Data-Driven Model Reduction using Deep Learning

Identification of MIMO Wiener-type Koopman Models for Data-Driven Model Reduction using Deep Learning

Koopman Form of Nonlinear Systems with Inputs

CKNet: A Convolutional Neural Network Based on Koopman Operator for Modeling Latent Dynamics from Pixels

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