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Autoencoder-Based and Physically Motivated Koopman Lifted States for Wind Farm MPC: A Comparative Case Study

Bindu Sharan, Antje Dittmer, Yongyuan Xu, Herbert Werner

Abstract

This paper explores the use of Autoencoder (AE) models to identify Koopman-based linear representations for designing model predictive control (MPC) for wind farms. Wake interactions in wind farms are challenging to model, previously addressed with Koopman lifted states. In this study we investigate the performance of two AE models: The first AE model estimates the wind speeds acting on the turbines these are affected by changes in turbine control inputs. The wind speeds estimated by this AE model are then used in a second step to calculate the power output via a simple turbine model based on physical equations. The second AE model directly estimates the wind farm output, i.e., both turbine and wake dynamics are modeled. The primary inquiry of this study addresses whether any of these two AE-based models can surpass previously identified Koopman models based on physically motivated lifted states. We find that the first AE model, which estimates the wind speed and hence includes the wake dynamics, but excludes the turbine dynamics outperforms the existing physically motivated Koopman model. However, the second AE model, which estimates the farm power directly, underperforms when the turbines' underlying physical assumptions are correct. We additionally investigate specific conditions under which the second, purely data-driven AE model can excel: Notably, when modeling assumptions, such as the wind turbine power coefficient, are erroneous and remain unchecked within the MPC controller. In such cases, the data-driven AE models, when updated with recent data reflecting changed system dynamics, can outperform physics-based models operating under outdated assumptions.

Autoencoder-Based and Physically Motivated Koopman Lifted States for Wind Farm MPC: A Comparative Case Study

Abstract

This paper explores the use of Autoencoder (AE) models to identify Koopman-based linear representations for designing model predictive control (MPC) for wind farms. Wake interactions in wind farms are challenging to model, previously addressed with Koopman lifted states. In this study we investigate the performance of two AE models: The first AE model estimates the wind speeds acting on the turbines these are affected by changes in turbine control inputs. The wind speeds estimated by this AE model are then used in a second step to calculate the power output via a simple turbine model based on physical equations. The second AE model directly estimates the wind farm output, i.e., both turbine and wake dynamics are modeled. The primary inquiry of this study addresses whether any of these two AE-based models can surpass previously identified Koopman models based on physically motivated lifted states. We find that the first AE model, which estimates the wind speed and hence includes the wake dynamics, but excludes the turbine dynamics outperforms the existing physically motivated Koopman model. However, the second AE model, which estimates the farm power directly, underperforms when the turbines' underlying physical assumptions are correct. We additionally investigate specific conditions under which the second, purely data-driven AE model can excel: Notably, when modeling assumptions, such as the wind turbine power coefficient, are erroneous and remain unchecked within the MPC controller. In such cases, the data-driven AE models, when updated with recent data reflecting changed system dynamics, can outperform physics-based models operating under outdated assumptions.
Paper Structure (13 sections, 24 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 13 sections, 24 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Wind Farm Control
  3. Koopman-based Linear system identification
  4. AE for Identification of the Koopman Matrix
  5. AE-based Wind Farm Model
  6. AE-based Koopman Model for Effective Wind Estimation
  7. AE-based Koopman Model for Farm Power Estimation
  8. Model Predictive control
  9. qLMPC Design using AE1 model
  10. KMPC Design using AE2 model
  11. Simulation Results
  12. Conclusion
  13. Autoencoder: Training

Figures (7)

  • Figure 1: Wind farm control and identified models AE1 and AE2. Diagram (a) showcases the centralized farm-level controller incorporating the AE1 model for wind speed estimation (blue) and the AE2 model for power estimation (red). Diagram (b) illustrates the estimation of wind farm power through the integration of the AE1 model and first principle turbine models. Diagram (c) shows the direct estimation of farm power using the AE2 model.
  • Figure 2: Identification of linear system representation of a nonlinear system by integrating Koopman and AE
  • Figure 3: Farm power reference tracking (K$_{24}$ qLMPC). Column 1 corresponds to scenarios 1 and column 2 corresponds to scenarios 2
  • Figure 4: Farm power reference tracking (K$_{\text{AE1(24)}}$ qLMPC), Column 1 corresponds to scenarios 1 and column 2 corresponds to scenarios 2
  • Figure 5: Farm power reference tracking (K$_{\text{AE2}}$ qLMPC), in both scenarios
  • ...and 2 more figures