# On the Potential of Reduced Order Models for Wind Farm Control: A Koopman Dynamic Mode Decomposition Approach

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Input Output Dynamic Mode Decomposition

- (1)
- Organize snapshot matrices: the series of data containing relevant information of the three dimensional vector flow field at each time instant k is taken and flatten into a single column vector ${\mathit{x}}_{k}$, referred to as a snapshot of data. All time instants are combined into one single matrix. The latter is divided into the data matrix
**X**and its time shifted version**X’**:$$\mathbf{X}=\left[\begin{array}{cccc}\mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \\ {\mathit{x}}_{1}& {\mathit{x}}_{2}& \cdots & {\mathit{x}}_{m-1}\\ \mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \end{array}\right]\phantom{\rule{28.45274pt}{0ex}}{\mathbf{X}}^{\prime}=\left[\begin{array}{cccc}\mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \\ {\mathit{x}}_{2}& {\mathit{x}}_{3}& \cdots & {\mathit{x}}_{m}\\ \mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \end{array}\right]$$$$\mathsf{\Psi}=\left[\begin{array}{cccc}\mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \\ {\mathit{u}}_{1}& {\mathit{u}}_{2}& \cdots & {\mathit{u}}_{m-1}\\ \mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \end{array}\right]\phantom{\rule{28.45274pt}{0ex}}\mathbf{Y}=\left[\begin{array}{cccc}\mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \\ {\mathit{y}}_{1}& {\mathit{y}}_{2}& \cdots & {\mathit{y}}_{m-1}\\ \mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \end{array}\right]$$ - (2)
- Take the singular value decomposition of X: by appropriately choosing a truncation value r of the number of singular values to retain,
**X**can be factorized as $\mathbf{U}\Sigma {\mathbf{V}}^{*}$, where $\mathbf{U}\in {\mathbb{R}}^{n\times r}$, $\Sigma \in {\mathbb{R}}^{r\times r}$, ${\mathbf{V}}^{*}\in {\mathbb{R}}^{r\times m-1}$ and ${}^{*}$ is the conjugate transpose. - (3)
- Formulate least squares problem by projecting system to reduced dimension space: Equation (1) is then rewritten in terms of the data matrices in (2) and the input and output snapshot matrices in (3) in the compact format:$$\left[\begin{array}{c}{\mathbf{X}}^{\prime}\\ \mathbf{Y}\end{array}\right]=\left[\begin{array}{cc}\mathbf{A}& \mathbf{B}\\ \mathbf{C}& \mathbf{D}\end{array}\right]\left[\begin{array}{c}\mathbf{X}\\ \mathsf{\Psi}\end{array}\right]$$The state is then projected onto the subspace defined by an orthonormal basis given by the POD modes of
**X**. The columns of**U**specify the orthonormal basis to project the high order state. The low order state matrices are then $\tilde{\mathbf{A}}={\mathbf{U}}^{*}\mathbf{A}\mathbf{U}$, $\tilde{\mathbf{B}}={\mathbf{U}}^{*}\mathbf{B}$, $\tilde{\mathbf{C}}=\mathbf{C}\mathbf{U}$ and $\tilde{\mathbf{D}}=\mathbf{D}$. Substituting in (4) and rearranging, the reduced order state matrices are obtained by minimizing the error of the Frobenius norm [22]:$$\underset{\left[\begin{array}{cc}\tilde{\mathbf{A}}& \tilde{\mathbf{B}}\\ \tilde{\mathbf{C}}& \tilde{\mathbf{D}}\end{array}\right]}{\mathrm{min}}{\u2225\left[\begin{array}{c}{\mathbf{X}}^{\prime}\\ \mathbf{Y}\end{array}\right]-\left[\begin{array}{cc}\mathbf{U}& \mathbf{0}\\ \mathbf{0}& \mathbf{I}\end{array}\right]\left[\begin{array}{cc}\tilde{\mathbf{A}}& \tilde{\mathbf{B}}\\ \tilde{\mathbf{C}}& \tilde{\mathbf{D}}\end{array}\right]\left[\begin{array}{cc}{\mathbf{U}}^{*}& \mathbf{0}\\ \mathbf{0}& \mathbf{I}\end{array}\right]\left[\begin{array}{c}\mathbf{X}\\ \mathsf{\Psi}\end{array}\right]\u2225}_{F}^{2}$$**X**and**X’**are, respectively, ${\mathbf{U}}^{*}{\mathbf{X}}^{\prime}$ and $\Sigma \mathbf{V}$:$${\Theta}_{\mathrm{IODMD}}=\left[\begin{array}{cc}\tilde{\mathbf{A}}& \tilde{\mathbf{B}}\\ \tilde{\mathbf{C}}& \tilde{\mathbf{D}}\end{array}\right]=\left[\begin{array}{c}{\mathbf{U}}^{*}{\mathbf{X}}^{\prime}\\ \mathbf{Y}\end{array}\right]{\left[\begin{array}{c}\Sigma \mathbf{V}\\ \mathsf{\Psi}\end{array}\right]}^{\u2020}$$

**W**are eigenvectors and $\mathsf{\Lambda}$ is a diagonal matrix containing the corresponding eigenvalues ${\mathbf{\lambda}}_{k}$. These can be investigated for fundamental properties of the underlying system such as growth modes and resonance frequencies. To approximate the eigenvalues and eigenvectors of

**A**without its explicit computation, an approximation is calculated based on the so called DMD modes, which correspond to the columns of $\mathsf{\Phi}$, $\mathsf{\Phi}$ =

**U**

**W**, as originally proposed in [24]. These representations of the high order eigenvectors based on the low order eigenvectors describe the dynamics observed in the time series in terms of oscillatory components.

#### 2.2. Koopman Operator Theory

**f (·)**represents the dynamics, defined on a state space

**M**. This induces a discrete-time dynamical system given by the flow map ${\mathbf{F}}_{t}:\mathbf{M}\to \mathbf{M}$, ${\mathit{x}}_{k+1}={\mathbf{F}}_{t}\left({\mathit{x}}_{k}\right)$. A function defined as $g:\mathbf{M}\to \mathbb{R}$ is called an observable of the system. The Koopman operator ${\mathcal{K}}_{t}$ acts on observable functions g as ${\mathcal{K}}_{t}=g\circ {\mathbf{F}}_{t}$ where ∘ is the composition operator, so that:

**F**corresponding to the observable $\mathit{g}$, associated with the kth Koopman eigenfunction, i.e., the weighting of each observable on the eigenfunction. The expression in (10) can be interpreted as a linear combination of the eigenfunctions ${\phi}_{k}$ of $\mathcal{K}$ where ${\mathit{v}}_{k}$ is the coefficient in the expansion. The nonlinear system can either be evolved in the original state space or in the measurement space as in (10).

- (1)
- IODMD using different observables: the IODMD algorithm is tested using different variables of the velocity flow field or combination of variables. The common variable used as a state, the streamwise velocity component, u, is replaced by others, such as the spanwise velocity component v, the vertical velocity component w or a non-linear combination of the three. This procedure attempts to find the coordinates best suited to describe the dynamics of AIC. The results of this formulation are referred to as IODMD${}_{xx}$, where $xx$ is the observable used. In terms of the IODMD described in Section 2.1, only the snapshot matrices organisation step differs, with the data matrices being generalized.$$\mathbf{X}=\left[\begin{array}{cccc}\mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \\ g\left({\mathit{x}}_{1}\right)& g\left({\mathit{x}}_{2}\right)& \cdots & g\left({\mathit{x}}_{m}\right)\\ \mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \end{array}\right]\phantom{\rule{28.45274pt}{0ex}}{\mathbf{X}}^{\prime}=\left[\begin{array}{cccc}\mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \\ g\left({\mathit{x}}_{1}^{\prime}\right)& g\left({\mathit{x}}_{2}^{\prime}\right)& \cdots & g\left({x}_{m}^{\prime}\right)\\ \mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \end{array}\right]$$
- (2)
- Augment state vector by another flow variable: the IODMD algorithm is tested by augmenting the data matrices with different observables simultaneously, as in [30]. This formulation is termed Extended Input Output Dynamic Mode Decomposition (EIODMD${}_{xx,yy}$) where $xx$ is the first observable and $yy$ the second observable used to augment the original snapshot matrix. In this approach, the data matrices are generalized to:$$\mathbf{X}=\left[\begin{array}{cccc}\mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \\ \mathit{g}\left({\mathit{x}}_{1}\right)& \mathit{g}\left({\mathit{x}}_{2}\right)& \cdots & \mathit{g}\left({\mathit{x}}_{m}\right)\\ \mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \end{array}\right]\phantom{\rule{28.45274pt}{0ex}}{\mathbf{X}}^{\prime}=\left[\begin{array}{cccc}\mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \\ \mathit{g}\left({\mathit{x}}_{1}^{\prime}\right)& \mathit{g}\left({\mathit{x}}_{2}^{\prime}\right)& \cdots & \mathit{g}\left({x}_{m}^{\prime}\right)\\ \mid & \mid & \phantom{\rule{1.em}{0ex}}& \mid \end{array}\right]$$
- (3)
- Augment the state vector by any variable: the third hypothesis focuses on augmenting the flow field information with turbine information, following the ideas laid out in [33]. The additional states are referred to as deterministic states ${\mathbf{X}}_{d}$ and the corresponding data matrices are organized as in (2). The least squares problem to solve boils down to:$${\Theta}_{\mathrm{EIODMD}}=\left[\begin{array}{c}{{\mathbf{X}}^{\prime}}_{d}\\ {\mathbf{U}}^{*}{\mathbf{X}}^{\prime}\\ \mathbf{Y}\end{array}\right]{\left[\begin{array}{c}{\mathbf{X}}_{d}\\ \Sigma \mathbf{V}\\ \mathsf{\Psi}\end{array}\right]}^{\u2020}$$

#### 2.3. Simulation

## 3. Results

#### 3.1. Input Output Dynamic Mode Decomposition with Different Observables

**A**, which, in turn, corresponds to a mode or coherent structure in space which inherits the dynamical properties indicated in Table 2. Figure 7, Figure 8 and Figure 9 allow to visualise these structures in different frequency ranges. Isosurfaces of various values have been chosen so that the structures can be visualised and interpreted qualitatively.

#### 3.2. Extended Input Output Dynamic Mode Decomposition with Flow Field Information

#### 3.3. Extended Input Output Dynamic Mode Decomposition with Turbine Information

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Input–output information for system identification (

**solid line**) and validation (

**dashed line**). Input data corresponds to the collective pitch blade angle of the upstream turbine (

**in orange**) and output data are the generator powers of both upstream and downstream turbines (

**in blue**). The last plot depicts the power gains and losses of the collective power throughout the simulations (

**in green**).

**Figure 2.**Fit of each model, as measured by VAF. From left to right, each model has increasing number of singular values (also referred to as modes in the context of DMD).

**Figure 3.**Best performing model validation results. Measurements from SOWFA are represented with a solid blue line and model response with a dashed green line.

**Figure 4.**Comparison of flow field reconstruction using reduced linear model (

**left**) and data from SOWFA (

**right**) at hub height for different time instants. Reconstruction obtained using validation data set.

**Figure 5.**Comparison of flow field reconstruction using reduced linear model (

**left**), data from SOWFA (

**middle**) and relative deviations at downstream rotor plane (

**right**) for different time instants. First row of snapshots corresponds to simulation instant of 16 min and 30 s represented in Figure 4. Higher deviations are noticeable when the travelling wake hits the downstream rotor, illustrated by the middle snapshot. Reconstruction obtained using validation data set.

**Figure 6.**Eigenvalues of the IODMD${}_{u}$. On the left, the discrete time eigenvalues $\lambda $ on the complex plane are represented relative to the unit circle (dashed line). On the right, the eigenvalues transformed to continuous time are represented. The area where the majority of modes are presented has been zoomed in.

**Figure 7.**DMD modes #1, #2 and #4. Mode #1 corresponds to a background mode that is not changing (i.e., it has zero eigenvalue). The negative value isosurfaces of mode #1 indicate the different regions marked by velocity deficits inbetween the upstream and downstream turbines.

**Figure 8.**DMD modes #6, #7 and #8, with natural frequencies comprised between 0.17 St and 0.26 St. The spatial patterns present here appear to also explain some of the dynamics more related with the downstream turbine, as opposite to background and low frequency modes. The positive value isosurfaces of modes #6 and #7 appear to explain the increased velocity near the downstream rotor.

**Figure 9.**DMD modes #11, #15 and #16, with natural frequencies comprised between 0.41 St and 0.80 St, showing pulsating structures inbetween upstream and downstream turbines. These higher frequency modes allow the reconstruction of the wake for faster pitch angle variations.

**Figure 10.**Comparison of IODMDu best performing model response (

**dashed green line**) with the EIODMD${}_{u,w}$ (

**dashed red line**) for turbine validation data retrieved from SOWFA (

**solid blue line**).

**Figure 11.**Comparison of IODMD${}_{u}$ (

**green solid line**) and EIODMD${}_{u,w}$ (

**red solid line**) model’s response in the frequency domain. The Nyquist frequency, defined as half of the sampling rate, is marked as a vertical black line.

**Table 1.**Models validation results using IODMD${}_{u}$ algorithm, where u is the streamwise velocity component. First column has the data chosen for the states. Second column has the fit criteria VAF (the second turbine generator power is the most relevant, hence acts as a primary decision criteria). Third column has the number of states of the model. Fourth column the fit of the upstream turbine generator power for the chosen model. Fifth and sixth columns present the average NRMSE for identification and validation data.

Model Properties | Model Obtained by IODMD | ||||
---|---|---|---|---|---|

Observable | $\mathrm{VAF}{\left(\mathrm{WT}2\right)}_{\mathrm{max}}$ | Model size | ${\mathrm{VAF}\left(\mathrm{WT}1\right)|}_{r}$ | $\overline{\mathrm{NRMSE}}$ (id) | $\overline{\mathrm{NRMSE}}$ (val) |

u | 73.59% | 38 | 98.08% | 3.66% | 3.75% |

**Table 2.**Dynamical properties of IODMD${}_{u}$ model. The first column entries corresponds to the numerology chosen for the modes, which are ordered by increasing natural frequency. The second column entries correspond to their order in terms of eigenvectors of state matrix $\tilde{\mathbf{A}}$. The third and fourth columns have information of the modes natural frequencies and fifth column the damping ratio.

Modes | Dynamical Properties | |||
---|---|---|---|---|

# | DMD | ${\omega}_{n}$ [Hz] | ${\omega}_{n}$ St [-] | $\xi $ |

1 | 27 | 0 | 0 | 1 |

2 | 34 | 0.0014 | 0.0283 | 1 |

3 | 37–38 | 0.0030 | 0.0585 | 0.7608 |

4 | 35–36 | 0.0032 | 0.0637 | 0.2041 |

5 | 32–33 | 0.0064 | 0.1268 | 0.3072 |

6 | 30–31 | 0.0088 | 0.1735 | 0.0275 |

7 | 28–29 | 0.0109 | 0.2162 | 0.1192 |

8 | 25–26 | 0.0129 | 0.2559 | 0.0497 |

9 | 23–24 | 0.0158 | 0.3124 | 0.0712 |

10 | 21–22 | 0.0194 | 0.3830 | 0.0140 |

11 | 19–20 | 0.0207 | 0.4087 | 0.0278 |

12 | 17–18 | 0.0247 | 0.4884 | 0.0358 |

13 | 15–16 | 0.0263 | 0.5204 | 0.0280 |

14 | 13–14 | 0.0301 | 0.5959 | 0.0066 |

15 | 11–12 | 0.0343 | 0.6787 | 0.0025 |

16 | 9–10 | 0.0404 | 0.7996 | 0.0026 |

17 | 7–8 | 0.0472 | 0.9331 | 0.0100 |

18 | 5–6 | 0.0560 | 1.1071 | 0.0040 |

19 | 3–4 | 0.1479 | 2.9251 | 0.0091 |

20 | 1–2 | 0.2249 | 4.4473 | 0.0482 |

**Table 3.**Models validation results using the IODMD algorithm with different observables. The information in the first 4 columns is the same as in Table 1. The last two columns show the improved performance in terms of fit and the improvements in terms of the size of model. Reduced models are less computationally expensive for the purposes of model predictive control, hence they are written in green. $\Delta $ VAF ${\left(\mathrm{WT}2\right)}_{\mathrm{max}}$ is the difference of fit with relation to the best model obtained using IODMD${}_{u}$. $\Delta $ size is the relative difference with relation to the best model obtained using IODMD${}_{u}$.

Model Properties | Model Obtained by IODMD_u | $\Delta $ IODMD_u | |||
---|---|---|---|---|---|

Observable | $\mathrm{VAF}{\left(\mathrm{WT}2\right)}_{\mathrm{max}}$ | Model size | ${\mathrm{VAF}\left(\mathrm{WT}1\right)|}_{r}$ | $\Delta $ VAF ${\left(\mathrm{WT}2\right)}_{\mathrm{max}}$ | $\Delta $ size |

v | 74.78% | 39 | 97.64% | +1.19% | +2.63% |

w | 81.71% | 39 | 96.99% | +8.12% | +2.63% |

${u}^{\prime}$ | 43.16% | 29 | 93.99% | −30.43% | −23.68% |

${v}^{\prime}$ | 43.09% | 49 | 96.71% | −30.50% | +28.95% |

${w}^{\prime}$ | 55.57% | 49 | 96.69% | −18.02% | +28.95% |

${u}^{\prime 2}$ | 51.83% | 57 | 95.58% | −21.76% | +50.00% |

${w}^{\prime 2}$ | 35.18% | 104 | 94.78% | −38.41% | +41.32% |

${u}^{2}$ | 74.71% | 24 | 96.76% | +1.12% | −36.84% |

${v}^{2}$ | 72.38% | 112 | 99.33% | −1.21% | +194.74% |

${w}^{2}$ | 77.67% | 106 | 99.29% | +4.08% | +178.95% |

$u\xb7v$ | 75.56% | 59 | 99.03% | +1.97% | +55.26% |

$u\xb7w$ | 76.54% | 35 | 97.42% | +2.95% | −7.89% |

$v\xb7w$ | 75% | 120 | 99.25% | +1.41% | +63.07% |

$u\xb7v\xb7w$ | 72.83% | 72 | 99.25% | −0.76% | +89.47% |

${u}^{2}+{w}^{2}$ | 75.67% | 24 | 96.79% | +2.08% | −36.84% |

${u}^{2}+{v}^{2}+{w}^{2}$ | 78.65% | 24 | 96.89% | +5.09% | −36.84% |

**Table 4.**Models validation results using the EIODMD algorithm with different observables. The second observables column has the state chosen to augment the initial data matrix with u information. The last two columns represent the average NRMSE of the first observable, using both identification and validation data.

Model Properties | Model Obtained by EIODMD | $\Delta $ IODMD${}_{\mathit{u}}$ | ||||||
---|---|---|---|---|---|---|---|---|

Observable | Second observables | $\mathrm{VAF}{\left(\mathrm{WT}2\right)}_{\mathrm{max}}$ | Model size | ${\mathrm{VAF}\left(\mathrm{WT}1\right)|}_{r}$ | $\Delta $ VAF ${\left(\mathrm{WT}2\right)}_{\mathrm{max}}$ | $\Delta $ size | $\overline{\mathrm{NRMSE}}$ (id) | $\overline{\mathrm{NRMSE}}$ (val) |

u | v | 79.00% | 32 | 96.70% | +5.41% | −15.79% | -0.05% | −0.07% |

u | w | 81.74% | 30 | 97.27% | +8.15% | −21.05% | −0.12% | −0.33% |

u | v, w | 79.76% | 28 | 97.22% | +6.17% | −2.19% | −0.08% | −0.13% |

**Table 5.**Models validation results using the EIODMD algorithm with different base observables and deterministic states.

Model Properties | Model Obtained by EIODMD | $\Delta $ IODMD${}_{\mathit{u}}$ | ||||||
---|---|---|---|---|---|---|---|---|

Observable | Deterministic | $\mathrm{VAF}{\left(\mathrm{WT}2\right)}_{\mathrm{max}}$ | Model size | ${\mathrm{VAF}\left(\mathrm{WT}1\right)|}_{r}$ | $\Delta $ VAF ${\left(\mathrm{WT}2\right)}_{\mathrm{max}}$ | $\Delta $ size | $\overline{\mathrm{NRMSE}}$ (id) | $\overline{\mathrm{NRMSE}}$ (val) |

u | ${\mathsf{\Omega}}_{1},{\mathsf{\Omega}}_{2}$ | 76.71% | 12 | 98.97% | +3.12% | −68.42% | +0.17% | +0.45% |

w | ${\mathsf{\Omega}}_{1},{\mathsf{\Omega}}_{2}$ | 76.79% | 30 | 98.93% | +3.20% | −21.05% | - | - |

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**MDPI and ACS Style**

Cassamo, N.; van Wingerden, J.-W. On the Potential of Reduced Order Models for Wind Farm Control: A Koopman Dynamic Mode Decomposition Approach. *Energies* **2020**, *13*, 6513.
https://doi.org/10.3390/en13246513

**AMA Style**

Cassamo N, van Wingerden J-W. On the Potential of Reduced Order Models for Wind Farm Control: A Koopman Dynamic Mode Decomposition Approach. *Energies*. 2020; 13(24):6513.
https://doi.org/10.3390/en13246513

**Chicago/Turabian Style**

Cassamo, Nassir, and Jan-Willem van Wingerden. 2020. "On the Potential of Reduced Order Models for Wind Farm Control: A Koopman Dynamic Mode Decomposition Approach" *Energies* 13, no. 24: 6513.
https://doi.org/10.3390/en13246513