A non-intrusive reduced-order model for wind farm wake analysis based on SPOD-DNN

Abstract

Wind farm wake modeling is of great significance for wind turbine layout optimization design and yaw control strategy. In this work, we combine deep neural network (DNN) with spectral proper orthogonal decomposition (SPOD) to discover dynamic characteristics of wake under different inflow conditions. Then an assessment of the proposed SPOD-DNN surrogate modeling method of parameterized fluid is performed by comparing the predicted results. Meanwhile, we demonstrate the robustness of the SPOD-DNN through a comparison with POD-DNN, where SPOD produces fewer modes than POD but can achieve the same cumulative contribution rate and wake prediction accuracy. In the end, the method is developed to predict the wake of single wind turbine in untrained inflow condition and Wake of six wind turbines with different yaw angles. The results reveals that the model has good generalization performance and can robustly reconstruct the wake of multiple wind turbines in different directions.

Keywords

Wind farm wake model large eddy simulation spectral proper orthogonal decomposition reduced-order model deep neural network

Introduction

Wind energy, as one renewable energy source, has become an important source of power generation. Especially, to achieve the goals of Paris Climate Agreement and net zero carbon emissions, many countries have set targets for wind energy generation, accelerating the development of wind energy Tarafdar Hagh and Khalili (2019). It is estimated that by 2050, wind power alone will provide about 35% of total electricity demand International Renewable Energy Agency (2019). Due to the limitation of energy storage system and substation equipment, the establishment of large-scale wind power plants is conducive to cost control. However, the rotation of wind turbine causes a long and complex vortex wake, and in wind farm, the wake effect of upstream wind turbines will seriously affect the dynamics and power generation efficiency of downstream wind turbines.

Generally speaking, wake effect is an aerodynamic phenomenon characterized by the decrease of wind speed and the increase of turbulence, which leads to the downstream wind turbines to capture less energy and bear more aerodynamic loads. For example, the comparative study of two inline wind turbine with the same rotor diameter shows that the power loss of downstream turbine reached 46% compared with the power output of single turbine under the same operating conditions Adaramola and Krogstad (2011). To mitigate the influence of wake effect, the optimize layout for wind turbine is usually implemented in the process of wind farm design stage while various control techniques are performed to reduce the influence of upstream wind turbine wake Fleming et al. (2015). The layout of wind turbines in the wind field not only considers the influence of topography, but also mainly considers the interference between wind turbine wake and wind turbine rotor. Large-scale wind farms usually arrange more wind turbines, but in this way, wind turbines often run in the wake of upstream or adjacent wind turbines, resulting in less captured wind energy. The model proposed in this paper can quickly predict the diffusion and recovery of wake, which has certain reference significance for the arrangement of wind turbine spacing and is helpful to optimize the layout of wind farm. At the same time, the wake not only affects the power output of the wind turbine, but also affects the structural fatigue of the wind turbine. In the wind farm, the downstream wind turbines are often seriously affected by the wake of the upstream wind turbines. By quickly predicting the wake of a single wind turbine, it is helpful for the downstream wind turbines to yaw and “avoid” the influence of the wake of the upstream wind turbines. Accurate prediction of wind turbine wake contributes to optimized layout of wind turbines and turbine control, significantly improving the power generation efficiency of large-scale wind farm Gao and Hong (2021), Dou et al. (2019).

Great efforts have been made in the numerical computation and analysis of wind turbine wake. The wake interaction is studied based on various low-fidelity and high-fidelity models Boersma et al. (2017). Some analysis models such as Jensen Park model Katic et al. (1986), Jensen (1983), Frandsen et al. (2006) model that are suitable for low fidelity or small-scale data analysis are applied to wind turbine wake computation, providing an effective way for layout optimization design of wind farms. Because the approaches involve sparse data and depend heavily on empirical parameters, their performance is limited by modeling errors. These low fidelity models can only be used for simple analysis of wind turbine wake. Currently, most investigations of the real-time dynamic wake simulation are carried out by solving Navier-Stokes equations with high-fidelity models that use actuator disks model (ADM) Calaf et al. (2010), Nilsson et al. (2015) or actuator lines model (ALM) Lu and Porté-Agel (2011), Churchfield et al. (2012a) to build wind turbine. Moreover, the accuracy difference of wind farm simulation with ADM and ALM methods were also investigated Witha et al. (2014), Martínez-Tossas et al. (2015). And the research of high-fidelity solver based on numerical solution is still quite active Sprague et al. (2020). However, numerical simulation takes a lot of computational time and cannot be used as the immediate feedback of control strategy. For instance, each 1000s large eddy simulation of a $3 km \times 3 km \times 3 km$ wind farm with six turbines takes around 60 hours using 512 processors on distributed computation Gebraad et al. (2016). Besides, high fidelity simulations generate a large amount of data which require in-depth analysis through developing better analysis tools. So, how to fully utilize these vast precious flow field data to analyze and understand turbine wake has become a subject of intense interest, especially with the advance of data science and intelligent learning.

Neural network is usually used as an effective means to study and analyze fluid problems because of the great potential of machine learning in data mining and reducing computational costs Brunton et al. (2019). Several groups have applied machine learning for wake prediction, including establishing the corresponding relationship between inflow parameters and wake using neural network Zilong et al. (2020), realizing the real-time simulation of distributed fluid wake using POD-LSTM model Jincheng and Xiaowei (2020a), and establishing a wind farm wake modeling method using DC-CGAN model Jincheng and Xiaowei (2020). However, it is difficult to directly process massive data by machine learning. For this reason, the research method combining machine learning with reduced-order model (ROM), which takes into account the advantages of data compression and data mining, has attracted more and more attention in the field of fluid mechanics Swischuk et al. (2019), Wang et al. (2019); Lee and You (2019). Lui and Wolf has proved that SPOD has a good application in the problems of circular cylinder flow and airfoil stall Lui and Wolf (2019). This inspired us to establish the surrogate model for wake prediction of wind turbine by using SPOD and POD methods. In this method, the flow field parameters are taken as input, the reduction coefficients are taken as training output, and the flow field is reconstructed by the reduction coefficients.

This work focuses on developing a new wind farm wake model based on non-intrusive ROM and high-fidelity CFD simulation. In the proposed modeling method, the high-fidelity data is first reduced to reduction coefficients by spectral proper orthogonal decomposition technique. Then, a regression model is constructed by deep feedforward neural networks to approximate the mapping of input parameters to reduction coefficients. Lastly, the flow field of corresponding flow parameters is reconstructed by the predicted reduction coefficients. Considering that SPOD can filter the temporal pattern, and redistribute the system energy to higher reduced-order modes to retain more information of fluid field Sieber et al. (2016), we choose SPOD as the tool of order reduction in this paper. Indeed, facing the complex dynamic system with huge amount of data, SPOD can effectively extract the mode of high energy ratio and compress the data dimension, which is extremely suitable for dealing with wind turbine wake. DNN is chosen because of its reliable performance in dealing with regression tasks. In this study, a series of wind turbine wake simulations are carried out under different inflow conditions and yawing conditions, and a high-fidelity LES database is generated. The ability of classical POD and SPOD in reconstructing wind turbine wake data is compared, and the superiority of SPOD in this respect is confirmed. After that, two cases of single wind turbine wake prediction and wind farm wake prediction were carried out.

The main contributions of this article are as follows:

Based on the high-fidelity dataset of wind turbine we computed, a new data-driven wake model is established. SPOD method is used to deal with the complex wake of wind turbines, which can effectively extract high-energy modes. DNN method is used to learn the transient characteristics of wake, which can provide accurate and stable long-term prediction.

Compared with the POD-DNN model, SPOD-DNN model can predict more accurately and stably by redistributing the mode energy to reduce the prediction targets.

Wake on untrained single turbine and six turbines is successfully real-time predicted with the presented model The results are also validated by comparing with high-fidelity SOWFA simulation. This fully show that the established model can accurately and steadily realize the real-time prediction of wind farm wake.

SPOD-DNN based surrogate modeling method

Based on finite-volume method of CFD, this work employs LES method to compute wind turbine wake under a series of yaw conditions, yielding the dataset. Detailed computing environment and computing parameters including regional geometry and boundary conditions will be given in Section 3. This section introduces a modeling method for approximate prediction of flow field data. The whole model is divided into two parts: offline prediction and online prediction. Figure 1 shows the flowchart of SPOD-DNN method, including data generation, DNN regression, and flow field forecast.

Figure 1.

The flowchart illustrating the proposed SPOD-DNN based surrogate modeling method.

Training dataset

The flow parameters $d_{i} = [{d^{i}}_{y}, {d^{i}}_{p}, {d^{i}}_{u}]$ , such as wind speed, turbulence, yaw angle, and the reduction coefficients of corresponding flow field $a^{i}$ , form a set of samples to train the surrogate model. For a set of design input variables $[d^{1}, d^{2}, . . ., d^{n}]$ , where $n$ represents the sample size. As shown in Figure 1, the flow field simulation on each group of inflow parameters $d^{i}$ is carried out to obtain the corresponding data $U^{i}$ , and then the data $U^{i}$ is processed with reduced-order method to obtain reduction coefficient $a^{i}$ , and the spanwise direction is $b^{i}$ . After the high-fidelity flow field data is generated, all inflow parameters are collected and reshaped as a input matrix $X$ of size $[N, N_{s}]$ , where $N_{s}$ represents the number of inflow parameter. Then all flow field data were applied to the reduced-order model, and all reduction coefficients are colloceted as a target matrix $Y$ of size $[N, N_{k}]$ , where $N_{k}$ represents the size of the reduction coefficients.

Data reduction

In this step, we need to reduce the collected high-fidelity flow field data to obtain the reduced coefficients and the corresponding forward and inverse transforms. The correlation matrix of POD is obtained by snapshot set Volkwein (2011)

R = X X^{T},

(1)

where $X$ represents $m$ flow filed snapshots form of $X = [x_{1}, x_{2}, \dots, x_{m}]$ . As the wake itself is complex, it is difficult to extract enough energy modes. Therefore, we introduce SPOD. SPOD is essentially a filter applied to the correlation matrix $R$ . It can filter the temporal pattern and redistribute the system energy to a higher pattern. Autocorrelation coefficients can be obtained from the correlation matrix $R$ , which represent the spectral contents of different time scales and wavelengths. Hence, the smoother modes can be obtained by strengthening the diagonal similarity of correlation matrix. Correlation matrix $\overset{\cdot}{R}$ is given as

{\overset{\cdot}{R}}_{i, j} = \sum_{k = - N_{f}}^{N_{f}} f_{k} R_{i + k, j + k} .

(2)

Here, $f_{k}$ is the filter function, and $N_{f}$ is the size of the filter window. According to the spectral prop orthogonal decomposition of Ribeiro and Wolf (2017). In this paper, the size of the filter window is 25%. When the window size is 20%–40%, it has a good noise filtering effect. If the window is too large, the stream information will be lost, while if it is less than 20%, more noise will appear.

f_{k} = 1 / (2 N_{f} + 1) .

(3)

After that, we can find the optimal basis vector by determining the eigenvalues and eigenvectors

R v_{k} = λ_{k} v_{k} .

(4)

Arrange the eigenvalue from the largest to the smallest $(λ_{1} \geq λ_{2} \geq . . . \geq λ_{m} \geq 0)$ , and the reduction basis $v_{k}$ corresponds to the sequences. Normally, we only retain $r$ modes to represent the flow field,

\sum_{i = 1}^{r} λ_{i} / \sum_{i = 1}^{s} λ_{i} \approx 1 .

(5)

The original flow field can be approximately represented as

\tilde{U} (p_{i}) \approx \sum_{k = 1}^{r} a_{k} (p_{i}) v_{k},

(6)

where the reduction coefficients $a_{k}$ are calculated by $a_{k} = 〈 xt, vk 〉$ and 〈, 〉 represents the inner product. The snapshot data and the reduction coefficients are time-related, but they are not shown to be temporal-dependent in equations for the convenience of description and representation. With these operations, we get the required training output matrix $Y$ , and the mapping relationship between flow parameters and the target. The offline prediction is completed.

Regression via DNN

After obtaining the training dataset, deep neural network (DNN) is used as the regression model in this work. As shown in the Figure 2, the correspondence between input and output can be expressed as

y_{i + 1} = σ (\sum w_{i} x_{i} + b),

(7)

where $σ$ represents the activation function, $x_{i}$ represents the neuron value of the ith layer, $w_{i}$ represents the weights of the ith layer, $b$ represents the biase. $x_{n}$ , $y_{n}$ , and $h_{n}$ in Figure 2 represent the input dimension, the output dimension, and the number of hidden layers. The process of offline training and online prediction are given in algorithm 1.

Figure 2.

A deep feedforward network with N inputs, several hidden layers, and one output layer with N outputs.

Algorithm 1:

Model generation process

1: % Offline training.
2: Generate

n

samples of the flow parameters [

d^{1}

d^{2}

,...,

d^{n}

].
3: Run CFD simulations to generate the corresponding flow field data [

U^{1}

U^{2}

,...,

U^{n}

].
4: Use POD/SPOD to obtain the reduced coefficients [

U^{1}

U^{2}

,...,

U^{n}

] and

v_{k}

.
5: Train the regression model using the training input [

d^{1}

d^{2}

,...,

d^{n}

] and the training target [

U^{1}

U^{2}

,...,

U^{n}

].
6: % Online prediction.
7: Set the input parameter of test

μ^{test}

.
8: Predict the reduced coefficients

U^{test}

.
9: Reconstructed flow field

U^{test}

The parameter update of DNN can be divided into two parts. The first step is forward propagation, which uses several weight coefficient matrices $W$ and offset vectors $b$ to perform a series of linear operations and activation operations with the input value vector $X$ . The second step is backward propagation. After the forward propagation computation is completed, the network is optimized by using the loss function depending on the result and the true value error, the $λ$ regularization term is further added to solve the over-fitting problem, and finally the required weight coefficient matrix $W$ and offset vector $b$ are obtained. Root mean square error is used as loss function. In algorithm 2, forward propagation and the training process of random gradient descent using back propagation algorithm is given.

Algorithm 2:

Deep feedforward network (DNN).

Input: Training data

X

, Target

Y

, network depth

L

, activation function

σ

, number of hidden units

n_{l}

for layer

l

, learning rate

α

, regularization parameter

λ

, maximum number of iterations

n_{i}

, exponential decay rate for the first moment estimates

β_{1}

, exponential decay rate for the second moment estimates

β_{2}

, and small constant for numerical stability

ε

.
Output:

W

b

Y

.
Random initialization:

W

b

.
Initialization of Adam parameters:

V_{w}^{l}

V_{b}^{l}

S_{w}^{l}

S_{b}^{l}

.
repeat
Forward propagation

a^{1}

x

for

l = 2 to L

a^{l}

σ

(

z^{l}

σ

(

W^{l}

a^{l - 1}

b^{l}

)
end
Backward propagation
for

l = L to 2 do

diag(

f_{l'}

(

z^{l}

))=

\frac{\partial a^{l}}{\partial z^{l}}

δ^{l - 1}

= diag(

f_{l'}

(

z^{l}

))

W^{l}

δ^{l}

W^{l}

←

W^{l}

−

α

(

δ^{l}

(

a^{l - 1})^{T})

b^{l}

←

b^{l}

−

α

δ^{l}

end
Adam optimization
for

l = 2 to L

V_{w}^{l}

β_{1}

S_{w}^{l}

+ (1-

β_{1}

)(

δ^{l}

(

a^{l - 1})^{T})

V_{b}^{l}

β_{1}

S_{b}^{l}

+ (1-

β_{1}

)

δ^{l}

S_{w}^{l}

β_{2}

S_{w}^{l}

+ (1-

β_{2}

)(

δ^{l}

((

a^{l - 1})^{T})^{2})

S_{b}^{l}

β_{2}

S_{b}^{l}

+ (1-

β_{2}

)(

δ^{l})^{2}

W^{l}

W^{l}

−

α

\frac{V_{w}^{l}}{\sqrt{S_{w}^{l}} + ε}

;

b^{l}

b^{l}

−

α

\frac{V_{b}^{l}}{\sqrt{S_{b}^{l}} + ε}

end
untilThe error rate on the verification set is no longer reduced

The training goal is to minimize the root mean square error (RMSE) between the predicted result and the simulated target. The neural network is trained using the Adam optimization algorithm in this paper Kingma and Ba (2014), which is based on PyTorch.

Activation function

In a neural network, the information computation among the layers in the network is linear, and the data probably contains linear and nonlinear relationships. Adding additional nonlinear activation functions can make the network capture more nonlinear information and strengthen the learning ability of the network. Commonly used activation functions are sigmoid, tanh, ReLU, etc. As an unsaturated activation function, ELU will not encounter the problem of gradient explosion or disappearance. In fact, compared with other activation functions (such as ReLU and variants, Sigmoid and hyperbolic tangent), the application of ELU as activation function has higher accuracy. The activation functions ELU is used in SPOD-DNN and the formulas are as follows.

ELU : g (x) = {\begin{matrix} x, x > 0 \\ α (e^{x} - 1), x \leq 0 \end{matrix}

(8)

Application in wind turbine wake prediction

This section develops a data-driven wind farm wake model by adopting the method introduced in Section 2. The generation of wind turbine wake data is introduced in Section 3.1. In section 3.2, the training and verification of SPOD-DNN model are described in detail, and the comparison with POD-DNN method is made. Finally, two cases are studied to verify the application of SPOD-DNN model in wake prediction of untrained wind turbines and wind farm in Section 3.3.

High-fidelity data generation

The high-fidelity flow field data are generated by SOWFA (Simulator for Wind Farm Applications), which is a numerical solver based on OpenFOAM developed by the National Renewable Energy Laboratory (NREL) of the United States Churchfield and Lee (2012). SOWFA has been sufficiently verified in previous researches, such as turbine dynamics studies Churchfield et al. (2012) and wind farm control Fleming et al. (2013).The NREL 5MW wind turbine is adopted in this paper, and the diameter of rotor is 126 m. Then a series of simulation with three wind turbines are carried out to capture the wake of downstream turbines. The sketch of flow domain is illustrated in Figure 3. The interval between two turbines is about five times of rotor diameter. For the grid, the whole field adopts the grid size of $10 m \times 10 m \times 10 m$ , while the two-stage mesh refinement is used around the wind turbine, and the grid size reaches $2.5 m \times 2.5 m \times 2.5 m$ . More details about the simulation can be found in Fleming et al. (2015). The total number of grids is about $2.7 \times 10^{7}$ . We have verified the validity of the grid. Please see the appendix for details.

Figure 3.

Sketch of the computational domain.

For the generated wake data to cover a wider range, three inflow velocity, the average free-stream wind velocity of $6 m / s$ , $8 m / s$ , and $10 m / s$ , are considered, and the hub-height turbulent intensity is 5%, respectively.Then yaw angles are randomly sampled from $[- 30, 30]$ . Finally, 90 simulations are carried out and 270 training samples are generated. For each case, $0.05 s$ time step is taken for $1000 s$ simulation. Then the wake velocity surface at the height of hub is extracted. There are 38,400 spatial points on each data surface. Besides, the flow field around the rotor is sampled and extracted the flow field data every $0.05 s$ , including the flow direction and the spanwise direction. The rotor rotates approximately $2.7$ between each snapshot. For a description of the time step, please refer to the appendix. Simulation data for the first $600 s$ is discarded because wind turbine wake has not been well established during this period. In particular, the data of three wind turbines are sampled separately for training and verification. The entire data generation process takes about $4.8 \times 10^{5}$ CPU hours, on a local cluster with $224$ CPU cores. The surrogate model established aims to effectively predict the two-dimensional average velocity field of turbine hub height in wind farm under the given inflow conditions and yaw angles of each turbine.

Model training and validation

The generated LES data contains 270 flow scenarios, and is divided into 210 training scenarios and 60 test scenarios for training and validation of which 30 were verification examples and 30 were test examples, and cross-verification was carried out. The results of error verification are shown in Table 1. The training data is fed into the neural network by a batch processing program. Specifically, the test data set is only utilized to verify the results after training in order to ensure no data leakage.

Table 1.

The optimal hyper-parameters of the two surrogate modeling methods.

Reduction method	$N_{r}$	$N_{h}$	$ε_{ROM}$	$ε_{total}$
SPOD(u)	20	50	0.0354	0.106
SPOD(v)	20	50	0.0279	0.042
POD(u)	50	50	0.0462	0.132
POD(v)	50	50	0.0294	0.047

The performance of DNN described in algorithm 1 depends to a great extent on the choice of hyperparameter, such as network depth, the number of hidden units in each layer, regularization parameters, and learning rate $α$ . To reduce the computation cost of parameter search, Adam optimization is applied Kingma and Ba (2014). The manual search is used for searching learning rate $α$ and the number of iterations $L$ Kelleher (2019). And the four-fold cross-validation technique is used to regulate the remaining hyperparameters. The final hyperparameters are shown in the algorithm 2.

The coefficient error of the model is defined as the RMSE between the prediction coefficients and the true coefficients:

ε_{ROM} = \frac{1}{N_{test} \times N_{r}} \sum_{i = 1}^{N_{test}} \sum_{j = 1}^{N_{r}} RMSE (a, \hat{a}) .

(9)

And the model error is defined as the average of the RMSE between the predicted flow field and the true flow field:

ε_{total} = \frac{1}{N_{test} \times N_{r}} \sum_{i = 1}^{N_{test}} \sum_{j = 1}^{N_{r}} RMSE (u, \hat{u}) .

(10)

After training, the predicted values are compared with the test data that is supposed to be unavailable during the training to evaluate its performance. Four typical test cases are shown in Figure 4. The flow conditions for these test studies are: Case1: average velocity = 6m/s, yaw = $- 14.6$ , Case2: average velocity = 6m/s, yaw = $12.8$ , Case3: average velocity = 6m/s, yaw = $7.9$ , and Case4: average velocity = 6m/s, yaw = $3.4$ . It can be seen that the flow field predictions in all cases are very consistent with the corresponding LES results, including flow velocity $u$ and spanwise velocity $v$ .

Figure 4.

The comparisons between the predictions by the developed SPOD-DNN model and the corresponding true values for four randomly selected test cases.

As shown in Figure 4, the main characteristics of wind turbine wake are accurately captured, for example, the wake deflection with change of yaw angle, the wake gradual recovery in the flow direction, the wake expansion in the spanwise direction, and the blocking effect of upstream wake. The root mean square error for the flow and spanwise direction velocity predictions is $0.106 m / s$ and $0.042 m / s$ . In Table 1, where $N_{r}$ represents the number of mode, $N_{h}$ represents the number of hidden layer, $ε_{ROM}$ represents the prediction coefficients and the true coefficients, $ε_{total}$ repersents the RMSE of the predicted flow field and the true flow field. As shown, we can clearly see the advantages of SPOD in dealing with multi-modal and multi-frequency fluid systems. After redistribution of energy, SPOD can achieve better performance with fewer modes. The problems of over-fitting and gradient explosion/disappearance can be effectively reduced.

To further examine the performance of surrogate model, the wind speed profiles at different positions are extracted. The results are given in Figure 5, including the predicted and the true values. Obviously, the wake flowing through the wind turbine can be observed from a “flat” profile to a “double-peak” profile and then gradually recover. When flow passing through the downstream wind turbine, it can be clearly observed that the wake turbulence increases and the recovery speed slows down. Moreover, the spanwise velocity fluctuation caused by wind turbine yaw can be observed. The predicted profiles at all locations match very well with the corresponding high-fidelity simulation results.

Figure 5.

The velocity profiles predicted by SPOD-DNN model for four randomly-selected test cases. The corresponding true values are also shown for comparisons: (a) Case1(u), (b) Case1(v), (c) Case2(u), (d) Case2(v), (e) Case3(u), (f) Case3(v), (g) Case4(u), and (h) Case4(v).

Two case study

Single turbine wake prediction

For proving the generalization performance of the model, we make a wake prediction of single wind turbine which is not in the training set. The average wind speed of the incoming flow is $12 m / s$ , and a series of yaw angles are considered. As shown in Figure 6, each column shows the wake of wind turbine at different yaw angles with same inflow wind profile, that is, $[- 30 °, - 15 °, 0 °, 15 °, 30 °]$ . It can be seen that with the change of yaw angle, the deflection of wake is successfully captured. Therefore, the prediction of the $12 m / s$ cases demonstrate that SPOD-DNN model can capture flow characteristics (such as yaw effect) to a great extent even in untrained flow scenarios.

Figure 6.

The flow fields predicted by the developed SPOD-DNN model at a series of turbine yaw angles and wind velocity 12 m/s.

Wind farm wake prediction

The purpose of this model is to further predict the wake of wind farm. Here, we take six 5MW wind turbines as an example for wind farm wake prediction. The high-fidelity simulation is still generated by SOWFA to ensure the reliability of the experiment. Sketch of the wind farm domain is shown in Figure 7. The grid scale of this case is $3 m \times 3 m \times 3 m$ , and the total number of meshes is $3.6 \times 10^{7}$ .

Figure 7.

Sketch of the wind farm domain.

Free wind condition with an average wind speed of $8 m / s$ is used. We mention that the data used to train SPOD-DNN includes yaw angle range of $[- 30 °, 30 °]$ , and in this case the yaw conditions are much more than this range. The yaw angles of the six wind turbines are respectively $[23.6 °, 2.6 °, - 36.2 °]$ and $[16.7 °, 5.4 °, 21.6 °]$ . Here, wind farm wake is realized by combining the prediction data of single wind turbine wake Zhang and Xiaowei (2021). As shown in Figure 8, the wake data predicted by SPOD-DNN is in good agreement with SOWFA results. The flow details, such as the blocking effect of turbine on upstream fluid flow and the wake bending caused by yaw, are effectively predicted.

Figure 8.

The prediction results by the developed SPOD-DNN model and SOWFA for the wind farm wake case.

In addition, we also conducted an example of six wind turbines with a longitudinal spacing of double diameter. The grid scale of this case and the total number of meshes are as the previous exampleis. Free wind condition with an average wind speed of $8 m / s$ is also used, and in this case the yaw angles of the six wind turbines are respectively $[- 7.7 °, - 28.6 °, 8.4 °]$ and $[26.4 °, - 9.4 °, - 3.2 °]$ . As shown in Figure 9, it can be seen that most of the flow details can be captured. But when the wind turbine in the lateral upstream has an impact on the downstream wind turbine, this impact is difficult to capture, as shown by the red rectangle in the Figure 9. This phenomenon shows that the developed model has certain limitations. Secondly, we think that this limitation is caused by the lack of corresponding training, but we have no better training means and methods for the time being, hoping to solve this problem in the future research.

Figure 9.

The prediction results by the developed SPOD-DNN model and SOWFA for the wind farm wake case under different longitudinal distances.

Moreover, we can observe that when the downstream wind turbine runs in full wake, the wind energy capture will be greatly reduced. If the downstream wind turbine takes timely avoidance measures in the wake change of the upstream wind turbine, the output and power generation efficiency of the wind farm can be effectively increased.

Conclusions

Based on SPOD and DNN, a new wind farm wake model is developed to predict the wake of wind turbines under different inflow conditions. Facing the complicated turbine wake dynamics, it is advantageous to use SPOD to create a stable and accurate ROM. Therefore, the correlation matrix (SPOD method) is applied with a filtering function to remove the high-frequency content existing in classic POD.

The established SPOD-DNN model can well predict the main wake characteristics, including the wake deflection under the change of yaw angle, the wake recovery and expansion. Compared with the POD-DNN model, the performance of SPOD-DNN model with prediction error of only $0.106 m / s$ and $0.042 m / s$ is obviously improved.

The case study of the untrained wind turbine with yaw angle and free-flowing wind speed shows that the SPOD-DNN model can well generalize the untrained flow scene, which proves the robustness of the current ROM constructed by DNN. Meanwhile, case study of a small wind farm show that the established model has good consistency with LES model, and can well capture the wake dynamics (including flow direction and spanwise velocity field). And the developed model still has room for improvement to capture the influence of lateral wake. Generally speaking, the established model successfully predicted the unsteady characteristics of turbine wake, which demonstrated the ability of the proposed method in modeling wind farm wake.

One direction for future work is to apply this new wake model to wake control, exploring some yaw control strategies to reduce the load on downstream wind turbines and maximize wind energy capture efficiency. The other direction is the prediction of the real-time power of the wind farm as a whole based on the wake prediction.

Footnotes

Appendix

In this part, we introduce the verification of the actuation line model. According to overview of the simulator for wind fram application, the computation results of the actuating line model will be greatly influenced by the scale of grid nodes $N$ distributed on the diameter of the wind turbine rotor and the Gaussian width $ε$ . However, the scale of grid nodes and the value of Gaussian width have not come up with a definitive guideline, so in order to improve the computation effect and accuracy, in this paper, we studies the scale of grid nodes and the value of Gaussian width on the diameter of wind turbine. The specific research idea is to simulate the combination of different grid nodes and different Gaussian width by means of constant wind velocity, and the final result is determined by the relative error of wind turbine output power $τ$ and the time consumed in computation $ε_{T}$ . The formula for computing the relative error of power is $τ$ = $(P - P_{0}) / P_{0}$ , where $P$ is the output power result obtained by simulation and $P_{0}$ is the output power data provided by NREL. The size of the calculation domain of the example is set to $1600 m \times 600 m \times 400 m$ , the calculation time is set to $500$ seconds, and the calculation time step is set to $0.05 s$ . The set working condition parameters and simulation results are shown in the Table A1.

It can be seen from the table that the relative error of power is negative when $ε$ is 1.5, and positive when $ε$ is 2, so it is preliminarily determined that $ε$ is between 1.5 and 2. Taking the consumption of computing resources as the evaluation standard, because the computing time is approximately proportional to the number of grids. When the scale of grid nodes $N$ distributed on the diameter of the wind turbine rotor is 2, 3, and 4 m, the calculation time is 16.72, 12.64, and 6.23 hours. Considering that the computation area at this time is only $1600 m \times 600 m \times 400 m$ and the computation time is $500 s$ , the computation area will be larger and the computation time will be longer in the computation related to the research content. Therefore, from the point of view of computation amount and computation resources, $N$ is determined, taking 2.5 m; As shown in Figure A1, the wake of tip vortex and root vortex can be clearly displayed.

The main content of this part is to test the convergence of the model reduction method. The data set to be tested consists of M = [1500; 1700; 1900; 2100; 2300; 2500] snapshot. The convergence of the first 18 modes is tested, as shown in Figure A2. It can be observed that when the number of snapshots is 1500, the first five pairs of modes have converged, and other modes need more snapshots to approximate well. When the number of snapshots reaches 2300, the change of smaller energy modes is already very small.

Acknowledgements

The author would like to thank Fengfeng Zhao, Jing Cao, and Boyang Song for the fruitful discussions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work has received funding from the National Natural Science Foundation of China No. 11502141. It was also supported by the Scientific Computing Research Technology Platform (SCRTP) at the Shanghai University of Electric Power.

ORCID iD

Li Xu

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