Updated thesis and added quantile performance plots to non autoregressive quantiles
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Example of the diffusion process. The image of a cat is generated by starting from noise and iteratively denoising the image.\relax }}{12}{figure.caption.6}\protected@file@percent }
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\newlabel{fig:diffusion_example}{{4}{12}{Example of the diffusion process. The image of a cat is generated by starting from noise and iteratively denoising the image.\relax }{figure.caption.6}{}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.4.3}Generation process}{12}{subsubsection.3.4.3}\protected@file@percent }
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\newlabel{fig:diffusion_process}{{\caption@xref {fig:diffusion_process}{ on input line 279}}{14}{Generation process}{figure.caption.7}{}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {3.5}Evaluation}{14}{subsection.3.5}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Visualization of the CRPS metric\relax }}{15}{figure.caption.8}\protected@file@percent }
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\newlabel{fig:crps_visualization}{{5}{15}{Visualization of the CRPS metric\relax }{figure.caption.8}{}}
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@@ -159,6 +159,8 @@ The quantile regression method can be used with both types of models. The autore
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The non-autoregressive model will output the quantiles for all quarters of the day based on the input features. The cumulative distribution functions all need to be reconstructed and samples can be drawn from each of the distributions. When sampling from the distributions at once, the samples are independent of each other. The sample for the next quarter does not depend on the sample of the previous quarter which can result in some unrealistic samples.
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The input features for autoregressive and non-autoregressive also differ. When forecasted features are used, the autoregressive model only uses the forecasted values for the next quarter while the non-autoregressive model uses the forecasted values for all quarters of the day. In theory, the autoregressive model should also be able to use forecasted values for quarters further in the future but this makes it harder to use in practice. When the last quarter of a day needs to be predicted, the forecasted values for the next day are needed which are not available. For simplicity, during this thesis, the autoregressive model will only be provided with the forecasted values for the next quarter.
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\subsection{Model Types}
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\subsubsection{Linear Model}
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A simple linear model can be used as a baseline to compare the more complex models. This model assumes a linear relation exists between the input features and the output. The relationship is modeled using the following formula:
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@@ -5,6 +5,8 @@ The electricity market is a complex system influenced by numerous factors. The r
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Forecasting the imbalance price is vital for market participants engaged in buying or selling electricity. It enables them to make informed decisions on the optimal times to buy or sell, aiming to maximize their profits. However, current industry practices often rely on simplistic policies, such as adhering to a fixed price for transactions. This approach is not optimal and overlooks the potential benefits of adaptive policies that consider the forecasted imbalance prices.
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The goal of this thesis is to generatively model the Belgian electricity market. This allows to reconstruct the imbalance price for a given day which can then be used by other simple policies to make decisions on when to buy or sell electricity. These policies can then be compared to the current industry practices to assess their performance.
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The goal of this thesis is to generatively model the Belgian electricity market. This allows the reconstruction of the imbalance price for a given day which can then be used by other simple policies to make decisions on when to buy or sell electricity. These policies can then be compared to the current industry practices to assess their performance.
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Forecasting the system imbalance will become increasingly important as the share of renewable energy sources continues to grow.
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Forecasting the system imbalance will become increasingly important as the share of renewable energy sources continues to grow.
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This thesis can be divided into two main parts. The first part focuses on modeling the Net Regulation Volume (NRV) of the Belgian electricity market for the next day. This modeling is conditioned on multiple inputs that can be obtained from Elia (TODO: add citation to the open data of Elia). The second part of the thesis focuses on optimizing a simple policy using the NRV generations for the next day. The policy tries to maximize profit by charging and discharging a battery and thereby buying and selling electricity on the market. Multiple models are trained and tested to model the NRV and compared to each other based on their profit optimization.
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@@ -41,33 +41,43 @@ TODO: ask Jonas: add urls to the correct data? via citation?
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A lot of data is available but only the most relevant data needs to be used. Experiments will be done to identify which data and features improve the NRV modeling. The data will be split into a training and test set. The training dataset starts depending on which data features are used but ends on 31-12-2022. The test set starts on 01-01-2023 and ends on (TODO: check the end date). This makes sure enough data is available to train the models and the test set is large enough to evaluate the models. The year 2023 is chosen as the test set because it is the most recent data available when the thesis experiments were conducted. Using data from 2022 in the test set also does not make a lot of sense because the trained models would be used to predict the future. Data from 2022 is not relevant anymore to evaluate the models.
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\subsubsection{Linear Model}
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% TODO: is it necessary to provide the parameter calculation?
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The linear model outputs the values for the chosen quantiles. The total amount of parameters depends on the input features and the number of chosen quantiles. Assuming the input features are the 96 previous NRV values and 13 quantiles are chosen, the total amount of parameters is $96 * 13 + 13 = 1261$. The linear model is trained using the Adam optimizer with a learning rate of 1e-4. Early stopping is used with a patience of 5 epochs. Different sets of input features are experimented with and are compared to each other based on the previously mentioned metrics. All results are shown in Table \ref{tab:autoregressive_linear_model_baseline_results}.
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\\\\
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% TODO: ask Jonas: add number of parameters to this table?
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\begin{table}[ht]
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\centering
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\begin{tabular}{@{}lcccccc@{}}
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\toprule
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& \multicolumn{2}{c}{MSE} & \multicolumn{2}{c}{MAE} & \multicolumn{2}{c}{CRPS} \\
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\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7}
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& Train & Test & Train & Test & Train & Test \\
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\midrule
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NRV & 35307.34 & 39222.41 & 146.38 & 152.49 & 88.46 & 91.56 \\
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NRV + Load & 34274.70 & 39266.29 & 144.20 & 152.54 & 85.84 & 90.36 \\
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NRV + PV & 33099.80 & 37489.68 & 141.57 & 149.32 & 84.87 & 89.42 \\
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NRV + Load + PV & 32777.86 & 37642.66 & 141.01 & 149.90 & 83.40 & 89.34 \\
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NRV + Load + Wind & 33009.05 & 39284.68 & 141.20 & 152.32 & 82.46 & 88.60 \\
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NRV + Load + PV + Wind & 29816.77 & 36134.87 & 133.89 & 146.22 & 78.19 & 84.56 \\
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NRV + Load + Wind + Net Position & 31634.27 & 37890.66 & 137.87 & 149.37 & 81.17 & 86.19 \\
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NRV + Load + PV + Wind + Net Position & 29034.53 & \textbf{35725.42} & 131.87 & \textbf{145.64} & 76.23 & \textbf{83.30} \\
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\bottomrule
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\end{tabular}
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\caption{Autoregressive linear model results}
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\label{tab:autoregressive_linear_model_baseline_results}
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\end{table}
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% TODO: explainedi nsection reference ?
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The simplest model to be trained for the NRV modeling is the linear model. The linear model is trained using the pinball loss function explained in the section above. The outputs of the model are values for the chosen quantiles. The linear model can be trained in an autoregressive and non-autoregressive way. Both methods will be compared to each other. The linear model is trained using the Adam optimizer with a learning rate of 1e-4. Early stopping is used with a patience of 5 epochs. The linear model is evaluated using the mean squared error (MSE), mean absolute error (MAE), and continuous ranked probability score (CRPS). The influence of the input features is also evaluated by training the models with different input feature sets.
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The linear model outputs the quantiles for the next quarter based on the given input features. The input features consist of previous history values of a certain feature or forecasts of a certain feature. The model, however, does not know which quarter of the day it is modeling. This is important information because as seen in Figure \ref{fig:nrv_mean_std_over_quarter}, the mean and standard deviation of the NRV values change over the day. The model should be able to capture this information given the quarter of the day.
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There is a big difference in the number of parameters between the autoregressive linear model and the non-autoregressive linear model. The autoregressive model only needs to output the NRV quantiles for one value while the non-autoregressive model needs to output the NRV quantiles for all the quarters of the day. Assuming thirteen quantiles are used, the autoregressive has 13 output parameters while the non-autoregressive model has 13 * 96 = 1248 output parameters. The total number of parameters for the autoregressive model is 13 * (number of input features + 1) while the total number of parameters for the non-autoregressive model is 13 * 96 * (number of input features + 1). Assuming only the NRV history of the previous day is used as input features, the autoregressive model has 1261 trainable parameters while the non-autoregressive model has 121056 parameters. This is a huge difference in the number of parameters and thus the complexity of the model.
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\begin{table}[ht]
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\centering
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\begin{tabular}{@{}lcccccc@{}}
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\toprule
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& \multicolumn{2}{c}{MSE} & \multicolumn{2}{c}{MAE} & \multicolumn{2}{c}{CRPS} \\
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\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7}
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& AR & NAR & AR & NAR & AR & NAR \\
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\midrule
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NRV & 39222.41 & 41219.98 & 152.49 & 152.26 & 91.56 & \textbf{73.97} \\
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NRV + Load & 39266.29 & 47045.17 & 152.54 & 163.24 & 90.36 & 79.72 \\
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% NRV + PV & 37489.68 & & 149.32 & & 89.42 & \\ No NAR experiment
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NRV + Load + PV & 37642.66 & 46404.63 & 149.90 & 161.82 & 89.34 & 79.74 \\
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NRV + Load + Wind & 39284.68 & 48148.10 & 152.32 & 164.84 & 88.60 & 79.51 \\
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NRV + Load + PV + Wind & 36134.87 & 50312.85 & 146.22 & 169.06 & 84.56 & 79.85 \\
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NRV + Load + Wind + NP & 37890.66 & 49442.48 & 149.37 & 167.90 & 86.19 & 76.72 \\
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NRV + Load + PV + Wind + NP & \textbf{35725.42} & 49132.26 & \textbf{145.64} & 167.37 & 83.30 & 78.75 \\
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\bottomrule
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\end{tabular}
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\caption{Linear model results}
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\label{tab:linear_model_baseline_results}
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\end{table}
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Comparing the results of the autoregressive and non-autoregressive linear models, it can be seen that the non-autoregressive model has a higher MSE and MAE on the test set. The CRPS is, however, lower for the non-autoregressive model. The CRPS is calculated using the outputted quantiles while the MSE and MAE are calculated by sampling from the reconstructed distributions. Because of error propagation in the autoregressive model, the outputted quantiles also contain more error which leads to a higher CRPS. The non-autoregressive model does not suffer from this problem. During the training of the autoregressive model, the model does not take into account that it will be used to generate full-day samples and thus the error is propagated. This is one possible explanation for the higher CRPS of the autoregressive model.
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The MSE and MAE of the non-autoregressive model are higher than the autoregressive model. This can be explained by the fact that the non-autoregressive model does not take into account the previous sampled value. Sampling is done for every quarter of the day independently. This can lead to large differences between the sampled values and thus can increase the MSE and MAE. The autoregressive model does take into account the previous sampled value and can adapt its quantile predictions based on this value so a smoother and more accurate sample can be generated.
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% TODO: Check listing of features -> hoofdletters en shit
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Another thing to note is the influence of the input features on the non-autoregressive linear model. When increasing the number of input features, the evaluation metrics are a lot worse in comparison with only using the NRV history of the previous day. A reason for this behavior could be that the model is not able to capture the patterns in the data because of the huge amount of input parameters. When using the NRV, Load, Photovoltanic power production, Wind power production, and the Net Position as input features, the non-autoregressive model has an input size of 864. This increases the complexity of the model as well. The total number of trainable parameters becomes 1,079,520. This is a huge number of parameters and the model is not able to learn the patterns in the data anymore.
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The performance of the autoregressive linear model, however, improves with the addition of more input features. When using the NRV, Load, Photovoltanic power production, Wind power production, and the Net Position as input features, the autoregressive model has an input size of 484. This is almost half the size of the non-autoregressive model. The total number of trainable parameters becomes 6,305 which is way less than the non-autoregressive model.
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An important thing to note is that the autoregressive model needs an additional feature to know which quarter of the day it is modeling. The quarter of the day also influences the value of the NRV. This can easily be seen in Figure \ref{fig:nrv_mean_std_over_quarter}. The figure shows the mean and standard deviation of the NRV values over the quarter of the day. These values change over the day which means the quarter is very valuable information for the model. The non-autoregressive on the other hand does not need this information because it models all the quarters of the day at once.
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\begin{figure}[ht]
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\centering
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@@ -76,136 +86,83 @@ The linear model outputs the quantiles for the next quarter based on the given i
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\label{fig:nrv_mean_std_over_quarter}
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\end{figure}
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Multiple methods exist to provide such information to the model. The quarter of the day can be provided as a one-hot encoded vector. The cyclic nature of the quarter would not be captured using a one-hot encoded vector. The vectors for quarter 0 and quarter 95 would be very different while they should be very close to each other. Other methods exist that do take the cyclic property of the quarter into account. Trigonometric functions can be used to provide the quarter of the day information. The quarter of the day can be mapped to a sine and cosine value which can be used as input features. The sine and cosine values are calculated as follows:
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Providing the autoregressive model with the quarter of the day can be done in multiple ways. The quarter of the day can be provided as a one-hot encoded vector. The cyclic nature of the quarter would not be captured using a one-hot encoded vector. The vectors for quarter 0 and quarter 95 would be very different while they should be very close to each other. Other methods exist that do take the cyclic property of the quarter into account. Trigonometric functions can be used to provide the quarter of the day information. The quarter of the day can be mapped to a sine and cosine value which can be used as input features. The sine and cosine values are calculated as follows:
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\begin{equation}
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\text{sin}(\frac{2\pi}{96} \times \text{quarter}) \quad \text{and} \quad \text{cos}(\frac{2\pi}{96} \times \text{quarter})
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\end{equation}
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The sine and cosine values are then concatenated with the input features. Another method that can be used is adding an embedding layer to the model. The discrete quarter of the day value can then be mapped to a vector. The embedding layer itself is learned during the training process which allows the model to learn patterns between quarters. The length of the embedding vector can be chosen and experimented with. The quarter-of-the-day information is then concatenated with the input features. Other information (eg. day of the week, month, year) can also easily be added to the model using this method by just increasing the size of the embedding layer. The results of the linear model with the quarter information are shown in Table \ref{tab:autoregressive_linear_model_quarter_embedding_baseline_results}. The results show that adding the quarter embedding to the model improves the performance of the linear model.
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\\\\
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The sine and cosine values are then concatenated with the input features. Another method that can be used is adding an embedding layer to the model. The discrete quarter of the day value can then be mapped to a vector. The embedding layer itself is learned during the training process which allows the model to learn patterns between quarters. The length of the embedding vector can be chosen and experimented with. The quarter-of-the-day information is then concatenated with the input features. Other information (eg. day of the week, month, year) can also easily be added to the model using this method by just increasing the size of the embedding layer. The results of the linear model with the quarter information are shown in Table \ref{tab:autoregressive_linear_model_quarter_embedding_baseline_results}.
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% TODO: Ask Jonas: Find cleaner way to present this table (remove repitition)
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% TODO: Add more time information like day of week, month
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\begin{table}[ht]
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\centering
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\begin{tabular}{@{}lcccccc@{}}
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\begin{tabular}{@{}lccc@{}}
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\toprule
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& \multicolumn{2}{c}{MSE} & \multicolumn{2}{c}{MAE} & \multicolumn{2}{c}{CRPS} \\
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\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7}
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& Train & Test & Train & Test & Train & Test \\
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& \multicolumn{1}{c}{MSE} & \multicolumn{1}{c}{MAE} & \multicolumn{1}{c}{CRPS} \\
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\midrule
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NRV & 35307.34 & 39222.41 & 146.38 & 152.49 & 88.46 & 91.56 \\
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NRV + Quarter Trigonometric & 35243.28 & 39069.96 & 146.18 & 152.06 & 87.84 & 90.90 \\
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NRV + Quarter Embedding \textbf{(2 dim)} & 34412.60 & \textbf{38216.27} & 144.35 & \textbf{150.41} & 86.59 & \textbf{89.69} \\
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NRV + Quarter Embedding \textbf{(5 dim)} & 34801.22 & 38617.17 & 145.22 & 151.20 & 86.64 & 89.72 \\
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NRV + Quarter Embedding \textbf{(8 dim)} & 34558.29 & 38423.30 & 144.73 & 150.89 & 86.59 & 89.81 \\
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NRV & 39222.41 & 152.49 & 91.56 \\
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NRV + QT & 39069.96 & 152.06 & 90.90 \\
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NRV + QE \textbf{(2 dim)} & \textbf{38216.27} & \textbf{150.41} & \textbf{89.69} \\
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NRV + QE \textbf{(5 dim)} & 38617.17 & 151.20 & 89.72 \\
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NRV + QE \textbf{(8 dim)} & 38423.30 & 150.89 & 89.81 \\
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\midrule
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NRV + Load + PV + Wind + Net Position & 29034.53 & 35725.42 & 131.87 & 145.64 & 76.23 & 83.30 \\
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NRV + Load + PV + Wind + Net Position \\ + Quarter Trigonometric & 28239.01 & 34783.13 & 130.30 & 143.98 & 76.59 & 84.21 \\
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NRV + Load + PV + Wind + Net Position \\ + Quarter Embedding \textbf{(2 dim)} & 28667.13 & 35746.01 & 131.20 & 146.01 & 77.28 & 85.54 \\
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NRV + Load + PV + Wind + Net Position \\ + Quarter Embedding \textbf{(5 dim)} & 27407.41 & \textbf{34031.71} & 128.31 & \textbf{142.29} & 72.06 & \textbf{79.99} \\
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NRV + Load + PV + Wind + NP & 35725.42 & 145.64 & 83.30 \\
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NRV + Load + PV + Wind + NP + QT & 34783.13 & 143.98 & 84.21 \\
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NRV + Load + PV + Wind + NP + QE \textbf{(2 dim)} & 35746.01 & 146.01 & 85.54 \\
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NRV + Load + PV + Wind + NP + QE \textbf{(5 dim)} & \textbf{34031.71} & \textbf{142.29} & \textbf{79.99} \\
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\bottomrule
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\end{tabular}
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\caption{Autoregressive linear model results with time features}
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\label{tab:autoregressive_linear_model_quarter_embedding_baseline_results}
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\end{table}
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Some examples of the sampled full NRV day samples are shown in figure \ref{fig:autoregressive_linear_model_samples}. The figure shows the real NRV values and the confidence intervals calculated based on 1000 full-day NRV samples. The mean of these samples is also plotted in the figure. The confidence intervals show the uncertainty of the NRV values. When the confidence interval is large, the model is not very certain about the NRV value and samples the NRV for that quarter with a high variance. The confidence intervals seen in the figure are quite narrow and do not always capture the real NRV value.
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% TODO: explain more what we see in the figure, we see the small peaks every hour. Talk about the problem of autoregression (error propagation)
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The results show that adding the quarter embedding to the model improves all evaluation metrics for the autoregressive linear model. The quarter embedding is a valuable feature for the model.
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[b]{0.49\textwidth}
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\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP_QE-Sample_864.png}
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\caption{Sample 1}
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\label{fig:autoregressive_linear_model_sample_1}
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\end{subfigure}
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\hfill
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\begin{subfigure}[b]{0.49\textwidth}
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\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP_QE-Sample_4320.png}
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\caption{Sample 2}
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\label{fig:autoregressive_linear_model_sample_2}
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\end{subfigure}
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\hfill
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\begin{subfigure}[b]{0.49\textwidth}
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\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP_QE-Sample_6336.png}
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\caption{Sample 3}
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\label{fig:autoregressive_linear_model_sample_3}
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\end{subfigure}
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\hfill
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\begin{subfigure}[b]{0.49\textwidth}
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\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP_QE-Sample_7008.png}
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\caption{Sample 4}
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\label{fig:autoregressive_linear_model_sample_4}
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\end{subfigure}
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\caption{Test examples of the autoregressive linear model. The plots show the confidence intervals calculated from 1000 generated full-day NRV samples. The samples were generated using input features NRV, Load, Wind, PV, Net Position and the quarter embedding.}
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\label{fig:autoregressive_linear_model_samples}
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\end{figure}
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Some examples of the generated full-day NRV samples are shown in Figure \ref{fig:autoregressive_linear_model_samples}. The examples are taken from the test set. The figure shows the confidence intervals of the NRV generations and the mean NRV prediction. The confidence intervals and mean are calculated based on 1000 generated full-day NRV samples. The samples were generated using the input features NRV, Load, Wind, PV, Net Position, and the quarter embedding for the autoregressive model.
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The linear model is a simple model and can be used as a baseline to compare with the more complex models. From the examples in figure \ref{fig:autoregressive_linear_model_samples}, it can be seen that the model is already modeling the NRV reasonably well. There is, however, room for improvement.
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\\\\
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% ----------- Non-autoregressive model -----------
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Until now, only the autoregressive linear model has been discussed. The non-autoregressive linear model can also be used to model the NRV and generate full-day samples. The model will now output the quantiles for every quarter of the day. The number of output values can be calculated as the number of quarters in a day multiplied by the number of quantiles. From this output, the cumulative distribution functions for every quarter of the day can be reconstructed. These functions can then be used to sample the NRV values for each quarter. There is a problem with this approach. The sampled NRV values are independent of each other. The NRV sample for the next quarter does not depend on what value was sampled for the quarter before.
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\\\\
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Training the non-autoregressive quantile model can be done in the same way as the autoregressive model. Now the pinball loss is calculated for every quarter of the day and the mean is taken over all the quarters. The models are also trained using the Adam optimizer with a learning rate of 1e-4. Early stopping is used with a patience of 5 epochs. Results of the non-autoregressive linear model are shown in Table \ref{tab:non_autoregressive_linear_model_baseline_results}.
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\begin{table}[ht]
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\centering
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\begin{tabular}{@{}lcccccc@{}}
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\toprule
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& \multicolumn{2}{c}{MSE} & \multicolumn{2}{c}{MAE} & \multicolumn{2}{c}{CRPS} \\
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\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7}
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& Train & Test & Train & Test & Train & Test \\
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\midrule
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NRV & 37690.02 & 41219.98 & 147.54 & 152.26 & 67.94 & 73.97 \\
|
||||
NRV + Load & 38461.93 & 47045.17 & 149.85 & 163.24 & 69.68 & 79.72 \\
|
||||
NRV + Load + PV & 37891.53 & 46404.63 & 148.95 & 161.82 & 69.02 & 79.74 \\
|
||||
NRV + Load + Wind & 38113.90 & 48148.10 & 148.76 & 164.84 & 68.21 & 79.51 \\
|
||||
NRV + Load + PV + Wind & 39072.94 & 50312.85 & 151.07 & 169.06 & 68.40 & 79.85 \\
|
||||
NRV + Load + Wind + Net Position & 39505.20 & 49442.48 & 151.65 & 167.90 & 68.69 & 76.72 \\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\caption{Non-Autoregressive linear model results}
|
||||
\label{tab:non_autoregressive_linear_model_baseline_results}
|
||||
\end{table}
|
||||
|
||||
In Figure \ref{fig:non_autoregressive_linear_model_samples}, some examples of the sampled full NRV day samples are shown. The confidence intervals are calculated based on 1000 full-day NRV samples. The mean of these samples is also plotted in the figure. The confidence intervals show the uncertainty of the NRV values. The same test set examples are used as the examples of the autoregressive linear model shown in Figure \ref{fig:autoregressive_linear_model_samples}.
|
||||
|
||||
\begin{figure}[ht]
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP_QE-Sample_864.png}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/naqr_linear_model_samples/NAQR_NRV_Load_Wind_PV_NP-Sample_864.png}
|
||||
\caption{Sample 1}
|
||||
\label{fig:non_autoregressive_linear_model_sample_1}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP_QE-Sample_4320.png}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/naqr_linear_model_samples/NAQR_NRV_Load_Wind_PV_NP-Sample_4320.png}
|
||||
\caption{Sample 2}
|
||||
\label{fig:non_autoregressive_linear_model_sample_2}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP_QE-Sample_6336.png}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/naqr_linear_model_samples/NAQR_NRV_Load_Wind_PV_NP-Sample_6336.png}
|
||||
\caption{Sample 3}
|
||||
\label{fig:non_autoregressive_linear_model_sample_3}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP_QE-Sample_7008.png}
|
||||
\caption{Autoregressive linear model}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/naqr_linear_model_samples/NAQR_NRV_Load_Wind_PV_NP-Sample_7008.png}
|
||||
\caption{Sample 4}
|
||||
\label{fig:non_autoregressive_linear_model_sample_4}
|
||||
\caption{Non-autoregressive linear model}
|
||||
\end{subfigure}
|
||||
\caption{Test examples of the non-autoregressive linear model. The plots show the confidence intervals calculated from 1000 generated full-day NRV samples. The samples were generated using input features NRV, Load, Wind, PV and the Net Position.}
|
||||
\label{fig:non_autoregressive_linear_model_samples}
|
||||
\end{figure}
|
||||
|
||||
\caption{Comparison of the autoregressive and non-autoregressive linear model samples.}
|
||||
\label{fig:linear_model_sample_comparison}
|
||||
\end{figure}
|
||||
|
||||
When looking at the examples in Figure \ref{fig:linear_model_sample_comparison}, it can be seen that the autoregressive linear model is already modeling the NRV quite well. The confidence intervals are quite small and the mean of the samples follows the trend of the real NRV. The mean of the samples, however, is way smoother than the real NRV. The real NRV has more peaks and fluctuations. The examples of the non-autoregressive model show another behavior. The confidence intervals are not as contained as the autoregressive model but fluctuates a lot more. A lot of peaks can be observed in the examples. The reason for this behavior is that the non-autoregressive model does not take into account the previous sampled value. The sampled value of the next quarter is not dependent on the sampled value of the previous quarter. This can lead to a large difference between these values which results in samples with a high variance. The mean of the samples of the non-autoregressive model, however, does not follow the trend of the real NRV as well as the autoregressive model. The mean stays in a narrow range around zero.
|
||||
|
||||
Some samples for the examples from the test set are visualized in Figure \ref{fig:linear_model_samples_comparison}. For the autoregressive model, the samples largely follow the trend of the real NRV while the non-autoregressive model has a lot of fluctuations and peaks. By visually looking at the samples themselves, the samples of the autoregressive model are more realistic than the samples of the non-autoregressive model. \\
|
||||
|
||||
Comparing the results from the autoregressive and non-autoregressive linear models, it is clear that the autoregressive model has lower MSE and MAE on the test set. The CRPS is, however, higher for the autoregressive model. The CRPS is calculated using the outputted quantiles while the MSE and MAE are calculated by sampling from the reconstructed distributions.
|
||||
% TODO: Is this reasoning correct? + Explain the results more we see in the table (Weird results :( )
|
||||
Because of error propagation in the autoregressive model, the outputted quantiles also contain more error which leads to a higher CRPS. The non-autoregressive model does not suffer from this problem. When comparing the examples from the autoregressive model in Figure \ref{fig:autoregressive_linear_model_samples} and the examples from the non-autoregressive model in Figure \ref{fig:non_autoregressive_linear_model_samples}, it can be seen that the non-autoregressive model has a higher variance in the samples. The confidence intervals are larger and have a lot more peaks. The autoregressive model examples are way more contained in the range of -500 to 500. Some full-day NRV samples are also compared in Figure \ref{fig:linear_model_sample_comparison}. The peaky behavior also appears in the samples from the non-autoregressive model. This is quite logical because the sampled value of the next quarter doesn't depend on the value sampled of the previous quarter. This explains why a lot of peaks can be observed. The autoregressive full-day NRV samples on the other hand, do depend on the previous sampled value. This leads to a smoother curve and fewer peaks.
|
||||
\\\\
|
||||
% figure with 2 models compared (left with 4 row examples and right with 4 row examples), autoregressive vs non-autoregressive
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
@@ -217,33 +174,33 @@ Because of error propagation in the autoregressive model, the outputted quantile
|
||||
\end{subfigure}
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP-QE-Example_4320_samples.png}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/naqr_linear_model_samples/NAQR_NRV_Load_Wind_PV_NP-Example_4320_samples.png}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP-QE-Example_6336_samples.png}
|
||||
\caption{Autoregressive linear model}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/naqr_linear_model_samples/NAQR_NRV_Load_Wind_PV_NP-Example_6336_samples.png}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/naqr_linear_model_samples/NAQR_NRV_Load_Wind_PV_NP-Example_4320_samples.png}
|
||||
\caption{Non-autoregressive linear model}
|
||||
\end{subfigure}
|
||||
% \begin{subfigure}[b]{0.49\textwidth}
|
||||
% \includegraphics[width=\textwidth]{images/quantile_regression/aqr_linear_model_samples/AQR_NRV_Load_Wind_PV_NP-QE-Example_7008_samples.png}
|
||||
%
|
||||
% \end{subfigure}
|
||||
% \hfill
|
||||
% \begin{subfigure}[b]{0.49\textwidth}
|
||||
% \includegraphics[width=\textwidth]{images/quantile_regression/naqr_linear_model_samples/NAQR_NRV_Load_Wind_PV_NP-Example_7008_samples.png}
|
||||
% \end{subfigure}
|
||||
|
||||
\caption{Comparison of the autoregressive and non-autoregressive linear models samples. The plots show ten samples of the full-day NRV for the autoregressive and non-autoregressive linear models. The samples were generated using input features NRV, Load, Wind, PV, and the Net Position. The autoregressive model also uses the quarter embedding with 5 dimensions.}
|
||||
\label{fig:linear_model_sample_comparison}
|
||||
\caption{Samples for two examples from the test set for the autoregressive and non-autoregressive linear model. The real NRV is shown in orange.}
|
||||
\label{fig:linear_model_samples_comparison}
|
||||
\end{figure}
|
||||
% TODO: Talk about the over/underestimation of the quantiles for the models. Plots have been made for this.
|
||||
Another way to evaluate the performance of the models is to look at the over/underestimation of the quantiles. For each day and every quarter in the test set, the quantiles are predicted by the model. Then for every quantile, it is checked how many times the real NRV is below the predicted quantile. For example, for the 10\% quantile, around 10\% of the real NRV values should be below the predicted quantile. This can be plotted for every quantile. These can be seen in Figure \ref{fig:linear_model_quantile_over_underestimation}. The plots show the over/underestimation of the quantiles for the autoregressive and non-autoregressive linear models.
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/quantile_performance/AQR_Quantile_Performance_Training.jpeg}
|
||||
\caption{Training set}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[b]{0.49\textwidth}
|
||||
\includegraphics[width=\textwidth]{images/quantile_regression/quantile_performance/AQR_Quantile_Performance_Test.jpeg}
|
||||
\caption{Test set}
|
||||
\end{subfigure}
|
||||
\caption{Over/underestimation of the quantiles for the autoregressive and non-autoregressive linear models.}
|
||||
\label{fig:linear_model_quantile_over_underestimation}
|
||||
\end{figure}
|
||||
|
||||
\subsubsection{Non-linear Model}
|
||||
In this context, a simple feedforward neural network is trained to predict the quantiles for the NRV. The quantiles are then used to reconstruct the cumulative distribution function (CDF) for the NRV of a quarter. Predictions for the NRV can then be sampled from this reconstructed CDF. The neural network is trained using the pinball loss function explained in section \ref{subsubsec:quantile_regression_training}.
|
||||
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|
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\abx@aux@cite{0}{dumas_deep_2022}
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@@ -18,13 +18,12 @@
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@@ -179,11 +179,7 @@
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\include{sections/literature_study}
|
||||
|
||||
% In introduction
|
||||
\section{TODO: Better title for this section}
|
||||
This thesis can be divided into two main parts. The first part focuses on modeling the Net Regulation Volume (NRV) of the Belgian electricity market for the next day. This modeling is conditioned on multiple inputs that can be obtained from Elia (TODO: add citation to the open data of Elia). The second part of the thesis focuses on optimizing a simple policy using the NRV generations for the next day. The policy tries to maximize profit by charging and discharging a battery and thereby buying and selling electricity on the market. Multiple models are trained and tested to model the NRV and compared to each other based on their profit optimization.
|
||||
|
||||
\input{sections/nrv_prediction}
|
||||
\input{sections/results}
|
||||
|
||||
|
||||
\section{Policies for battery optimization}
|
||||
|
||||
@@ -19,13 +19,12 @@
|
||||
\contentsline {section}{\numberline {5}Literature Study}{18}{section.5}%
|
||||
\contentsline {subsection}{\numberline {5.1}Electricity Price Forecasting}{18}{subsection.5.1}%
|
||||
\contentsline {subsection}{\numberline {5.2}Policies for Battery Optimization}{19}{subsection.5.2}%
|
||||
\contentsline {section}{\numberline {6}TODO: Better title for this section}{20}{section.6}%
|
||||
\contentsline {section}{\numberline {7}Results \& Discussion}{20}{section.7}%
|
||||
\contentsline {subsection}{\numberline {7.1}Data}{20}{subsection.7.1}%
|
||||
\contentsline {subsubsection}{\numberline {7.1.1}Linear Model}{22}{subsubsection.7.1.1}%
|
||||
\contentsline {subsubsection}{\numberline {7.1.2}Non-linear Model}{25}{subsubsection.7.1.2}%
|
||||
\contentsline {subsubsection}{\numberline {7.1.3}GRU Model}{30}{subsubsection.7.1.3}%
|
||||
\contentsline {subsection}{\numberline {7.2}Diffusion}{32}{subsection.7.2}%
|
||||
\contentsline {section}{\numberline {8}Policies for battery optimization}{32}{section.8}%
|
||||
\contentsline {subsection}{\numberline {8.1}Baselines}{32}{subsection.8.1}%
|
||||
\contentsline {subsection}{\numberline {8.2}Policies using NRV predictions}{32}{subsection.8.2}%
|
||||
\contentsline {section}{\numberline {6}Results \& Discussion}{20}{section.6}%
|
||||
\contentsline {subsection}{\numberline {6.1}Data}{20}{subsection.6.1}%
|
||||
\contentsline {subsubsection}{\numberline {6.1.1}Linear Model}{21}{subsubsection.6.1.1}%
|
||||
\contentsline {subsubsection}{\numberline {6.1.2}Non-linear Model}{26}{subsubsection.6.1.2}%
|
||||
\contentsline {subsubsection}{\numberline {6.1.3}GRU Model}{30}{subsubsection.6.1.3}%
|
||||
\contentsline {subsection}{\numberline {6.2}Diffusion}{32}{subsection.6.2}%
|
||||
\contentsline {section}{\numberline {7}Policies for battery optimization}{32}{section.7}%
|
||||
\contentsline {subsection}{\numberline {7.1}Baselines}{32}{subsection.7.1}%
|
||||
\contentsline {subsection}{\numberline {7.2}Policies using NRV predictions}{32}{subsection.7.2}%
|
||||
|
||||
Reference in New Issue
Block a user