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Reports/Thesis/sections/results.tex
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\input{sections/results/models/comparison}
\newpage
\section{Policies for battery optimization}
\subsection{Policies for battery optimization}
The goal of this thesis is to model the NRV data and use this to optimize the buying and selling of electricity to make a profit. Different models and methods can be used to model the NRV data which can all result in different results. To evaluate the performance of the models, the generated profit on the test set can be used as a metric. First of all, baselines are needed to be able to compare the models to if adding NRV predictions to the policies improves the profit. The baselines are already discussed in the background section. It is very important to compare the baselines and other policies fairly. The profit depends on the number of charge cycles that are used. The more charge cycles a policy uses, the more profit it will be able to make. Using too many charge cycles is bad for the health of the battery. A penalty parameter can be used to penalize the policy when too many charge cycles are used in a day. To fairly compare the policies with different models and baselines, a maximum number of charge cycles is determined for the test period. The test period starts on 01-01-2023 and ends on (TODO: check the end date). Assuming a maximum of 400 charge cycles can be used in a year, only 293 charge cycles can be used during the test period. The penalty parameter is optimized using a simple gradient descent approach to make sure only 293 charge cycles are used during the test period. The profit is then calculated using the optimized penalty parameter.
To evaluate the policies, a battery of 2 MWh is used with a maximum charge and discharge power of 1 MW. The battery is charged and discharged in quarter-hour intervals at the price of that quarter-hour.

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Reports/Thesis/sections/results/policies/baselines.tex
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\subsection{Baselines}
\subsubsection{Baselines}
As discussed earlier, the most simple baseline that can be used is choosing two fixed thresholds. One threshold is used to buy electricity and the other threshold is used to sell electricity. The thresholds can be chosen based on the available historical data. A simple grid search is done over multiple combinations of thresholds. For each combination, the penalty parameter is optimized to make sure only 400 charge cycles are used in a year. The thresholds that achieve the highest profit on the training set are then used to evaluate the policy on the test set. The thresholds found that maximize the profit on the training set are 100€/MWh for buying and 200€/MWh for selling. Evaluating these thresholds on the test set results in a profit of €266,294.15 but using 492.0 charge cycles. This can not be used to fairly compare other policies. The thresholds can also be determined on the test set itself to find what the maximum possible profit is for the fixed thresholds policy. This can not be used in practice because the thresholds are determined on future data. It is, however, useful for the comparison of the other policies. The best thresholds found on the test set are 200€/MWh for buying and 250€/MWh for selling. The profit achieved using these thresholds is €143,004.34 with 287.12 charge cycles.
Another baseline that can be used is using the real \ac{NRV} data of the previous day to determine the buying and selling thresholds for the next day. The \ac{NRV} data of the previous day can be seen as a prediction for the \ac{NRV} data of the next day. Now, the thresholds are not fixed anymore but are determined based on the \ac{NRV} predictions. The thresholds are determined separately for each day using a simple grid search which allows the policy to adapt to the changing \ac{NRV} data. The penalty parameter is also optimized to make sure only around 283 charge cycles are used. The profit achieved using this policy is €198,807.09 using 283.5 charge cycles.

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Reports/Thesis/sections/results/policies/nrv_samples_policy.tex
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\subsection{Policy using generated NRV samples}
\subsubsection{Policy using generated NRV samples}
The generated full-day samples can be used to improve the profit of the policy. For each day, the generated samples can be used to determine the buying and selling thresholds. Assume that there are 100 generated full-day NRV samples for the day for which the profit needs to be optimized. The thresholds are determined for each generated sample separately using a simple grid search. All these thresholds can then be reduced by taking the mean to get one value for the buying threshold and one value for the selling threshold. Again, the penalty parameter is optimized for the test set to make sure around 283 charge cycles are used for fair comparison.
A low CRPS value does not necessarily mean the policy will generate a high profit. Because of this, the CRPS metric can not be used to evaluate the model during the training phase and use this metric to do early stopping. To fairly evaluate and compare the models, a validation set is split off from the training set. The validation set is used to evaluate the profit of the policy during the training and use this to do early stopping. The last two months of the training set are used as the validation set. This range starts on 01-11-2022 and ends on 31-12-2022. Two months are chosen to make sure enough data is available to have a good estimate of the profit while making sure the validation set is not too large. The policy can be evaluated quite fast on the validation set which is feasible to do during the training after a certain number of epochs.
@@ -6,23 +6,91 @@ A low CRPS value does not necessarily mean the policy will generate a high profi
\begin{table}[ht]
\centering
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{lcccccc}
\begin{tabular}{lccccc}
\toprule
Model & Type & CRPS & Profit (€) & Charge Cycles & Penalty \\
Layers & Steps & Test CRPS & Test Profit (€) & Test Charge Cycles & Test Penalty \\
\midrule
\acs{AQR} & Non-Linear & 79.22 & 189,717.74 & 282.06 & 566.4 \\
\acs{AQR} & GRU & 80.92 & 188,973.84 & 283.06 & 536.06 \\
Diffusion & Non-Linear & 80.30 & 191,553.58 & 282.56 & 553.38 \\
\multicolumn{6}{l}{\textbf{Only NRV}} \\
\midrule
256 - 256 & 5 & 275.03 & 191,221.97 & 282.50 & 315.6875 \\
256 - 256 & 20 & 113.27 & 215,946.13 & 283.13 & 421.8125 \\
256 - 256 & 50 & 139.61 & \textbf{218,170.75} & 283.00 & 449.2750 \\
512 - 512 & 50 & 167.23 & 209,625.07 & 282.25 & 449.1875 \\
1024 - 1024 & 50 & 100.72 & 217,560.20 & 283.75 & 455.5625 \\
256 - 256 & 80 & 201.15 & 209,761.89 & 283.50 & 457.2500 \\
\midrule
\multicolumn{6}{l}{\textbf{All Features}} \\
\midrule
256 - 256 & 20 & 108.84 & \textbf{218,141.31} & 283.94 & 428.6875 \\
256 - 256 & 20 & 105.31 & 215,862.35 & 283.06 & 440.2500 \\
512 - 512 & 20 & 103.41 & 216,411.79 & 282.56 & 450.3125 \\
1024 - 1024 & 20 & \textbf{100.36} & 215,686.32 & 282.69 & 463.6875 \\
256 - 256 & 50 & 117.81 & 216,632.39 & 282.75 & 421.3125 \\
512 - 512 & 50 & 180.83 & 210,769.03 & 282.06 & 446.4375 \\
1024 - 1024 & 50 & 179.59 & 212,793.94 & 282.88 & 454.5000 \\
256 - 256 & 80 & 242.68 & 205,374.94 & 283.13 & 451.3125 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\caption{Comparison of the different models using the profit metric. The best-performing models for a certain type are selected based on the profit.}
\caption{Comparison of diffusion models using different hyperparameters. Early stopping is done based on the profit using the validation set.}
\label{tab:diffusion_policy_comparison}
\end{table}
From the results shown in Table \ref{tab:diffusion_policy_comparison} multiple conclusions can be made. First of all, a lower CRPS metric does not correlate with a higher profit. The CRPS metric captures how well the NRV distribution is modeled. The profit metric captures how well the policy can make a profit using the generated samples. A lower CRPS means a better modeling of the NRV but in the table, it can be seen that the lowest CRPS is achieved by the model with layers 1024 - 1024 and 20 steps. This model also uses all features as input. The model achieves a CRPS of 100.36 with a profit of €215,686.32. The highest profit, however, is achieved by the model with layers 256 - 256 and using 50 steps. This model only uses the NRV as input. This model achieves a CRPS of 139.61 with a profit of €218,170.75. The CRPS here is higher which means a worse modeling of the NRV but the profit is higher.
Some examples of the generated samples from the model with the lowest CRPS and the model with the highest profit are shown in Figure \ref{fig:diffusion_policy_comparison_high_low_crps}. A significant difference in the confidence intervals can be observed. The left model clearly shows a better modeling of the NRV compared to the right model. The right model has confidence intervals that range from the minimum to the maximum value of the NRV. There is a high variance in the generated samples. This shows that the policy does not only benefit from good modeling of the NRV but also from a high variance in the generated samples. The model with the highest profit also only uses the NRV as input. This shows that more features do not necessarily result in a higher profit and that better modeling of the NRV does not necessarily improve the achievable profit of the policy.
\begin{figure}[H]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{images/diffusion/policies/comparison/Testing_864_Low_CRPS.jpeg}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{images/diffusion/policies/comparison/Testing_864_High_CRPS.jpeg}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{images/diffusion/policies/comparison/Testing_7008_Low_CRPS.jpeg}
\caption{Lowest CRPS}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{images/diffusion/policies/comparison/Testing_7008_High_CRPS.jpeg}
\caption{Highest profit}
\end{subfigure}
\caption{Comparison of the two samples from the model with the lowest CRPS and the model with the highest profit. }
\label{fig:diffusion_policy_comparison_high_low_crps}
\end{figure}
A comparison of the baselines and the best-performing models is shown in Table \ref{tab:policy_comparison}. The best-performing model is the diffusion model with two layers consisting of 256 hidden units. Only the NRV values of yesterday are used as input features and 50 steps were used. The profit achieved using this model is €218,170.75 with 283.00 charge cycles. This is an improvement of 9.74\% compared to the baseline that uses the NRV of yesterday as a prediction. When the policy is evaluated using the real NRV data for the evaluated day, a total profit of €230,317.84 is achieved. This is the maximum profit that can be achieved using the simple policy that determines a buying and selling threshold for each day. The best-performing diffusion model achieves a profit of €218,170.75. This means that 94.78\% of the maximum profit is achieved using the diffusion model. This is a significant improvement compared to the baseline that uses the NRV of yesterday as a prediction. This baseline achieves a profit of €198,807.09 which is 94.72\% of the maximum profit. This shows that integrating the use of multiple full-day NRV samples into the policy can improve the profit significantly.
\begin{table}[H]
\centering
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{lccc}
\toprule
Model & Test Profit (€) & Test Charge Cycles & Yesterday Baseline Improvement \\
\midrule
\multicolumn{4}{l}{\textbf{Baselines}} \\
\midrule
Fixed thresholds & 143,004.34 & 287.12 & \\
Yesterday NRV & 198,807.09 & 283.5 & \\
Perfect NRV & 230,317.84 & 282.5 & \\
\midrule
\multicolumn{4}{l}{\textbf{Models}} \\
\midrule
Diffusion (256 - 256, 50 steps, only NRV) & 218,170.75 & 283.00 & +9.74\% \\
\bottomrule
\end{tabular}
\end{adjustbox}
\caption{Comparison of the different models using the CRPS, profit, charge cycles and penalty. The best-performing models for a certain type are selected based on the profit.}
\label{tab:policy_comparison}
\end{table}
% TODO: Explain the results of the different models
% TODO: Compare with baselines
\newpage
% TODO: explain further
An observation that can be made is that the metrics used to evaluate the NRV predictions, do not necessarily correlate with the profit that can be made using the predictions. This means the best model in terms of the metrics does not necessarily result in the best profit. During the training, the profit should be used as a metric to evaluate the models but this becomes computationally expensive. Because of this, the policy now will only be evaluated after the training is done. The profit can give a better insight if the model improves the profit compared to the baselines.
\section{Conclusion}
In this thesis, generative methods are explored to model the NRV data of the Belgian electricity market. These methods are then used to improve the decision-making to charge and discharge a battery to make a profit.

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\contentsline {subsubsection}{\numberline {6.2.3}GRU Model}{33}{subsubsection.6.2.3}%
\contentsline {subsection}{\numberline {6.3}Diffusion}{37}{subsection.6.3}%
\contentsline {subsection}{\numberline {6.4}Comparison}{41}{subsection.6.4}%
\contentsline {section}{\numberline {7}Policies for battery optimization}{44}{section.7}%
\contentsline {subsection}{\numberline {7.1}Baselines}{44}{subsection.7.1}%
\contentsline {subsection}{\numberline {7.2}Policy using generated NRV samples}{45}{subsection.7.2}%
\contentsline {section}{\numberline {A}Appendix}{49}{appendix.A}%
\contentsline {subsection}{\numberline {6.5}Policies for battery optimization}{44}{subsection.6.5}%
\contentsline {subsubsection}{\numberline {6.5.1}Baselines}{44}{subsubsection.6.5.1}%
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\contentsline {section}{\numberline {7}Conclusion}{48}{section.7}%
\contentsline {section}{\numberline {A}Appendix}{51}{appendix.A}%