Gaussian Process Regression with a Hybrid Risk Measure for Dynamic Risk Management in the Electricity Market

Abstract

In this work, we introduce an innovative approach to managing electricity costs within Germany’s evolving energy market, where dynamic tariffs are becoming increasingly normal. In line with recent German governmental policies, particularly the Energiewende (Energy Transition) and European Union directives on clean energy, this work introduces a risk management strategy based on a combination of the well-known risk measures of the Value at Risk (VaR) and Conditional Value at Risk (CVaR). The goal is to optimize electricity procurement by forecasting hourly prices over a certain horizon and allocating a fixed budget using the aforementioned measures to minimize the financial risk. To generate price predictions, a Gaussian process regression model is used. The aim of this hybrid approach is to design a model that is easily understandable but allows for a comprehensive evaluation of potential financial exposure. It enables consumers to adjust their consumption patterns or market traders to invest and allows more cost-effective and risk-aware decision-making. The potential of our approach is shown in a case study based on the German market. Moreover, by discussing the political and economical implications, we show how the implementation of our method can contribute to the realization of a sustainable, flexible, and efficient energy market, as outlined in Germany’s Renewable Energy Act.

1. Introduction

Germany’s energy market is undergoing a profound transformation, with stakeholders seeking to reduce carbon emissions and an ever-increasing share of renewable energy sources. As a result, electricity prices have become increasingly volatile (cf. AL-Alimi et al. 2023; Alberini et al. 2023 and references therein), presenting challenges to consumers and energy suppliers alike. In this context, scientific research offers tools for consumers and policymakers to better understand and manage price fluctuations, e.g., by designing adequate forecasting or volatility models. Accurate forecasting helps in minimizing costs (or maximizing profits) under volatile conditions. Moreover, price models help to learn more about the inherent dynamics (cf. Narajewski and Ziel 2020; Weron and Ziel 2019, chp. 35). Additionally, robust risk management frameworks are developed to mitigate the financial impacts of price spikes and unpredictable market behavior (cf. Conejo et al. 2005). Such research promotes market transparency and stability, benefiting stakeholders and ensuring consumer welfare through a resilient electricity system.

The literature on electricity price prediction is vast, but, broadly speaking, it can be subdivided into three categories: statistical/probabilistic methods (cf. Cornell et al. 2024; Massidda et al. 2024), machine Learning/deep learning methods (cf. Poggi et al. 2023; Zamee et al. 2024), and hybrid methods that combine both approaches (cf. Jiang et al. 2023; Xiong and Qing 2023). As electricity prices are highly volatile and complex, fitting a model is challenging due to the multiple seasonal patterns, non-stationarity, heteroskedasticity, non-linear autoregressive dependence structures, and sudden changes in price patterns. Furthermore, numerous external factors, such as the electricity demand, weather conditions, and commodity prices (e.g., coal), impact the prices. Modeling these complex interactions is (computationally) demanding, and simplifying the models by reducing the factors may compromise the prediction accuracy. Therefore, a balanced approach that incorporates sufficient variables without losing accuracy is crucial. Classical time series models such as the autoregressive integrated moving average (ARIMA) and the seasonal ARIMA (SARIMA), which is its extension for seasonal data, have been frequently used for electricity price forecasting (cf. Conejo et al. 2005; Contreras et al. 2003). These methods form the foundation for time series analysis but assume linear relationships between current and past values, limiting their ability to capture non-linear dependencies in the data (cf. Hamilton 2020). In contrast, machine learning and deep learning techniques, such as convolutional neural networks (CNN) and long short-term memory (LSTM) networks, offer greater flexibility and scalability in capturing complex relationships in the data (cf. Bozlak and Yaşar 2024; Zhang et al. 2020). However, these methods often require large datasets to generalize effectively and are prone to overfitting when applied to noisy or sparse data, such as hourly time series (cf. Guo et al. 2024). In this work, motivated by Das et al. (2024), we employ Gaussian process regression (GPR), a kernel-based approach, to forecast electricity prices, leveraging its ability to handle non-linear dependencies without relying on explicit feature engineering. By selecting appropriate kernels to define input variable similarities, GPR effectively models complex, non-linear relationships, making it a powerful tool for time series forecasting, particularly in volatile, high-frequency data like electricity prices (cf. Jin and Xu 2024a, 2024b; Yadav et al. 2024).

Predicting electricity prices accurately allows market participants to make informed decisions regarding their financial strategies. In Chen et al. (2023), the researchers emphasize the importance of integrating predictive models with risk mitigation techniques such as the VaR or CVaR, particularly in trading scenarios for microgrid clusters. Their approach highlights how uncertainty in renewable energy and price fluctuations can be effectively managed through predictive modeling coupled with advanced risk measures. Moreover, the literature illustrates that integrating predictions with optimization frameworks has proven effective in real-world electricity market applications (cf. Agakishiev et al. 2025; Janczura and Wójcik 2022; Syuhada et al. 2023 and references therein). For instance, leveraging predictive techniques to forecast prices, followed by risk-adjusted portfolio optimization, enables the better handling of price volatility and reduces financial uncertainty. Such methodologies enhance the trading efficiency and promote market stability by managing downside risks and ensuring smoother operations in electricity markets. The term downside risk refers to the risk of losses or unfavorable outcomes, specifically capturing scenarios where the returns fall below a certain threshold or benchmark. Unlike variance, which treats both positive and negative deviations from the mean equally, downside risk focuses exclusively on the negative deviations that are most relevant to risk-averse decision-making. In this study, we use the VaR and CVaR as the downside risk measures.

In financial risk management, the VaR is one of the most widely used risk measures used to quantify the potential loss over a given time horizon at a specific confidence level (cf. Dempster 2002; McNeil et al. 2005). Simply speaking, it quantifies a loss level that is not exceeded in $\alpha \%$ of all potential scenarios. Despite being very popular in research and practice (cf. Basak and Shapiro 2001; Chen et al. 2024; Demirdöğen 2024; Munari et al. 2024), the VaR’s limitations have been extensively studied (cf. Duffie and Pan 1997; Emmer et al. 2001; Gaivoronski and Pflug 2005; Grootveld and Hallerbach 2000) and became apparent during the 2008 financial crisis: it does not provide any insight into what happens if this loss barrier is exceeded, cf. Larsen et al. (2002). In mathematical terms, the tail distribution is not considered sufficiently. As a consequence, the conditional VaR or expected shortfall was introduced (cf. McNeil et al. 2005), which basically quantifies the expected loss that one has to face if the abovementioned $\alpha$% barrier is exceeded. In this work, we propose a method for the optimization of electricity procurement in a dynamic pricing framework, focusing on minimizing the financial risk while ensuring cost efficiency. The approach integrates a hybrid risk management technique that combines both the VaR and CVaR to construct a model that allocates a given budget across different time points (hours) within a certain period using predicted hourly prices and weights. As a benchmark, we consider a scenario that allocates the budget equally over the chosen time horizon. Being aware that this strategy is not optimal (e.g., it does not consider dynamic weight adjustment over the week based on new price information), we see it as an initial step to determine whether a more active, i.e., more risky, strategy is beneficial. We also compare our proposed optimization framework with the existing optimization framework proposed by Rockafellar and Uryasev (2000). The objective function optimizes the trade-off between the expected return and downside risk, with the VaR and CVaR as the primary risk measures. By assigning weights to each hourly return, the model captures granular price behavior and tail events, offering a refined approach to managing risk, as the solution of the optimization problem optimally allocates the weights $\mathbf{w}$ across the hourly prices of the week. The optimized return $R\left(\mathbf{w}\right)$ represents the expected return of this portfolio, while the computed VaR and CVaR provide measures of the financial risk associated with this allocation. Using these risk measures, we can better assess potential downside risks and adjust the portfolio to ensure that it remains within the desired risk tolerance.

In this study, we simulate a household with the flexibility to adjust its demand within a period of one week, providing insights into the effectiveness of such a strategy under varying conditions. We address the problem of portfolio optimization under a fixed budget, with the aim of minimizing the risk while maximizing the returns. This integration of downside risk measures allows for a more robust decision-making process, tailoring the strategy to account for potential losses while exploiting opportunities for favorable returns. We base our strategy on point forecasts, which have been applied for (energy) time series forecasting in various articles (cf. Dybowski and Roberts 2000; Jensen et al. 2024). For example, Berrisch and Ziel (2022) proposed a concept for the forecasting of electricity prices. The allocation of the budget to the particular hour is dependent on the predicted return, which is calculated using the predicted price.

This paper is structured as follows. Based on a few precluding remarks in Section 2, the problem is formulated in Section 3. We explain our price prediction in Section 4. Eventually, our approach is tested and discussed in Section 5. Section 6 includes potential economical and political implications, as well as critical reflections on our approach. Section 7 concludes the paper. The mathematical reasoning and explanations are explained in Appendix A.

2. Precluding Remarks

In standard portfolio theory, a portfolio is an aggregation of multiple assets, where diversification reduces risk. Here, we define our portfolio as a time-based allocation strategy for a single asset (hourly electricity prices throughout the week), focusing on optimizing the returns at different hours. This time-based portfolio is a practical adaptation to our context. It reflects the dynamic nature of energy pricing across all points of time, each with its unique risk and return characteristics. Similarly, the return, in traditional financial contexts, represents the profit or loss relative to the initial investment over time. We define the return as the hourly log-based change in the electricity price. The log-return approach is consistent with financial theory, making returns additive over time, which is especially relevant for high-frequency data such as hourly electricity prices. On the German market, due to the lack of storage capacities, electricity prices are sometimes negative in times of considerable oversupply. This is typically when renewable electricity production exceeds the demand or in times of low demand, such as the Christmas holidays (cf. Halbrügge et al. 2023; Hilger et al. 2024; Loizidis et al. 2024). In this situation, computing the log-return is impossible, so, in this case, we transform the prices to the positive by adding a constant and compensate for this step by rescaling. Typically, the portfolio return is the weighted average return between assets. Here, we apply a similar concept but across time, defining the portfolio return as the weighted sum of the hourly returns based on our allocation.

Throughout this paper, we consider an investment horizon of one week (including weekends) and hourly granularity, i.e., 24 hours (h) per day. We assume that we have a fixed budget for power sourcing and that the demand is 100% flexible to be shifted across the time span of one week. Thereby, we aim to distribute the budget across each hour of a week in an optimal way. Considering every hour of the week as an investment opportunity, the combination of all these hours over a week forms our portfolio. The weights allocated to specific hours determine the proportion of the budget invested at each specific hour and sum to the total budget available. The goal is to assign higher weights to hours where the predicted returns are favorable, while minimizing exposure to high-risk hours, effectively managing the risk through the VaR and CVaR. In this optimization problem, the portfolio represents the aggregated exposure to the predicted returns across all hours of the week.

Let $\mathbf{R}\in {\mathbb{R}}^{T\times D}$ be the matrix of predicted returns for the week, with elements $r_{t,d}$ representing the return at hour t on day d. Let $\mathbf{W}\in {\mathbb{R}}^{T\times D}$ be the matrix of portfolio weights, where each $w_{t,d}\in \mathbf{W}$ represents the amount of the total budget allocated to the time point t on day d. The total budget available per week is given by B. The portfolio return $R\left(\mathbf{w}\right)$, over one week is given by the weighted sum of the predicted returns:

$$R\left(\mathbf{w}\right)={\displaystyle \sum _{t=1}^T\sum _{d=1}^D}{w}_{t,d}{r}_{t,d},\mathrm{where}\mathbf{w}\mathrm{is}\mathrm{a}\mathrm{column}\mathrm{of}\mathbf{W}.$$

Thereby, we define the losses ${\ell }_{t,d}$ as the negative of the returns, i.e.,

$${\ell }_{t,d}=-w_{t,d}{r}_{t,d}.$$

The VaR at confidence level $\alpha$ is based on the loss distribution. It quantifies the threshold that is exceeded with a probability of ($1-\alpha$)%. Mathematically speaking, the VaR for a portfolio is defined as the $(1-\alpha )$ quantile of the loss distribution, i.e., the $\alpha$ quantile of the profit distribution (see Equation (1)).

$${\mathrm{VaR}}_{\alpha }\left(\mathbf{w}\right)=inf\left\{\ell \in \mathbb{R}\mid \mathbb{P}\left(L\right(\mathbf{w})\le \ell )\ge \alpha \right\},$$

(1)

where $L\left(\mathbf{w}\right)$ is the distribution of losses from the portfolio. The $\alpha$% Conditional Value at Risk, again, is computed as the expected loss exceeding the VaR (see Equation (2)).

$${\mathrm{CVaR}}_{\alpha }\left(\mathbf{w}\right)=\mathbb{E}[L\left(\mathbf{w}\right)\mid L\left(\mathbf{w}\right)\ge {\mathrm{VaR}}_{\alpha }\left(\mathbf{w}\right)].$$

(2)

The value ℓ in the VaR definition is mathematically analogous to the loss defined by the left tail of the return distribution. It represents the threshold below which the loss exceeds with a given probability. Specifically, in the context of the VaR, ℓ is the quantile of the loss distribution at the confidence level $\alpha$. Loss, in this case, is often defined as the negative of the return, $L=-R$, where R represents the portfolio return. The left tail of the return distribution refers to the region of the worst (most negative) returns, and the VaR at level $\alpha$ captures the threshold below which the loss is expected to occur with probability $1-\alpha$. This means that ℓ, i.e., the loss threshold, corresponds to the quantile of the return distribution at $1-\alpha$, or, equivalently, it is the point where the cumulative distribution of the loss reaches $\alpha$. Thus, ℓ serves as a critical point in the left tail of the return distribution, marking the maximum potential loss that will not be exceeded with probability $\alpha$.

3. Problem Formulation

In the context of electricity markets, price fluctuations may have considerable budget impacts. Using the VaR helps to control for typical price variations within the specified confidence interval, while the CVaR protects against extreme unexpected spikes in the cost. Jointly they offer a comprehensive picture of the portfolio’s risk profile and help to design an allocation strategy that is both protective against the regular volatility and resilient to extreme outliers. This is why we use both measures to account for both the likelihood of losses and the magnitude of extreme losses, respectively. We minimize the risk of the portfolio using both the VaR and CVaR, while ensuring that the portfolio is fully invested:

$$\begin{array}{c}\hfill \underset{\mathbf{w}}{min}\left\{-\mathbb{E}\left[R\left(\mathbf{w}\right)\right]+{\lambda }_{\mathrm{VaR}}·{\mathrm{VaR}}_{\alpha }\left(R\left(\mathbf{w}\right)\right)+{\lambda }_{\mathrm{CVaR}}·{\mathrm{CVaR}}_{\alpha }\left(R\left(\mathbf{w}\right)\right)\right\}\end{array}$$

(3)

subject to

$$\begin{array}{ccc}\hfill \sum _{t=1}^T\sum _{d=1}^D{w}_{t,d}& =& B\hfill \\ \hfill w_{t,d}& \ge & 0\forall t=1,\cdots ,T,d=1,\cdots ,D\hfill \\ \hfill w_{t,d}& \le & \gamma B\forall t=1,\cdots ,T,d=1,\cdots ,D\hfill \\ \hfill \mathrm{Var}\left(R\right(\mathbf{w}\left)\right)& \le & {\sigma }_{\max }^2\hfill \end{array}$$

(4)

where $\mathbf{w}=(w_1,w_2,\dots ,w_n)$ is the vector of portfolio weights for n assets, and ${\lambda }_{\mathrm{VaR}}$ and ${\lambda }_{\mathrm{CVaR}}$ are the penalty weights for the VaR and CVaR, respectively, allowing us to balance risk minimization and return maximization. Variable B denotes the total budget, whereby the sum of the portfolio weights must equal B; $\gamma$ is the proportion of the budget allowed to be allocated to each asset, preventing over-concentration.

The combination of the VaR and CVaR has been explored in the literature, where the VaR is often used to impose regulatory limits, and the CVaR is used to improve risk management by mitigating the risk of extreme losses. Notable works, such as those by (Rockafellar and Uryasev 2002), demonstrate that this dual approach provides a balance between minimizing the tail risk (via CVaR) and ensuring compliance with VaR-based regulations. However, the inclusion of the VaR introduces non-convexity to the optimization problem, making it challenging to solve efficiently. Given this complexity, it is natural to consider whether a convex reformulation could be used instead. In this context, Rockafellar and Uryasev (2000) proposed a convex formulation for CVaR optimization by avoiding the direct inclusion of the VaR. While convex reformulations have clear computational advantages, our approach deliberately retains the non-convex formulation to preserve the complementary benefits of the VaR and CVaR, allowing us to address both regulatory constraints and tail-risk mitigation simultaneously. We discuss this alternative approach in Section 5.1 and Appendix A, where the convex formulation is evaluated for comparison. Despite being non-convex, the use of local optimization methods such as the sequential quadratic programming approach remains a practical and effective solution for such cases. Although non-convexity may lead to the possibility of converging to a local minimum rather than the global optimum, this approach is used in the literature for similar risk management problems (cf. Ben-Tal and Nemirovski 2002), where the primary objective is to find sufficient solutions within reasonable computational time and resources. Additionally, the problem was solved under typical market conditions, with the results showing feasible and robust portfolio allocations that adhered to risk management principles. The focus of the optimization is to balance risk minimization (through VaR and CVaR) and return maximization, which is often sufficiently addressed through local minima in practice, especially when the solver is initialized effectively, as discussed in Chopra and Ziemba (1993). Furthermore, while local minima can potentially be a concern, the practical significance of the results is not diminished. The portfolio solution obtained remains interpretable, actionable, and aligned with regulatory constraints, which is typically sufficient for risk management. While more advanced measures, such as the GlueVaR or other distortion risk measures, could enhance the modeling of extreme risk scenarios, our approach prioritizes practical interpretability and computational efficiency (cf. Krężołek and Trzpiot 2018; Zhao and Yin 2024). The inclusion of weighting parameters ${\lambda }_{VaR}$ and ${\lambda }_{CVaR}$ allows the tailoring of the risk–return tradeoff without additional complexity. Although the exploration of alternative risk measures, such as the GlueVaR, presents a valuable avenue for future research, the current formulation effectively meets the objectives of this study, offering a robust and adaptable solution for portfolio optimization in energy markets. Thus, while acknowledging the theoretical limitations of local minima, we argue that the results are practical and sufficiently optimal within the problem’s constraints, providing valuable insights for portfolio optimization.

The objective of Problem (3) is to maximize the returns while penalizing the VaR and CVaR. The parameters ${\lambda }_{\mathrm{VaR}}$ and ${\lambda }_{\mathrm{CVaR}}$ control the trade-off between maximizing returns and minimizing risk. The first constraint ensures that the sum of the portfolio equals the total available budget; the second constraint assumes that that one cannot allocate negative weights in the portfolio, which would be the case if battery storage is allowed. Over-concentration in any single asset is ensured by demanding that no single asset can hold $\gamma$ proportion of the total budget. The total volatility of the portfolio is controlled by setting an upper bound on the variance of the portfolio return. By limiting the variance, we ensure that the portfolio’s overall risk (in terms of volatility) is kept within an acceptable level. This control mechanism is particularly useful in achieving stable returns, as it prevents excessive fluctuations in portfolio value. Furthermore, it complements the management of the VaR and CVaR, which focus on extreme losses over a given time horizon. While variance limits address the overall distribution of returns, the VaR and CVaR provide additional safeguards by quantifying and controlling the tail risk. By combining these risk measures, the optimization process achieves a balanced portfolio, where both the total risk and tail risk are constrained, leading to more robust and stable returns.

4. Electricity Price Prediction

Gaussian process regression is a non-parametric Bayesian approach. It can be viewed either as a standard regression or as a functional form, where a Gaussian process is a distribution over functions. Here, we do not delve into the theoretical aspects of the Gaussian process, but we use it as a prediction method for hourly electricity prices. For a tutorial on how to use GPR, please refer to Schulz et al. (2018). We briefly introduce Gaussian process regression as follows.

A continuous-time stochastic process $X=\{X_{\mathbf{t}}:\mathbf{t}\in T\}$ is said to be a Gaussian process if, for any ${\mathbf{t}}_1,\cdots ,{\mathbf{t}}_n\in T$, $\tilde{X}=\{X_{{\mathbf{t}}_1},\cdots ,X_{{\mathbf{t}}_n}\}$ is multivariate Gaussian, meaning that the joint distribution of $\tilde{X}$ is given by

$$f_{\tilde{X}}(x_1,\cdots ,x_n)={\displaystyle \frac{exp\left(-\frac{1}{2}{(\mathbf{x}-\mathit{\mu })}^{\top }{\mathsf{\Sigma }}^{-1}(\mathbf{x}-\mathit{\mu })\right)}{\sqrt{{\left(2\pi \right)}^n\left|\mathsf{\Sigma }\right|}}},$$

where $\mathit{\mu }$ and $\mathsf{\Sigma }$ are the mean vector and the covariance matrix of $\tilde{X}$. In practice, we generally assume the mean vector to be zero, and the Gaussian process in this case is fully explained by the covariance matrix. The covariance matrix is obtained by choosing an appropriate covariance function that fits to the characteristics of the dataset.

Assuming that the prices are the realization of a Gaussian process, we predict the prices as follows. Let us assume that we have observations $\tilde{X}=\{X_{{\mathbf{t}}_1},\cdots ,X_{{\mathbf{t}}_k}\}$ at $\{{\mathbf{t}}_1,\cdots ,{\mathbf{t}}_k\}$, whose mean is $\mathit{\mu }$ and whose covariance function K is generating a covariance matrix $\mathsf{\Sigma }$. We wish to predict the prices $\widehat{X}=\{X_{{\mathbf{t}}_{k+1}},\cdots ,X_{{\mathbf{t}}_{k+m}}\}$ at $\{{\mathbf{t}}_{k+1},\cdots ,{\mathbf{t}}_{k+m}\}$. From Rasmussen and Williams (2005), we can see that the prediction at instances $\{{\mathbf{t}}_{k+1},\cdots ,{\mathbf{t}}_m\}$ is given by

$${\mathit{\mu }}^{*}={\mathit{\mu }}_{\widehat{X}}+{\mathsf{\Sigma }}_{\widehat{X},X}{\left({\mathsf{\Sigma }}_X+{\sigma }_n\mathbf{I}\right)}^{-1}\left(\mathbf{X}-{\mathit{\mu }}_X\right),$$

and the uncertainty associated with the prediction is given by

$${\mathsf{\Sigma }}^{*}={\mathsf{\Sigma }}_{\widehat{X}}-{\mathsf{\Sigma }}_{\widehat{X},X}{\left({\mathsf{\Sigma }}_X+{\sigma }_n\mathbf{I}\right)}^{-1}{\mathsf{\Sigma }}_{X,\widehat{X}}.$$

For a detailed application of Gaussian process regression for electricity price prediction, please refer to Das et al. (2024), where the covariance function is the sum of a squared exponential function and a rational quadratic function, which reads as follows:

$$C({\mathbf{t}}_i,{\mathbf{t}}_j)=\underset{\mathrm{Squared}\mathrm{Exponential}}{\underbrace{{\sigma }_{se}^{2}exp\left(-{\displaystyle \frac{\left|\right|{\mathbf{t}}_i-{\mathbf{t}}_j{\left|\right|}^2}{2{\ell }_{se}^2}}\right)}}+\underset{\mathrm{Rational}\mathrm{Quadratic}}{\underbrace{{\sigma }_{rq}^2{\left(1+{\displaystyle \frac{\left|\right|{\mathbf{t}}_i-{\mathbf{t}}_j{\left|\right|}^2}{2\alpha {\ell }_{rq}^2}}\right)}^{-\alpha }}},$$

where $\left|\right|{\mathbf{t}}_{\mathbf{i}}-{\mathbf{t}}_{\mathbf{j}}\left|\right|$ is the Euclidean distance between points ${\mathbf{t}}_i$ and ${\mathbf{t}}_j$, and ${\sigma }_{se}$ and ${\ell }_{se}$ are the parameters for the squared exponential part. Moreover, ${\sigma }_{rq}$, ${\ell }_{rq}$ and $\alpha$ are the parameters for the rational quadratic part. All of these parameters of the covariance function are estimated via maximum likelihood estimation.

5. Case Study

As per the report of the European Union1, in 2023, the average German household costs for electricity amounted to about EUR 100–150 per month. Taking this budget and assuming a sufficiently large battery at hand, we analyze the potential profits given a weekly budget of EUR 30 and the currently available possibility for end users to purchase power given the market prices. Note again that this is only a very simplified example, but it helps to show the potential of our proposed model.

5.1. Numerical Results

The prediction of hourly electricity prices is performed by calibrating a three-dimensional Gaussian process using three input features, namely the hourly time points, hourly residual load, and hourly total renewable production. As output, the hourly price is used. As shown in (Das et al. 2024), the GPR model is calibrated using a rolling window approach comprising the previous 100 days to predict one week into the future. The data used in this work are publicly available and provided by the German Central Infrastructure Authority via their website www.smard.de/en (accessed on 18 November 2024).

For evaluation purposes, we test our approach on the weeks of 2023 given an assumed budget of $B=30$ EUR/week. First, the GPR model is calibrated and predictions are made. Second, the predicted prices are used for the optimization problem described in Equations (3) and (5). Since we are evaluating the efficiency of the optimization problem (3), we also use the real data to check the performance of our approach. As an initial result, we present a comparison between the predicted and actual weekly returns in Figure 1a, while, in Figure 1b, the corresponding root mean squared error (RMSE) is displayed. Both the predicted and actual returns are calculated as weekly averages. The predictive performance of the model was evaluated over 51 weeks, where it showed strong alignment with the real returns in 48 weeks. While some discrepancies occurred in weeks with high volatility or market shocks, these can be attributed to inherent noise or outliers in the data, which are common challenges in energy markets. To assess the overall performance of the weekly returns, we computed the change in the weekly return as follows:

$$\mathrm{Change}\mathrm{in}\mathrm{Return}={\displaystyle \frac{\mathrm{final}\mathrm{value}-\mathrm{initial}\mathrm{value}}{\mathrm{initial}\mathrm{value}}}$$

where the final value is the last return of the week and the initial value is the first return of the week. It should be noted that the returns are hourly; hence, there are 168 returns in a week. In Figure 1c, the changes in the weekly returns for real and predicted returns are compared across 51 weeks. It is observed that, for 46 weeks, the changes in returns in the real and predicted data are closely aligned, indicating the model’s ability to track portfolio performance trends accurately. However, in 5 weeks, there is a noticeable deviation, which could suggest the presence of external market influences, anomalies, or limitations in the predictive model.

Similarly, to evaluate the concentration of the portfolio, we calculated the coefficient coefficient given by

$$CC={\left({\displaystyle \sum _{i=1}^M}{w}_i\right)}^{-1}$$

where $w_i$ is the weight assigned to the returns and $M=168$ for a week. The coefficient concentration was introduced by the Brande Institute (cf. Mainik et al. 2015) via inverting the Herfindahl–Hirschman Index (HHI). A higher $CC$ value for a given week indicates that the electricity prices are more diversified, meaning that the price fluctuations are more evenly spread across the hours of the week, suggesting a higher level of competition in price movements. In other words, there is less dominance by any particular hour, and price variations are more evenly distributed across the week. Conversely, a lower $CC$ value for a week indicates higher concentration of price movements, where a few specific hours (or times) dominate the overall price pattern for that week. Specifically, in Figure 1d, there are a few weeks when the $CC$ is comparatively higher than during the other weeks, which indicates that there was less concentration in the price changes, meaning that different times of the week experienced relatively equal volatility or variation in prices. This suggests that certain times of the week drive the price fluctuations more strongly than others, leading to less competition or the dominance of specific periods. However, the concentration coefficient values exhibit better alignment across all 51 weeks in comparison to the weekly change values. This highlights the model’s consistent capability to maintain the portfolio’s intended concentration structure, ensuring that the diversification strategy is not compromised, regardless of fluctuations in returns.

The alignment in the concentration coefficient values underscores the robustness of the model in preserving portfolio concentration, an essential factor in effective risk management. On the other hand, the minor misalignment in the change in returns points to areas for potential refinement, such as addressing market dynamics or incorporating additional features into the predictive framework. Despite these discrepancies, the overall results demonstrate that the model is effective in capturing the primary patterns of portfolio performance while maintaining structural integrity.

To evaluate our optimization approach, we use the Sharpe ratio (SR), which is defined according to Pav (2021):

$$\mathrm{SR}={\displaystyle \frac{\mathbb{E}[R_p-R_f]}{{\sigma }_p}},$$

(5)

where ${\sigma }_p$ is the standard deviation of the returns, $R_p$ is the hourly predicted returns, and $R_f$ is the risk-free interest rate. In general, $R_f\ne 0$, and, since we are using hourly returns, we fix $R_f=0.0001$.

We compare our optimization framework with the model proposed in Rockafellar and Uryasev (2000), where the authors proposed a convex optimization framework. The formulation of the problem in the convex optimization setting is shown in Appendix A in detail. To evaluate the performance of our model, we compare the weight distribution across each of the weeks via the optimization method. Figure 2a illustrates that the proposed model in (3) and (4) exhibits more balanced and even allocation across different hours of the week. This pattern reflects the model’s flexibility in adapting to varying returns and risk conditions, ensuring a diversified allocation strategy that minimizes the concentration risk and accommodates dynamic market changes. Meanwhile, Figure 2b demonstrates that the benchmark model performs sharper and more concentrated allocations, with significant weights assigned to specific hours.

While this strategy may effectively target high-return or low-risk intervals, it may lack the adaptability required to handle unexpected market fluctuations or extreme conditions, especially in volatile environments such as hourly electricity pricing.

To ensure a comprehensive evaluation of our proposed non-convex optimization framework compared to the convex framework, we not only analyzed the weight distributions but also extended our analysis to the Sharpe ratio. The weight distributions, as discussed earlier, highlight the differences in the allocation strategies between the two approaches, with our model exhibiting a greater capacity for nuanced budget allocation across hours based on the predicted returns and risk preferences. This distinct allocation strategy has direct implications for the portfolio performance, particularly in terms of risk-adjusted returns, which we evaluate through the Sharpe ratio. To compare the Sharpe ratios from our model with those from the convex optimization framework, we first addressed the challenge of 18 combinations of ${\lambda }_{VaR}$ and ${\lambda }_{CVaR}$. For each combination, we calculated the weekly Sharpe ratios, resulting in 18 arrays, each containing 51 values. To simplify the comparison, we averaged the Sharpe ratios across the combinations for each week, producing a single array of 51 values that preserved the performance trends while reducing the dimensionality. The comparison shown in Figure 3a,b reveals that the number of averaged Sharpe ratios from our approach exceeding those from the convex optimization framework is higher for both the real and predicted data. This indicates that, on average, our non-convex framework achieves superior risk-adjusted returns compared to the convex framework. The decision to incorporate flexible risk-weighting parameters (${\lambda }_{VaR}$ and ${\lambda }_{CVaR}$) in our model likely contributes to this advantage, as it allows for the more tailored balancing of the risk and return, particularly suited to the dynamic and volatile nature of hourly electricity prices.

Moreover, the fact that this trend is observed consistently across both the real and predicted data underscores the robustness of our approach. By averaging the Sharpe ratios across the combinations, we ensure that our analysis reflects the model’s overall effectiveness, rather than being overly influenced by specific parameter settings. These results demonstrate the practical and theoretical strength of our framework in managing volatility and optimizing the risk-adjusted returns in electricity markets, while also emphasizing its adaptability to varying market conditions. Following this analysis, we turn to a critical benchmark for the evaluation of our approach: the equal weight allocation strategy. We compare the results for the SR, VaR, and CVaR of our strategy against a myopic approach, which means an equally allocated budget. For visual comparison, we have generated the plots with ${\lambda }_{VaR}={\lambda }_{CVaR}=0.5$, but we evaluate our approach for different combinations of ${\lambda }_{VaR}\mathrm{and}{\lambda }_{CVaR}$, which we explain in Section 5.2. The weekly average values of the VaR, CVaR, and Sharpe ratio are shown in

Table 1. Combined performance across different ${\lambda }_{VaR}$ and ${\lambda }_{CVaR}$ settings.

Setting	${\mathit{VaR}}_{\mathit{OPT}}$	${\mathit{VaR}}_{\mathit{INT}}$	${\mathit{CVaR}}_{\mathit{OPT}}$	${\mathit{CVaR}}_{\mathit{INT}}$	${\mathit{SR}}_{\mathit{OPT}}$	${\mathit{SR}}_{\mathit{INT}}$
Performance for ${\lambda }_{VaR}=0.1$ and ${\lambda }_{CVaR}=0.9$
Real Data	−1.2340	−$0.0054$	$\mathbf{0.0020}$	$0.0131$	$\mathbf{0.0843}$	−$0.0289$
Predicted Data	−0.5429	−$0.0089$	$\mathbf{0.0032}$	$0.0134$	$\mathbf{0.1038}$	−$0.0526$
Performance for ${\lambda }_{VaR}=0.3$ and ${\lambda }_{CVaR}=0.7$
Real Data	−1.2473	−$0.0054$	$\mathbf{0.0020}$	$0.0131$	$\mathbf{0.0858}$	−$0.0289$
Predicted Data	−0.5503	−$0.0089$	$\mathbf{0.0041}$	$0.0134$	$\mathbf{0.1031}$	−$0.0526$
Performance for ${\lambda }_{VaR}=0.5$ and ${\lambda }_{CVaR}=0.5$
Real Data	−1.2475	−$0.0054$	$\mathbf{0.0020}$	$0.0131$	$\mathbf{0.0858}$	−$0.0289$
Predicted Data	−0.5522	−$0.0089$	$\mathbf{0.0047}$	$0.0134$	$\mathbf{0.1024}$	−$0.0526$
Performance for ${\lambda }_{VaR}=0.7$ and ${\lambda }_{CVaR}=0.3$
Real Data	−1.2615	−$0.0054$	$\mathbf{0.0014}$	$0.0131$	$\mathbf{0.0882}$	−$0.0289$
Predicted Data	−0.5529	−$0.0089$	$\mathbf{0.0051}$	$0.0134$	$\mathbf{0.1019}$	−$0.0526$
Performance for ${\lambda }_{VaR}=0.9$ and ${\lambda }_{CVaR}=0.1$
Real Data	−1.2577	−$0.0054$	$\mathbf{0.0020}$	$0.0131$	$\mathbf{0.0858}$	−$0.0289$
Predicted Data	−0.5532	−$0.0089$	$\mathbf{0.0055}$	$0.0134$	$\mathbf{0.1016}$	−$0.0526$

, which shows the comparison for a few selected ${\lambda }_{VaR}$ and ${\lambda }_{CVaR}$. In each selected combination of ${\lambda }_{VaR}$ and ${\lambda }_{CVaR}$, the optimized values are better than the initial values. In Table 1, we denote by $Va{R}_{OPT}$ and $CVa{R}_{OPT}$ the optimized values of the VaR and CVaR, while $Va{R}_{INT}$ and $CVa{R}_{INT}$ refer to the VaR and CVaR when the budget is allocated equally. For the Sharpe ratio, we use the same same subscript for the optimized values and initial values. Similarly, the boldface numerical values in the table represent the lesser values for the VaR and CVaR and higher values for the Sharpe ratio.

As displayed in Figure 4a,b, we compare the performance of our optimization approach on both real and predicted return data. In both of the cases, the Sharpe ratio with optimization is significantly improved.

To check the robustness of the approach, we checked the weekly VaR and CVaR values for real and predicted data. For both of the datasets, we compare the results with the equally allocated weight case and the optimized weight case, and the results are displayed in Figure 5 and Figure 6. For both real and predicted returns, the VaR and CVaR for the optimized case performed better than the initial allocation in which the budget was equally allocated to each hour.

Since both the VaR and CVaR are measures of downside risks, lower values of the VaR and CVaR indicate lower risk exposure and thus are considered preferable. In this case, the optimization successfully minimized the VaR and CVaR, demonstrating reduced average losses in extreme cases. In addition, this study validated the methodology using both predicted and real data. While predicted data provide forward-looking insights, real data serve as a retrospective benchmark to evaluate the optimization’s effectiveness. The consistency observed in the performance improvements across these datasets underscores the reliability of the optimization framework, even when applied to predictive scenarios. The use of downside risk metrics, specifically the VaR and CVaR, further enhances the robustness of the results. These metrics, which quantify the potential losses at varying confidence levels, ensure that portfolio optimization effectively addresses financial risks. The reduction in the VaR and CVaR in the optimized portfolio compared to the initial allocation demonstrates the approach’s ability to mitigate the worst-case losses, a critical requirement in volatile electricity markets.

5.2. Discussion

To evaluate the effect of varying ${\lambda }_{VaR}$ and ${\lambda }_{CVaR}$ in the optimization setup, we test a range of ${\lambda }_{VaR}$ and ${\lambda }_{CVaR}$ values, namely between 0.1 and 0.9 with a step size of 0.1. Thereby, we ensure that ${\lambda }_{VaR}+{\lambda }_{CVaR}=1$; as a result, we have 18 such combinations. In the case of an equally allocated budget, the comparison metrics, i.e., $Va{R}_{INT}$, $CVa{R}_{INT}$, and the $S{R}_{INT}$ ratio, remain the same, since they do not depend on the penalties with respect to the VaR and CVaR. Figure 7a and Figure 8a show that the minimum values of the VaR and CVaR from the predicted returns are attained at $({\lambda }_{VaR},{\lambda }_{CVaR})=(0.9,0.1)\mathrm{and}(0.1,0.9)$, whereas Figure 7b and Figure 8b show that, for real returns, the minimum is attained at $({\lambda }_{VaR},{\lambda }_{CVaR})=(0.7,0.3)$ and (0.3,0.7). Similarly, Figure 9a,b highlight that the Sharpe ratio reaches the maximum for the predicted returns at $({\lambda }_{VaR},{\lambda }_{CVaR})=(0.1,0.9)\mathrm{and}(0.9,0.1)$ and for the real returns at $({\lambda }_{VaR},{\lambda }_{CVaR})=(0.2,0.8)\mathrm{and}(0.8,0.2)$.

Table 2. Comparison with respect to different combinations of ${\lambda }_{VaR}$ and ${\lambda }_{CVaR}$.

	${\mathit{VaR}}_{\mathit{INT}}$	$max\left({\mathit{VaR}}_{\mathit{OPT}}\right)$	${\mathit{CVaR}}_{\mathit{INT}}$	$max\left({\mathit{CVaR}}_{\mathit{OPT}}\right)$	${\mathit{SR}}_{\mathit{INT}}$	$min\left({\mathit{SR}}_{\mathit{OPT}}\right)$
Real	−0.0054	−1.2340	0.0131	0.0020	−0.0289	0.0843
Predicted	−0.0089	−0.5427	0.0134	0.0055	−0.0526	0.1016

shows that the VaR, CVaR, and Sharpe ratio in the optimized case are better than in the equal budget allocation case, even when considering the worst combination of ${\lambda }_{VaR}\mathrm{and}{\lambda }_{CVaR}$. The findings provide evidence for the effectiveness of our proposed portfolio optimization method by demonstrating its robustness across multiple performance metrics. Moreover, its adaptability to both real and predicted returns and its consistent superiority over equal budget allocation are shown. The results further highlight the adaptability of the method, as the optimal penalty combinations differ between real and predicted returns. Additionally, the Sharpe ratio, a key indicator for portfolio efficiency, attains its maximum values at distinct combinations for real and predicted returns, demonstrating the framework’s ability to adapt to different data scenarios. Finally, the comparison with the equal budget allocation case confirms the framework’s overall superiority, as it achieves better VaR, CVaR, and Sharpe ratio values even under suboptimal penalty parameter settings. This holistic evaluation establishes the reliability and practicality of the method in constructing well-diversified and high-performing portfolios.

6. Critical Reflections and Practical Implications

6.1. Economical and Social Implications

By leveraging GPR for accurate electricity price forecasting and incorporating robust portfolio optimization, our proposed methodology enables industries to better manage their operational costs by reducing the exposure to price volatility. The integration of downside risk metrics like the VaR and CVaR ensures that energy-intensive industries can minimize the potential financial losses during price surges, thereby stabilizing their cash flows, which frees up capital for investment in sustainable operations. From a broader perspective, reliable price forecasts allow both renewable and flexible conventional energy producers (such as gas-fired power plants) to optimize their generation schedules, aligning production with peak price hours. This enhances the profitability in the first step but will reduce the intraday price difference in the long run. Reliable forecasts also facilitate the integration of additional renewable energy sources into the grid, reducing the reliance on fossil fuels and advancing environmental sustainability. Furthermore, by mitigating financial risks, this approach contributes to industrial stability, potentially preserving jobs and driving economic growth in energy-dependent sectors, ultimately benefiting society at large.

6.2. Policy Implications

More accurate electricity price forecasting and backtests, as performed in this study, will help policymakers to evaluate the impacts of regulatory interventions such as price caps or subsidies for renewable energy with greater precision. Additionally, the enhanced predictability of electricity prices supports the design of dynamic tariff structures, incentivizing demand-side management and reducing peak load stresses on the grid. This framework also provides quantitative evidence for the promotion of renewable energy investments, as better risk management reduces the financial barriers for developers. Overall, the integration of this methodology into policymaking processes will enhance decision-making, foster sustainable energy transitions, and ensure equitable access to energy resources for all stakeholders.

6.3. Critical Reflections

While the proposed approach demonstrates robust performance in forecasting electricity prices and optimizing risk-adjusted returns, there are areas for future enhancement. The use of Gaussian process regression (GPR) with selected kernels provides strong predictive power based on the current dataset’s characteristics. However, these kernels may lose their effectiveness if the underlying market dynamics evolve—such as changes in price distribution, regulatory shifts, or the impact of energy transition policies—requiring kernel reevaluation. This highlights the need for the periodic reassessment of the model assumptions and the potential exploration of adaptable or hybrid frameworks to sustain the predictive accuracy. Additionally, the framework currently assumes static optimization with predefined risk metrics like the VaR and CVaR, which simplifies real-world factors such as transaction costs and liquidity constraints. Incorporating these elements into future iterations will improve the applicability and scalability of the model.

Despite these limitations, the methodology and predictive models used in this study can be extended to other electricity markets. While the market dynamics—such as regulatory frameworks, energy mixes, and volatility patterns—may vary across regions, the core framework of portfolio optimization, combined with predictive models and risk management measures like the VaR and CVaR, can be adapted to markets with similar volatility and risk characteristics. Researchers in other regions could benefit from calibrating the model to local energy price data and adjusting the risk metrics to regional preferences. The flexibility of the approach allows for adaptation to different energy markets, thus enhancing its utility in optimizing resource allocation and managing risk globally, even in markets distinct from the German energy market.

7. Conclusions

This work presents a robust portfolio optimization framework that dynamically adjusts the weights to maximize the predicted returns while adhering to budget and non-negativity constraints. By incorporating a predictive model alongside risk measures such as the VaR, CVaR, and variance, the methodology effectively manages hourly electricity price fluctuations, extreme price spikes, and overall stability, demonstrating reliable performance across varied market conditions. The results demonstrate that our framework outperforms both equal weight allocation and the convex formulation in terms of weight distribution adaptability, Sharpe ratio performance, and the ability to handle market-specific characteristics such as volatility and extreme price variations. By leveraging Gaussian process regression (GPR) for price prediction, our methodology aligns resource allocation dynamically with the predicted returns and market risks, enabling informed decision-making under uncertain conditions. However, the framework’s reliance on accurate predictions makes it vulnerable to market noise and outliers, and its assumption of static risk preferences may not fully reflect the dynamic market conditions. Additionally, while the non-convex formulation provides dynamic flexibility, it may converge to local minima, making the initial conditions and solver selection critical. Future work could mitigate these limitations by incorporating adaptive risk preferences, exploring alternative risk metrics, enhancing the predictive accuracy through advanced machine learning models, and extending the methodology to multi-period optimization or other energy markets with distinct characteristics, further improving its flexibility and applicability in dynamic financial environments.