The profit-maximizing trading in energy markets is a highly active research field. Therefore, multiple literature reviews are dedicated to this topic (e.g., Fathima and Palanisamy
2015; Rahman et al.
2015; Weitzel and Glock
2018). We focus our literature review on the trading in multiple markets. In most countries, multiple electricity markets coexist which are used sequentially by the market participants. Since the integrated or coordinated trading in these markets can dramatically increase the complexity of the decision problem, most authors focus on a single market setting, while other markets are (if at all) only included implicitly (e.g., Jiang and Powell
2015; Gönsch and Hassler
2016; Zhou et al.
2016; Franz et al.
2020; Ghavidel et al.
2020; Finnah and Gönsch
2021). Like in sequential auction markets, a product can be traded multiple times in a continuous intraday market. The key difference is that in continuous intraday markets, the current energy price is observable. While this opportunity is often ignored and products are only traded once, some papers model multiple trades (e.g., Aïd et al.
2016; Bertrand and Papavasiliou
2020; Boukas et al.
2020).
Löhndorf et al. (
2013) formulate the decision problem of an owner of a hydro storage with stochastic inflow on the German day-ahead and continuous intraday market as a dynamic program and use an approximate dual dynamic programming approach to solve it. The energy price processes are based on a few fundamentals, such as the mean temperature, total solar power generation, and gas price. To apply their heuristic, the underlying stochastic processes are discretized. Meanwhile, Ding et al. (
2015) use stochastic programming to (re-) optimize the management of a wind turbine combined with an energy storage on the Spanish day-ahead, intraday, and real-time market. Here, they employ four different timescales down to one minute to capture the dynamics of the real-time market. Ding et al. (
2015) do not optimize all considered markets integrated but use the different stochastic programs in a receding horizon manner. The authors assume that the next market’s price is deterministic. This dramatically decreases the complexity, as the price components of the bids are not needed. Furthermore, Crespo-Vazquez et al. (
2018) focus on a wind power producer with an energy storage on two Spanish auction markets and an implicit modeled balancing market with hourly products; the other auction markets are ignored. The decision problem is formulated as a stochastic program with an optimization horizon of 24 hours. Again, the price component of the bids is not modeled. Heredia et al. (
2018) use multistage stochastic programming (MSSP) to optimize the management of a wind turbine combined with an energy storage. The considered power producer maximizes the daily profits on two Spanish auction markets and a reserve power market. Therefore, the Spanish energy markets are not modeled in full complexity. Rintamäki et al. (
2020) considers the integrated trading in two energy markets of an energy producer with controllable load. The problem is modeled as a bi-level program. The upper level is a stochastic program that models the integrated trading, itself, while the lower-level problems are the dispatching problems on the day-ahead and intraday market. Sequential bidding in a day-ahead and a balancing market is considered in, e.g., Boomsma et al. (
2014); Kumbartzky et al. (
2017); Kongelf et al. (
2019); Mazzi et al. (
2019). Most authors restrict themselves to a small number of markets. This is because the computational burden of MSSPs increases exponentially in the number of markets. Therefore, the Spanish electricity market, which consists of seven sequential markets, cannot be handled in full complexity using this approach. The first to handle all seven Spanish markets is Wozabal and Rameseder (
2020). In contrast with the rest of the literature, the authors model the trading problem for the Spanish electricity market as a dynamic program, in which each decision stage corresponds to one market. Since the computational burden of dynamic programs increases only linearly in the number of stages/markets, Wozabal and Rameseder (
2020) can solve the trading problem efficiently but heuristically with an approximate dual dynamic programming approach. Wozabal and Rameseder (
2020) propose two model variants: one without updating the power production forecast and one with updates. We refer to the more interesting model with production forecast updates. As no energy storage is modeled, each day can be optimized separately. However, Wozabal and Rameseder (
2020) modeled the dynamic program for all 24 h at once, even if the hours could be modeled independently, as well. To determine the bids for each stage (market), different quantities are mapped to the hour-dependent price points selected previously. The authors demonstrate the influence of the underlying price process (e.g., distribution of noise) and include risk aversion by optimizing the nested Conditional Value-at-Risk. Since the state space must include all Markovian features of the underlying price and production processes, Wozabal and Rameseder (
2020) use only the most recent market prices and production forecast as state-dependent features. Due to missing production data, the authors focus primarily on a setting without updating the power production forecast in their numerical study.
It is common in the literature that the prices and volumes of the bids are not optimized simultaneously. This is because the simultaneous price and volume decisions result in nonlinear and non-concave decision problems. Therefore, most authors decide on either the prices
\(p^i\) or the volumes
\(x^i\), while the counterpart is given by parameters (e.g., Morales et al. (
2010); Löhndorf et al. (
2013); Boomsma et al. (
2014); Guerrero-Mestre et al. (
2016); Mazzi et al. (
2019); Wozabal and Rameseder (
2020)). This reduces the computational burden but leads to sub-optimal decisions. A consequence is that the modeled decision problem must be (partially) replaced by the sample average approximation (SAA), which lowers the quality of the solution again. The computational burden of the SAA formulation increases in the number of samples used, which are needed to capture state-of-the-art stochastic processes with their correlations.
In summary, to the best of our knowledge, no paper solves the energy trading problem in sequential markets analytically. In contrast, for single market settings analytical solutions exist in the literature (e.g., Kim and Powell (
2011); Densing (
2013); Aïd et al. (
2016)). Kim and Powell (
2011) model the hour-ahead market trading of an energy storage combined with a wind farm as a dynamic program but ignore price–volume bids. For the analytical solution, Kim and Powell (
2011) need assumptions regarding the stochastic processes of the wind power production and the energy prices. Densing (
2013) analytically solve the price–volume bidding of an energy storage in an auction market. For this, the lower and upper bound of the energy storage is ignored and only the expectation of the storage level is constrained. Aïd et al. (
2016) minimize the imbalance cost in the continuous intraday market under stochastic demand, production, and prices. Further, a controllable thermal power plant is integrated. Aïd et al. (
2016) analytically solve a relaxed problem that allows negative production. The solution of the relaxed problem is used to solve the non-relaxed problem heuristically.