(305g) Scenario Generation for Day-Ahead Scheduling Problems Using Normalizing Flows

Flexible operation of industrial processes requires planning ahead by solving scheduling problems con-
sidering relevant uncertainties (Zhang et al., 2016). For instance, the inherent uncertainty of renewable
electricity sources must be considered in decision-making for the operation of energy systems, e.g., through
stochastic programming based on discrete scenarios (Morales et al., 2013; Mitsos et al., 2018). Meanwhile,
the market structure of the modern day-ahead markets (European Power Exchange, 2021) gives a fixed
format of 24 h scheduling horizons. Thus, the stochastic programming problems for day-ahead scheduling
require scenarios that cover the time interval from midnight to midnight. Such full-day scenarios can be
generated using multivariate modeling approaches that predict scenarios of the series of realizations over
the 24 h horizon in a vector format (Cramer et al., 2022). These scenarios reflect typical fluctuations of,
e.g., wind power generation. In operational problems such as day-ahead scheduling, the scenarios should
be specific to the given day while also accurately representing the underlying distributions of the time
series to make profitable and feasible decisions (Morales et al., 2013). Furthermore, different scenario sets
sampled from the same model should yield stable objective values (Kaut and Wallace, 2003). Generating
such day-ahead scenarios requires sampling from the high dimensional distribution of time series intervals
that are highly dependent on external factors. We use the deep generative model called normalizing flow
to generate day-ahead scenarios. The normalizing flow models a high-dimensional distribution using a
nonlinear transformation of a multivariate Gaussian based on an invertible neural network (INN). The
inverse of the INN gives an explicit expression for the probability density function (PDF), which enables
training by log-likelihood maximization (Papamakarios et al., 2021). Furthermore, normalizing flows
seamlessly incorporate external information to learn conditional distributions (Winkler et al., 2019). We
apply the normalizing flow to sample scenarios of wind power generation for a fictional wind farm in
eastern Germany. The conditional distributions allow us to use wind speed forecasts to generate day-
ahead wind power generation scenarios that are specifically tailored to the given day. We then apply
the generated scenarios in a stochastic day-ahead wind electricity producer bidding problem based on
Garcia-Gonzalez et al. (2008) and Conejo et al. (2010) and derive statistical results over the full test year
2019. Our analysis shows that conditional scenario generation via normalizing flows results in a loss of
11% in profit relative to the solution using perfect information while using historical scenarios results in
over 80% lower profit. We also perform a statistical investigation of the stability defined by Kaut and
Wallace (2003). Using five normalizing flow scenarios leads to a lower variance in the objective function
than using tenor more scenarios from other scenario generation methods.

Bibliography
Conejo, A. J., Carrión, M., and Morales, J. M. (2010). Decision Making Under Uncertainty in Electricity
Markets, volume 1. Springer Science & Buisness Media, Boston, MA, 1 edition.
Cramer, E., Paeleke, L., Mitsos, A., and Dahmen, M. (2022). Normalizing flow-based day-ahead wind
power scenario generation for profitable and reliable delivery commitments by wind farm operators.
Computers & Chemical Engineering, 166:107923.
European Power Exchange (2021). EPEX SPOT documentation. http://www.epexspot.com/en/
extras/download-center/documentation. Accessed: 2021-05-30.
Garcia-Gonzalez, J., de la Muela, R. M. R., Santos, L. M., and Gonzalez, A. M. (2008). Stochastic joint
optimization of wind generation and pumped-storage units in an electricity market. IEEE Transactions
on Power Systems, 23(2):460–468.
Kaut, M. and Wallace, S. W. (2003). Evaluation of scenario-generation methods for stochastic program-
ming. Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für
Mathematik, Berlin, Germany.
Mitsos, A., Asprion, N., Floudas, C. A., Bortz, M., Baldea, M., Bonvin, D., Caspari, A., and Schäfer,
P. (2018). Challenges in process optimization for new feedstocks and energy sources. Computers &
Chemical Engineering, 113:209–221.
Morales, J. M., Conejo, A. J., Madsen, H., Pinson, P., and Zugno, M. (2013). Integrating Renewables in
Electricity Markets: Operational Problems, volume 205. Springer Science & Business Media, Boston,
MA, 1 edition.
Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S., and Lakshminarayanan, B. (2021).
Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research,
22(57):1–64.
Winkler, C., Worrall, D., Hoogeboom, E., and Welling, M. (2019). Learning likelihoods with conditional
normalizing flows. arXiv preprint arXiv:1912.00042.
Zhang, Q., Cremer, J. L., Grossmann, I. E., Sundaramoorthy, A., and Pinto, J. M. (2016). Risk-based
integrated production scheduling and electricity procurement for continuous power-intensive processes.
Computers & Chemical Engineering, 86:90–105.