# Energy Systems Operation under Uncertainty#

Part of a series: Uncertainty in Future Energy Systems.

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## Demand response of industrial processes#

In Germany, industry consumes about a third of the total produced energy [Energiebilanzen, 2021]. Flexible industrial processes can adapt their production load and energy consumption in a so-called demand response [Zhang and Grossmann, 2016], and, hence, adjust their energy demand to the current energy supply in the electricity grid. Demand response is incentivized by time-changing electricity prices, leading to an over-production in low-priced hours and an under-production in high-priced hours. Overall, demand response can lower the electricity cost of flexible industrial processes and help to balance the power grid. Demand response is particularly interesting for energy-intensive processes for which electricity is a major cost factor.

## Scheduling optimization#

The operation of an industrial processes can be determined by means of numerical optimization. In scheduling optimization, scheduling-relevant decisions are determined such that an objective function, e.g., the electricity cost or carbon emissions, is minimized while ensuring the operability and safety of the process.
For the optimization, scheduling-relevant characteristics must be considered such as process oversizing, minimal part-load, product storage capacity, ramping limitations, shutdown capabilities, and efficiency losses at off-design operation [Germscheid *et al.*, 2022, Schäfer *et al.*, 2020].
These process characteristics can be described numerically by a set of equations and constraints.

A simple scheduling optimization can be set up as follows:

Here, the total operation cost over \(T\) time steps is minimized. The optimal electricity purchase \(p_t\) is determined given the cost of electricity at different time steps \(c_t\). The process considers oversizing \(P_{max}\), minimal part-load capacity \(P_{min}\), and ramping limitations \(R_{max}\) from one time step to the next.

## Operation under uncertainty#

For the operation of an industrial process, uncertainties considering electricity supply, product demand, and energy prices have to be taken into account.
The uncertainty can affect both the objective function and the model constraints, e.g., the cost function and the operational constraints due to oversizing, minimal part-load and ramping in the example above.
Hence, constraints may have to hold for a probability distribution and the objective function may describe a probability distribution.
Methods that can tackle the uncertainty in optimization problems are chance constraint optimization, robust optimization, risk-neutral stochastic programming, and risk-averse stochastic programming.
Chance constraints allow for constraint violation with a certain probability. For example, an on-site electricity generator with uncertain production level, e.g., wind and solar energy, may supply electricity to an industrial process in 90% of the time to ensure a certain level of self-sufficiency.
In contrast, robust programming does not allow for any violation of constraints and, thus, ensures feasibility and operability for all realizations of the uncertain parameter.
For example, the national power grid with uncertain supply and demand must be functioning at all times calling for robust decisions regarding the electricity distribution and generation.
For stochastic programming, the probability distribution is optimized by considering statistical measures.
For risk-neutral stochastic programming, the expected cost of the objective function is optimized.
In contrast, risk-averse stochastic programming tackles risk associated to a set of worst-case scenarios by optimizing a risk measure such as the Conditional Value-at-Risk that is the expected cost of the (1-alpha) probable worst-case scenarios.
For example, stochastic programming can be used to tackle electricity price uncertainty in scheduling optimization, see, e.g., [Germscheid *et al.*, 2022].

## Challenges of optimization under uncertainty#

Optimization under uncertainty is connected to several challenges. The uncertain parameter can be described by a set of scenarios, i.e., realizations of the uncertain parameter. Suitable scenario generation and validation methods need to be used.
Furthermore, optimizing under uncertainty is connected to a high computational effort calling for both scenario reduction methods, see, e.g., [Heitsch and Römisch, 2003], and decomposition approaches, see, e.g., [Benders, 1962, Carøe and Schultz, 1999, Rockafellar and Wets, 1991, Van Slyke and Wets, 1969].
Finally, the benefit of considering uncertainty in the optimization has to be shown by evaluating, e.g., the metrics *value of stochastic solution and expected value of perfect information* [Birge and Louveaux, 2011].

## References#

- Ben62
J. F. Benders. Partitioning Procedures for Solving Mixed-Variable Programming Problems.

*Numerische Mathematik 4*, pages 238–252, 1962.- BL11
John R. Birge and François Louveaux.

*Introduction to Stochastic Programming*. Springer Series in Operations Research and Financial Engineering. Springer New York, 2011. ISBN 978-1-4614-0236-7. URL: http://link.springer.com/10.1007/978-1-4614-0237-4, doi:10.1007/978-1-4614-0237-4.- CaroeS99
Claus C. Carøe and Rüdiger Schultz. Dual decomposition in stochastic integer programming.

*Operations Research Letters*, 24(1-2):37–45, feb 1999. doi:10.1016/S0167-6377(98)00050-9.- Ene21
Umweltbundesamt auf Basis Arbeitsgemeinschaft Energiebilanzen. Endenergieverbrauch 2020 nach Sektoren und Energieträgern. 2021. URL: https://www.umweltbundesamt.de/daten/energie/energieverbrauch-nach-energietraegern-sektoren#allgemeine-entwicklung-und-einflussfaktoren.

- GMD22(1,2)
Sonja H. M. Germscheid, Alexander Mitsos, and Manuel Dahmen. Demand response potential of industrial processes considering uncertain short‐term electricity prices.

*AIChE Journal*, 2022. doi:10.1002/aic.17828.- HRomisch03
Holger Heitsch and Werner Römisch. Scenario Reduction Algorithms in Stochastic Programming.

*Computational Optimization and Applications*, 24(2):187–206, 2003. doi:10.1023/A:1021805924152.- RW91
R. T. Rockafellar and Roger J.-B. Wets. Scenarios and Policy Aggregation in Optimization Under Uncertainty.

*Mathematics of Operations Research*, 16(1):119–147, feb 1991. doi:10.1287/moor.16.1.119.- SchaferDM20
Pascal Schäfer, Torben M. Daun, and Alexander Mitsos. Do investments in flexibility enhance sustainability? A simulative study considering the German electricity sector.

*AIChE Journal*, 66(11):1–14, 2020. doi:10.1002/aic.17010.- ZG16
Qi Zhang and Ignacio E. Grossmann. Planning and Scheduling for Industrial Demand Side Management: Advances and Challenges. In

*Alternative Energy Sources and Technologies*, pages 383–414. Springer International Publishing, 2016. doi:10.1007/978-3-319-28752-2_14.- VanSlykeW69
R. M. Van Slyke and Roger Wets. L -Shaped Linear Programs with Applications to Optimal Control and Stochastic Programming.

*SIAM Journal on Applied Mathematics*, 17(4):638–663, jul 1969. doi:10.1137/0117061.

## Contributors#

Philipp Böttcher

Manuel Dahmen