**Project Heads**

René Henrion, Dietmar Hömberg

**Project Members**

Nina Kliche

**Project Duration**

16.06.2022 − 15.06.2025

**Located at**

WIAS

Battery degradation can be quantified in terms of capacity loss and resistance increase and is accelerated e.g. the higher the depth-of-discharge and the battery operating temperature is. This asks for a suitable battery operation strategy.

Typical sources of uncertainty for a stand-alone mini-grid are energy supply by renewables, demand of energy and ambient temperature. These uncertainties naturally translate into battery dynamics, namely battery state-of-charge and battery temperature which are described by means of differential equations. The pre-scribed battery operation strategies asks these quantities to stay within reasonable and degradation-aware ranges in a probabilistic framework. To handle the uncertainties, the concept of so-called joint chance constraints is deployed: the relevant quantities are asked to stay within their bounds up to some given probability level. This offers an attractive solution method striking balance between reliability and costs. In this sense, the problem of finding an energy management strategy for minimizing the daily operational costs of a stand-alone mini-grid under uncertainty will be tackled.

In a first step, a probabilistic optimal control problem is set up and solved numerically. For this purpose, a pre-defined mini-grid design is taken into account, i.e. a pre-defined layout as well as a pre-defined battery operation strategy. This yields an optimal control problem of the generic form

\[

\min_{u=(P_{DG}, P_{dump}, P_{BTMS})} \quad \int_{t_0}^{t_f} FC \big( P_{DG}(t) \big) \; \mathrm{d}t

\]

subject to a state equation

\[ x'(t) = F(t,x,u,\xi), \quad x(0)=x_0 \]

and probabilistic state constraints

\[ \mathbb{P} \Big( x(t;u,\xi)\in [x^{\min}, x^{\max} \; \forall t\in [t_0,t_f] ] \Big) \geq p \]

where the variable \(x\) denotes the state variable, \(u\) is the control and \(\xi\) corresponds to the uncertainty. As becomes evident, reliable energy management including battery operation results in treating chance constraints in the framework of differential equations. The resulting energy management on a sunny day is depicted in the below Figures.

**Ongoing work**

**External Website**

**Related Publications
**

- M. H. Farshbaf-Shaker, R. Henrion, and D. Hömberg. Properties of chance constraints in infinite dimensions with an application to pde constrained optimization. Set-Valued and Variational Analysis, 26(4):821–841, 2018.
- M. Hassan Farshbaf-Shaker, Martin Gugat, Holger Heitsch, and René Henrion. Optimal neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints. SIAM Journal on Control and Optimization, 58(4):2288–2311, 2020.
- T. González Grandón, R. Henrion, and P. Pérez-Aros. Dynamic probabilistic constraints under continuous random distributions. Mathematical Programming, 196(1):1065–1096, 2022.
- Wim van Ackooij, René Henrion, and Pedro Pérez-Aros. Generalized gradients for probabilistic/robust (probust) constraints. Optimization, 69(7- 8):1451–1479, 2020.
- J. Schmalstieg, S. Käbitz, M. Ecker, and D. U. Sauer. A holistic aging model forLi(NiMnCo)O2 based 18650 lithium-ion batteries. Journal of Power Sources, 257:325–334, 2014.