Project
AA4-10
Modelling and Optimization of Weakly Coupled Minigrids under Uncertainty
Project Heads
René Henrion, Dietmar Hömberg
Project Members
Nina Kliche
Project Duration
16.06.2022 − 15.06.2025
Located at
WIAS
Description
Ensuring access to affordable, reliable, sustainable and modern energy for all is one of the United Nations’ seventeen Sustainable Development Goals for 2030. In rural areas of African sub-Saharan countries, electrification is still at low speed. In Ethiopia for instance, 70% of the people living in re- mote rural areas have no access to electricity. As a viable solution mini-grids have emerged as the extension of the main grid can be extremely expensive and cannot even guarantee good quality of service. Mini-grids typically consist of several decentralised energy generation sources, a battery energy storage system (BESS), a backup diesel generator (DG), inverters, meters and controller units. They are either connected to or isolated from the main grid and distribute energy at a community level. The mini-grid designer now must find a compromise between efficient management of resources, a low cost and high resilience. In the following, a stand-alone mini-grid is taken into consideration where the weakly coupled mini-grid is part of future work.
There is a vast literature on modeling and optimization related to mini-grids which often miss involving battery degradation, which is of major importance as the battery is a key component in terms of capital and operational expenditures, and the presence of uncertainty and their impact on optimal decision making.
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
In a next step, battery degradation will be included. This will naturally induce a two-time scale optimal control problem for designing mini-grids. The design problem not only includes sizing the BESS as well as all other power components, but it also includes finding optimal operation strategies that strike balance between investment costs, operating costs and minimizing degradation.
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.