Mathematical models

In vertically integrated systems the strategic electrical network management is performed in an integrated fashion by the monopolist, whereas in those market based, this problem is responsibility of another entity usually called the Transmission System Operator (TSO). The transmission network is the nervous system of any EES and the strategic network management poses very challenging issues. Basically in the long term perspective, the main goal are:

In order to approach the three above-mentioned classes of problems, many tools described in the short term management are obsviously used. Their usage is somehow differently oriented however:

Differently from the short term management, in the strategic view, these problems are solved in models equipped with an upper lever set of decision variables, that indicate the virtual presence (or dismission) or a certain set of new branches, special devices and some representation of their costs. Additionally in recent times, due to the increasing capability of storage mainly at the distribution level, also the size, types and siting of such storage equipment contributes to the set of the decision variables. From a methodological standpoint, very often these prescriptive problems are tackled in a what if analysis fashion, or with approaches that fall in the broad definition of optimization with costly function. In other words the upper level decision variables are seen as a set on which the decision maker does sensitivity or in a more sophisticated approaches as a set of variables whose change on cascade produces another optimization problem with other - more operating - variables. However especially in the scientific literature there are attempts to model the whole problem with all variables at the same level.

Indeed, it is very important to consider the operation and scheduling of generation and storage units already at design phase to determine the most convenient combination (i.e. minimum objective function) of technology selection and size. This is especially true when dealing with selection, sizing and unit commitment of long-term, or seasonal, energy storage. Long-term storage systems have recently caught much attention due to their ability to compensate the seasonal intermittency of renewable energy sources. However, compensating renewable fluctuations at the seasonal scale is particularly challenging: on the one hand, a few systems, such as hydro storage, hydrogen storage and large thermal storage can be used to this purpose; on the other hand, the optimization problem is complicated due to the different periodicities of the involved operation cycles, i.e. from daily to yearly. This implies long time horizons with fine resolution which, in its turn, translates into very large optimization problems. Furthermore, such systems often require the integration of different energy carriers, e.g. electricity, heat and hydrogen. Exploiting the interaction between different energy infrastructure, in the so-called multi-energy systems (MES), allows to improve the technical, economic and environmental performance of the overall system [3].

In this framework, including the unit commitment problem already at design phase implies taking into account the expected profiles of electricity and gas prices, weather conditions, and electricity and thermal demands along entire years. Moreover, the technical features of conversion and storage units should be accurately described. The resulting optimization problem can be described through a mixed integer nonlinear program (MINLP), which is often simplified in a mixed integer linear problem (MILP) due to the global optimality guarantees and the effectiveness of the available commercial solvers (e.g. CPLEX, Gurobi, Mosek, etc.). In this context, integer variables are generally implemented to describe the number of installed units for a given unit, whereas binary variables are typically used to describe the on/off status of a certain technology. Furthermore, decomposition approaches relying on meta-heuristic algorithms for unit selection and sizing have been proposed. A comprehensive review of MINLP, MILP and decomposition approaches for the design of MES including storage technologies has been carried out by [4]. However, independently of the implemented approach, significant model simplifications are required to maintain the tractability of the problem. Such simplifications include limiting the number of considered technologies, restricting technology installation to a subset of locations, analyzing entire years based on seasonal design days or weeks, or aggregating the hours of each day into a few periods.

References:

[1] D. Bienstock and A. Verma. The n - k problem in power grids: New models, formulations and numerical experiments. 1386 Siam J. on Optimization, 2011.

[2] Mostafa Nick, Rachid Cherkaoui, Mario Paolone. Optimal siting and sizing of distributed energy storage systems via alternating direction method of multipliers. International Journal of Electrical Power & Energy Systems. 03/2015.

[3]   P. Mancarella, MES (multi-energy systems): An overview of concepts and evaluation models, Energy. 65 (2014) 1–17.

[4]   C. Elsido, A. Bischi, P. Silva, E. Martelli, Two-stage MINLP algorithm for the optimal synthesis and design of networks of CHP units, Energy. 121 (2017) 403–426.

Contributor

Prof. Laureano Escudero, Universidad Rey Juan Carlos

Dr Fabrizio Lacalandra, QuanTek

Dr Paolo Gabrielli, ETH Zürich