## 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:

- Reinforce the networks itself by constructing new branches and possibly dismissing old ones. Additionally the decision of installing network technologies, together with their siting, can be considered in the grid reinforcement process. These technologies could enhance the operation/controllability of the grid and include: Phasor Measurement Units (
*PMUs*), Wide-Area Measurement Systems (*WAMS*), and*notably*Flexible Alternating Current Transmission Systems (*FACTS*). It is in fact known that power flowing on individual lines cannot be controlled, though these devices are improving the ability of system operators to do so (possibly in conjunction with OTS related operations). - Energy Storage System siting and sizing, say deciding where locate and the size of an ESS (e.g. 2).
- Smart grids design. The actual design of a smart grid in a certain local area is another universe of related (optimization) problems, and surely includes also previous two class of problems.

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:

**Load Flow**: LF is actually*not*an optimization problem, it is (just) a calculation of the power flowing along an electrical network where we have fixed the generation schedule and the load in the several nodes of the grid. While being not an optimization problem, it gives evidence on the networks operating points under different conditions. Under strategic perspective LF can be used integrated in a what if analysis tool.**Optimal Power Flow**: The OPF problem deals with the optimization of the generating cost, and possibly hydro resources, considering the electricity grid. In considering the grid OPF takes into account the non linear Kirchhoff laws and the restrictions on power flow on each branch (transmission line) and voltage angles. Typically the generation cost optimization is performed considering all the units status (on or off)*fixed*to a feasible status otherwise found. As for the LF, also OPF under strategic perspective can be used integrated in a what if analysis tool.To date there are many formulations of OPF from the first one appeared in the sixties, they basically fall into two broad classes:*The Direct Current (DC) model*: here the network structure is taken into account, including the capacity of the transmission links, but a simplified version of Kirchhoff laws is used so that the corresponding constraints are still linear.*The Alternative Current (AC) model:*here the full version of Kirchhoff laws is used, leading to highly nonlinear and nonconvex constraints. To cope with these difficulties a recent interesting avenue of research concerns the fact that the non-convex AC constraints can be written as quadratic relations. In particular quadratic relaxation approach have been proposed which builds upon the narrow bounds observed on decision variables (e.g. phase angle differences, voltage magnitudes) involved in power systems providing a formulation of the AC power flows equations that can be better incorporated into UC models with discrete variables. Again in the long term, the level of details of the OPF models can be adjusted depending on the goals.

**Security Constrained UC (SCUC):**SCUC is an integrated problem, say an integration of OPF and UC. So from one side one wants to consider a detailed set of constraints from power plants and from the other the physics of the grid itself as in an OPF. The inclusion of the status variables as in an ordinary UC further complicates the problem.**N-k OPF/SCUC/security:**This problem is an example of how things are decoupled in power systems. The issue here is to find a least cost schedule of production and flows that is also resistant to unpredictable fault of one of the component (power plant, network branch etc.). The n-1 security problem refers to a single fault. From a methodological standpoint one could consider n-k SCUC as an integrated problem, and some modeling proposal in this direction have been presented. In practice TSO tend to decouple OPF or SCUC from n-k, solving this latter problem by adding security requirements to an already*quasi-*fixed solution from SCUC (e.g. 1)

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.

**References:**

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

- 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.

## Optimization methods