In vertically integrated systems the short term electrical network manag= ement is performed in an integrated fashion by the monopolist, whereas in t= hose market based, this problem is responsibility of another entity us= ually called the Transmission System Operator (TSO). The transmission netwo= rk is the nervous system of any EES and the network management poses very c= hallenging issues. The list of the relevant problems in this field we will = describe is as follows:

**Load Flow**: LF is actually*not*an optimization= problem, it is (just) a calculation of the power flowing along an electric= al 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 give= s evidence on the networks operating points under different conditions.**Optimal Power Flow**: The first historical problem in ne= twork management is called Optimal Power Flow (OPF). 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 a= ccount the non linear Kirchhoff laws and the restrictions on power flow on = each branch (transmission line) and voltage angles. Typically the generatio= n cost optimization is performed considering all the units status (on or of= f)*fixed*to a feasible status otherwise found. To date there= are many formulations of OPF from the first one appeared in the sixties, t= hey basically fall into two broad classes:*The Direct Current (DC) model*: here the network structure is t= aken into account, including the capacity of the transmission links, but a = simplified version of Kirchhoff laws is used so that the corresponding cons= traints are still linear.*The Alternative Current (AC) model:*here the full version o= f Kirchhoff laws is used, leading to highly nonlinear and nonconvex constra= ints. To cope with these difficulties a recent interesting avenue of resear= ch concerns the fact that the non-convex AC constraints can be written as q= uadratic relations. In particular quadratic relaxation approach have been p= roposed which builds upon the narrow bounds observed on decision variables = (e.g. phase angle differences, voltage magnitudes) involved in power system= s providing a formulation of the AC power flows equations that can be bette= r incorporated into UC models with discrete variables (e.g. 3).

**Security Constrained UC (SCUC):**SCUC is an integrat= ed 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 othe= r 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 o= f 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 un= predictable fault of one of the component (power plant, network branch etc.= ). The n-1 security problem refers to a single fault. From a methodolo= gical standpoint one could consider n-k SCUC as an integrated problem, and = some modeling proposal in this direction have been presented. In practice T= SO tend to decouple OPF or SCUC from n-k, solving this latter problem by ad= ding security requirements to an already*quasi-*fixed solution from= SCUC (e.g. 4)**.****Optimal Network Islanding (ONI):**When things go wro= ng and the schedule from n-k security arrangements fails to be sufficient, = there is another*real time*issue: how to disconnect pieces (island= ) of the networks so that cascade failure is minimized**.**By= isolating the faulty part of the network, the total load disconnected in t= he event of a cascading failure could be reduced. Controlled islanding or s= ystem splitting is therefore attracting an increasing amount of attention. = The problem is how to efficiently split the network into islands that are b= alanced in load and generation, and have stable steady-state operating poin= ts. This is a considerable challenge, since the search space of line cutset= s grows exponentially with the networks nodes number. The critical issue he= re is the time, since as a conseguence of a severe fault, few minutes or ev= en seconds are available to perform actions aimed at triggering these spits= .**Optimal Transmission Switching (OTS):**This problem rec= ently emerged mostly because of the increasing penetration of renewables (s= olar or wind with virtually zero cost) power plants. Network must be design= ed with redundancies in order to cope with failure, but on considering a st= eady state operation these redundancies can result in bottleneck. OTS probl= em is an example of braess's paradox where r= emoving some branch can actually be beneficial in terms of overall cost of = supply (e.g. 5). From a technical point of view the reason here is that rem= oving a branch also removes a potential-like constraint on the nodes voltag= e angle. In fact even if a simple DC model is considered the power flowing = on a branch is subjected to be proportional to the differences of voltage a= ngles at sending and receiving nodes (ultimately the kirchhoff's laws). Esp= ecially in fully (but not necessarely) renewable penetrated systems, the co= ncentration of these power plant can be quite high (i.e. they are clustered= where sun or wind is high), this in turn means that in these areas network= conditions on some branch may easily became severe. Like in the SCUC probl= ems, OTS could - and should - be considered as a single integrated problem = together with SCUC/OPF itself. In practice, at the present time and knowled= ge, the complexity is too high for an optimization problem that can deal fo= r a real sized grid with 1) a full AC representation of the grid, 2) a deta= iled representation of the power plant restrictions, 3) the degree of freed= om of tripping some line out in order to reduce costs and possibly the inhe= rent reduction of n-k like-security standard of the system. As a consequenc= e, scientific literature has concentrated in the last years in simplified, = PoC-like, models while from an industrial perspective some TSO uses some he= uristic approach.**Smart grids operations:**The smar= t grid paradigm in the electricity contest has emerged in the last year= s as somehow natural evolution of the classical EES dating to more than an = 100 years ago. There many similar definitions of smart grid but central poi= nts are 1) the smart capability of operating 2) interacting and 3) integrat= ing customers into the system by means of more and more sophisticated IT fa= cilities. Therefore some problems remain centralized but there is an active= partecipation of the final users. Because of this interactions of decision= s, coupled with a more and more penetration of renewables production, a bro= ad class of optimization problems emerged. For instance, the distribution s= ystem in classic models has always been seen as a passive element. However = with the increase of distributed production (mainly solar but also wind) an= d most importantly some form of distributed storage (batteries, compressed air storage, but also Plug in hyb= rid electric vehicle, PHEV) the fluxes of energy can be dynamically reverse= d creating opportunities for optimization and adding complexity to the oper= ation of the whole system. Also, a partial load shifting from peak hours to= off peak hours is now a reality at least in principle. However when we spe= ak of Smart Grids the relevat portion of the network involved is at the dis= tribution level (althought transmission one is of course involved as well).= Also, as appears from the definition, the smart grid paradigm involves cha= nges in many structural part of the distribution system as a whole. Many of= the atomic problems explained before are still of interest (mainly LF, OPF= , ONI and OTS) together with more specific problems such as optimal PHEV ch= arging dynamics expecially when price elasticity is high (e.g. 6)**.<= /strong>**

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