Unit Commitment (UC) has a research history of more than 50 years. UC mo= dels being large-scale mixed-integer nonlinear optimization problems soluti= on approaches always have been inspired by ideas from different subdiscipli= nes of optimization, with permanently adjusting =E2=80=9Clarge-scale=E2=80= =9D to bigger and bigger numbers. In recent years, the integration of UC in= to energy optimization models which, themselves, already are large-scale, e= .g., power flow or uncertainty management in production and trading, became= a focal research topic.

In their UC literature synopsis [14] from 1994, Sheble and Fahd review t= he first 25 years of the development and identify approaches some of which = later became major pathways in algorithmic unit commitment. On the one hand= , they address heuristics, such as Exhaustive Enumeration, Priority Lists, = or Simulated Annealing, as well as mathematically rigorous methods from sub= disciplines of optimization as there are Dynamic Programming, (Mixed-Intege= r) Linear Programming, Network Flows, and Lagrangian Methods. The computati= onally more demanding rigorous methods, on the other hand, yield provably o= ptimal solutions or at least lower bounds allowing for gap estimates betwee= n objective function values of the best feasible solution found so far and = lower bounds generated in the course of the algorithm.

**Lagrangian Relaxation - as it was**

In [14] the authors grant a =E2=80=9Cclear consensus presently tending t= oward the Lagrangian Relaxation (LR) over other methodologies=E2=80=9D. Ind= eed, still today LR offers flexible possibilities for relaxing constraints = complicating the model, however, at the cost of having to solve repeatedly = =E2=80=9Cclose cousins=E2=80=9D to the relaxed problem. The key featu= res of LR applied to UC have been and still are:

(i) relaxation of constraints inter-linking units, e.g., load coverage o= r reserve requirements, and arrival at single-unit subproblems,

(ii) addition of the relaxed constraints, together with Lagrange multipl= iers, to the objective, so that the resulting problem is easier to solve th= an the original,

(iii) solution of the convex, non- smooth Lagrangian dual whose objectiv= e-function value calculation benefits from reduction to solving single-unit= subproblems and whose optimal value forms a lower bound to the optimal val= ue of the UC problem,

(iv) application of Lagrangian heuristics to obtain =E2=80=9Cpromising= =E2=80=9D feasible primal solutions from the results of the dual optimizati= on.

**Lagrangian Relaxation - as it is**

Fueled by improved bundle-trust subgradient methods for the Lagrangian d= ual and by permanent progress in =E2=80=9Coff-the-shelve=E2=80=9D mixed-int= eger linear programming (MILP) software, up to the advent of market deregul= ation, two basic aproaches developed which still today are widely used:

(i) LR, often in conjunction with heuristic methods for finding =E2=80= =9Cpromising=E2=80=9D feasible solutions,

(ii) direct solution (by branch-and-bound) of MILP formulations of UC by= =E2=80=9Coff-the-shelve=E2=80=9D solvers.

**Lagrangian Relaxation - as it will be**

Rather than the transition to different time horizons, from short via me= dium to long-term, the new economic environment in the course of energy mar= ket deregulation poses research necessities and provides incentive to integ= rate UC and ED with load flow and uncertainty treatment, [5]. The latter is= intended in the widest sense, from handling stochasticity to topics of mat= hematical equilibria in the context of power trading and bidding into power= markets. In particular, this means to integrate UC into models which alrea= dy are complex themselves.

**Power Flow - Integrating UC and AC Load Flow**

This was considered utopic throughout the =E2=80=9CEarly Days=E2=80=9D, = but now became possible by studying the quadratic nonconvex AC load flow eq= uations from the viewpoint of semidefinite optimization. In [6] after relax= ation of the rank condition the solution to the dual of the remaining conve= x model allows to retrieve a primal solution often meeting the relaxed rank= condition, and thus enabling to solve nonconvex power flow optimization pr= oblems to global optimality.

**Power Flow =E2=80=93 DC Model and Ohmic Losses**

The DC Load Flow Model provides a linear approximation of its AC counter= part by resorting to linear relations and avoiding variables in the space o= f complex numbers, see [3, 4]. The Ohmic Losses approximation, [12], provid= es the possibility to include power losses within the DC-approximation of a= n AC power system. Precise modeling of power losses turns out instrumental = in congestion management when load dispatches or even commitments of units = have to be revised to increase throughput of the grid under increased inflo= ws of renewables.

Despite its success in combinatorial optim= ization, cutting plane methods based on polyhedral studies, either applied = directly or enhancing branch-and-bound came to the fore in UC a bit more th= an 10 years ago, only. At this time, market deregulation enforced the need = of solving UC in a competitive environment under incomplete information. In= this way, solving UC problems became a subroutine in the treatment of more= complex decision problems in electricity supply. Today, tight formula= tions for crucial model ingredients and for complete polytopes arising in U= C are available. In particular, [7] = give a best-possible formulation of minimum up and down times for power-gen= erating units. [11] give an alt= ernative formulation with additional binary variables. [17] extend this to = handle maximum up and down times. [11= ] provide techniques for handling slow- and quick-start units (start= up and shutdown power trajectories for slow-start units, and startup and sh= utdown capabilities for quick-start units)

In the presence of uncertainty, UC either lives in a non-competitive or = competitive environment. The former concerns the time before, the lat= ter since deregulation of energy markets. Before deregulation load has been= the dominating uncertainty prone entity, [15]. After deregulation UC-relev= ant sources of uncertainty have spread considerably: power input from renew= ables, power prices determined by bidding into power exchanges, competitors= =E2=80=99 actions at electricity markets. Yet, UC is understood in a = broader context than before. It rather is the scheduling of decentralized p= ower supply with its small generating facilities than commitment of thermal= let alone nuclear generation units. While the mathematical apparatus is fa= irly well developed for exogenous uncertainty, the situation is completely = different for endogenous uncertainty, i.e., with decision dependent probabi= lity distributions. In case uncertainty is captured by probability me= asures, stochastic integer programming offers methodology for handling UC, = both algorithmically and regarding structural understanding, [2, 13, 15].

The interaction of growing distributed generation, with increased consum= er flexibility and volatility of the input of renewable energy require a de= mand side management with unit commitment in a focal role and without negle= cting the remaining determinates of the generation system. The market is in= vited to provide incentives for market participants to engage in Demand Sid= e Management. Vice versa, engagement of market participants must be c= arried out in rational manner which, in turn, brings to the fore research a= t the interface of energy science and mathematics, with many open problems = up to the present day. Last but not least, uncertainty of crucial model dat= a remains a particular challenge.

[1] Bertocchi, M.I., Consigli, G., Dempster, M.A.H. (Eds.) : Stochastic = optimization methods in finance and energy, Springer, New York, 2011.

[2] Car=C3=B8e, C.C.; Schultz R.: A Two-Stage Stochastic Program for Uni= t Commitment Under Uncertainty in a Hydro-Thermal Power System, ZIB Preprin= t 98-11, Konrad-Zuse-Zentrum fur Informationstechnik (ZIB) Berlin, 1998.

[3] Frank, S., Steponavice, I., Rebennack, S.: A primer on optimal power= flow: A bibliographic survey - (i) formulations and deterministic methods,= Energy Systems 3(2012), 221-258.

[4] Frank, S., Steponavice, I., Rebennack, S.: A primer on optimal power= flow: A bibliographic survey - (ii) non-deterministic and hybrid methods, = Energy Systems 3(2012), 259-289.

[5] Gabriel, S.A., Conejo, A.J., Fuller, J.D., Hobbs, B.F., Ruiz, C.: Co= mplementarity Mdeli in Energy Markets, Springer, New York 2013.

[6] Lavaei, J., Low, S. H. : Zero duality gap in optimal power flow prob= lem, IEEE Transactions on Power Systems 27(2011), 92-107.

[7] Lee, J., Leung, J., Margot, F.: Min-up/min-down polytopes, Discrete = Optimization 1(2004), 7785.

[11] Morales-Espana, G., Gentile, C, Ramos, A.: Tight MIP formulations o= f the power-based unitcommitment problem, OR Spectrum (to appear 2015).

[12] Sanchez-Martin, P., Ramos, A.: . Modeling transmission ohmic losses= in a stochastic bulk production cost model, Instituto de Investigacion Tec= nologica, Universidad Pontificia Comillas, Madrid, 1997.

[13] Schultz, R.: Stochastic programming with integer variables, Mathema= tical Programming 97(2003), 285-309.

[14] Sheble, G.B., Fahd, G.N.: Unit commitment literature synopsis, IEEE= Transactions on Power Systems 9(1994), 128-135.

[15] Takriti, S., Birge, J.R., Long, E.: A stochastic model for the unit= commitment problem, IEEE Transactions on Power Systems 11(1996), 1497-1508= .

[16] D. Rajan and S. Takriti. Minimum up/down polytopes of the = unit commitment problem with start-up costs. Technical report, IBM Research= Report RC23628, 2005.

[17] Queyranne, M., Wolsey, L.A. = (2017). "Tight MIP formulations for bounded up/down times and interval-depe= ndent start-ups". Mathematical Programming 164 129=E2=80=93155.

**Contributors**:

Dr Michael Diekerhof, RWTH Aachen University

Dr Mehdi Madani, Universit=C3=A9 catholique de Louvain

Prof. Jon Lee, University of Michigan