One of the major and difficult problems in the energy area that the Euro= pean Union (EU) is facing today consists of estimating the timing for clean= power generation technologies and electricity transmission expansion netwo= rk at a pan-European level in a long term (e.g., 30 years time horizon). EU= has established aggressive pollutant emission reduction targets: a 20% (re= s. 27%) reduction in greenhouse gases with respect to 1990 levels by 2020 (= res. 2030) and an objective of 80% reduction by 2050. See Eur= ostat.

Mathematical optimization models and algorithms for problem solving to a= ddress the above challenges in the electricity open market [14] are essenti= al computerized tools for helping in the decision making for estimating the= following key issues: feasible type and mix of power generation sources, r= anging from less coal, nuclear and combined cycle gas turbine to more renew= able energy sources (RES), namely hydroelectric, wind, solar, photovoltaic = and biomass, among others; and timing for power generation plant / farm sit= e locations and dimensions. The solution should maximize different types of= utility criteria and quantifying the benefits of using cleaner, safer and = efficient (cheaper) energy.

In the past (say, up to 25 years ago) practically all over the world the= energy sector was a very centralized one where the Generation Companies (e= lectricity generating companys) had a reduced decision-making on generation= expansion capacity planning. The energy prices were centrally decided as w= ell as the main geographical areas where to service the energy demand. So, = the maximization of profit functions was out of question and, then, the mai= n goal was to minimize the net present value (NPV) of the global cost for t= he planning of the site, location and capacity of new energy generation sou= rces in, first, hydropower, second, variety of thermal plants, and also in = the too-much controversial nuclear generation. In the latter energy source,= not too-much room was left for the modeling tools to help in the decision = making. On the other hand, the claims of other stakeholders (mainly, enviro= nmentalist ones) were not a strong issue given the strong regulation of the= sector. The modeling could consider uncertainty on the main parameters tha= t, by definition of the non-open market and, then, its strong regulation, w= as reduced to macro-economic and demographic factors that influence the ene= rgy demand to serve and the generation disruption. On the other side, the s= tate-of-the-art on theory, modeling and algorithms for dealing with mathema= tical optimization under uncertainty (i.e., stochastic optimization) was no= t as advanced as it is today and, sure, will be in the future. Additionally= , the HW/SW computer power was so low (until, say, 10-15 years ago) that th= e gigantic models that were needed to provide solutions to help the decisio= n-making could not be considered.

Today the situation has drastically changed. The hardware/software = computer power capability is very high, and it seems that its exponential g= rowth will continue at least in the near future. On the other hand, some st= ochastic optimization tools could be considered today as a sort of commodit= ies ready to be used. Additionally, the energy sector is very different fro= m what it was in the past. It continues being a crucial sector for the Euro= pean economy as for the rest of the world. However, its market, without bei= ng fully an open one (something that, by definition, probably it cannot be)= , it allows higher freedom to the GenCo for performing strategic energy gen= eration capacity expansion planning in a long horizon. First, there is enou= gh freedom on deciding the amount of power generation from the different pl= ants / farms and, on the other hand, the energy price is not (fully, at lea= st) decided by regulation. Second, the electricity generating company = has very much freedom on deciding the location, capacity and timing on new = generation sources. Third, the macro-economic and demographic factors are n= ot the only main factor to influence the demand, but the competitors' strat= egies are a major source of uncertainty. Fourth, the power to be generated = by some news RES, mainly wind and, in a lesser extent, solar and photovolta= ic sources, is subject to a high uncertainty (that, on the other hand, it i= s difficult to formulate). Fifth, given the type of new energy sources, the= re is more variety on the location sites and generation capacity, which all= ows considering more opportunities for strategic planning. And, sixth, ther= e are other stakeholders (environmentalists, among others) having different= goals (whose directions are not the same sense as the electricity generati= ng company) that, in some way, have to be considered in the decision making= , plus some Government and EC directives, etc. All of that induce to consid= er multiobjective optimization.

So, the electricity generating company's aim has been moved from cost mi= nimization to expected profit maximization along the time horizon. One of t= he interesting disciplines for problem solving is stochastic optimization, = where the uncertainty of the main elements is represented (i.e., quantified= ) by a finite set of scenarios, ether in a two stage or a multistage scenar= io tree. Anyway, the problem modeling and algorithm development are big cha= llenges, given the gigantic character of the problem. As an additional diff= iculty, big GenCos can influence some of the main uncertain parameters in t= he problem; say the energy price, among others. Then, the probability and v= alue along the time horizon can be influenced by the decision-maker. In thi= s case, the so-called decision-dependent probabilities [15,18,19] (also cal= led endogenous uncertainty) should be considered. One of the potential tool= s consists of using stochastic mixed 0-1 quadratic mechanisms; see [10], to= be added to the modeling schemes presented next.

Multistage stochastic mixed 0-1 optimization modeling for risk managemen= t should be considered. There are different approaches (some of them very r= ecent ones) for energy generation planning, see [1,2,7,13,21,22,23], among = others. The main parameters in the problem are uncertain and, so, a set of = scenarios should be generated by considering the realization of the followi= ng parameters, at least: availability (and price, in case) of raw material = for power generation: gas, fuel, water, wind, solar, etc.; electricity dema= nd and prices at focal nodes in the energy network; operating hours per per= iod of power generation technologies; CO2 emission permits; green Certifica= tes prices for buying and selling in the market, and allowed bounds; power = generation costs of different technologies; electricity loss of candidate p= ower generation technologies; investment allocation bounding of the cost fo= r the total power generation.

There is not a unique function-criterion to consider. Rather, it is a mu= lticriteria problem, since the model considers the maximization of the NPV = of the expected profit of the investment and consumer stakeholders=E2=80=99= goals over the scenarios along the time horizon, subject to risk reduction= of the negative impact of the solution to be provided by the optimization = system in non-wanted scenarios, plus utility objectives of other stakeholde= rs. Those other objectives include the power share of cleaner, safer and ef= ficient energy accessible to all consumption nodes, cost investment from pr= ivate and public institutions, generation network reliability, EC directive= s and EU Governments on environmental issues and others.

One of the difficult problems to deal with is the generation of a set of= scenarios to represent the realization of the uncertainty as structured in= a multistage scenario tree. Hence, a node in the tree for a given stage is= related (in a one-to-one correspondence) to a group of scenarios that up t= o the stage have the same values in the uncertain parameters. Then, the sol= ution for those scenarios should be unique for all stages up to the stage w= here the node in the tree belongs to, i.e., the so-named nonanticipativity = principle is satisfied.

In the so-called Risk Neutral approach (RN), the function to maximize co= nsists of the NPV of the expected profit along the time horizon over the sc= enarios with the following elements related to a given GenCo the energy net= work: revenue from sale of electricity, revenue from sale (or, alternativel= y, cost from purchase) of Green Certificates, penalization of CO2 emissions= , variable generation cost of thermal power plants, variable generati= on cost of renewable energy source power plants / farms (wind, solar, photo= voltaic, biomass, etc.), periodic debt repayment of the investment on the n= ew power plants / farms and new hydro power turbines, fixed power generatio= n cost of available new plants / farms and new hydro power turbines, = etc.

The gigantic character of the problem can be assessed by considering its= dynamic setting (say, 30 years time horizon), the number and dimensions of= replicated networks (i.e., hyper hydro valleys) in the time horizon for so= me big generator companies (e.g., EdF has 20+ valleys, some with 50+ elemen= ts, see [4]), multiple choices in time and space of location and capacity d= ecisions for the energy generation system of the given GenCo (current and c= andidate power generation plants / farms), and the representative scenario = tree to consider for the uncertain parameters.

The aim of the Risk Neutral type model performs the maximization of the = NPV of the expected profit. The main drawback of this popular strategy is t= hat it ignores the variability of the profit over the scenarios, in particu= lar the =E2=80=9Cleft=E2=80=9D tail of the profits of the non-wanted scenar= ios. For the problems with so high variability, there are some risk averse = approaches that, additionally, deal with risk management. Among them, the s= o-called time-inconsistent stochastic dominance (TSD) measure reduces the r= isk of the negative impact of the solution in non-wanted scenarios in a bet= ter way than others under some circumstances. See [3] for a computational c= omparison of some risk averse strategies, see also [5].

The TSD measure presented in [9] for stochastic problems as a mixture of= first- and second-order stochastic dominance strategies. It is a multistag= e extension of the two-stage strategies introduced in [16,17], plus the con= sideration of hedging the solution against some types of negative impacts i= n non-wanted scenarios at selected stages along the given time horizon.

Then, the maximization of the NPV of the expected profit is subject to a= scheme for risk management that consists of appending to the model a set o= f TSD constraints for given profiles at a stage subset for each function (i= ncluding the objective one), such that a profile is given by the 4-tupla: t= hreshold on the function value; maximum target shortfall on reaching the th= reshold that is allowed for each node in the scenario tree related to any o= f those stages; bound target on the probability of failure on reaching the = threshold; and bound target on the expected shortfall.

As an alternative, the time consistent strategy proposed in [12] is so-c= alled the expected stochastic dominance (ESD) measure. In ESD, however, the= profiles are associated with the nodes of a modeler-driven stage subset fo= r each function, where a profile consists of the 4-tupla: threshold of the = function to be satisfied by any scenario in the group with one-to-one corre= spondence to the node; maximum shortfall of the value of the function that = is allowed for any of those scenarios; upper bound target on the expected d= eficit (shortfall) on reaching the threshold that is allowed for that group= of scenarios; and upper bound on the probability of failing to satisfy the= threshold.

The rationale [20] behind a time-consistent risk averse measure is that = the solution value to be obtained for the successor set of a give node in t= he scenario tree for the related time consistent submodel, see [12], 'solve= d' at the stage to whom the node belongs to, should have the same value as = in the original model 'solved' at the beginning of the time horizon. See in= [20] and references therein the time consistent version of CVaR (Condition= al Value-at-Risk), a very popular risk averse measure.

It is worth to point out that, by construction, the time-consistent vers= ion of risk averse measure does not avoid the risk on non-desired shortfall= s on reaching the thresholds for the given functions at intermediate stages= of the (long) time horizon for the energy generation expansion planning pr= oblem. It is a challenge for problem solving, but both time-consistent and = time-inconsistent versions of risk averse measures should be jointly consid= ered in the same model.

The gigantic but well structured multicriteria multistage stochastic non= linear mixed integer (SMINO) problem with risk management cannot be solved = up to optimality, see [8], such that a realistic approach could consist of = a combination of the following elements: sample scenario schemes, iterative= algorithms for solving SMINO by sequential mixed 0- linear one; node-based= decomposition algorithms; stochastic mixed 0-1 bilinear optimization solve= rs; and high performance computing. See in [11] a review of decomposition a= lgorithms for multistage stochastic problem solving.

**References**

[1] W. van Ackooij, R. Henrion, A. Moelle and, R. Zorgati. Joint chance =
constrained programming for hydro reservoir management. *Optimization En=
gineering*, 15:509-531, 2014.

[2] W. van Ackooij and C. Sagastizabal. Constrained bundle methods=
for upper inexact oracles with application to join chance constrained ener=
gy problems. *SIAM Journal on **Optimization*, 24:733-76=
5, 2014.

[3] A. Alonso-Ayuso, F. Carvallo, L.F. Escudero, M. Guignard, J. Pi, and=
R. Puranmalka, A. Weintraub. On the optimization of copper extraction in m=
ining under uncertainty in copper prices. *European Journal of Operation=
al Research*, 233:711-726, 2014.

[4] S. Charousset. A review of the main mixed-integer nonlinear optimiza= tion problems regarding electricity generation management at Electrici= te de France. Cost Workshop on Mixed-Integer NonLinear Programming. EC= Cost Action TD1207, Paris, 2013.

[5] A.J. Conejo, M. Carrion and J.M. Morales. Decision making under unce= rtainty in electricity markets. Springer, 2010.

[6] G. Emiel and C. Sagastizabal. Incremental like bundle methods with a=
pplications to energy planning. *Computational Optimization and Applicat=
ions*, 46:305-332, 2009.

[7] L.F. Escudero, J.L. de la Fuente, C. Garcia and F.J. Prieto, Hydropo=
wer generation management under uncertainty via scenario analysis and =
parallel computation*. IEEE **Transactions on Power Systems<=
/em>, 11:683-690, 1996.*

[8] L.F Escudero, A. Garin, M. Merino and G. Perez. A decomposition fram= ework for solving dynamic MINLP problems under uncertainty. Workshop o= n Mixed Integer

Nonlinear Programming, Workshop on Mixed Integer Nonlinear Programming,&= nbsp;WMINLP2014. http://minlp.cheme.cmu.edu/2014/index= .html. Carnegie Mellon University, Pittsburgh (PA, USA), 2014.

[9] L.F. Escudero, A. Garin, M. Merino and G. Perez. On time stochastic =
dominance induced by mixed-integer linear recourse in multistage stochastic=
programs. *European Journal of Operational Research*, 249:164-176, =
2016.

[10] L.F Escudero, A. Garin, M. Merino and G. Perez. On multistage mixed= 0-1 optimization under a mixture of Exogenous and Endogenous Uncertainty i= n a risk averse environment Mini-symposium on Exogenous and Endogenous unce= rtainty in Stochastic Programming. 13h Tri-annual International Conferecne = on Stochastic Programming (ICSP). Bergamo (Italy), 2013.

[11] L.F Escudero, A. Garin and A. Unzueta. Cluster Lagrangean decomposi= tion for time stochastic dominance induced by mixed integer-linear recourse= in multistage stochastic optimization. Submitted.

[12] L.F. Escudero, J.F. Monge and D. Romero Morales. On time-consistent= stochastic dominance risk averse measure for Tactical Supply Chain Plannin= g under uncertainty. Submitted.

[13] L.F. Escudero, I. Paradinas, J. Salmeron and M. Sanchez, SEGEM: A S=
imulation approach for Electric Generation Management. *IEEE Transaction=
s on Power Systems*, 13:738-748, 1998.

[14] L.F. Escudero and M.V.F. Pereira. New trends and OR/MS opportunitie=
s in the electricity open market. *OR/MS Today*, 27,2:42-46, 2000.

[15] V. Goel and I. Grossmann. A class of stochastic programs with=
decision dependent uncertainty. *Mathematical Programming*, 108: 35=
5-294, 2006.

[16] R. Gollmer, U. Gotzes and R. Schultz. A note on second-order =
stochastic dominance constraints induced by mixed-integer linear recourse. =
*Mathematical Programming*, Ser. A, 126:179=E2=80=93190, 2011.

[17] R. Gollmer, F. Neise and R. Schultz. Stochastic programs with=
first-order stochastic dominance constraints induced by mixed-integer line=
ar recourse. *SIAM Journal on Optimization*, 19:552-571, 2008.

[18] V. Gupta and I.E. Grossmann. Solution strategies for multistage sto=
chastic programming with endogenous uncertainties. *Computers & Chem=
ical Engineering*, 35:2235-2247, 2011.

[19] L. Hellemo, P.I. Barton and A. Tomasgard Stochastic Programming wit= h Decision-Dependent Probabilities. Submitted.

[20] T. Homem-de-Mello and B.K. Pagnoncelli. Risk aversion in mult=
istage stochastic programming: A modeling and algorithmic perspective. *=
European Journal of Operational Research*, 249:188-199, 2016.

[21] M.V.F. Pereira and L.M.V.G. Pinto.. Multi-stage stochastic optimiza=
tion applied to energy planning. *Mathematical Programming*, 52:359-=
375, 1991.

[22] C. Sagastizabal. Divide to conquer: Decomposition methods for energ=
y optimization. *Mathematical Programming Ser. B*, 134:187-222, 2012=
.

[23] M.T. Vespucci, M. Bertocchi, S. Zigrino and L.F. Escudero. Stochast= ic optimization models for the power generation capacity expansion problem = with risk management. doi:10.1109/EEM.2013.6607352. 10th International Conf= erence IEEE on the European Energy Market EEM13, Stockholm (Sweden), 2013

In monopolistic environment for energy transmission network capacities w=
ere designed with a wide safety margin, so for a long time expansion planni=
ng in electrical energy systems was concentrated on generation expansion pl=
anning (GEP) with the goal to cover cumulative demand uncertainty based on =
averaged historic demand data. These were modeled as stochastic optimizatio=
n problems with a one dimensional demand distribution represented by 2-stag=
e or multi-stage scenario trees that were generated by Monte Carlo methods.=
The models went to the limit of computational possibilities at any point i=
n time,

included binary decision variable, with a risk neutral approach =
and, then, only expected values in the objective function where considered =
in the time horizon over the scenarios. Very limited use was made of risk a=
verse measures.

In order to solve the large scale problems, decompos=
ition methods played a central role, in particular the following methodolog=
ies:

- Two-stage Benders Decomposition (BD) for linear problems [11]. See [8,3= 7,55], among many others.
- Multistage Benders Decomposition (BD) methodology for linear problems. = See [13] among others.
- Two-stage Lagrangean Decomposition (LD) heuristic methodology. Se= e [14,1718,29,28,33,41,42], among many others. See also [2, 56] for t= wo surveys on the state-of-the-art of two-stage stochastic unit commitment,= and using LD with bundle methods. See also [1,16,51] two-stage LD ap= proaches with bundle methods applied to energy problems.
- Multistage Clustering Lagrangean Decomposition (MCLD) heuristic methodo= logy. See [21,22,23,38], among some others.
- Regularization methods. See [9, 33, 39, 49, 50, 53], among others.
- Progressive Hedging algorithm (PHA) for multistage primal decomposition= . See [47,57], among others.
- Multistage Stochastic Dynamic Programming (SDP). See [4,15,25,26,30,32,= 40,44,45,48,54], among others.
- Multistage cluster primal decomposition. See [5,7,10,19,24,38,43,52,58]= , among others.
- Parallelized decomposition algorithms. See [3,4,5,6,9,12,34,39,43,48,52= ,58], among others.

Today, new power production possibilities, technological developments an=
d deregulation bring along several new sources of uncertainty with highly d=
iffering levels of variability. In addition to traditional demand these are=
foremost dependencies on wind, market prices, mobile electricity consumers=
like cars, power exchanges on international level, local energy producers =
on distribution network level and, to a lesser extent, solar radiation. Thi=
s introduces complex and volatile load and demand structures that pose a se=
vere challenge for strategic planning in production and transmission and, o=
n a shorter time scale, in distribution. Networks may now be equipped with =
new infrastructure like Phase Measurement Units (PMUs) and other informatio=
n technology in order to improve their cost efficiency. At the same time th=
ese upgraded networks should ensure high standards in reliability in their =
daily use and resilience against natural or human caused disasters. Compani=
es now have teams devoted to the task of generating suitable planning data.=

In optimization models, the emphasis has shifted to high dimensiona=
l stochastic data and to considering risk reduction measures instead of exp=
ected values. Computationally integrated models considering all relevant as=
pects are out of scope. Even for simplified models it is often difficult or=
not known how to provide stochastic data of sufficient quality [63]. Alter=
natives are then:

- robust optimization, where distributions are replaced by "easier" uncer= tainty sets [59],
- fuzzy methods, where uncertainties are replaced by a kind of interval a= rithmetic equipped with scenario dependent probabilities [60],
- information gap decision theory that aims at hedging against info= rmation errors [61,62].

Methods for solving these stochastic optimization problems with binary d=
ecision variables employ the same decomposition approaches listed above, bu=
t much more care needs to be devoted to the properties of the decomposition=
. For risk averse measures in multistage models, methods are distinguished =
regarding their "time consistency" or "time inconsistency". So far, stochas=
tic dynamic programming approaches are the most suitable ones for dealing w=
ith the time consistency property of risk measures, so that the original st=
ochastic problem may be decomposed more easily via scenario clustering and =
cluster dependent risk levels.

In power generation optimization mode=
ls for big companies the following are the issues of relevance, mainly addr=
essed in the context of market competition:

- when and where to install how much new production capacity, mainly cons= idering wind generators and thermal plants (decisions on nuclear power are = political)
- how to extend or renew hydro plants and where to install what pumping c= apacities. Today, solar power is typically handled at the level of di= stribution networks.

In contrast, competition is not an issue for transmission and distributi= on network operators. Regulations on efficiency, reliability and resilience= levels are the driving force in the following problems:

- when and where to install how much network capacity and information equ= ipment,
- reducing transmission losses,
- reducing distribution losses (technical and detecting non-technical one= s).

Challenges today and for the future comprise:

- The robust approach allows for safe optimization with uncertain data.&n= bsp; What information can be extracted from these robust solutions e.g. on = which additional data would be needed to improve the quality of the model?<= /li>
- Several risk averse measures have been proposed, each with&= nbsp; its advantages and disadvantages. How to make use of them in the best= way?
- How to deal with endogenous uncertainty, i.e., with optimizing big play= er decisions that influence the probability distributions that are optimize= d over?
- How to construct hierarchical decomposition approaches in a consistent = way?
- How to make use of high performance computing (HPC, multi-core or Distr= ibuted) in decomposition approaches?
- How to integrate chance constraints (ICC), e.g. with respect to reliabi= lity or resilience?

General goals for future models include: increasing the level of integra= tion; bringing models closer to reality by avoiding the excessive lineariza= tion of nonlinear aspects; reducing the gap between methods used in academi= a and those applied in practice; making use of new monitoring devices and c= ommunication systems; exploring the chances of cooperation between electric= and other energy commodity systems.

**References**

[1] W van Ackooij and C. Sagastizabal=
. Constrained bundle methods for upper inexact oracles with application to =
join chance constrained energy problems. SIAM Journal on Optimization, 24:7=
33-765, 2014.

[2] W. van ackooij and J. Malick. Decomposition algori=
thm for large-scale two-stage unitcommitment. Annals of Operatiosn Research=
, 238:587-613, 2016.

[3] U. Aldasoro, L.F. Escudero, M. Merino and G=
. Perez. An algorithmic framework for solving large-scale multistage stocha=
stic mixed 0-1 problems with nonsymmetric scenario trees. Part II: Parallel=
ization. Computers & Operations Research, 40:2950-2960, 2013.

[4=
] U. Aldasoro, L.F. Escudero, M. Merino, J.F. Monge and G. Perez. On parall=
elization of a Stochastic Dynamic Programming algorithm for solving large-s=
cale mixed 0-1 problems under uncertainty. TOP, 23:703-742, 2015.

[5=
] U. Aldasoro, L.F. Escudero, M. Merino, J.F. Monge and, G. Perez. A parall=
el Branch-and-Fix Coordination based matheuristic algorithm for solving lar=
ge-scale multistage stochastic mixed 0-1 problems. Submitted.

[6] T.=
Al-Khamisl and R. MHallah. A two-stage stochastic programming model for th=
e parallel machine scheduling problem with machine capacity. Computers &=
; Operations Research, 38:17471759, 2011.

[7] A. Alonso-Ayuso, L.F. =
Escudero and M.T. Ortu=C3=B1o. BFC, a Branch-and-Fix Coordination algorithm=
ic framework for solving stochastic 0-1 programs. European Journal of Opera=
tional Research, 151:503-519, 2003.

[8] L. Aranburu, L.F. Escudero, =
M.A. Garin, G. Perez, A so-called Cluster Benders Decomposition approach fo=
r solving two-stage stochastic linear programs, TOP, 20:279-295, 2012.

<=
br>[9] T. Asamov and W.B. Powell. Regularized Decomposition of HighDimensio=
nal Multistage Stochastic Programs with Markov Uncertainty. arXiv: 1505,022=
27, 2015.

[10] C. Beltran-Royo, L.F. Escudero, J.F. Monge and R.E. R=
odriguez-Ravines. An effective heuristic for multistage stochastic linear o=
ptimization. Computers & Operations Research, 51:237-250, 2014.

=
[11] J.F. Benders. Partitioning procedures for solving mixed variables prog=
ramming problems. Numerische Mathematik, 14:238-252, 1962.

[12] P. B=
eraldi, L. Grandinetti, R. Musmanno and C. Triki. Parallel algorithms to so=
lve two-stage stochastic linear programs with robustness constraints. Paral=
lel Computing, 26:18891908, 2000.

[13] J.R. Birge. Decomposition and=
partitioning methods for multi-stage stochastic linear programs. Operation=
s Research, 33:989-1007, 1985.

[14] C.C. Car=C3=B8e and R. Schultz. =
Dual decomposition in stochastic integer programming. Operations Research L=
etters, 24:37-45, 1999.

[15] M.P. Cristobal, L.F. Escudero and J.F. =
Monge. On Stochastic Dynamic Programming for solving large-scale tactical p=
roduction planning problems. Computers & Operations Research, 36:2418-2=
428, 2009.

[16] G. Emiel and C. Sagastizabal. Incremental like bundl=
e methods with applications to energy planning. Computational Optimization =
and Applications, 46:305-332, 2009.

[17] L.F. Escudero, A. Garin, M.=
Merino, and G. Perez. A general algorithm for solving two-stage stochastic=
mixed 0-1 first-stage problems. Computers & Operations Research, 36:25=
90-2600, 2009.

[18] L.F. Escudero, A. Garin, G. Perez, and A. Unzuet=
a. Scenario cluster decomposition of the Lagrangian dual in two-stage stoch=
astic mixed 0-1 optimization. Computers & Operations Research, 40:362-3=
77, 2013.

[19] L.F. Escudero, A. Gar=C3=ADi, M. Merino and G. Perez.=
An algorithmic framework for solving large-scale multistage stochastic mix=
ed 0-1 problems with nonsymmetric scenario trees. Computers & Operation=
s Research, 39:1133-1144, 2012.

[20] L.F Escudero, A. Garin, M. Meri=
no, and G. Perez. A decomposition framework for solving dynamic MINLP probl=
ems under uncertainty. Workshop on Mixed Integer Nonlinear Programming, Wor=
kshop on Mixed Integer Nonlinear Programming, WMINLP2014. http://minlp.cheme.cmu.edu/2014/index.html. Carnegie Mellon Univers=
ity, Pittsburgh (PA, USA), 2014.

[21] L.F. Escudero, A. Garin and A.=
Unzueta. Cluster Lagrangean decomposition in multistage stochastic optimiz=
ation. Computers & Operations Research, 67:48-62, 2016.

[22] L.F=
. Escudero, M.A. Garn, C. Pizarro and A. Unzueta. On the strongest Lagrange=
an bounds for stochastic location-assignment problems. Submitted.

[2=
3] L.F. Escudero, M.A. Garin and A. Unzueta. Cluster Lagrangean decompositi=
on for time stochastic dominance induced by mixed integer-linear recourse i=
n multistage stochastic optimization. Submitted.

[24] L.F. Escudero,=
A. Garin, M. Merino, and G. Perez. On time stochastic dominance induced by=
mixed-integer linear recourse in multistage stochastic programs. European =
Journal of Operational Research, 249:164-176, 2016.

[25] L.F. Escude=
ro, J.F. Monge and D. Romero-Morales. An SDP approach for multiperiod mixed=
0=E2=80=931 linear programming models with stochastic dominance constraint=
s for risk management. Computers & Operations Research, 58:32-40, 2015.=

[26] L.F. Escudero, J.F. Monge and D. Romero-Morales. On time-consi=
stent stochastic dominance risk averse measure for Tactical Supply Chain Pl=
anning under uncertainty. Submitted.

[27] L.F. Escudero, J.F. Monge,=
D. Romero-Morales and J.Wang. Expected Future Value Decomposition Based Bi=
d Price Generation for Large-Scale Network Revenue Management. Transportati=
on Science, 47:181-197, 2013.

[28] R. Gollmer, F. Neise and R. Schul=
tz. Stochastic programs with first-order stochastic dominance constraints i=
nduced by mixed-integer linear recourse. SIAM Journal on Optimization, 19:5=
52-571, 2008.

[29] R. Gollmer, U. Gotzes and R. Schultz. A note on s=
econd-order stochastic dominance constraints induced by mixed-integer linea=
r recourse. Mathematical Programming Ser.A, 126:179-190, 2011.

[30] =
V. Guigues. SDDP for some interstage dependent risk averse problems and app=
lication to hydrothermal planning Computational Optimization & Applicat=
ions, 57:167-203, 2014.

[31] E. Klerides and E. Hadjiconstantinou. A=
decomposition-based stochastic programming approach for the project schedu=
ling problem under time/cost trade-off setting and uncertain durations. Com=
puters & Operations Research, 37:2131-2140, 2010.

[32] V. Kozmik=
and D. P. Morton. Risk-averse stochastic dual dynamic programming. Optimiz=
ation Online, org/DB HTML/2013/02/3794, 2013.

[33] Z.Li and M. Ierap=
etritou. Capacity expansion planning through augmented Lagrangian optimizat=
ion and scenario decomposition. AIChE Journal, 58:871-883, 2012.

[34=
] J. Linderoth, A. Shapiro, and S.Wright. The empirical behavior of samplin=
g methods for stochastic programming. Annals of Operations Research, 142:21=
5241, 2006.

[35] G. Lulli and S. Sen. A branch and price algorithm f=
or multistage integer programs with application to stochastic batch sizing =
problems. Management Science, 50:787-796, 2004.

[36] G. Lulli and S.=
Sen. A heuristic procedure for stochastic integer programs with complete r=
ecourse. European Journal of Operational Research, 879-890, 2006.

[3=
7] S. Lumbreras and A. Ramos. Optimal design of the electrical layout of an=
offshore wind farm applying decomposition strategies. IEEE Transactions on=
Power Systems, 28:1434-1441, 2013.

[38] D. Mahlke. A scenario tree-=
based decomposition for solving multistage stochastic programs with applica=
tions in energy production. Springer, 2011.

[39] J.M. Mulvey and A. =
Ruszczynski. A new scenario decomposition method for large scale stochastic=
optimization. Operations Research, 43:477-490, 1995.

[40] M.P. Nowa=
k and W. Roemisch. Stochastic Lagrangean relaxation applied to power schedu=
ling under uncertainty. Annals of Operations Research, 100:251-272, 2000.**43:477-490, 1995.[41] F. Oliveira, V. Gupta, S. Hamacher, and I.E=
. Grossmann. A Lagrangean decomposition approach for oil supply chain inves=
tment planning under uncertainty with risk considerations. Computers and Ch=
emical Engineering, 50:184-195, 2013.[42] W. Oliveira, C. Sagastiza=
bal, and S. Scheimberg. Inexact bundle methods for two-stage stochastic pro=
gramming. SIAM Journal on Optimization, 21:517-544, 2011.[43] A. Pa=
ges-Bernaus, G. Perez-Valdes and A. Tomasgard. A parallelised distributed i=
mplementation of a Branch and Fix Coordination algorithm. European Journal =
of Operational Research, 244:77-85, 2013.[44] M.V.F. Pereira and L.=
M.V.G. Pinto. Stochastic optimization of a multireservoir hydroelectric sys=
tem, a decomposition approach. Water Resources Research, 21:779=E2=80=93792=
, 1985.[45] M.V.F. Pereira and L.M.V.G. Pinto. Multistage stochasti=
c optimization applied to energy planning. Mathematical Programming, 52:359=
=E2=80=93375, 1991.[46] A.R. de Queiroz and D.P. Morton. Sharing cu=
ts under aggregated forecasts when decomposing multi-stage stochastic progr=
ams. Operations Research Letters, 31:311-316, 2013.[47] R.T. Rockaf=
ellar and R.J-B. Wets. Scenario and policy aggregation in optimisation unde=
r uncertainty. Mathematics of Operations Research, 16:119147, 1991.=
[48] A. Ruszczynski. Parallel decomposition of multistage stochastic progra=
mming problems Mathematical Programming, 58:201-228, 1993.[49] A. R=
uszczynski. On the convergence of an augmented lagrangian decomposition met=
hod for sparse convex optimization. Mathematics of Operations Research, 634=
-566, 1995.[50] A. Ruszczynski and A. Swietanowski. Accelerating th=
e regularized decomposition method for two stage stochastic linear problems=
. SIAM Journal on Optimization, 24:127-153, 2014.[51] C. Sagastizab=
al. Divide to conquer: Decomposition methods for energy optimization. Mathe=
matical Programming Ser. B, 134:187-222, 2012.[52] B. Sandikci and =
O.Y. Ozaltin. A scalable bounding method for multi-stage stochastic integer=
programs. Technical report, Working paper 14-21, Booth School of Business,=
University of Chicago, Chicago, IL, USA, 2014. URL http:/=
/dx.doli.org/10.21.39/ssrn.26666.[53] S. Sen and Z. Zhou. Multi=
stage stochastic decomposition: a bridge between stochastic programming and=
approximate dynamic programming. SIAM Journal on Optimization, 24:1278-153=
, 2014.[54] A. Shapiro, W. Tekaya, J.P. da Costa and M. Pereira. Ri=
sk neutral and risk averse stochastic dual dynamic programming method. Euro=
pean Journal of Operational Research, 224:375-391, 2013.[55] R. van=
Slyke R. and R.B.Wets. L-shaped linear programs with application to optima=
l control and stochastic programming. SIAM Journal on Applied Mathematics, =
17:638-663, 1969.[56] M. Tahanan, W. van Ackooij, A. Frangioni and =
F. Lacalandra. Large-scale unit commitment under uncertainty. 4OR, 13:115-1=
71, 2015.[57] J.P.Watson and D.Woodruff. Progressive hedging innova=
tions for a class of stochastic mixed-integer resource allocation problems.=
Computational Management Science, 8:355370, 201[58] G.L. Zenarosa,=
O.A. Prokopyev, and A.J. Schaefer. Scenario-tree decomposition: Bounds for=
multistage stochastic mixed-integer programs. Technical report, Technical =
paper, Department of Industrial Engineering, University of Pittsburgh, Pitt=
sburgh, PA, USA, 2014. URL http://dx.doli.org/10.21.39/ssr=
n.26666.[59] Ben-Tal, Aharon, and Arkadi Nemirovski. "Robust op=
timization=E2=80=93methodology and applications." Mathematical Programming =
92.3 (2002): 453-480.[60] Zadeh, Lotfi Asker. "Fuzzy sets as a basi=
s for a theory of possibility." Fuzzy sets and systems 1.1 (1978): 3-28.[61] Ben-Haim, Yakov. Info-gap decision theory: decisions under severe=
uncertainty. Academic Press, 2006.[62] A. Rabiee, A. Soroudi and A=
. Keane, "Information Gap Decision Theory Based OPF With HVDC Connected Win=
d Farms," in IEEE Transactions on Power Systems, vol. 30, no. 6, pp. 3396-3=
406, Nov. 2015.[63] Soroudi, Alireza, and Turaj Amraee. "Decision m=
aking under uncertainty in energy systems: state of the art." Renewable and=
Sustainable Energy Reviews 28 (2013): 376-384**

The GEP can be mathematically formulated as a high dimensional, nonlinea= r, nonconvex, mix-integer and highly constrained optimization problem with = the least cost of the investment as the optimization criterion. The complex= ity of the problem rapidly increases if many practical constraints are take= n into account.

Methods to solve the GEP can be generally categorized into two types: tr= aditional mathematical programming methods and methods based on heuristic t= echniques. The traditional mathematical methods include dynamic programming= (DP), mix-integer programming (MIP), branch and bound, Benders=E2=80=99 de= composition, network flow methods, and others. These methods are usually st= rict in mathematics, strong in numerical manipulations, weak in handling qu= alitative constraints, and slower in computational performance [1]. The heu= ristic techniques mainly include the application of Artificial Intell= igence (AI) approaches, such as GAs, DE, EP, evolutionary strategy (ES), AC= O, PSO, TS, SA, and a hybrid approach [2]. The main advantage of the heuris= tic techniques is their flexibility for handling numerous qualitative const= raints that are common in the GEP in the deregulated power industry.

*The application of PSO for solving GEP has been reported in the literatu=
re [2]=E2=80=93[4]. The application and comparison of metaheuristic techniq=
ues to solve the GEP has been presented in [2], where PSO is compared with =
eight other metaheuristic techniques in terms of the success rate and execu=
tion time. The performance of the metaheuristic techniques is improved by t=
he application of a virtual mapping procedure, intelligent initial populati=
on generation, and penalty factor approach. Optimal and near-optimal soluti=
ons are reached within a reasonable amount of time. In addition, Kannan et al. *[5] have presented the application of the PSO technique and it=
s variants to the GEP. In this case, three different test cases are used fo=
r comparison with DP. The comparisons are based on computation time, succes=
s rate, and error limit. The virtual mapping procedure is also used here. F=
or all test cases, the PSO technique and its variants produce better result=
s in much less time compared with DP. Among the PSO variants, it was observ=
ed that the constriction factor approach performed comparatively better.

Additionally, Sensarna *et al. *[3] have presented the applicatio=
n of PSO to GEP with a parametric approach for planning and transmission ex=
pansion including topology optimization, capital expenses, power losses, ne=
twork security, and revenue. Likewise, Slochanal *et al. *[4] have p=
resented the application of PSO to the generation expansion planning proble=
m in a competitive environment.

[1] F. Wu, Z. Yen, Y. Hou, and Y. Ni, =E2=80=9CApplications of AI techni=
ques to generation planning and investment,=E2=80=9D in *Proc. IEEE Powe=
r Eng. Soc. General Meeting*, Jun. 2004, vol. 1, pp. 936=E2=80=93940.

[2] S. Kannan, S. Slochanal, and N. Padhy, =E2=80=9CApplication and comp=
arison of metaheuristic techniques to generation expansion planning problem=
,=E2=80=9D *IEEE Trans. Power Syst.*, vol. 20, no. 1, pp. 466=E2=80=
=93475, Feb. 2005.

[3] P. Sensarna, M. Rahmani, and A. Carvalho, =E2=80=9CA comprehensive m=
ethod for optimal expansion planning using particle swarm optimization,=E2=
=80=9D in *Proc. IEEE Power Eng. Soc. Winter Meeting 2002*, Jan. 200=
2, vol. 2, pp. 1317=E2=80=931322.

[4] S. Slochanal, S. Kannan, and R. Rengaraj, =E2=80=9CGeneration expans=
ion planning in the competitive environment,=E2=80=9D in *Proc. 2004 Int=
. Conf. Power Syst. Technol.*, Nov. 2004, pp. 1546=E2=80=931549.

[5] S. Kannan, S. Slochanal, and N. Padhy, =E2=80=9CApplication of parti=
cle swarm optimization technique and its variants to generation expansion p=
roblem,=E2=80=9D *ELSERVIER Electric Power Syst. Res.*, vol. 70, no.=
3, pp. 203=E2=80=93210, Aug. 2004

On the software side, general purpose stochastic optimization software s= till seems far away. Planning models are highly problem oriented and off-th= e-shelf packages are not available. Companies use modeling languages like G= AMS, AMPL, AIMMS, Python together with standard solvers to develop problem = specific approaches.

Prof. Laureano Escudero, Universidad Rey = Juan Carlos

Dr Alireza Soroudi, University Colle= ge Dublin (UCD)

Prof. Christoph Helmberg, TU Chemnitz

Dr. =C5=BDeljko Kanovi=C4=87, University =
of Novi Sad