IBM ILOG CPLEX Optimizer provides flexible, high-performance mathematica= l programming solvers for linear programming, mixed integer programming, qu= adratic programming, and

quadratically constrained programming problems. For linear programming, = the algorithms include primal simplex algorithm, the dual simplex algorithm= , the network simplex algorithm, as well as a barrier method. For mixed int= eger programming models, CPLEX uses branch-and-cut algorithm. CPLEX can sol= ve both convex and non-convex quadratic to global optimality. CPLEX has bot= h barrier and simplex algorithms for solving convex quadratic programs and = a barrier algorithm for solving non-convex problems.

**Reference**:

http://www-01.ibm.com/so= ftware/commerce/optimization/cplex-optimizer/

CPLEX has been applied to various problems arising in the energy sector:=

**Resource Planning Models** address the supply and demand =
side investment decisions an energy supplier makes to ensure that it can sa=
tisfy customer demand. The objective is to minimize the total cost of build=
ing and operating production facilities to serve forecasted loads over a mu=
ltiyear planning period.

**References**:

Mirzaesmaeeli, H., Elkamel, A., Douglas, P. L., Croiset, E., & Gupta= , M. (2010). A multi-period optimization model for energy planning with CO2= emission consideration. Journal of environmental management, 91(5), 1063-1= 070.

Omu, A., Choudhary, R., & Boies, A. (2013). Distributed energy resou= rce system optimisation using mixed integer linear programming. Energy Poli= cy, 61, 249-266.

Zhang, B. J., & Hua, B. (2007). Effective MILP model for oil refiner= y-wide production planning and better energy utilization. Journal of Cleane= r Production, 15(5), 439-448.

**Unit Commitment/Economic Dispatch Models** are used to sc=
hedule hourly production of thermal power stations for periods up to about =
a week in advance. The objective is to minimize the short-term costs of ope=
rating the generators to serve forecasted customer loads. The costs include=
both fuel costs and start-up costs. The constraints represent the requirem=
ent to serve hourly customer loads, various reserve requirements, minimum u=
ptimes and downtimes for generators, and ramping limits for generators.

**References**:

Frangioni, A., Gentile, C., & Lacalandra, F. (2009). Tighter approxi= mated MILP formulations for unit commitment problems. Power Systems, IEEE T= ransactions on, 24(1), 105-113.

Carri=C3=B3n, M., & Arroyo, J. M. (2006). A computationally efficien= t mixed-integer linear formulation for the thermal unit commitment problem.= Power Systems, IEEE Transactions on, 21(3), 1371-1378.

Padhy, N. P. (2004). Unit commitment-a bibliographical survey. Power Sys= tems, IEEE Transactions on, 19(2), 1196-1205.

Yamin, H. Y. (2004). Review on methods of generation scheduling in elect= ric power systems. Electric Power Systems Research, 69(2), 227-248.

Hedman, K. W., Ferris, M. C., O'Neill, R. P., Fisher, E. B., & Oren,= S. S. (2010). Co-optimization of generation unit commitment and transmissi= on switching with N-1 reliability. Power Systems, IEEE Transactions on, 25(= 2), 1052-1063.

Bukhsh W. A., Zhang C., Pinson P. (2016): An integrated multiperiod OPF = model with demand response and renewable generation uncertainty. Smart Grid= , IEEE Transactions on (2016). DOI: 10.1109/TSG.2015.2502723

**Hydro/Thermal Scheduling Models** are used to determine t=
he use of water resources in power systems with a lot of hydroelectric gene=
ration. The objective is to minimize the short-term costs of operating the =
power plants to serve forecasted customer loads. The costs include thermal =
power plants=E2=80=99fuel costs, since hydro generation typically has negli=
gible direct costs. The constraints represent the requirement to serve cust=
omer load per hour, the availability and flow of water through the supporti=
ng water network, various reserve requirements, various restrictions on wat=
er flows and reservoir volumes reflecting environmental regulations.

**References**:

Nowak, M. P., & R=C3=B6misch, W. (2000). Stochastic Lagrangian relax= ation applied to power scheduling in a hydro-thermal system under uncertain= ty. Annals of Operations Research, 100(1-4), 251-272.

Shawwash, Z. K., Siu, T. K., & Russell, S. D. (2000). The BC Hydro s= hort term hydro scheduling optimization model. Power Systems, IEEE Transact= ions on, 15(3), 1125-1131.

Chang, G. W., Aganagic, M., Waight, J. G., Medina, J., Burton, T., Reeve= s, S., & Christoforidis, M. (2001). Experiences with mixed integer line= ar programming based approaches on short-term hydro scheduling. Power Syste= ms, IEEE Transactions on, 16(4), 743-749.

**Optimal Power Flow/Security Constrained Dispatch Models**=
are used to determine the flows of power along the various transmission pa=
ths in a power network for the purpose of evaluating the feasibility, relia=
bility and security of the power system. The objective is to minimize the o=
perating cost of serving the load. The constraints represent the requiremen=
t to serve instantaneous customer load at all nodes of the network, the gen=
erator capacity limits, conservation of power flow and voltage laws governi=
ng the physical power flows (which may be represented nonlinearly), and var=
ious reserve, security and reliability criteria.

**References**:

Alguacil, N., & Conejo, A. J. (2000). Multiperiod optimal power flow= using Benders decomposition. Power Systems, IEEE Transactions on, 15(1), 1= 96-201.

Jabr, R. (2008). Optimal power flow using an extended conic quadratic fo= rmulation. Power Systems, IEEE Transactions on, 23(3), 1000-1008.

Alsac, O., Bright, J., Prais, M., & Stott, B. (1990). Further develo= pments in LP-based optimal power flow. Power Systems, IEEE Transactions on,= 5(3), 697-711.

**Contract and Risk Management Models** enable energy and p=
ower companies to implement profitable bidding strategies while limiting pr=
ice and volume risks to acceptable levels. The objective is to determine vo=
lume and price for possible energy transactions and emission credits bought=
or sold in order to maximize expected net returns. The constraints represe=
nt forward price curve uncertainty and volatility, and impose limits on val=
ue at risk and conditional value at risk.

**References**:

Arroyo, J. M., & Galiana, F. D. (2005). Energy and reserve pricing i= n security and network-constrained electricity markets. Power Systems, IEEE= Transactions on, 20(2), 634-643.

Li, T., Shahidehpour, M., & Li, Z. (2007). Risk-constrained bidding = strategy with stochastic unit commitment. Power Systems, IEEE Transactions = on, 22(1), 449-458.

Al-Awami, A. T., & Sortomme, E. (2012). Coordinating vehicle-to-grid= services with energy trading. Smart Grid, IEEE Transactions on, 3(1), 453-= 462.

Angarita, J. M., & Usaola, J. G. (2007). Combining hydro-generation = and wind energy: Biddings and operation on electricity spot markets. Electr= ic Power Systems Research, 77(5), 393-400.

**Contingency Planning Models** are used to determine the r=
ecourse in cases of outages. Sometimes, these involve transmission switchin=
g, load shedding, and controlled islanding.

**References**:

Bienstock, D. (2016): Electrical Transmission System Cascades and Vulner= ability: An Operations Research Viewpoint. Springer-MOS Series on Optimizat= ion, ISBN 978-1-611974-15-7.

Trodden, P. A., Bukhsh W. A., Grothey A., McKinnon K. I. M. (2014): Opti= mization-based islanding of power networks using piecewise linear AC power = flow. Power Systems, IEEE Transactions, 29(), 1212-1220.

Witthaut, D., & Timme, M. (2015). Nonlocal effects and countermeasur= es in cascading failures. Phys. Rev. E 92(3):032809. DOI: 10.1103/PhysRevE.= 92.032809

**Pooling and Blending Models** are bilinear network flow p=
roblem on an arbitrary directed graph. Given a list of available suppliers =
with raw materials containing known specifications, the objective is to min=
imize the cost of mixing these materials in intermediate pools so as to mee=
t the demand and specifications requirements at multiple final blends.

**References**:

Misener, R., & Floudas, C. A. (2010). Global optimization of large-s= cale generalized pooling problems: quadratically constrained MINLP models. = Industrial & Engineering Chemistry Research, 49(11), 5424-5438.

Gupte, A., Ahmed, S., Dey, S. S., & Cheon, M. S. (2013). Pooling pro= blems: relaxations and discretizations. School of Industrial and Systems En= gineering, Georgia Institute of Technology, Atlanta, GA. and ExxonMobil Res= earch and Engineering Company, Annandale, NJ.

**Contributors**:

Dr Bissan Ghaddar, University of Waterloo

Dr Jakub Marecek, IBM