Skip to main content

Advertisement

SpringerLink
Log in

Large-scale unit commitment under uncertainty: an updated literature survey

  • SI: 4OR Surveys
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper “Large-scale Unit Commitment under uncertainty: a literature survey” that appeared in 4OR 13(2):115–171 (2015); this version has over 170 more citations, most of which appeared in the last 3 years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Abbreviations

UC:

Unit commitment problem

UUC:

UC problem under uncertainty

bUC:

Basic UC problem (common modeling assumptions)

ED:

Economic dispatch

EIE:

Energy intensive enterprise

GENCO:

GENeration COmpany

TSO:

Transmission system operator

MP:

Monopolistic producer

PE:

Power exchange

PEM:

PE manager

OTS:

Optimal transmission switching

UCOTS:

UC with OTS

MSG:

Minimal stable generation

OPF:

Optimal power flow

ROR:

Run-of-river hydro unit

DR:

Demand response

\(X_1\) :

Set of technically feasible production schedules

\(X_2\) :

Set of system wide constraints

\(\mathcal {T}\) :

Set of time steps

MILP:

Mixed-integer linear programming

MIQP:

Mixed-integer quadratic programming

DP:

Dynamic programming

SDDP:

Stochastic dual DP

B&B, B&C, B&P:

Branch and bound (cut, price respectively)

AL:

Augmented Lagrangian

LR:

Lagrangian relaxation

LD:

Lagrangian dual

CP:

Cutting plane

SO:

Stochastic optimization

SD:

Scenario decomposition

UD:

Unit decomposition (also called space decomposition or stochastic decomposition)

RO:

Robust optimization

IUC:

Interval unit commitment

CCO:

Chance-constrained optimization

ICCO:

Chance-constrained optimization with individual probabilistic constraints

JCCO:

Chance-constrained optimization with joint probabilistic constraints

References

  • Abdelaziz, A. Y., Kamh, M. Z., Mekhamer, S. F., & Badr, M. A. L. (2010). An augmented hopfield neural network for optimal thermal unit commitment. International Journal of Power System Optimization, 2(1), 37–49.

    Google Scholar 

  • Adam, L., & Branda, M. (2016). Nonlinear chance constrained problems: Optimality conditions, regularization and solvers. Journal of Optimization Theory and Applications, 170(2), 419–436.

    Google Scholar 

  • Adam, L., Branda, M., Heitsch, H., & Henrion, R. (2018). Solving joint chance constrained problems using regularization and benders’ decomposition. Preprint available on Researchgate, 1–20.

  • Adam, L., Henrion, R., & Outrata, J. (2017). On M-stationarity conditions in mpecs and the associated qualification conditions. Mathematical Programming, 1–31.

  • Adelhütte, D., Aßmann, D., Gonzàlez-Gradòn, T., Gugat, M., Heitsch, H., Henrion, R., et al. (2018). Joint model of probabilistic (probust) constraints with application to gas network optimization, 1–33 (preprint).

  • Aganagic, M., & Mokhtari, S. (1997). Security constrained economic dispatch using nonlinear dantzig-wolfe decomposition. IEEE Transactions on Power Systems, 12(1), 105–112.

    Google Scholar 

  • Aghaei, J., Nikoobakht, A., Siano, P., Nayeripour, M., Heidari, A., & Mardaneh, M. (2016). Exploring the reliability effects on the short term ac security-constrained unit commitment: A stochastic evaluation. Energy, 114, 1016–1032.

    Google Scholar 

  • Ahmadi, H., & Ghasemi, H. (2014). Security-constrained unit commitment with linearized system frequency limit constraints. IEEE Transactions on Power Systems, 29(4), 1536–1545.

    Google Scholar 

  • Ahmadi-Khatir, A., Bozorg, M., & Cherkaoui, R. (2013). Probabilistic spinning reserve provision model in multi-control zone power system. IEEE Transactions on Power Systems, 28(3), 2819–2829.

    Google Scholar 

  • Aïd, R., Guigues, V., Ndiaye, P. M., Oustry, F., & Romanet, F. (2006). A value-at-risk approach for robust management of electricity power generation. Rapport de recherche, IMAG-LMC (submitted).

  • Al-Kalaani, Y., Villaseca, F. E., & Renovich, F, Jr. (1996). Storage and delivery constrained unit commitment. IEEE Transactions on Power Systems, 11(2), 1059–1066.

    Google Scholar 

  • Alqurashi, A., Etemadi, A. H., & Khodaei, A. (2016). Treatment of uncertainty for next generation power systems: State-of-the-art in stochastic optimization. Electric Power System Research, 141(1), 233–245.

    Google Scholar 

  • Alvarez, G. E., Marcovecchio, M. G., & Aguirre, P. A. (2018). Security-constrained unit commitment problem including thermal and pumped storage units: An MILP formulation by the application of linear approximations techniques. Electric Power Systems Research, 154, 67–74.

    Google Scholar 

  • Amiri, M., & Khanmohammadi, S. (2013). A primary unit commitment approach with a modification process. Applied Soft Computing, 13(2), 1007–1015.

    Google Scholar 

  • An, Y., & Zeng, B. (2015). Exploring the modeling capacity of two-stage robust optimization: Variants of robust unit commitment model. IEEE Transactions on Power Systems, 30(1), 109–122.

    Google Scholar 

  • Anders, G. J. (1981). Genetration planning model with reliability constraints. IEEE Transactions on Power Apparatus and Systems, PAS–100(12), 4901–4908.

    Google Scholar 

  • Anderson, E. J., & Philpott, A. B. (2002). Optimal offer construction in electricity markets. Mathematics of Operations Research, 27(1), 82–100.

    Google Scholar 

  • Annakkage, U. D., Numnonda, T., & Pahalawaththa, N. C. (1995). Unit commitment by parallel simulated annealing. IEE Proceedings-Generation, Transmission and Distribution, 142(6), 595–600.

    Google Scholar 

  • Anstine, L. T., Burke, R. E., Casey, J. E., Holgate, R., John, R. S., & Stewart, H. G. (1963). Application of probability methods to the determination of spinning reserve requirements for the Pennsylvania–New Jersey–Maryland interconnection. IEEE Transactions on Power Apparatus and Systems, 82(68), 726–735.

    Google Scholar 

  • Anstreicher, K. M., & Wolsey, L. A. (2009). Two “well-known” properties of subgradient optimization. Mathematical Programming, 120(1), 213–220.

    Google Scholar 

  • Aoki, K., Itoh, M., Satoh, T., Narah, K., & Kanezashi, M. (1989). Optimal long-term unit commitment in large scale systems including fuel constrained thermal and pumped storage hydro. IEEE Transactions on Power Systems, 4(3), 1065–1073.

    Google Scholar 

  • Aoki, K., Satoh, T., Itoh, M., Ichimori, T., & Masegi, K. (1987). Unit commitment in a large-scale power system including fuel constrained thermal and pumped-storage hydro. IEEE Transactions on Power Systems, 2(4), 1077–1084.

    Google Scholar 

  • Apparigliato, R. (2008). Règles de décision pour la gestion du risque: Application á la gestion hebdomadaire de la production électrique. Ph.D. thesis, École Polytechnique.

  • Archibald, T. W., Buchanan, C. S., McKinnon, K. I. M., & Thomas, L. C. (1999). Nested benders decomposition and dynamic programming for reservoir optimisation. The Journal of the Operational Research Society, 50(5), 468–479.

    Google Scholar 

  • Ardakani, A. J., & Bouffard, F. (2013). Identification of umbrella constraints in DC-based security-constrained optimal power flow. IEEE Transactions on Power Systems, 28(4), 3924–3934.

    Google Scholar 

  • Arnold, T., Henrion, R., Möller, A., & Vigerske, S. (2014). A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints. Pacific Journal of Optimization, 10, 5–20.

    Google Scholar 

  • Arroyo, J. M., & Conejo, A. J. (2000). Optimal response of a thermal unit to an electricity spot market. IEEE Transactions on Power Systems, 15(3), 1098–1104.

    Google Scholar 

  • Arroyo, J. M., & Conejo, A. J. (2004). Modeling of start-up and shut-down power trajectories of thermal units. IEEE Transactions on Power Systems, 19(3), 1562–1568.

    Google Scholar 

  • Asensio, M., & Contreras, J. (2016). Stochastic unit commitment in isolated systems with renewable penetration under CVaR assessment. IEEE Transactions on Smart Grid, 7(3), 1356–1367.

    Google Scholar 

  • Astorino, A., Frangioni, A., Gaudioso, M., & Gorgone, E. (2011). Piecewise quadratic approximations in convex numerical optimization. SIAM Journal on Optimization, 21(4), 1418–1438.

    Google Scholar 

  • Atakan, S., Lulli, G., & Sen, S. (2018). A state transition mip formulation for the unit commitment problem. IEEE Transactions on Power Systems, 33(1), 736–748.

    Google Scholar 

  • Attouch, H., Bolte, J., Redont, P., & Soubeyran, A. (2010). Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Lojasiewicz inequality. Mathematics of Operations Research, 35(2), 438–457.

    Google Scholar 

  • Babonneau, F., Vial, J. P., & Apparigliato, R. (2010). Robust optimization for environmental and energy planning (chapter 3 in [74]). Volume 138 of international series in operations research & management science. Berlin: Springer.

    Google Scholar 

  • Bacaud, L., Lemaréchal, C., Renaud, A., & Sagastizábal, C. (2001). Bundle methods in stochastic optimal power management: A disaggregate approach using preconditionners. Computation Optimization and Applications, 20(3), 227–244.

    Google Scholar 

  • Bahiense, L., Maculan, N., & Sagastizábal, C. (2002). The volume algorithm revisited: Relation with bundle methods. Mathematical Programming, 94(1), 41–69.

    Google Scholar 

  • Baillo, A., Ventosa, M., Rivier, M., & Ramos, A. (2004). Optimal offering strategies for generation companies operating in electricity spot markets. IEEE Transactions on Power Systems, 19(2), 745–753.

    Google Scholar 

  • Bakirtzis, E. A., & Biskas, P. N. (2017). Multiple time resolution stochastic scheduling for systems with high renewable penetration. IEEE Transactions on Power Systems, 32(2), 1030–1040.

    Google Scholar 

  • Balas, E., Ceria, S., & Cornuéjols, G. (1993). A lift-and-project cutting plane algorithm for mixed 0–1 programs. Mathematical Programming, 58(1–3), 295–324.

    Google Scholar 

  • Baldick, R. (1995). The generalized unit commitment problem. IEEE Transactions on Power Systems, 10(1), 465–475.

    Google Scholar 

  • Baldwin, C. J., Dale, K. M., & Dittrich, R. F. (1959). A study of the economic shutdown of generating units in daily dispatch. Transactions of the American Institute of Electrical Engineers Power Apparatus and Systems, Part III, 78(4), 1272–1282.

    Google Scholar 

  • Bandi, C., & Bertsimas, D. (2012). Tractable stochastic analysis in high dimensions via robust optimization. Mathematical Programming, 134(1), 23–70.

    Google Scholar 

  • Barahona, F., & Anbil, R. (2000). The volume algorithm: Producing primal solutions with a subgradient method. Mathematical Programming, 87(3), 385–399.

    Google Scholar 

  • Bard, J. F. (1988). Short-term scheduling of thermal-electric generators using lagrangian relaxation. Operations Research, 36(5), 765–766.

    Google Scholar 

  • Baringo, L., & Conejo, A. J. (2011). Offering strategy via robust optimization. IEEE Transactions on Power Systems, 26(3), 1418–1425.

    Google Scholar 

  • Batut, J., & Renaud, A. (1992). Daily scheduling with transmission constraints: A new class of algorithms. IEEE Transactions on Power Systems, 7(3), 982–989.

    Google Scholar 

  • Bechert, T. E., & Kwatny, H. G. (1972). On the optimal dynamic dispatch of real power. IEEE Transactions on Power Apparatus and Systems, PAS–91(1), 889–898.

    Google Scholar 

  • Bellman, R. E., & Dreyfus, S. E. (1962). Applied dynamic programming. Princeton: Princeton University Press.

    Google Scholar 

  • Belloni, A., Diniz, A. L., Maceira, M. E., & Sagastizábal, C. (2003). Bundle relaxation and primal recovery in unit-commitment problems: The Brazilian case. Annals of Operations Research, 120(1–4), 21–44.

    Google Scholar 

  • Beltran, C., & Heredia, F. J. (2002). Unit commitment by augmented lagrangian relaxation: Testing two decomposition approaches. Journal of Optimization Theory and Applications, 112(2), 295–314.

    Google Scholar 

  • Ben-Salem, S. (2011). Gestion Robuste de la production électrique à horizon court-terme. Ph.D. thesis, Ecole Centrale Paris.

  • Ben-Tal, A., Bertsimas, D., & Brown, D. (2010). A soft robust model for optimization under ambiguity. Operations Research, 58(4), 1220–1234.

    Google Scholar 

  • Ben-Tal, A., El Ghaoui, L., & Nemirovski, A. (2009). Robust Optimization. Princeton: Princeton University Press.

    Google Scholar 

  • Ben-Tal, A., Goryashko, A., Guslitzer, E., & Nemirovski, A. (2003). Adjustable robust counterpart of uncertain linear programs. Mathematical Programming, Series A, 99, 351–376.

    Google Scholar 

  • Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769–805.

    Google Scholar 

  • Ben-Tal, A., & Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operations Research Letters, 25(1), 1–13.

    Google Scholar 

  • Ben-Tal, A., & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, Series A, 88, 411–424.

    Google Scholar 

  • Ben-Tal, A., & Nemirovski, A. (2009). On safe tractable approximations of chance-constrained linear matrix inequalities. Mathematics of Operations Research, 34(1), 1–25.

    Google Scholar 

  • Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4(1), 238–252.

    Google Scholar 

  • Bendotti, P., Fouilhoux, P., & Rottner, C. (2017a). On the complexity of the unit commitment problem. Technical report, EDF R&D. Optimization online report.

  • Bendotti, P., Fouilhoux, P., & Rottner, C. (2017b). Orbitopal fixing for the full orbitope and application to the unit commitment problem. Technical report, EDF R&D. Optimization online report.

  • Bendotti, P., Fouilhoux, P., & Rottner, C. (2018). The min-up/min-down unit commitment polytope. Journal of Combinatorial Optimization, 1–35.

  • Benth, F. E., Kiesel, R., & Nazarova, A. (2012). A critical empirical study of three electricity spot price models. Energy Economics, 34(5), 1589–1616.

    Google Scholar 

  • Beraldi, P., Conforti, D., & Violi, A. (2008). A two-stage stochastic programming model for electric energy producers. Computers & Operations Research, 35, 3360–3370.

    Google Scholar 

  • Bertsekas, D., Lauer, G., Sandell-Jr, N. R., & Posbergh, T. A. (1983). Optimal short-term scheduling of large-scale power systems. IEEE Transactions on Automatic Control, 28(1), 1–11.

    Google Scholar 

  • Bertsekas, D. P. (1999). Nonlinear programming (2nd ed.). Belmont: Athena Scientific.

    Google Scholar 

  • Bertsekas, D. P. (2005). Dynamic programming & optimal control (3rd ed., Vol. I). Belmont: Athena Scientific.

    Google Scholar 

  • Bertsekas, D. P. (2012). Dynamic programming & optimal control, vol II: Approximate dynamic programming (4th ed.). Belmont: Athena Scientific.

    Google Scholar 

  • Bertsimas, D., Brown, D., & Caramanis, C. (2011). Theory and applications of robust optimization. SIAM Review, 53(3), 464–501.

    Google Scholar 

  • Bertsimas, D., Litvinov, E., Sun, X. A., Zhao, J., & Zheng, T. (2013). Adaptive robust optimization for the security constrained unit commitment problem. IEEE Transactions on Power Systems, 28(1), 52–63.

    Google Scholar 

  • Bertsimas, D., & Sim, M. (2003). Robust discrete optimization and network flows. Mathematical Programming, 98(1), 49–71.

    Google Scholar 

  • Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35–53.

    Google Scholar 

  • Bienstock, D. (2013). Progress on solving power flow problems. Optima, 93, 1–8.

    Google Scholar 

  • Bienstock, D., Chertkov, M., & Harnett, S. (2014). Chance-constrained optimal power flow: Risk-aware network control under uncertainty. SIAM Review, 56(3), 461–495.

    Google Scholar 

  • Bienstock, D., & Verma, A. (2011). The n-k problem in power grids: New models, formulations and numerical experiments. SIAM Journal on Optimization.

  • Billinton, R., Karki, B., Karki, R., & Ramakrishna, G. (2009). Unit commitment risk analysis of wind integrated power systems. IEEE Transactions on Power Systems, 24(2), 930–939.

    Google Scholar 

  • Billinton, R., & Karki, R. (1999). Capacity reserve assessment using system well-being analysis. IEEE Transactions on Power Systems, 14(2), 433–438.

    Google Scholar 

  • Birge, J. R., & Louveaux, F. (1988). A multicut algorithm for two-stage stochastic linear programs. European Journal of Operational Research, 34(3), 384–392.

    Google Scholar 

  • Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer.

    Google Scholar 

  • Blanco, I., & Morales, J. M. (2017). An efficient robust solution to the two-stage stochastic unit commitment problem. IEEE Transactions on Power Systems, 32(6), 4477–4488.

    Google Scholar 

  • Bompard, E., & Ma, Y. (2012). Models of strategic bidding in electricity markets under network constraints. In A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis, & M. V. F. Pereira (Eds.), Handbook of networks in power systems I (pp. 3–39). Heidelberg: Springer.

    Google Scholar 

  • Bond, S. D., & Fox, B. (1986). Optimal thermal unit scheduling using improved dynamic programming algorithm. IEEE Proceedings C, 133(1), 1–5.

    Google Scholar 

  • Bonnans, J. F., Gilbert, J. C., Lemaréchal, C., & Sagastizábal, C. (2006). Numerical optimization: Theoretical and practical aspects (2nd ed.). Berlin: Springer.

    Google Scholar 

  • Borghetti, A., D’Ambrosio, C., Lodi, A., & Martello, S. (2008). A MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir. IEEE Transactions on Power Systems, 23(3), 1115–1124.

    Google Scholar 

  • Borghetti, A., Frangioni, A., Lacalandra, F., Lodi, A., Martello, S., Nucci, C. A., et al. (2001). Lagrangian relaxation and Tabu search approaches for the unit commitment problem. In IEEE power tech proceedings, 2001 Porto (Vol. 3).

  • Borghetti, A., Frangioni, A., Lacalandra, F., & Nucci, C. A. (2003). Lagrangian heuristics based on disaggregated bundle methods for hydrothermal unit commitment. IEEE Transactions on Power Systems, 18, 313–323.

    Google Scholar 

  • Borghetti, A., Frangioni, A., Lacalandra, F., Nucci, C. A., & Pelacchi, P. (2003). Using of a cost-based unit commitment algorithm to assist bidding strategy decisions. In A. Borghetti, C. A. Nucci, & M. Paolone (Eds.), Proceedings IEEE 2003 powertech bologna conference (Vol. 547).

  • Bouffard, F., & Galiana, F. D. (2008). Stochastic security for operations planning with significant wind power generation. IEEE Transactions on Power Systems, 23(2), 306–316.

    Google Scholar 

  • Brandenberg, R., Huber, M., & Silbernagl, M. (2017). The summed start-up costs in a unit commitment problem. EURO Journal Computional Optimization, 5(1), 203–238.

    Google Scholar 

  • Briant, O., Lemaréchal, C., Meurdesoif, Ph, Michel, S., Perrot, N., & Vanderbeck, F. (2008). Comparison of bundle and classical column generation. Mathematical Programming, 113(2), 299–344.

    Google Scholar 

  • Bruninx, K., & Delarue, E. (2017). Endogenous probabilistic reserve sizing and allocation in unit commitment models: Cost-effective, reliable, and fast. IEEE Transactions on Power Systems, 32(4), 2593–2603.

    Google Scholar 

  • Büsing, C., & D’Andreagiovanni, F. (2012). New results about multi-band uncertainty in robust optimization. In: R. Klasing (Ed.), Experimental algorithms-SEA 2012. Volume 7276 of LNCS (pp. 63–74).

    Google Scholar 

  • Büsing, C., & D’Andreagiovanni, F. (2013). Robust optimization under multi-band uncertainty-part I: theory. Technical report ZIB-report 13-10. Berlin: Zuse-Institut (ZIB).

  • Calafiore, G. C., & Campi, M. C. (2005). Uncertain convex programs: Randomized solutions and confidence levels. Mathematical Programming, 102(1), 25–46.

    Google Scholar 

  • Calafiore, G. C., & El Ghaoui, L. (2006). On distributionally robust chance-constrained linear programs. Journal of Optimization Theory and Applications, 130(1), 1–22.

    Google Scholar 

  • Cardozo, C., Capely, L., & Dessante, P. (2017). Frequency constrained unit commitment. Energy Systems, 8(1), 31–56.

    Google Scholar 

  • Cardozo, C., Capely, L., & van Ackooij, W. (2016). Placement journalier de la production avec contrainte sur le creux de fréquence en cas de perte d’un groupe par décomposition de benders. In SYMPOSIUM DE GENIE ELECTRIQUE.

  • Cardozo, C., van Ackooij, W., & Capely, L. (2018). Cutting plane approaches for the frequency constrained economic dispatch problems. Electrical Power Systems Research, 156(1), 54–63.

    Google Scholar 

  • Cardozo Arteaga, C. (2016). Optimisation of power system security with high share of variable renewables: Consideration of the primary reserve deployment dynamics on a frequency constrained unit commitment model. Ph.D. thesis, Université Paris Saclay.

  • Carøe, C. C., Ruszczyński, A., & Schultz, R. (1997). Unit commitment under uncertainty via two-stage stochastic programming. In Proceedings of NOAS 1997.

  • Carøe, C. C., & Schultz, R. (1998). A two-stage stochastic program for unit-commitment under uncertainty in a hydro-thermal power system. Technical report, ZIB.

  • Carpentier, P., Cohen, G., Culioli, J. C., & Renaud, A. (1996). Stochastic optimization of unit commitment: A new decomposition framework. IEEE Transactions on Power Systems, 11(2), 1067–1073.

    Google Scholar 

  • Carrión, M., & Arroyo, J. M. (2006). A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem. IEEE Transactions on Power Systems, 21(3), 1371–1378.

    Google Scholar 

  • Catalão, J. P. S., Mariano, S. J. P. S., Mendes, V. M. F., & Ferreira, L. A. F. M. (2006). Parameterisation effect on the behavior of a head-dependent hydro chain using a nonlinear model. Electric Power Systems Research, 76, 404–412.

    Google Scholar 

  • Catalão, J. P. S., Mariano, S. J. P. S., Mendes, V. M. F., & Ferreira, L. A. F. M. (2010). Nonlinear optimization method for short-term hydro scheduling considering head-dependency. European Transactions on Electrical Power, 20, 172–183.

    Google Scholar 

  • Cerisola, S. (2004). Benders decomposition for mixed integer problems: Application to a medium term hydrothermal coordination problem. Ph.D. thesis, Instituto Investigación Tecnológica Madrid.

  • Cerisola, S., Baíllo, A., Fernández-López, J. M., Ramos, A., & Gollmer, R. (2009). Stochastic power generation unit commitment in electricity markets: A novel formulation and a comparison of solution methods. Operations Research, 57(1), 32–46.

    Google Scholar 

  • Cerjan, M., Marcic, D., & Delimar, M. (2011). Short term power system planning with water value and energy trade optimisation. In International conference on the European energy market (EEM).

  • Chandrasekaran, K., & Simon, S. P. (2012a). Multi-objective scheduling problem: Hybrid approach using fuzzy assisted cuckoo search algorithm. Swarm and Evolutionary Computation, 5, 1–12.

    Google Scholar 

  • Chandrasekaran, K., & Simon, S. P. (2012b). Network and reliability constrained unit commitment problem using binary real coded firefly algorithm. Electrical Power and Energy Systems, 43, 921–932.

    Google Scholar 

  • Chang, G. W., Aganagic, M., Waight, J. G., Medina, J., Burton, T., Reeves, S., et al. (2001). Experiences with mixed integer linear programming based approaches on short-term hydro scheduling. IEEE Transactions on Power systems, 16(4), 743–749.

    Google Scholar 

  • Chen, C.-H., Chen, N., & Luh, P. B. (2017). Head dependence of pump-storage-unit model applied to generation scheduling. IEEE Transactions on Power Systems, 32(4), 2869–2877.

    Google Scholar 

  • Chen, X., Sim, M., & Sun, P. (2007). A robust optimization perspective on stochastic programming. Operations Research, 55(6), 1058–1071.

    Google Scholar 

  • Chen, L., Zheng, T., Mei, S., Xue, X., Liu, B., & Lu, Q. (2016). Review and prospect of compressed air energy storage system. Journal of Modern Power Systems and Clean Energy, 4(4), 529–541.

    Google Scholar 

  • Cheung, K., Gade, D., Monroy, C. S., Ryan, S. M., Watson, J.-P., Wets, R. J.-B., et al. (2015). Toward scalable stochastic unit commitment-part 2: Solver configuration and performance assessment. Energy Systems, 6(3), 417–438.

    Google Scholar 

  • Chinneck, J. W., & Ramadan, K. (2000). Linear programming with interval coefficients. The Journal of the Operational Research Society, 51(2), 209–220.

    Google Scholar 

  • Chitra-Selvi, S., Kumundi-Devi, R. P., & Asir-Rajan, C. C. (2009). Hybrid evolutionary programming approach to multi-area unit commitment with import and export constraints. International Journal of Recent Trends in Engineering, 1(3), 223–228.

    Google Scholar 

  • Cohen, A. I., & Wan, S. H. (1987). A method for solving the fuel constrained unit commitment problem. IEEE Transactions on Power Systems, 2(3), 608–614.

    Google Scholar 

  • Cohen, A. I., & Yoshimura, M. (1983). A branch-and-bound algorithm for unit commitment. IEEE Transactions on Power Apparatus and Systems, 102(2), 444–451.

    Google Scholar 

  • Cohen, G. (1980). Auxiliairy problem principle and decomposition of optimization problems. Journal of Optimization Theory and Applications, 32(3), 277–305.

    Google Scholar 

  • Cohen, G., & Zhu, D. L. (1983). Decomposition-coordination methods in large-scale optimization problems: The non-differentiable case and the use of augmented Lagrangians. Large Scale Systems, Theory and Applications, 1.

  • Conejo, A. J., Carrión, M., & Morales, J. M. (2010). Decision making under uncertainty in electricity markets. Volume 153 of international series in operations research & management science (1st ed.). Berlin: Springer.

    Google Scholar 

  • Conejo, A. J., Contreras, J., Arroyo, J. M., & de la Torre, S. (2002). Optimal response of an oligopolistic generating company to a competitive pool-based electric power market. IEEE Transactions on Power Systems, 17(2), 424–430.

    Google Scholar 

  • Conejo, A. J., Nogales, F. J., & Arroyo, J. M. (2002). Price-taker bidding strategy under price uncertainty. IEEE Transactions on Power Systems, 17(4), 1081–1088.

    Google Scholar 

  • Conejo, A. J., & Prieto, F. J. (2001). Mathematical programming and electricity markets. TOP, 9(1), 1–53.

    Google Scholar 

  • Constantinescu, E. M., Zavala, V. M., Rocklin, M., Lee, S., & Anitescu, M. (2011). A computational framework for uncertainty quantification and stochastic optimization in unit commitment with wind power generation. IEEE Transactions on Power Systems, 26(1), 431–441.

    Google Scholar 

  • Corchero, C., Mijangos, E., & Heredia, F.-J. (2013). A new optimal electricity market bid model solved through perspective cuts. TOP, 21(1), 84–108.

    Google Scholar 

  • Cour des Comptes. (2012). Les coûts de la filière électronucléaire. Technical report, Cour des Comptes.

  • Dal’Santo, T., & Costa, A. S. (2016). Hydroelectric unit commitment for power plants composed of distinct groups of generating units. Electric Power Systems Research, 137(1), 16–25.

    Google Scholar 

  • Daly, P., Flynn, D., & Cunniffe, N. (2015). Inertia considerations within unit commitment and economic dispatch for systems with high non-synchronous penetrations. In PowerTech, 2015 IEEE Eindhoven (pp. 1–6).

  • d’Ambrosio, C., Lodi, A., & Martello, S. (2010). Piecewise linear approxmation of functions of two variables in MILP models. Operations Research Letters, 38, 39–46.

    Google Scholar 

  • Daniildis, A., & Lemaréchal, C. (2005). On a primal-proximal heuristic in discrete optimization. Mathematical Programming Series A, 104, 105–128.

    Google Scholar 

  • d’Antonio, G., & Frangioni, A. (2009). Convergence analysis of deflected conditional approximate subgradient methods. SIAM Journal on Optimization, 20(1), 357–386.

    Google Scholar 

  • Dantzig, G. B., & Wolfe, P. (1960). The decomposition principle for linear programs. Operations Research, 8, 101–111.

    Google Scholar 

  • Dasgupta, D., & McGregor, D. R. (1994). Thermal unit commitment using genetic algorithms. IEEE Proceedings-Generation, Transmission and Distribution, 141(5), 459–465.

    Google Scholar 

  • David, A. K., & Wen, F. (2001). Strategic bidding in competitive electricity markets: A literature survey. Proceedings IEEE PES Summer Meeting, 4, 2168–2173.

    Google Scholar 

  • de Farias, D. P., & Van Roy, B. (2003). The linear programming approach to approximate dynamic programming. Operations Research, 51(6), 850–865.

    Google Scholar 

  • de la Torre, S., Arroyo, J. M., Conejo, A. J., & Contreras, J. (2002). Price maker self-scheduling in a pool-based electricity market: A mixed-integer LP approach. IEEE Transactions on Power Systems, 17(4), 1037–1042.

    Google Scholar 

  • de Oliveira, W., & Sagastizábal, C. (2014). Level bundle methods for oracles with on demand accuracy. Optimization Methods and Software, 29(6), 1180–1209.

    Google Scholar 

  • de Oliveira, W., Sagastizábal, C., & Lemaréchal, C. (2014). Convex proximal bundle methods in depth: A unified analysis for inexact oracles. Mathematical Programming Series B, 148, 241–277.

    Google Scholar 

  • de Oliveira, W., Sagastizábal, C. A., & Scheimberg, S. (2011). Inexact bundle methods for two-stage stochastic programming. SIAM Journal on Optimization, 21(2), 517–544.

    Google Scholar 

  • Demartini, G., De Simone, T. R., Granelli, G. P., Montagna, M., & Robo, K. (1998). Dual programming methods for large-scale thermal generation scheduling. IEEE Transactions on Power Systems, 13, 857–863.

    Google Scholar 

  • Dempe, S., & Dutta, J. (2012). Is bilevel programming a special case of a mathematical program with complementarity constraints? Mathematical Programming, 131(1), 37–48.

    Google Scholar 

  • Dempe, S., Kalashnikov, V., Pérez-Valdés, G. A., & Kalashnykova, N. (2015). Bilevel programming problems: Theory, algorithms and applications to energy networks. Energy systems. Berlin: Springer.

    Google Scholar 

  • Dentcheva, D. (2009). Optimisation models with probabilistic constraints. In A. Shapiro, D. Dentcheva, & A. Ruszczyński (Eds.), Lectures on stochastic programming: Modeling and theory. Volume 9 of MPS-SIAM series on optimization (pp. 87–154). Philadelphia: SIAM and MPS.

    Google Scholar 

  • Dentcheva, D., & Römisch, W. (1998). Optimal power generation under uncertainty via stochastic programming. In K. Marti & P. Kall (Eds.), Stochastic programming methods and technical applications. Volume 458 of lecture notes in economics and mathematical systems (pp. 22–56). Berlin: Springer.

    Google Scholar 

  • Dieu, V. N., & Ongsakul, W. (2008). Ramp rate constrained unit commitment by improved priority list and augmented lagrange hopfield network. Electric Power Systems Research, 78(3), 291–301.

    Google Scholar 

  • Di Lullo, M. (2013). Modelli di ottimizzazione per lo unit commitment con optimal transmission switching: Analisi e implementazione. Master’s thesis, Facoltá di Ingegneria dell’Informazione, Informatica e Statistica, Universitá di Roma La Sapienza, Piazzale Aldo Moro, 5 00185, Roma.

  • Dillon, T. S., Edwin, K. W., Kochs, H. D., & Taud, R. J. (1978). Integer programming approach to the problem of optimal unit commitment with probabilistic reserve determination. IEEE Transactions on Power Apparatus and Systems, PAS–97(6), 2154–2166.

    Google Scholar 

  • Dillon, T. S., & Egan, G. T. (1976). The application of combinatorial methods to the problems of maintenance scheduling and unit commitment in large power system. In 1st IFAC symposium on large scale systems theory and applications, Udine.

  • Ding, X., Lee, W.-J., Jianxue, W., & Liu, L. (2010). Studies on stochastic unit commitment formulation with flexible generating units. Electric Power Systems Research, 80, 130–141.

    Google Scholar 

  • Diniz, A. L., & Henrion, R. (2017). On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions. Energy Systems, 8(1), 149–167.

    Google Scholar 

  • Diongue, A. K. (2005). Modélisation longue mémoire multivariée : Applications aux problématiques du producteur d’EDF dans le cadre de la libéralisation du marché Européen de l’électricité. Ph.D. thesis, ENS Cachan.

  • du Merle, O., Goffin, J.-L., & Vial, J.-P. (1998). On improvements to the analytic center cutting plane method. Computational Optimization and Applications, 11, 37–52.

    Google Scholar 

  • Dubost, L., Gonzalez, R., & Lemaréchal, C. (2005). A primal-proximal heuristic applied to french unitcommitment problem. Mathematical Programming, 104(1), 129–151.

    Google Scholar 

  • Duo, H., Sasaki, H., Nagata, T., & Fujita, H. (1999). A solution for unit commitment using lagrangian relaxation combined with evolutionary programming. Electric Power Systems Research, 51(1), 71–77.

    Google Scholar 

  • Dupačová, J., Gröwe-Kuska, N., & Römisch, W. (2003). Scenario reduction in stochastic programming: An approach using probability metrics. Mathematical Programming, 95(3), 493–511.

    Google Scholar 

  • Durga Hari Kiran, B., & Kumari, M  Sailaja. (2016). Demand response and pumped hydro storage scheduling for balancing wind power uncertainties: A probabilistic unit commitment approach. International Journal of Electrical Power and Engineering Systems, 81, 114–122.

    Google Scholar 

  • Dvorkin, Y., Pandžić, H., Ortega-Vazquez, M. A., & Kirschen, D. S. (2015). A hybrid stochastic/interval approach to transmission-constrained unit commitment. IEEE Transactions on Power Systems, 30(2), 621–631.

    Google Scholar 

  • Ea, K. (2012). The electricity spot markets prices modeling: Proposal for a new mathematical formulation taking into account the market player strategy. In International conference on the European energy market (EEM).

  • Eichhorn, A., Heitsch, H., & Römisch, W. (2010). Stochastic optimization of electricity portfolios: Scenario tree modeling and risk management. In S. Rebennack, P. M. Pardalos, M. V. F. Pereira, & N. Iliadis (Eds.), Handbook of power systems II (pp. 405–432). Berlin: Springer.

    Google Scholar 

  • Ela, E., Gevorgian, V., Tuohy, A., Kirby, B., Milligan, M., & O’Malley, M. (2014a). Market designs for the primary frequency response ancillary service-part I: Motivation and design. IEEE Transactions on Power Systems, 29(1), 421–431.

    Google Scholar 

  • Ela, E., Gevorgian, V., Tuohy, A., Kirby, B., Milligan, M., & O’Malley, M. (2014b). Market designs for the primary frequency response ancillary service-part II: Case studies. IEEE Transactions on Power Systems, 29(1), 432–440.

    Google Scholar 

  • El Ghaoui, L., & Lebret, H. (2006). Robust solutions to least-squares problems with uncertain data. SIAM Journal on Matrix Analysis and Applications, 18(4), 1035–1064.

    Google Scholar 

  • El Ghaoui, L., Oustry, F., & Lebret, H. (1998). Robust solutions to uncertain semidefinite programs. SIAM Journal on Optimization, 9(1), 33–52.

    Google Scholar 

  • Erkmen, I., & Karatas, B. (1994). Short-term hydrothermal coordination by using multi-pass dynamic programming with successive approximation. In 7th Mediterranean electrotechnical conference 1994 (Vol. 3, pp. 925–928).

  • Eyer, J., & Corey, G. (2010). Energy storage for the electricity grid: Benefits and market potential assessment guide. Technical report, Sandia National Laboratories, Albuquerque, New Mexico.

  • Fábián, C. I. (2013). Computational aspects of risk-averse optimisation in two-stage stochastic models. Technical report, Institute of Informatics, Kecskemét College, Hungary. Optimization online report.

  • Fan, L., Pan, K., Guan, Y., Chen, Y., & Wang, X. (2016). Strengthened MILP formulation for combined-cycle units. IEEE (submitted).

  • Fan, W., Guan, X., & Zhai, Q. (2002). A new method for unit commitment with ramping constraints. Electric Power Systems Research, 62(3), 215–224.

    Google Scholar 

  • Farhat, I. A., & El-Hawary, M. E. (2009). Optimization methods applied for solving the short-term hydrothermal coordination problem. Electric Power Systems Research, 79, 1308–1320.

    Google Scholar 

  • Farshbaf-Shaker, M. H., Henrion, R., & Hömberg, D. (2017). Properties of chance constraints in infinite dimensions with an application to PDE constrained optimization. Set Valued and Variational, Analysis, 1–21.

  • Fattahi, S., Ashraphijuo, M., Lavaei, J., & Atamtúrk, A. (2017). Conic relaxations of the unit commitment problem. Energy, 134, 1079–1095.

    Google Scholar 

  • Fattahi, S., Lavaei, J., & Atamturk, A. (2017). A bound strengthening method for optimal transmission switching in power systems. Technical report.

  • Feizollahi, M. J., Costley, M., Ahmed, S., & Grijalva, S. (2015). Large-scale decentralized unit commitment. Electrical Power and Energy Systems, 73, 97–106.

    Google Scholar 

  • Feltenmark, S., & Kiwiel, K. C. (2000). Dual applications of proximal bundle methods, including lagrangian relaxation of nonconvex problems. SIAM Journal on Optimization, 10(3), 697–721.

    Google Scholar 

  • Feng, Y., & Ryan, S. M. (2016). Solution sensitivity-based scenario reduction for stochastic unit commitment. Computional Management Science, 1(13), 29–62.

    Google Scholar 

  • Ferreira, L. A. F. M. (1994). On the convergence of the classic hydro-thermal coordination algorithm. IEEE Transactions on Power Systems, 9, 1002–1008.

    Google Scholar 

  • Finardi, E. C., & Scuzziato, M. R. (2013). Hydro unit commitment and loading problem for day-ahead operation planning problem. Electrical Power and Energy Systems, 44, 7–16.

    Google Scholar 

  • Finardi, E. C., & Scuzziato, M. R. (2014). A comparative analysis of different dual problems in the Lagrangian relaxation context for solving the hydro unit commitment problem. Electric Power Systems Research, 107, 221–229.

    Google Scholar 

  • Finardi, E. C., & Da Silva, E. L. (2006). Solving the hydro unit commitment problem via dual decomposition and sequential quadratic programming. IEEE Transactions on Power Systems, 21(2), 835–844.

    Google Scholar 

  • Finardi, E. C., Takigawa, F. Y. K., & Brito, B. H. (2016). Assessing solution quality and computational performance in the hydro unit commitment problem considering different mathematical programming approaches. Electric Power Systems Research, 136, 212–222.

    Google Scholar 

  • Fischetti, M., & Monaci, M. (2009). Light robustness. In R. K. Ahuja, R. Möhring, & C. Zaroliagis (Eds.), Robust and online large-scale optimization. Volume 5868 of LNCS (pp. 61–84).

    Google Scholar 

  • Fisher, E. B., O’Neill, R. P., & Ferris, M. C. (2008). Optimal transmission switching. IEEE Transactions on Power Systems, 23(3), 1346–1355.

    Google Scholar 

  • Fisher, M. L. (1973). Optimal solution of scheduling problems using lagrange multipliers: Part I. Operations Research, 21(5), 1114–1127.

    Google Scholar 

  • Fleten, S.-E., & Kristoffersen, T. K. (2008). Short-term hydropower production planning by stochastic programming. Computers & Operations Research, 35, 2656–2671.

    Google Scholar 

  • Fonoberova, M. (2010). Algorithms for finding optimal flows in dynamic networks. In S. Rebennack, P. M. Pardalos, M. V. F. Pereira, & N. Iliadis (Eds.), Handbook of power systems II (pp. 31–54). Berlin: Springer.

    Google Scholar 

  • Fotuhi-Firuzabad, M., & Billinton, R. (2000). A reliability framework for generating unit commitment. Electric Power Systems Research, 56(1), 81–88.

    Google Scholar 

  • Frangioni, A. (2002). Generalized bundle methods. SIAM Journal on Optimization, 13(1), 117–156.

    Google Scholar 

  • Frangioni, A. (2005). About Lagrangian methods in integer optimization. Annals of Operations Research, 139(1), 163–193.

    Google Scholar 

  • Frangioni, A., & Gentile, C. (2006). Perspective cuts for a class of convex 0–1 mixed integer programs. Mathematical Programming, 106(2), 225–236.

    Google Scholar 

  • Frangioni, A., & Gentile, C. (2006). Solving non-linear single-unit commitment problems with ramping constraints. Operations Research, 54(4), 767–775.

    Google Scholar 

  • Frangioni, A., & Gentile, C. (2015). Technical report R. 15–06, Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, C.N.R.

  • Frangioni, A., Gentile, C., & Lacalandra, F. (2008). Solving unit commitment problems with general ramp contraints. International Journal of Electrical Power and Energy Systems, 30, 316–326.

    Google Scholar 

  • Frangioni, A., Gentile, C., & Lacalandra, F. (2009). Tighter approximated MILP formulations for unit commitment problems. IEEE Transactions on Power Systems, 24(1), 105–113.

    Google Scholar 

  • Frangioni, A., Gentile, C., & Lacalandra, F. (2011). Sequential Lagrangian-MILP approaches for unit commitment problems. International Journal of Electrical Power and Energy Systems, 33, 585–593.

    Google Scholar 

  • Frangioni, A., & Gorgone, E. (2012). Generalized bundle methods for sum-functions with “easy” components: Applications to multicommodity network design. Technical report 12-12, Dipartimento di Informatica, Università di Pisa.

  • Fu, Y., Li, Z., & Wu, L. (2013). Modeling and solution of the large-scale security-constrained unit commitment. IEEE Transactions on Power Systems, 28(4), 3524–3533.

    Google Scholar 

  • Fu, Y., & Shahidehpour, M. (2007). Fast SCUC for large-scale power systems. IEEE Transactions on Power Systems, 22(4), 2144–2151.

    Google Scholar 

  • Fu, Y., Shahidehpour, M., & Li, Z. (2005). Long-term security-constrained unit commitment: Hybrid dantzig-wolfe decomposition and subgradient approach. IEEE Transactions on Power Systems, 20(4), 2093–2106.

    Google Scholar 

  • Gabriel, S. A., Conejo, A. J., Fuller, J. D., Hobbs, B. F., & Ruiz, C. (2013). Complementarity modeling in energy markets. Volume 180 of International series in operations research & management science (1st ed.). Berlin: Springer.

    Google Scholar 

  • García-González, J., San Roque, A. M., Campos, F. A., & Villar, J. (2007). Connecting the intraday energy and reserve markets by an optimal redispatch. IEEE Transactions on Power Systems, 22(4), 2220–2231.

    Google Scholar 

  • Garver, L. L. (1962). Power generation scheduling by integer programming-development of theory. Transactions of the American Institute of Electrical Engineers Power Apparatus and Systems, Part III, 81(3), 730–734.

    Google Scholar 

  • Gazafroudi, A. S., Shafie-khah, M., Abedi, M., Hosseinian, S. H., Dehkordi, G. H. R., Goel, L., et al. (2017). A novel stochastic reserve cost allocation approach of electricity market agents in the restructured power systems. Electric Power Systems Research, 152, 223–236.

    Google Scholar 

  • Ge, W. (2010). Ramp rate constrained unit commitment by improved priority list and enhanced particle swarm optimization. International Conference on Computational Intelligence and Software Engineering (CiSE), 2010, 1–8.

    Google Scholar 

  • Geng, Z., Conejo, A. J., Chen, Q., & Kang, C. (2018). Power generation scheduling considering stochastic emission limits. Electrical Power and Energy Systems, 95(1), 374–383.

    Google Scholar 

  • Georges, D. (1994). Optimal unit commitment in simulations of hydrothermal power systems: An augmented Lagrangian approach. Simulation Practice and Theory, 1(4), 155–172.

    Google Scholar 

  • Gil, H. A., Gómez-Quiles, C., Gómez-Exposito, A., & Santos, J. R. (2012). Forecasting prices in electricity markets: Needs, tools and limitations. In A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis, & M. V. F. Pereira (Eds.), Handbook of networks in power systems I (pp. 123–150). Heidelberg: Springer.

    Google Scholar 

  • Gill, P. E., Murray, W., & Wright, M. H. (1982). Practical optimization (1st ed.). London: Emerald Group Publishing Limited.

    Google Scholar 

  • Gjengedal, T. (1996). Emission constrained unit-commitment (ECUC). IEEE Transaction on Energy Conversion, 11(1), 132–138.

    Google Scholar 

  • Gollmer, R., Moller, A., Nowak, M. P., Romisch, W., & Schultz, R. (1999). Primal and dual methods for unit commitment in a hydro-thermal power system. In Proceedings 13th power systems computation conference (pp. 724–730).

  • Gradón, T. González, Heitsch, H., & Henrion, R. (2017). A joint model of probabilistic/robust constraints for gas transport management in stationary networks. Computational Management Science, 14, 443–460.

    Google Scholar 

  • Gooi, H. B., Mendes, D. P., Bell, K. R. W., & Kirschen, D. S. (1999). Optimal scheduling of spinning reserve. IEEE Transactions on Power Systems, 14(4), 1485–1492.

    Google Scholar 

  • Gotzes, C., Heitsch, H., Henrion, R., & Schultz, R. (2016). On the quantification of nomination feasibility in stationary gas networks with random load. Mathematical Methods of Operations Research, 84, 427–457.

    Google Scholar 

  • Gröwe, N., Römisch, W., & Schultz, R. (1995). A simple recourse model for power dispatch under uncertain demand. Annals of Operations Research, 59(1), 135–164.

    Google Scholar 

  • Gröwe-Kuska, N., Kiwiel, K. C., Nowak, M. P., Römisch, W., & Wegner, I. (2002). Power management in a hydro-thermal system under uncertainty by Lagrangian relaxation. In C. Greengard & A. Ruszczyński (Eds.), Decision making under uncertainty. Volume 128 of The IMA volumes in mathematics and its applications (pp. 39–70). New York: Springer.

    Google Scholar 

  • Guan, X., Luh, P. B., Houzhong, Y., & Amalfi, J. A. (1991). Environmentally constrained unit commitment. In Power industry computer application conference, Baltimore, MD.

  • Guan, X., Luh, P. B., & Zhang, L. (1995). Nonlinear approximation method in Lagrangian relaxation-based algorithms for hydrothermal scheduling. IEEE Transactions on Power Systems, 10, 772–778.

    Google Scholar 

  • Guan, X., Luh, P. B., Yan, H., & Rogan, P. (1994). Optimization-based scheduling of hydrothermal power systems with pumped-storage units. IEEE Transactions on Power Systems, 9, 1023–1031.

    Google Scholar 

  • Guan, X., Ni, E., Li, R., & Luh, P. B. (1997). An optimization-based algorithm for scheduling hydrothermal power systems with cascaded reservoirs and discrete hydro constraints. IEEE Transactions on Power Systems, 12, 1775–1780.

    Google Scholar 

  • Guan, Y., & Wang, J. (2014). Uncertainty sets for robust unit commitment. IEEE Transactions on Power Systems, 29(3), 1439–1440.

    Google Scholar 

  • Guignard, M. (2003). Lagrangean relaxation. TOP, 11(2), 151–228.

    Google Scholar 

  • Guignard, M., & Kim, S. (1987). Lagrangian decomposition: A model yielding stronger Lagrangian bounds. Mathematical Programming, 39, 215–228.

    Google Scholar 

  • Guigues, V. (2009). Robust product management. Optimization and Engineering, 10(4), 505–532.

    Google Scholar 

  • Guigues, V. (2013). SDDP for some interstage dependent risk-averse problems and application to hydro-thermal planning. Computational Optimization and Applications, 10(4), 505–532.

    Google Scholar 

  • Habibollahzadeh, H., & Bubenko, J. A. (1986). Application of decomposition techniques to short-term operation planning of hydrothermal power system. IEEE Transactions on Power Systems, 1(1), 41–47.

    Google Scholar 

  • Hanasusanto, G. A., Roitch, V., Kuhn, D., & Wiesemann, W. (2015). A distributionally robust perspective on uncertainty quantification and chance constrained programming. Mathematical Programming Series B, 151, 35–62.

    Google Scholar 

  • Hantoute, A., Henrion, R., & Pérez-Aros, P. (2018). Subdifferential characterization of continuous probability functions under gaussian distribution. Mathematical Programming, 1–28.

  • Hara, K., Kimura, M., & Honda, N. (1966). A method for planning economic unit commitment and maintenance of thermal power systems. IEEE Transactions on Power Apparatus and Systems, PAS–85(5), 427–436.

    Google Scholar 

  • Harris, C. (2011). Electricity markets: Pricing, structures and economics. Volume 565 of the Wiley finance series. Hoboken: Wiley.

    Google Scholar 

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

    Google Scholar 

  • Hedman, K. W., O’Neill, R. P., Fisher, E. B., & Oren, S. S. (2009). Optimal transmission switching with contingency analysis. IEEE Transactions on Power Systems, 24(3), 1577–1586.

    Google Scholar 

  • Hedman, K. W., Oren, S. S., & O’Neill, R. P. (2011a). Optimal transmission switching: Economic efficiency and market implications. Journal of Regulatory Economics, 40(3), 111–140.

    Google Scholar 

  • Hedman, K. W., Oren, S. S., & O’Neill, R. P. (2011b). A review of transmission switching and network topology optimization. In 2011 IEEE power and energy society general meeting (pp. 1–7). IEEE.

  • Heitsch, H., & Römisch, W. (2003). Scenario reduction algorithms in stochastic programming. Computation Optimization and Applications, 24(2–3), 187–206.

    Google Scholar 

  • Heitsch, H., & Römisch, W. (2009). Scenario tree reduction for multistage stochastic programs. Computational Management Science, 6(2), 117–133.

    Google Scholar 

  • Heitsch, H., & Römisch, W. (2011). Scenario tree generation for multi-stage stochastic programs. In M. Bertocchi, G. Consigli, & M. A. H. Dempster (Eds.), Stochastic optimization methods in finance and energy: New financial products and energy market strategies. Volume 163 of international series in operations research & management science (pp. 313–341). Berlin: Springer.

    Google Scholar 

  • Henrion, R. (2004). Introduction to chance constraint programming. Tutorial paper for the Stochastic Programming Community HomePage. http://www.wias-berlin.de/people/henrion/ccp.ps.

  • Henrion, R. (2010). Optimierungsprobleme mit wahrscheinlichkeitsrestriktionen: Modelle, struktur, numerik. Lecture notes (pp. 43).

  • Henrion, R., Küchler, C., & Römisch, W. (2008). Discrepancy distances and scenario reduction in two-stage stochastic integer programming. Journal of Industrial and Management Optimization, 4, 363–384.

    Google Scholar 

  • Henrion, R., Küchler, C., & Römisch, W. (2009). Scenario reduction in stochastic programming with respect to discrepancy distances. Computational Optimization and Applications, 43, 67–93.

    Google Scholar 

  • Henrion, R., & Möller, A. (2012). A gradient formula for linear chance constraints under Gaussian distribution. Mathematics of Operations Research, 37, 475–488.

    Google Scholar 

  • Henrion, R., & Römisch, W. (1999). Metric regularity and quantitative stability in stochastic programs with probabilistic constraints. Mathematical Programming, 84, 55–88.

    Google Scholar 

  • Henrion, R., & Römisch, W. (2004). Hölder and lipschitz stability of solution sets in programs with probabilistic constraints. Mathematical Programming, 100, 589–611.

    Google Scholar 

  • Henrion, R., & Strugarek, C. (2008). Convexity of chance constraints with independent random variables. Computational Optimization and Applications, 41, 263–276.

    Google Scholar 

  • Henrion, R., & Strugarek, C. (2011). Convexity of chance constraints with dependent random variables: The use of copulae. In M. Bertocchi, G. Consigli, & M. A. H. Dempster (Eds.), Stochastic optimization methods in finance and energy: New Financial products and energy market strategies, international series in operations research and management science (pp. 427–439). New York: Springer.

    Google Scholar 

  • Heredia, F. J., & Nabona, N. (1995). Optimum short-term hydrothermal scheduling with spinning reserve through network flows. IEEE Transactions on Power Systems, 10, 1642–1651.

    Google Scholar 

  • Higgs, H., & Worthington, A. (2008). Stochastic price modeling of high volatility, mean-reverting, spike-prone commodities: The australian wholesale spot electricity market. Energy Economics, 30(6), 3172–3185.

    Google Scholar 

  • Hijazi, H. L., Coffrin, C., & Van Hentenryck, P. (2013). Convex quadratic relaxations of nonlinear programs in power systems (submitted).

  • Hobbs, W. J., Hermon, G., Warner, S., & Shelbe, G. B. (1988). An enhanced dynamic programming approach for unit commitment. IEEE Transactions on Power Systems, 3(3), 1201–1205.

    Google Scholar 

  • Hobbs, B. F., Rothkopf, M., O’Neill, R. P., & Chao, H. P. (2001). The next generation of electric power unit commitment models. International series in operations research & management science (Vol. 36). Berlin: Springer.

    Google Scholar 

  • Hreinsson, K., Vrakopoulou, M., & Andersson, G. (2015). Stochastic security constrained unit commitment and non-spinning reserve allocation with performance guarantees. Electrical Power and Energy Systems, 72, 109–115.

    Google Scholar 

  • Hsu, Y. Y., Su, C.-C., Lin, C.-J., & Huang, C.-T. (1991). Dynamic security constrained multi-area unit commitment. IEEE Transactions on Power Systems, 6(3), 1049–1055.

    Google Scholar 

  • Huang, K. Y., Yang, H. T., & Huang, C. L. (1998). A new thermal unit commitment approach using constraint logic programming. IEEE Transactions on Power Systems, 13(3), 936–945.

    Google Scholar 

  • Huang, Y., Zheng, Q. P., & Wang, J. (2014). Two-stage stochastic unit commitment model includingnon-generation resources with conditional value-at-risk constraints. Electric Power Systems Research, 116(1), 427–438.

    Google Scholar 

  • Jabr, R. A. (2005). Robust self-scheduling under price uncertainty using conditional value-at-risk. IEEE Transactions on Power Systems, 20(4), 1852–1858.

    Google Scholar 

  • Jabr, R. A. (2006). Radial distribution load flow using conic programming. IEEE Transactions on Power Systems, 21(3), 1458–1459.

    Google Scholar 

  • Jabr, R. A. (2008). Optimal power flow using an extended conic quadratic formulation. IEEE Transactions on Power Systems, 23(3), 1000–1008.

    Google Scholar 

  • Jabr, R. A. (2010). Recent developments in optimal power flow modeling. In S. Rebennack, P. M. Pardalos, M. V. F. Pereira, & N. Iliadis (Eds.), Handbook of power systems II (pp. 3–30). Berlin: Springer.

    Google Scholar 

  • Jabr, R. A. (2012). Tight polyhedral approximation for mixed-integer linear programming unit commitment formulations. IET Generation, Transmission & Distribution, 6(11), 1104–1111.

    Google Scholar 

  • Jabr, R. A. (2013). Adjustable robust OPF with renewable energy sources. IEEE Transactions on Power Systems, 28(4), 4741–4751.

    Google Scholar 

  • Jia, J., & Guan, X. (2011). MILP formulation for short-term scheduling of cascaded reservoirs with head effects. In 2011 2nd International conference on artificial intelligence, management science and electronic commerce (AIMSEC) (pp. 4061–4064).

  • Jiang, R., Guan, Y., & Watson, J.-P. (2016). Cutting planes for the multistage stochastic unit commitment problem. Mathematical Programming, 157(1), 121–151.

    Google Scholar 

  • Jiang, R., Wang, J., & Guan, Y. (2012). Robust unit commitment with wind power and pumped storage hydro. IEEE Transactions on Power Systems, 27(2), 800–810.

    Google Scholar 

  • Jiang, R., Wang, J., Zhang, M., & Guan, Y. (2013). Two-stage minimax regret robust unit commitment. IEEE Transactions on Power Systems, 28(3), 2271–2013.

    Google Scholar 

  • Jiang, R., Zhang, M., Li, G., & Guan, Y. (2014). Two-stage network constrained robust unit commitment problem. European Journal of Operational Research, 234(1), 751–762.

    Google Scholar 

  • Jin, H., Li, Z., Sun, H., Guo, Q., Chen, R., & Wang, B. (2017). A robust aggregate model and the two-stage solution method to incorporate energy intensive enterprises in power system unit commitment. Applied Energy, 206, 1364–1378.

    Google Scholar 

  • Johnson, R. C., Happ, H. H., & Wright, W. J. (1971). Large scale hydro-thermal unit commitment-method and results. IEEE Transactions on Power Apparatus and Systems, PAS–90(3), 1373–1384.

    Google Scholar 

  • Juste, K. A., Kita, H., Tanaka, E., & Hasegawa, J. (1999). An evolutionary programming solution to the unit commitment problem. IEEE Transactions on Power Systems, 14(4), 1452–1459.

    Google Scholar 

  • Kaibel, V., Peinhardt, M., & Pfetsch, M. E. (2011). Orbitopal fixing. Discrete Optimization, 8(4), 595–610.

    Google Scholar 

  • Kalantari, A., & Galiana, F. D. (2015). Generalized sigma approach to unit commitment with uncertain wind power generation. Electrical Power and Energy Systems, 65(1), 367–374.

    Google Scholar 

  • Kall, P., & Mayer, J. (2005). Stochastic linear programming: Models, theory and computation. International series in operations research and management science (1st ed.). Berlin: Springer.

    Google Scholar 

  • Kazemzadeh, N., Ryan, S . M., & Hamzeei, M. (2017). Robust optimization vs. stochastic programming incorporating risk measures for unit commitment with uncertain variable renewable generation. Energy Systems, To Appear, 1–25.

    Google Scholar 

  • Kelley, J. E. (1960). The cutting-plane method for solving convex programs. Journal of the Society for Industrial and Applied Mathematics, 8(4), 703–712.

    Google Scholar 

  • Kerr, R. H., Scheidt, J. L., Fontanna, A. J., & Wiley, J. K. (1966). Unit commitment. IEEE Transactions on Power Apparatus and Systems, PAS–85(5), 417–421.

    Google Scholar 

  • Keyhani, A., Marwali, M. N., & Dai, M. (2010). Integration of Green and Renewable Energy in Electric Power Systems (1st ed.). Hoboken: Wiley.

    Google Scholar 

  • Kia, M., Nazar, M. S., Sepasian, M. S., Heidari, A., & Siano, P. (2017). An efficient linear model for optimal day ahead scheduling of CHP units in active distribution networks considering load commitment programs. Energy, 139, 798–817.

    Google Scholar 

  • Kiwiel, K. C. (2012). Bundle methods for convex minimization with partially inexact oracles. Computational Optimization and Applications

  • Kocuk, B., Dey, S. S., & Sun, X. A. (2017). New formulation and strong misocp relaxations for AC optimal transmission switching problem. IEEE Transactions on Power Systems, 32(6), 4161–4170.

    Google Scholar 

  • Kocuk, B., Jeon, H., Dey, S. S., Linderoth, J., Luedtke, J., & Sun, X. A. (2016). A cycle-based formulation and valid inequalities for DC power transmission problems with switching. Operations Research, 64(4), 922–938.

    Google Scholar 

  • Korad, K., & Hedman, A. S. (2013). Robust corrective topology control for system reliability. IEEE Transactions on Power Systems, 28(4), 1346–1355.

    Google Scholar 

  • Kort, B. W., & Bertsekas, D. P. (1972). A new penalty function method for constrained optimization. IEEE Conference on Decision and Control, 1972, 162–166.

    Google Scholar 

  • Kuloor, S., Hope, G. S., & Malik, O. P. (1992). Environmentally constrained unit commitment. IEE Proceedings C: Generation, Transmission and Distribution, 139(2), 122–128.

    Google Scholar 

  • Kwon, R. H., & Frances, D. (2012). Optimization-based bidding in day-ahead electricity auction markets: A review of models for power producers. In A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis, & M. V. F. Pereira (Eds.), Handbook of networks in power systems I (pp. 41–60). Heidelberg: Springer.

    Google Scholar 

  • Laia, R., Pousinho, H. M. I., Melício, R., Mendes, V. M. F., & Collares-Pereira, M. (2014). Stochastic unit commitment problem with security and emissions constraints. In L. M. Camarinha-Matos, N. S. Barrento, & R. Mendonça (Eds.), Technological innovation for collective awareness systems. Volume 423 of IFIP advances in information and communication technology (pp. 388–397).

    Google Scholar 

  • Laporte, G., & Louveaux, F. V. (1993). The integer l-shaped method for stochastic integer programs with complete recourse. Operations Research Letters, 13(3), 133–142.

    Google Scholar 

  • Lauer, G. S., Sandell, N. R., Bertsekas, D. P., & Posbergh, T. A. (1982). Solution of large-scale optimal unit commitment problems. IEEE Transactions on Power Apparatus and Systems, PAS–101(1), 79–86.

    Google Scholar 

  • Lavaei, J., & Low, S. (2012). Zero duality gap in optimal power flow problem. IEEE Transactions on Power Systems, 27(1), 92–107.

    Google Scholar 

  • Le, K. D., Jackups, R. R., Feinstein, J., & Griffith, J. S. (1990). Operational aspects of generation cycling. IEEE Transactions on Power Systems, 5(4), 1194–1203.

    Google Scholar 

  • Lee, C., Liu, C., Mehrotra, S., & Shahidehpour, M. (2014). Modeling transmission line constraints in two-stage robust unit commitment problem. IEEE Transactions on Power Systems, 29(3), 1221–1231.

    Google Scholar 

  • Lee, F. N. (1988). Short-term thermal unit commitment-a new method. IEEE Transactions on Power Systems, 3(2), 421–428.

    Google Scholar 

  • Lee, F. N. (1991). The application of commitment utilization factor (CUF) to thermal unit commitment. IEEE Transactions on Power Systems, 6(2), 691–698.

    Google Scholar 

  • Lee, F. N., & Feng, Q. (1992). Multi-area unit commitment. IEEE Transactions on Power Systems, 7(2), 591–599.

    Google Scholar 

  • Lee, F. N., Huang, J., & Adapa, R. (1994). Multi-area unit commitment via sequential method and a DC power flow network model. IEEE Transaction on Power Systems, 9(1), 297–287.

    Google Scholar 

  • Lemaréchal, C. (1975). An extension of davidon methods to nondifferentiable problems. Mathematical programming study, 3, 95–109.

    Google Scholar 

  • Lemaréchal, C. (2001). Lagrangian relaxation. In M. Jünger & D. Naddef (Eds.), Computational combinatorial optimization: Optimal or provably near-optimal solutions. Volume 9 of Lecture notes in computer science (pp. 112–156). Berlin: Springer.

    Google Scholar 

  • Lemaréchal, C., Nemirovskii, A., & Nesterov, Y. (1995). New variants of bundle methods. Mathematical Programming, 69(1), 111–147.

    Google Scholar 

  • Lemaréchal, C., & Renaud, A. (2001). A geometric study of duality gaps, with applications. Mathematical Programming, 90, 399–427.

    Google Scholar 

  • Lemaréchal, C., & Sagastizábal, C. (1994). An approach to variable metric bundle methods. Lecture Notes in Control and Information Science, 197, 144–162.

    Google Scholar 

  • Lemaréchal, C., & Sagastizábal, C. (1995). Application of bundle methods to the unit-commitment problem. Rapport Technique Nb 0184 INRIA (pp. 1–19).

  • Leveque, F. (2002). Competitive electricity markets and sustainability. Cheltenham: Edward Elgar Pub.

    Google Scholar 

  • Li, C., Johnson, R. B., & Svoboda, A. J. (1997). A new unit commitment method. IEEE Transaction on Power Systems, 12(1), 113–119.

    Google Scholar 

  • Li, G., Lawarree, J., & Liu, C. C. (2010). State-of-the-art of electricity price forecasting in a grid. In S. Rebennack, P. M. Pardalos, M. V. F. Pereira, & N. Iliadis (Eds.), Handbook of power systems II (pp. 161–188). Berlin: Springer.

    Google Scholar 

  • Li, J., Wen, J., & Han, X. (2015). Low-carbon unit commitment with intensive wind power generation and carbon capture power plant. Journal of Modern Power System and Clean Energy, 3(1), 63–71.

    Google Scholar 

  • Li, T., & Shahidehpour, M. (2005). Strategic bidding of transmission-constrained GENCOs with incomplete information. IEEE Transactions on Power Systems, 20(1), 437–447.

    Google Scholar 

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

    Google Scholar 

  • Li, Z., & Shahidehpour, M. (2003). Generation scheduling with thermal stress constraints. IEEE Transactions on Power Systems, 18(4), 1402–1409.

    Google Scholar 

  • Liang, R.-H., & Kang, F.-C. (2000). Thermal generating unit commitment using an extended mean field annealing neural network. IEE Proceedings-Generation, Transmission and Distribution, 147(3), 164–170.

    Google Scholar 

  • Lin, W.-M., Cheng, F.-S., & Tsay, M.-T. (2002). An improved tabu search for economic dispatch with multiple minima. IEEE Transactions on Power Systems, 17(1), 108–112.

    Google Scholar 

  • Liu, C., Shahidehpour, M., & Wu, L. (2010). Extended benders decomposition for two-stage scuc. IEEE Transactions on Power Systems, 25(2), 1192–1194.

    Google Scholar 

  • Liu, C., Wang, J., & Ostrowski, J. (2012). Heuristic prescreening switchable branches in optimal transmission switching. IEEE Transactions on Power Systems, 27(4), 2289–2290.

    Google Scholar 

  • Liu, C., Wang, J., & Ostrowski, J. (2012). Static security in multi-period transmission switching. IEEE Transactions on Power Systems, 27(4), 1850–1858.

    Google Scholar 

  • Liu, G., & Tomsovic, K. (2015). Robust unit commitment considering uncertain demand responseg. Electric Power Systems Research, 119(1), 126–137.

    Google Scholar 

  • Liu, X., Küçükyavuz, S., & Luedtke, J. (2016). Decomposition algorithm for two-stage chance constrained programs. Mathematical Programming Series B, 157(1), 219–243.

    Google Scholar 

  • Løkketangen, A., & Woodruff, D. L. (1996). Progressive Hedging and Tabu search applied to mixed integer (0,1) multistage stochastic programming. Journal of Heuristics, 2(2), 111–128.

    Google Scholar 

  • Louveaux, F. V., & Schultz, R. (2003). Stochastic integer programming. In A. Ruszczyński & A. Shapiro (Eds.), Stochastic programming. Volume 10 of Handbooks in operations research and management science (Chapter 4). Amsterdam: Elsevier.

    Google Scholar 

  • Lu, B., & Shahidehpour, M. (2005). Unit commitment with flexible generating units. IEEE Transactions on Power Systems, 20(2), 1022–1034.

    Google Scholar 

  • Lucas, J.-Y., & Triboulet, T. (2012). Hybridization of augmented Lagrangian and genetic algorithm for day-to-day unit commitment problem. In META 12: International conference on metaheuristics and nature inspired computing.

  • Luedtke, J. (2014). A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Mathematical Programming, 146(1–2), 219–244.

    Google Scholar 

  • Luedtke, J., & Ahmed, S. (2008). A sample approximation approach for optimization with probabilistic constraints. SIAM Journal on Optimization, 19, 674–699.

    Google Scholar 

  • Luenberger, D. G., & Ye, Y. (2010). Linear and nonlinear programming. Volume 116 of International series in operations research & management science (3rd ed.). Berlin: Springer.

    Google Scholar 

  • Luh, P. B., Wang, Y., & Zhao, X. (1999). Lagrangian relaxation neural network for unit commitment. In IEEE power engineering society 1999 winter meeting (Vol. 1, pp. 490–495).

  • Luh, P. B., Zhang, D., & Tomastik, R. N. (1998). An algorithm for solving the dual problem of hydrothermal scheduling. IEEE Transactions on Power Systems, 13, 593–600.

    Google Scholar 

  • Lujano-Rojas, J. M., Osório, G. J., & Catal ao, J. P. S. (2016). New probabilistic method for solving economic dispatch and unit commitment problems incorporating uncertainty due to renewable energy integration. Electrical Power and Energy Systems, 78(1), 61–71.

    Google Scholar 

  • Lyon, J. D., Zhang, M., & Hedman, K. W. (2016). Capacity response sets for security-constrained unit commitment with wind uncertainty. Electric Power Systems Research, 136(1), 21–30.

    Google Scholar 

  • Madrigal, M., & Quintana, V. H. (2000). An interior-point/cutting-plane method to solve unit commitment problems. IEEE Transactions on Power Systems, 15(3), 1022–1027.

    Google Scholar 

  • Magnago, F. H., Alemany, J., & Lin, J. (2015). Impact of demand response resources on unit commitment and dispatch in a day-ahead electricity market. Electrical Power and Energy Systems, 68(1), 142–149.

    Google Scholar 

  • Makkonen, S., & Lahdelma, R. (2006). Non-convex power plant modelling in energy optimisation. European Journal of Operational Research, 171, 1113–1126.

    Google Scholar 

  • Mantawy, A. H., Abdel-Magid, Y. L., & Selim, S. Z. (1998). A simulated annealing algorithm for unit commitment. IEEE Transactions on Power Systems, 13(1), 197–204.

    Google Scholar 

  • Mantawy, A. H., Soliman, S. A., & El-Hawary, M. E. (2002). A new Tabu search algorithm for the long-term hydro scheduling problem. In LESCOPE 02 Large engineering systems conference on power engineering 2002 (pp. 29–34).

  • Marchand, A., Gendreau, M., Blais, M., & Emiel, G. (2018). Fast near-optimal heuristic for the short-term hydro-generation planning problem. IEEE Transactions on Power Systems, 33(1), 227–235.

    Google Scholar 

  • Merlin, A., & Sandrin, P. (1983). A new method for unit commitment at Electricité de France. IEEE Transactions on Power Apparatus and Systems, PAS–102, 1218–1225.

    Google Scholar 

  • Mezger, A. J., & de Almeida, K. C. (2007). Short term hydrothermal scheduling with bilateral transactions via bundle method. Electrical Power and Energy Systems, 29, 387–396.

    Google Scholar 

  • Minoux, M. (2009). Solving some multistage robust decision problems with huge implicitly defined scenario trees. Algorithmic Operations Research, 4(1), 1–18.

    Google Scholar 

  • Minoux, M. (2014). Two-stage robust optimization, state-space representable uncertainty and applications. RAIRO-Operations Research, 48, 455–475.

    Google Scholar 

  • Miranda, J., Wanga, A., Botterud, R., Bessa, H., Keko, L., Carvalho, D., et al. (2011). Wind power forecasting uncertainty and unit commitment. Applied Energy, 88, 4014–4023.

    Google Scholar 

  • Mokhtari, S., Sing, J., & Wollenberg, B. (1988). A unit commitment expert system. IEEE Transactions on Power Systems, 3(1), 272–277.

    Google Scholar 

  • Molzahn, D. K., Holzer, J. T., Lesieutre, B. C., & DeMarco, C. L. (2013). Implementation of a large-scale optimal power flow solver based on semidefinite programming. IEEE Transactions on Power Systems, 28(4), 3987–3998.

    Google Scholar 

  • Momoh, J. A., Adapa, R., & El-Hawary, M. E. (1999a). A review of selected optimal power flow literature to 1993. I. Nonlinear and quadratic programming approaches. IEEE Transactions on Power Systems, 14, 96–104.

    Google Scholar 

  • Momoh, J. A., Adapa, R., & El-Hawary, M. E. (1999b). A review of selected optimal power flow literature to 1993. II. Newton, linear programming and interior point methods. IEEE Transactions on Power Systems, 14, 105–111.

    Google Scholar 

  • Moradi, S., Khanmohammadi, S., Hagh, M. T., & Mohammadi-ivatloo, B. (2015). A semi-analytical non-iterative primary approach based on priority list to solve unit commitment problem. Energy, 88, 244–259.

    Google Scholar 

  • Morales, J. M., Conejo, A. J., & Pérez-Ruiz, J. (2009). Economic valuation of reserves in power systems with high penetration of wind power. IEEE Transactions on Power Systems, 24(2), 900–910.

    Google Scholar 

  • Morales-España, G., Gentile, C., & Ramos, A. (2015). Tight mip formulations of the power-based unit commitment problem. OR Spectrum, 37(1), 929–950.

    Google Scholar 

  • Morales-España, G., Latorre, J. M., & Ramos, A. (2013). Tight and compact MILP formulation for the thermal unit commitment problem. IEEE Transactions on Power Systems, 28(4), 4897–4908.

    Google Scholar 

  • Morales-España, G., Latorre, J. M., & Ramos, A. (2013). Tight and compact MILP formulation of start-up and shut-down ramping in unit commitment. IEEE Transactions on Power Systems, 28(2), 1288–1296.

    Google Scholar 

  • Morales-España, G., Lorca, A., & de Weerdt, M. M. (2018). Robust unit commitment with dispatchable wind power. Electric Power Systems Research, 155(1), 58–66.

    Google Scholar 

  • Morales-España, G., Ramírez-Elizondo, L., & Hobbs, B. F. (2017). Hidden power system inflexibilities imposed by traditional unit commitment formulations. Applied Energy, 191(1), 223–238.

    Google Scholar 

  • Morales-España, G., Ramos, A., & García-González, J. (2014). An MIP formulation for joint market-clearing of energy and reserves including ramp scheduling. IEEE Transactions on Power Systems, 29(1), 476–488.

    Google Scholar 

  • Mori, H., & Matsuzaki, O. (2001). Embedding the priority list into Tabu search for unit commitment. In IEEE power engineering society winter meeting, 2001 (Vol. 3, pp. 1067–1072).

  • Moura, P . S., & de Almeida, A. T. (2010). Large scale integration of wind power generation. In S. Rebennack, P. M. Pardalos, M. V. F. Pereira, & N. Iliadis (Eds.), Handbook of power systems I (pp. 95–120). Berlin: Springer.

    Google Scholar 

  • Muche, T. (2014). Optimal operation and forecasting policy for pump storage plants in day-ahead markets. Applied Energy, 113, 1089–1099.

    Google Scholar 

  • Muckstadt, J. A., & Koenig, S. A. (1977). An application of lagrangian relaxation to scheduling in power-generation systems. Operations Research, 25(3), 387–403.

    Google Scholar 

  • Muckstadt, J. A., & Wilson, R. C. (1968). An application of mixed-integer programming duality to scheduling thermal generating systems. IEEE Transactions on Power Apparatus and Systems, PAS–87(12), 1968–1978.

    Google Scholar 

  • Muñoz, A., Sánchez Úbeda, E . F., Cruz, A., & Marín, J. (2010). Short-term forecasting in power systems: A guided tour. In S. Rebennack, P. M. Pardalos, M. V. F. Pereira, & N. Iliadis (Eds.), Handbook of power systems II (pp. 129–160). Berlin: Springer.

    Google Scholar 

  • Murillo-Sanchez, C., & Thomas, R. J. (1998). Thermal unit commitment including optimal AC power flow constraints. In Thirty-first Hawaii international conference on system sciences (Vol. 3).

  • Nasrolahpour, E., & Ghasemi, H. (2015). A stochastic security constrained unit commitment model forreconfigurable networks with high wind power penetration. Electric Power Systems Research, 121(1), 341–350.

    Google Scholar 

  • Nayak, R., & Sharma, J. D. (2000). A hybrid neural network and simulated annealing approach to the unit commitment problem. Computers and Electrical Engineering, 26(6), 461–477.

    Google Scholar 

  • Nemirovski, A., & Shapiro, A. (2004). Scenario approximations of chance constraints (pp. 1–45). http://www.optimization-online.org/DB_HTML/2004/11/1000.html (preprint).

  • Nemirovski, A., & Shapiro, A. (2006a). Convex approximations of chance constrained programs. SIAM Journal of Optimization, 17(4), 969–996.

    Google Scholar 

  • Nemirovski, A., & Shapiro, A. (2006b). Scenario approximations of chance constraints. In G. Calafiore & F. Dabbene (Eds.), Probabilistic and randomized methods for design under uncertainty (1st ed., pp. 3–47). Berlin: Springer.

    Google Scholar 

  • Nesterov, Y. (2009). Primal-dual subgradient methods for convex problems. Mathematical Programming, 120(1), 221–259.

    Google Scholar 

  • Nguyen-Huu, A. (2012). Valorisation financière sur les marchés d’électricité. Ph.D. thesis, Paris Dauphine.

  • Ni, E., Guan, X., & Li, R. (1999). Scheduling hydrothermal power systems with cascaded and head-dependent reservoirs. IEEE Transactions on Power Systems, 14, 1127–1132.

    Google Scholar 

  • Ni, E., Luh, P. B., & Rourke, S. (2004). Optimal integrated generation bidding and scheduling with risk management under a deregulated power market. IEEE Transactions on Power Systems, 19(1), 600–609.

    Google Scholar 

  • Nilsson, O., & Sjelvgren, D. (1996). Mixed-integer programming applied to short-term planning of a hydro-thermal system. IEEE Transactions on Power Systems, 11(1), 281–286.

    Google Scholar 

  • Nogales, F. J., Contreras, J., Conejo, A. J., & Espínola, R. (2002). Forecasting next-day electricity prices by time series models. IEEE Transactions on Power Systems, 17(2), 342–348.

    Google Scholar 

  • Nowak, M. P. (2000). Stochastic Lagrangian relaxation in power scheduling of a hydrothermal system under uncertainty. Ph.D. thesis, Humboldt University Berlin.

  • Nowak, M. P., & Römisch, W. (2000). Stochastic Lagrangian relaxation applied to power scheduling in a hydro-thermal system under uncertainty. Annals of Operations Research, 100(1–4), 251–272.

    Google Scholar 

  • Nowak, M. P., Schultz, R., & Westphalen, M. (2005). A stochastic integer programming model for incorporating day-ahead trading of electricity into hydro-thermal unit commitment. Optimization and Engineering, 6, 163–176. https://doi.org/10.1007/s11081-005-6794-0.

    Article  Google Scholar 

  • Nürnberg, R., & Römisch, W. (2003). A two-stage planning model for power scheduling in a hydro-thermal system under uncertainty. Optimization and Engineering, 3, 355–378.

    Google Scholar 

  • Oliveira, A. R. L., Soares, S., & Nepomuceno, L. (2005). Short term hydroelectric scheduling combining network flow and interior point approaches. Electrical Power and Energy Systems, 27, 91–99.

    Google Scholar 

  • O’Neill, R. P., Hedman, K. W., Krall, E. A., Papavasiliou, A., & Oren, S. S. (2010). Economic analysis of the n-1 reliable unit commitment and transmission switching problem using duality concepts. Energy Systems, 1(2), 165–195.

    Google Scholar 

  • Oren, S. S., Svoboda, A. J., & Johnson, R. B. (1997). Volatility of unit commitment in competitive electricity markets. International Conference on System Sciences, 5, 594–601.

    Google Scholar 

  • Ortega-Vazquez, M., & Kirschen, D. S. (2009). Estimating the spinning reserve requirements in systems with significant wind power generation penetration. IEEE Transactions on Power Systems, 24(1), 114–124.

    Google Scholar 

  • Ostrowski, J., Anjos, M. F., & Vannelli, A. (2015). Modified orbital branching for structured symmetry with an application to unit commitment. Mathematical Programming, 150(1), 99–129.

    Google Scholar 

  • Ostrowski, J., Anjos, M. F., & Vannelli, A. (2012). Tight mixed integer linear programming formulations for the unit commitment problem. IEEE Transactions on Power Systems, 27(1), 39–46.

    Google Scholar 

  • Ostrowski, J., Linderoth, J., Rossi, F., & Smriglio, S. (2011). Modified orbital branching for structured symmetry with an application to unit commitment. Mathematical Programming, 126(1), 147–178.

    Google Scholar 

  • Ostrowski, J. Vannelli, A., & Anjos, M. F. (2010). Groupe d’études et de recherche en analyse des décisions. Symmetry in scheduling problems. Groupe d’études et de recherche en analyse des décisions.

  • Ostrowski, J., & Wang, J. (2012). Network reduction in the transmission-constrained unit commitment problem. Computers & Industrial Engineering, 63(1), 702–707.

    Google Scholar 

  • Ostrowski, J., Wang, J., & Liu, C. (2012). Exploiting symmetry in transmission lines for transmission switching. IEEE Transactions on Power Systems, 27(3), 1708–1709.

    Google Scholar 

  • Oudjane, N., Collet, J., & Duwig, V. (2006). Some non-Gaussian models for electricity spot prices. In 9th International conference on probabilistic methods applied to power systems.

  • Outrata, J. V. (1990). On the numerical solution of a class of Stackelberg games. ZOR Mathematical Methods of Operations Research, 34, 255–277.

    Google Scholar 

  • Ouyang, Z., & Shahidehpour, M. (1991). An intelligent dynamic programming for unit commitment application. IEEE Transactions on Power Systems, 6(3), 1203–1209.

    Google Scholar 

  • Ouyang, Z., & Shahidehpour, M. (1992). A hybrid artificial neural network-dynamic programming approach to unit commitment. IEEE Transactions on Power Systems, 7(1), 236–242.

    Google Scholar 

  • Ozturk, U. A., Mazumdar, M., & Norman, B. A. (2004). A solution to the stochastic unit commitment problem using chance constrained programming. IEEE Transactions on Power Systems, 19(3), 1589–1598.

    Google Scholar 

  • Padhy, N. P. (2004). Unit commitment: A bibliographical survey. IEEE Transaction On Power Systems, 19(2), 1196–1205.

    Google Scholar 

  • Palamarchuk, S. I. (2012). Compromise scheduling of bilateral contracts in electricity market environment. In A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis, & M. V. F. Pereira (Eds.), Handbook of networks in power systems I (pp. 241–262). Heidelberg: Springer.

    Google Scholar 

  • Pandžić, H., Dvorkin, Y., Qiu, T., Wang, Y., & Kirschen, D. S. (2016). Towards cost-efficient and reliable unit commitment under uncertainty. IEEE Transactions on Power Systems, 31(2), 970–982.

    Google Scholar 

  • Pang, C. K., & Chen, H. C. (1976). Optimal short-term thermal unit commitment. IEEE Transactions on Power Apparatus and Systems, 95(4), 1336–1346.

    Google Scholar 

  • Pang, C. K., Sheble, G. B., & Albuyeh, F. (1981). Evaluation of dynamic programming based methods and multiple area representation for thermal unit commitments. IEEE Transactions on Power Apparatus and Systems, PAS–100(3), 1212–1218.

    Google Scholar 

  • Papavasiliou, A., & Oren, S. S. (2012). A stochastic unit commitment model for integrating renewable supply and demand response. In Invited panel paper, Proceeding of the IEEE PES GM, San Diego, CA, July 24–28, 2012.

  • Papavasiliou, A., & Oren, S. S. (2013). A comparative study of stochastic unit commitment and security-constrained unit commitment using high performance computing. In Proceeding of the European control conference ECC 2013.

  • Papavasiliou, A., Oren, S. S., & O’Neill, R. (2011). Reserve requirements for wind power integration: A scenario-based stochastic programming framework. IEEE Transactions on Power Systems, 26(4), 2197–2206.

    Google Scholar 

  • Papavasiliou, A., Oren, S. S., & O’Neill, R. (2013). Multi-area stochastic unit commitment for high wind penetration in a transmission constrained network. Operations Research, 61(3), 578–592.

    Google Scholar 

  • Papavasiliou, A., Oren, S. S., Yang, Z., Balasubramanian, P., & Hedman, K. W. (2013). An application of high performance computing to transmission switching. In IREP bulk power system dynamics and control symposium, Rethymnon, Greece.

  • Papavasilou, A., Oren, S. S., & Rountree, B. (2015). Applying high performance computing to transmission-constrained stochastic unit commitment for renewable energy integration. IEEE Transactions on Power Systems, 30(3), 1109–1120.

    Google Scholar 

  • Parrilla, E., & García-González, J. (2006). Improving the B&B search for large-scale hydrothermal weekly scheduling problems. Electrical Power and Energy Systems, 28, 339–348.

    Google Scholar 

  • Pedregal, D. J., Contreras, J., & Sanchez de la Nieta, A. A. (2012). Ecotool: A general matlab forecasting toolbox with applications to electricity markets. In A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis, & M. V. F. Pereira (Eds.), Handbook of networks in power systems I (pp. 151–171). Heidelberg: Springer.

    Google Scholar 

  • Peng, T., & Tomsovic, K. (2003). Congestion influence on bidding strategies in an electricity market. IEEE Transactions on Power Systems, 18(3), 1054–1061.

    Google Scholar 

  • Pepper, W., Ring, B. J., Read, E. G., & Starkey, S. R. (2012). Short-term electricity market prices: A review of characteristics and forecasting methods. In A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis, & M. V. F. Pereira (Eds.), Handbook of networks in power systems II (pp. 3–36). Heidelberg: Springer.

    Google Scholar 

  • Pereira, M. V., Granville, S., Fampa, M. H. C., Dix, R., & Barroso, L. A. (2005). Strategic bidding under uncertainty: A binary expansion approach. IEEE Transactions on Power Systems, 11(1), 180–188.

    Google Scholar 

  • Pereira, M. V. F., & Pinto, L. M. V. G. (1983). Application of decomposition techniques to the mid- and short-term scheduling of hydrothermal systems. IEEE Transactions on Power Apparatus and Systems, PAS–102(11), 3611–3618.

    Google Scholar 

  • Philpott, A., & Schultz, R. (2006). Unit commitment in electricity pool markets. Mathematical Programming: Series B, 108, 313–337.

    Google Scholar 

  • Piekutowki, M., Litwinowcz, T., & Frowd, R. (1994). Optimal short-term scheduling for a large-scale cascaded hydro system. IEEE Transactions on Power Systems, 9(2), 805–811.

    Google Scholar 

  • Pineau, P. O., & Murto, P. (2003). An oligopolistic investment model of the finnish electricity market. Annals of Operations Research, 121(1–4), 123–148.

    Google Scholar 

  • Polyak, B. T. (1977). Subgradient methods: A survey of soviet research. In C. Lemaréchal & R. Mifflin (Eds.), Nonsmooth optimization. IIASA proceedings series. Oxford: Pergamon Press.

    Google Scholar 

  • Pozo, D., & Contreras, J. (2013). A chance-constrained unit commitment with an \(n-k\) security criterion and significant wind generation. IEEE Transactions on Power Systems, 28(3), 2842–2851.

    Google Scholar 

  • Pozo, D., Contreras, J., & Sauma, E. E. (2014). Unit commitment with ideal and generic energy storage units. IEEE Transactions on Power Systems, 29(6), 2974–2984.

    Google Scholar 

  • Prékopa, A. (1995). Stochastic Programming. Dordrecht: Kluwer.

    Google Scholar 

  • Prékopa, A. (2003). Probabilistic programming. In A. Ruszczyński & A. Shapiro (Eds.), Stochastic programming. Volume 10 of handbooks in operations research and management science (pp. 267–351). Amsterdam: Elsevier.

    Google Scholar 

  • Prékopa, A., Rapcsák, T., & Zsuffa, I. (1978). Serially linked reservoir system design using stochastic programming. Water Resources Research, 14, 672–678.

    Google Scholar 

  • Price, J. E. (2007). Market-based price differentials in zonal and LMP market designs. IEEE Transaction on Power Systems, 22(4), 1486–1494.

    Google Scholar 

  • Rahmaniani, R., Crainic, T. G., Gendreau, M., & Rei, W. (2017). The benders decomposition algorithm: A literature review. European Journal of Operational Research, 259(3), 801–817.

    Google Scholar 

  • Rajan, C. C. A., & Mohan, M. R. (2004). An evolutionary programming-based tabu search method for solving the unit commitment problem. IEEE Transactions on Power Systems, 19(1), 577–585.

    Google Scholar 

  • Rajan, C. C. A., Mohan, M. R., & Manivannan, K. (2003). Neural-based tabu search method for solving unit commitment problem. IEEE Proceedings-Generation, Transmission and Distribution, 150(4), 469–474.

    Google Scholar 

  • Rajan, C. C. A., Selvi, S. C., & Kumudini Devi, R. P. (2012). Multi-area unit commitment with transmission losses using evolutionary iteration particle swarm optimization approach. European Journal of Scientific Research, 76(4), 672–691.

    Google Scholar 

  • Rajan, D., & Takriti, S. (2005). Minimum up/down polytopes of the unit commitment problem with start-up costs. Technical report, IBM.

  • Ramos, A., Cerisola, S., Latorre, J. M., Bellido, R., Perea, A., & Lopez, E. (2012). A decision support model for weekly operation of hydrothermal systems by stochastic nonlinear optimization. In M. Bertocchi, G. Consigli, & M. A. H. Dempster (Eds.), Stochastic optimization methods in finance and energy: New financial products and energy market strategies (pp. 143–162). Berlin: Springer.

    Google Scholar 

  • Razaviyayn, M., Hong, M., & Luo, Z-Q. (2012). A unified convergence analysis of block successive minimization methods for nonsmooth optimization. Technical report, University of Minnesota, Twin Cites.

  • Read, E. G. (2010). Co-optimization of energy and ancillary service markets. In S. Rebennack, P. M. Pardalos, M. V. F. Pereira, & N. Iliadis (Eds.), Handbook of power systems I (pp. 307–330). Berlin: Springer.

    Google Scholar 

  • Redondo, N. J., & Conejo, A. J. (1999). Short-term hydro-thermal coordination by lagrangian relaxation: solution of the dual problem. IEEE Transactions on Power Systems, 14, 89–95.

    Google Scholar 

  • Reliability Test System Task Force. (1999). The IEEE reliability test system. IEEE Transactions on Power Systems, 14(3), 1010–1020.

    Google Scholar 

  • Restrepo, J. F., & Galiana, F. D. (2005). Unit commitment with primary frequency regulation constraints. IEEE Transactions on Power Systems, 20(4), 1836–1842.

    Google Scholar 

  • Restrepo, J. F., & Galiana, F. D. (2011). Assessing the yearly impact of wind power through a new hybrid deterministic/stochastic unit commitment. IEEE Transactions on Power Systems, 26(1), 401–410.

    Google Scholar 

  • Rocha, P., & Das, T. K. (2012). Finding joint bidding strategies for day-ahead electricity and related markets. In A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis, & M. V. F. Pereira (Eds.), Handbook of networks in power systems I (pp. 61–88). Heidelberg: Springer.

    Google Scholar 

  • Rockafellar, R. T., & Wets, R. J.-B. (1991). Scenarios and policy aggregation in optimization under uncertainty. Mathematics of Operations Research, 16(1), 119–147.

    Google Scholar 

  • Rockafellar, R. T., & Uryas’ev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–42.

    Google Scholar 

  • Rockafellar, R. T., & Uryas’ev, S. (2002). Conditional value-at-risk for general distributions. Journal of Banking & Finance, 26, 1443–1471.

    Google Scholar 

  • Römisch, W. (2003). Stability of stochastic programming problems. In A. Ruszczyński & A. Shapiro (Eds.), Stochastic programming. Volume 10 of Handbooks in operations research and management science (Chapter 8). Amsterdam: Elsevier.

    Google Scholar 

  • Römisch, W., & Schultz, R. (1991). Distribution sensitivity for certain classes of chance-constrained models with application to power dispatch. Journal of Optimization Theory and Applications, 71, 569–588.

    Google Scholar 

  • Römisch, W., & Schultz, R. (1993). Stability of solutions for stochastic programs with complete recourse. Mathematics of Operations Research, 18, 590–609.

    Google Scholar 

  • Römisch, W., & Schultz, R. (1996). Decomposition of a multi-stage stochastic program for power dispatch. SUPPL, 3, 29–32.

    Google Scholar 

  • Römisch, W., & Vigerske, S. (2010). Recent progress in two-stage mixed-integer stochastic programming with applications to power production planning. In S. Rebennack, P. M. Pardalos, M. V. F. Pereira, & N. Iliadis (Eds.), Handbook of power systems I (pp. 177–208). Berlin: Springer.

    Google Scholar 

  • Ruiz, P. A., Philbrick, C. R., & Sauer, P. W. (2010). Modeling approaches for computational cost reduction in stochastic unit commitment formulations. IEEE Transactions on Power Systems, 25(1), 588–589.

    Google Scholar 

  • Ruiz, P. A., Philbrick, C. R., Zak, E. J., Cheung, K. W., & Sauer, P. W. (2009). Uncertainty management in the unit commitment problem. IEEE Transactions on Power Systems, 24(2), 642–651.

    Google Scholar 

  • Ruiz, P. A., Rudkevich, A., Caramanis, M. C., Goldis, E., Ntakou, E., & Philbrick, C. R. (2012). Reduced MIP formulation for transmission topology control. In Allerton conference, 2012 (pp. 1073–1079). IEEE.

  • Ruszczyński, A. (1995). On convergence of an augmented Lagrangian decomposition method for sparse convex optimization. Mathematics of Operations Research, 20(3), 634–656.

    Google Scholar 

  • Ruszczyński, A. (2003). Decomposition methods. In A. Ruszczyński & A. Shapiro (Eds.), Stochastic programming. Volume 10 of handbooks in operations research and management science (pp. 141–211). Amsterdam: Elsevier.

    Google Scholar 

  • Ruszczyński, A., & Shapiro, A. (2009a). Multi-stage problems (Chapter 3). In A. Shapiro, D. Dentcheva & A. Ruszczyński (Eds.), Lectures on stochastic programming. Modeling and theory, Volume 9 of MPS-SIAM series on optimization. Philadelphia: SIAM and MPS.

  • Ruszczyński, A., & Shapiro, A. (2009b). Two stage problems (Chapter 2). In A. Shapiro, D. Dentcheva & A. Ruszczyński (Eds.), Lectures on stochastic programming. Modeling and theory, Volume 9 of MPS-SIAM series on optimization. Philadelphia: SIAM and MPS.

  • Ruzic, S., & Rajakovic, R. (1998). Optimal distance method for lagrangian multipliers updating in short-term hydro-thermal coordination. IEEE Transactions on Power Systems, 13, 1439–1444.

    Google Scholar 

  • Sagastizábal, C. (2012). Divide to conquer: Decomposition methods for energy optimization. Mathematical Programming, 134(1), 187–222.

    Google Scholar 

  • Sahebi, M. M. R., & Hosseini, S. H. (2014). Stochastic security constrained unit commitment incorporating demand side reserve. Electrical Power and Energy Systems, 56(1), 175–184.

    Google Scholar 

  • Sahraoui, Y., Bendotti, P., & D’Ambrosio, C. (2017). Real-world hydro-power unit-commitment: Dealing with numerical errors and feasibility issues. Energy, 1–14.

  • Salam, M. S., Hamdan, A. R., & Nor, K. M. (1991). Integrating an expert system into a thermal unit-commitment algorithm. IEE Proceedings Generation, Transmission and Distribution, 138(6), 553–559.

    Google Scholar 

  • Salam, S., Nor, K. M., & Hamdan, A. R. (1997). Comprehensive algorithm for hydrothermal coordination. IEE Transactions on Generation Transmission and Distribution, 144, 482–488.

    Google Scholar 

  • Salam, S., Nor, K. M., & Hamdan, A. R. (1998). Hydrothermal scheduling based lagrangian relaxation approach to hydrothermal coordination. IEEE Transactions on Power Systems, 13, 226–235.

    Google Scholar 

  • Saravanan, B., Das, S., Sikri, S., & Kothari, D. P. (2013). A solution to the unit commitment problem: A review. Frontiers in Energy, 7(2), 223–236.

    Google Scholar 

  • Saravanan, B., Mishra, S., & Nag, D. (2014). A solution to stochastic unit commitment problem for a wind-thermal system coordination. Frontiers in Energy, 8(2), 192–200.

    Google Scholar 

  • Sari, D., Lee, Y., Ryan, S. M., & Woodruff, D. (2016). Statistical metrics for assessing the quality of wind power scenarios for stochastic unit commitment. Wind Energy, 19(5), 873–893.

    Google Scholar 

  • Sari, D., & Ryan, S. M. (2017). Statistical reliability of wind power scenarios and stochastic unit commitment cost. Energy Systems, 1–26.

  • Sarić, A. T., & Stankovic, A. M. (2007). Finitely adaptive linear programming in robust power system optimization. In 2007 IEEE Lausanne PowerTech (pp. 1302–1307).

  • Sasaki, H., Watanabe, M., Kubokawa, J., Yorino, N., & Yokoyama, R. (1992). A solution method of unit commitment by artificial neural networks. IEEE Transactions on Power Systems, 7(3), 974–981.

    Google Scholar 

  • Sauma, E., Jerardino, S., Barria, C., Marambio, R., Brugman, A., & Mejia, J. (2012). Electric interconnections in the andes community: Threats and opportunities. In A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis, & M. V. F. Pereira (Eds.), Handbook of networks in power systems I (pp. 345–366). Heidelberg: Springer.

    Google Scholar 

  • Schultz, R., Nowak, M., Nürnberg, R., Römisch, W., & Westphalen, M. (2003). Stochastic programming for power production and trading under uncertainty. In W. Jvsger & H.-J. Krebs (Eds.), Mathematics: Key technology for the future (pp. 623–636). Berlin: Springer.

    Google Scholar 

  • Schulze, T., Grothey, A., & McKinnon, K. (2017). A stabilised scenario decomposition algorithm applied to stochastic unit commitment problems. European Journal of Operational Research, 261(1), 247–259.

    Google Scholar 

  • Schulze, T., & McKinnon, K. (2016). The value of stochastic programming in day-ahead and intra-day generation unit commitment. Energy, 101(1), 592–605.

    Google Scholar 

  • Scuzziato, M. R., Finardi, E. C., & Frangioni, A. (2018). Comparing spatial and scenario decomposition for stochastic hydrothermal unit commitment problems. IEEE Transactions on Sustainable Energy.

  • Séguin, S., Fleten, S.-E., Côté, P., Pichler, A., & Audet, C. (2017). Stochastic short-term hydropower planning with inflow scenario trees. European Journal of Operational Research, 259(1), 1156–1168.

    Google Scholar 

  • Sendaula, M. H., Biswas, S. K., Eltom, A., Parten, C., & Kazibwe, W. (1991). Application of artificial neural networks to unit commitment. Proceedings of the First International Forum on Applications of Neural Networks to Power Systems, 1991, 256–260.

    Google Scholar 

  • Senthil-Kumar, S., & Palanisamy, V. (2007). A dynamic programming based fast computation hopfield neural network for unit commitment and economic dispatch. Electric Power Systems Research, 77(8), 917–925.

    Google Scholar 

  • Shafie-Khah, M., Parsa Moghaddam, M., & Sheikh-El-Eslami, M. K. (2011). Unified solution of a non-convex SCUC problem using combination of modified branch-and-bound method with quadratic programming. Energy Conversion and Management, 52(12), 3425–3432.

    Google Scholar 

  • Shahidehpour, M., Yamin, H., & Li, Z. (2002). Market operations in electric power systems: Forecasting, scheduling, and risk management. Hoboken: Wiley.

    Google Scholar 

  • Sharaf, T. A. M., & Berg, G. J. (1982). Voltampere reactive compensation using chance-constrained programming. IEEE Proceedings C Generation, Transmission and Distribution, 129(1), 24–29.

    Google Scholar 

  • Shaw, J. J., Gendron, R. F., & Bertsekas, D. P. (1985). Optimal scheduling of large hydrothermal power systems. IEEE Power Engineering Review, PER–5(2), 32.

    Google Scholar 

  • Sheble, G. B., & Fahd, G. N. (1994). Unit commitment literature synopsis. IEEE Transactions on Power Systems, 9(1), 128–135.

    Google Scholar 

  • Sheble, G. B., Maifeld, T. T., Brittig, K., Fahd, G., & Fukurozaki-Coppinger, S. (1996). Unit commitment by genetic algorithm with penalty methods and a comparison of Lagrangian search and genetic algorithm-economic dispatch example. International Journal of Electrical Power & Energy Systems, 18(6), 339–346.

    Google Scholar 

  • Sher, M., & Banerjee, A. (2014). Solving unit commitment problem with parallel computing. In J. Xu, J. Fry, B. Lev, & A. Hajiyev (Eds.), Proceedings of the seventh international conference on management science and engineering management. Volume 242 of lecture notes in electrical engineering, (pp. 1165–1173).

  • Sherali, H. D., & Adams, W. P. (1998). A reformulation-linearization technique for solving discrete and continuous nonconvex problems. Nonconvex optimization and its applications. Berlin: Springer.

    Google Scholar 

  • Sherali, H. D., & Fraticelli, B. M. P. (2002). A modification of Benders’ decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse. Journal of Global Optimization, 22, 319–342.

    Google Scholar 

  • Shi, J., & Oren, S. S. (2018). Stochastic unit commitment with topology control recourse for power systems with large-scale renewable integration. IEEE Transactions on Power Systems, 33(3), 3315–3324.

    Google Scholar 

  • Shiina, T. (1999). Numerical solution technique for joint chance-constrained programming problem: An application to electric power capacity expansion. Journal of the Operations Research Society of Japan, 42(2), 128–140.

    Google Scholar 

  • Shiina, T., & Birge, J. R. (2004). Stochastic unit commitment problem. International Transactions in Operational Research, 11(1), 19–32.

    Google Scholar 

  • Shiina, T., Yurugi, T., Morito, S., & Imaizumi, J. (2016). Unit commitment by column generation. In M. Lübbecke, A. Koster, P. Letmathe, R. Madlener, B. Peis, & G. Walther (Eds.), Operations research proceedings 2014 (pp. 559–565).

    Google Scholar 

  • Siahkali, H., & Vakilian, M. (2010). Stochastic unit commitment of wind farms integrated in power system. Electric Power Systems Research, 80(9), 1006–1017.

    Google Scholar 

  • Sifuentes, W., & Vargas, A. (2007a). Hydrothermal scheduling using Benders decomposition: Accelerating techniques. IEEE Transactions on Power Systems, 22, 1351–1359.

    Google Scholar 

  • Sifuentes, W., & Vargas, A. (2007b). Short-term hydrothermal coordination considering an AC network modeling. International Journal of Electrical Power & Energy Systems, 29, 488–496.

    Google Scholar 

  • Simopoulos, D. N., Kavatza, S. D., & Vournas, C. D. (2006). Unit commitment by an enhanced. IEEE Transactions on Power Systems, 21(1), 68–76.

    Google Scholar 

  • Singhal, P. K., & Sharma, R. N. (2011). Dynamic programming approach for large scale unit commitment problem. International Conference on Communication Systems and Network Technologies (CSNT), 2011, 714–717.

    Google Scholar 

  • Siu, T. K., Nash, G. A., & Shawwash, Z. K. (2001). A practical hydro, dynamic unit commitment and loading model. IEEE Transactions on Power Systems, 16(2), 301–306.

    Google Scholar 

  • Snyder, W. L., Powell, H. D., & Rayburn, J. C. (1987). Dynamic programming approach to unit commitment. IEEE Transactions on Power Systems, 2(2), 339–348.

    Google Scholar 

  • Song, Y., & Luedtke, J. (2015). An adaptive partition-based approach for solving two-stage stochastic programs with fixed recourse. SIAM Journal on Optimization, 25(3), 1344–1367.

    Google Scholar 

  • Soroudi, A., Rabiee, A., & Keane, A. (2017). Contents lists available at sciencedirect electric power systems research information gap decision theory approach to deal with wind power uncertainty in unit commitment. Electric Power Systems Research, 26(145), 137–148.

    Google Scholar 

  • Steber, D., Pruckner, M., Schlund, J., Bazan, P., & German, R. (2018). Including a virtual battery storage into thermal unit commitment. Computer Science Research and Development, 33(1–2), 223–229.

    Google Scholar 

  • Street, A., Oliveira, F., & Arroya, J. M. (2011). Contingency-constrained unit commitment with \(n-k\) security criterion: A robust optimization approach. IEEE Transactions on Power Systems, 26(3), 1581–1590.

    Google Scholar 

  • Suazo-Martinez, C., Pereira-Bonvallet, E., Palma-Behnke, R., & Zhang, X. P. (2014). Impacts of energy storage on short term operation planning under centralized spot markets. IEEE Transactions on Smart Grid, 5(2), 1110–1118.

    Google Scholar 

  • Sudhakaran, M., & Ajay-D-Vimal Raj, P. (2010). Integrating genetic algorithms and Tabu search for unit commitment. International Journal of Engineering, Science and Technology, 2(1), 57–69.

    Google Scholar 

  • Surowiec, T.M. (2010). Explicit Stationarity Conditions and Solution Characterization for Equilibrium Problems with Equilibrium Constraints. Ph.D. thesis, Humboldt-Universität zu Berlin, 1.

  • Tahanan, M., van Ackooij, W., Frangioni, A., & Lacalandra, F. (2015). Large-scale unit commitment under uncertainty: A literature survey. 4OR, 13(2), 115–171.

    Google Scholar 

  • Takigawa, F. Y. K., da Silva, E. L., Finardi, E. C., & Rodrigues, R. N. (2012). Solving the hydrothermal scheduling problem considering network constraints. Electric Power Systems Research, 88, 89–97.

    Google Scholar 

  • Takigawa, F. Y. K., Finardi, E. C., & da Silva, E. L. (2013). A decomposition strategy to solve the short-term hydrothermal scheduling based on lagrangian relaxation. Journal of Algorithms and Optimization, 1(1), 13–24.

    Google Scholar 

  • Takriti, S., & Birge, J. R. (2000). Using integer programming to refine lagrangian-based unit commitment solutions. IEEE Transactions on Power Systems, 15(1), 151–156.

    Google Scholar 

  • Takriti, S., Birge, J. R., & Long, E. (1996). A stochastic model for the unit commitment problem. IEEE Transactions on Power Systems, 11, 1497–1508.

    Google Scholar 

  • Takriti, S., Krasenbrink, B., & Wu, L. S. Y. (2000). Incorporating fuel constraints and electricity spot prices into the stochastic unit commitment problem. Operations Research, 48(2), 268–280.

    Google Scholar 

  • Taktak, R., & d’Ambrosio, C. (2016). An overview on mathematical programming approaches for the deterministic unit commitment problem in hydro valleys. Energy Systems.

  • Taverna, A. (2017). Benders decomposition on large-scale unit commitment problems for medium-term power systems simulation. In: A. Fink, A. Fügenschuh, & M. Geiger (Eds.), Operations research proceedings 2016 (pp. 179–184).

    Google Scholar 

  • Teng, F., Trovato, V., & Strbac, G. (2016). Stochastic scheduling with inertia-dependent fast frequency response requirements. IEEE Transactions on Power Systems, 1–8.

  • Tong, S. K., & Shahidehpour, M. (1989). Combination of lagrangian-relaxation and linear-programming approaches for fuel-constrained unit-commitment problems. IEEE Proceedings Generation, Transmission and Distribution, 136(3), 162–174.

    Google Scholar 

  • Triki, C., Beraldi, P., & Gross, G. (2005). Optimal capacity allocation in multi-auction electricity markets under uncertainty. Computers & Operations Research, 32, 201–217.

    Google Scholar 

  • Triki, C., Conejo, A. J., & Garcés, L. P. (2011). Short-term trading for electricity producers. In M. Bertocchi, G. Consigli, & M. A. H. Dempster (Eds.), Stochastic optimization methods in finance and energy: New financial products and energy market strategies. Volume 163 of international series in operations research & management science (pp. 181–202). Berlin: Springer.

    Google Scholar 

  • Trukhanova, S., Ntaimo, L., & Schaefer, A. (2010). Adaptive multicut aggregation for two-stage stochastic linear programs with recourse. European Journal of Operational Research, 206(2), 395–406.

    Google Scholar 

  • Tseng, C. L., Li, C. A., & Oren, S. S. (2000). Solving the unit commitment problem by a unit decommitment method. Journal of Optimization Theory and Applications, 105(3), 707–730.

    Google Scholar 

  • Tseng, P. (2001). Convergence of a block coordinate descent method for nondifferentiable minimization. Journal of Optimization Theory and Applications, 109(3), 475–494.

    Google Scholar 

  • Tumuluru, V. K., Huang, Z., & Tsang, D. H. K. (2014). Unit commitment for systems with significant wind penetration. Electrical Power and Energy Systems, 57, 222–231.

    Google Scholar 

  • Tuohy, A., Meibom, P., Denny, E., & O’Malley, M. J. (2009). Unit commitment for systems with significant wind penetration. IEEE Transactions on Power Systems, 24(2), 592–601.

    Google Scholar 

  • Turgeon, A. (1978). Optimal scheduling of thermal generating units. IEEE Transactions on Automatic Control, 23(6), 1000–1005.

    Google Scholar 

  • Uçkun, C., Botterud, A., & Birge, J. R. (2016). An improved stochastic unit commitment formulation to accommodate wind uncertainty. IEEE Transactions on Power Systems, 31(4), 2507–2517.

    Google Scholar 

  • Upahyay, A., Hu, B., Li, J., & Wu, L. (2016). A chance-constrained wind range quantification approach to robust SCUC by determining dynamic uncertainty intervals. CSEE Journal of Power and Energy Systems, 2(1), 54–64.

    Google Scholar 

  • Valenzuela, J., & Mazumdar, M. (2003). Commitment of electric power generators under stochastic market prices. Operations Research, 51(6), 880–893.

    Google Scholar 

  • Valenzuela, J., & Smith, A. E. (2002). A seeded memetic algorithm for large unit commitment problems. Journal of Heuristics, 8(2), 173–195.

    Google Scholar 

  • van Ackooij, W. (2014). Decomposition approaches for block-structured chance-constrained programs with application to hydro-thermal unit commitment. Mathematical Methods of Operations Research, 80(3), 227–253.

    Google Scholar 

  • van Ackooij, W. (2015). Eventual convexity of chance constrained feasible sets. Optimization (A Journal of Mathematical Programming and Operations Research), 64(5), 1263–1284.

    Google Scholar 

  • van Ackooij, W. (2017). A comparison of four approaches from stochastic programming for large-scale unit-commitment. EURO Journal on Computational Optimization, 5(1), 119–147.

    Google Scholar 

  • van Ackooij, W., Aleksovska, I., & Munoz Zuniga, M. (2017). (Sub-)Differentiability of probability functions with elliptical distributions. Set Valued and Variational Analysis (pp. 1–24).

  • van Ackooij, W., Danti Lopez, I., Frangioni, A., Lacalandra, F., & Tahanan, M. (2018). Large-scale unit commitment under uncertainty: An updated literature survey. Technical report, Dipartimento di Informatica, Università di Pisa.

  • van Ackooij, W., de Boeck, J., Detienne, B., Pan, S., & Poss, M. (2018). Optimizing power generation in the presence of micro-grids. Submitted preprint (pp. 1–23).

  • van Ackooij, W., & de Oliveira, W. (2016). Convexity and optimization with copulae structured probabilistic constraints. Optimization: A Journal of Mathematical Programming and Operations Research, 65(7), 1349–1376.

    Google Scholar 

  • van Ackooij, W., & de Oliveira, W. (2017). DC programming techniques with inexact subproblems’ solution for general DC programs. Submitted manuscript, 1–27.

  • van Ackooij, W., de Oliveira, W., & Song, Y. (2018). An adaptive partition-based level decomposition for solving two-stage stochastic programs with fixed recourse. Informs Journal on Computing, 30(1), 57–70.

    Google Scholar 

  • van Ackooij, W., Finardi, E. C., & Matiussi Ramalho, G. (2018). An exact solution method for the hydrothermal unit commitment under wind power uncertainty with joint probability constraints. Submitted Preprint, 1–11.

  • van Ackooij, W., & Frangioni, A. (2018). Incremental bundle methods using upper models. SIAM Journal on Optimization, 28(1), 379–410.

    Google Scholar 

  • van Ackooij, W., Frangioni, A., & de Oliveira, W. (2016). Inexact stabilized Benders’ decomposition approaches: With application to chance-constrained problems with finite support. Computational Optimization And Applications, 65(3), 637–669.

    Google Scholar 

  • van Ackooij, W., & Henrion, R. (2014). Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions. SIAM Journal on Optimization, 24(4), 1864–1889.

    Google Scholar 

  • van Ackooij, W., & Henrion, R. (2017). (Sub-) Gradient formulae for probability functions of random inequality systems under Gaussian distribution. SIAM Journal on Uncertainty Quantification, 5(1), 63–87.

    Google Scholar 

  • van Ackooij, W., Henrion, R., Möller, A., & Zorgati, R. (2010). On probabilistic constraints induced by rectangular sets and multivariate normal distributions. Mathematical Methods of Operations Research, 71(3), 535–549.

    Google Scholar 

  • van Ackooij, W., Henrion, R., Möller, A., & Zorgati, R. (2011). Chance constrained programming and its applications to energy management. In I. Dritsas (Ed.), Stochastic optimization: Seeing the optimal for the uncertain (pp. 291–320). INTECH.

  • van Ackooij, W., Henrion, R., Möller, A., & Zorgati, R. (2014). Joint chance constrained programming for hydro reservoir management. Optimization and Engineering, 15, 509–531.

    Google Scholar 

  • van Ackooij, W., Laguel, Y., Malick, J., Matiussi Ramalho, G., & de Oliveira, W. (2018). On transconcavity and probability constraints. Submitted preprint, 1–24.

  • van Ackooij, W., Lebbe, N., & Malick, J. (2017). Regularized decomposition of large-scale block-structured robust optimization problems. Computational Management Science, 14(3), 393–421.

    Google Scholar 

  • van Ackooij, W., & Malick, J. (2016). Decomposition algorithm for large-scale two-stage unit-commitment. Annals of Operations Research, 238(1), 587–613.

    Google Scholar 

  • van Ackooij, W., & Malick, J. (2017). Eventual convexity of probability constraints with elliptical distributions. Mathematical Programming, 1–20.

  • van Ackooij, W., & Oudjane, N. (2015). Prise en compte des incertitudes dans la gestion de production court-terme: retrospection. Technical report H-R36-2013-04179-FR, EDF R&D, 2.

  • van Ackooij, W., & Wirth, J. (2007). Un jeu d’acteurs n-zones pour SSPS. synthèse et propositions. Technical report H-R33-2006-03913-FR, EDF R&D, 2.

  • van Slyke, R. M., & Wets, R. J.-B. (1969). L-shaped linear programs with applications to optimal control and stochastic programming. SIAM Journal of Applied Mathematics, 17, 638–663.

    Google Scholar 

  • Ventosa, M., Baíllo, A., Ramos, A., & Rivier, M. (2005). Electricity market modeling trends. Energy Policy, 33(7), 897–913.

    Google Scholar 

  • Victoire, T. A. A., & Jeyakumar, A. E. (2005). Unit commitment by a Tabu-search-based hybrid-optimisation technique. IEE Proceedings-Generation, Transmission and Distribution, 152(4), 563–574.

    Google Scholar 

  • Vieira, B., Viana, A., Matos, M., & Pedrosa, J. P. (2016). A multiple criteria utility-based approach for unit commitment with wind power and pumped storage hydro. Electric Power Systems Research, 131(1), 244–254.

    Google Scholar 

  • Villumsen, J. C., & Philpott, A. B. (2011). Column generation for transmission switching of electricity networks with unit commitment. Proceedings of the International Multiconference of Engineers and Computer Scientists, 2.

  • Vucetic, S., Tomsovic, K., & Obradovic, Z. (2001). Discovering price-load relationships in California’s electricity market. IEEE Transactions on Power Systems, 16(2), 280–286.

    Google Scholar 

  • Wallace, S. W., & Fleten, S.-E. (2003). Stochastic programming models in energy (Chapter 10). In A. Ruszczynski & A. Shapiro (Eds.), Stochastic programming. Volume 10 of Handbooks in operations research and management science (pp. 637–677). Amsterdam: Elsevier.

    Google Scholar 

  • Walsh, M. P., & O’Malley, M. J. (1997). Augmented hopfield network for unit commitment and economic dispatch. IEEE Transactions on Power Systems, 12(4), 1765–1774.

    Google Scholar 

  • Wang, B., & Hobbs, B. F. (2016). Real-time markets for flexiramp: A stochastic unit commitment-based analysis. IEEE Transactions on Power Systems, 31(2), 846–860.

    Google Scholar 

  • Wang, C., & Fu, Y. (2016). Fully parallel stochastic security-constrained unit commitment. IEEE Transactions on Power Systems, 31(5), 3561–3571.

    Google Scholar 

  • Wang, C., & Shahidehpour, M. (1993). Effects of ramp-rate limits on unit commitment and economic dispatch. IEEE Transactions on Power Systems, 8(3), 1341–1350.

    Google Scholar 

  • Wang, J., Shahidehpour, M., & Li, Z. (2008). Security-constrained unit commitment with volatile wind power generation. IEEE Transactions on Power Systems, 23(3), 1319–1327.

    Google Scholar 

  • Wang, J., Wang, J., Liu, C., & Ruiz, J. P. (2013). Stochastic unit commitment with sub-hourly dispatch constraints. Applied Energy, 105, 418–422.

    Google Scholar 

  • Wang, J., Wang, X., & Wu, Y. (2005). Operating reserve model in the power market. IEEE Transactions on Power Systems, 20(1), 223–229.

    Google Scholar 

  • Wang, L., Mazumdar, M., Bailey, M. D., & Valenzuela, J. (2007). Oligopoly models for market price of electricity under demand uncertainty and unit reliability. European Journal of Operational Research, 181(3), 1309–1321.

    Google Scholar 

  • Wang, Q., Guan, Y., & Wang, J. (2012). A chance-constrained two-stage stochastic program for unit commitment with uncertain wind power output. IEEE Transactions on Power Systems, 27(1), 206–215.

    Google Scholar 

  • Wang, Q., Watson, J.-P., & Guan, Y. (2013). Two-stage robust optimization for \(n-k\) contingency-constrained unit commitment. IEEE Transactions on Power Systems, 28(3), 2366–2375.

    Google Scholar 

  • Wang, S. J., Shahidehpour, M., Kirschen, D. S., Mokhtari, S., & Irisarri, G. D. (1995). Short-term generation scheduling with transmission and environmental constraints using an augmented Lagrangian relaxation. IEEE Transactions on Power Systems, 10(3), 1294–1301.

    Google Scholar 

  • Wang, Y., Xia, Q., & Kang, C. (2011). Unit commitment with volatile node injections by using interval optimization. IEEE Transactions on Power Systems, 26(3), 1705–1713.

    Google Scholar 

  • Warrington, J., Goulart, P., Mariéthoz, S., & Morari, M. (2013). Policy-based reserves for power systems. IEEE Transactions on Power Systems, 28(4), 4427–4437.

    Google Scholar 

  • Wen, F., & David, A. K. (2001). Optimal bidding strategies and modeling of imperfect information among competitive generators. IEEE Transactions on Power Systems, 16(1), 15–21.

    Google Scholar 

  • Wolfe, P. (1975). A method of conjugate subgradients for minimizing nondifferentiable functions. Mathematical Programming Study, 3, 143–173.

    Google Scholar 

  • Wong, K. P., & Wong, Y. W. (1994). Genetic and genetic/simulated-annealing approaches to economic dispatch. IEEE Proceedings-Generation, Transmission and Distribution, 141(5), 507–513.

    Google Scholar 

  • Wong, K. P., & Wong, Y. W. (1996). Combined genetic algorithm/simulated annealing/fuzzy set approach to short-term generation scheduling with take-or-pay fuel contract. IEEE Transactions on Power Systems, 11(1), 128–136.

    Google Scholar 

  • Wong, S., & Fuller, J. D. (2007). Pricing energy and reserves using stochastic optimization in an alternative electricity market. IEEE Transactions on Power Systems, 22(2), 631–638.

    Google Scholar 

  • Wood, A. J., & Wollemberg, B. F. (1996). Power generation operation and control. Hoboken: Wiley.

    Google Scholar 

  • Wu, H., Shahidehpour, M., Li, Z., & Tian, W. (2014). Chance-constrained day-ahead scheduling in stochastic power system operation. IEEE Transactions on Power Systems, 29(4), 1583–1591.

    Google Scholar 

  • Wu, L. (2013). An improved decomposition framework for accelerating LSF and BD based methods for network-constrained UC problems. IEEE Transactions on Power Systems, 28(4), 3977–3986.

    Google Scholar 

  • Wu, L. (2011). A tighter piecewise linear approximation of quadratic cost curves for unit commitment problems. IEEE Transactions on Power Systems, 26(4), 2581–2583.

    Google Scholar 

  • Wu, L., & Shahidehpour, M. (2010). Accelerating the benders decomposition for network-constrained unit commitment problems. Energy Systems, 1, 339–376.

    Google Scholar 

  • Wu, L., Shahidehpour, M., & Li, T. (2007). Stochastic security-constrained unit commitment. IEEE Transactions on Power Systems, 22(2), 800–811.

    Google Scholar 

  • Wu, L., Shahidehpour, M., & Li, Z. (2012). Comparison of scenario-based and interval optimization approaches to stochastic SCUC. IEEE Transactions on Power Systems, 27(2), 913–921.

    Google Scholar 

  • Wu, L., Shahidehpour, M., & Tao, L. (2007). Stochastic security-constrained unit commitment. IEEE Transactions on Power Systems, 22(2), 800–811.

    Google Scholar 

  • Wu, Z., Zeng, P., Zhang, X.-P., & Zhou, Q. (2016). A solution to the chance-constrained two-stage stochastic program for unit commitment with wind energy integration. IEEE Transactions on Power Systems, 31(6), 4185–4196.

    Google Scholar 

  • Xiong, P., & Jirutitijaroen, P. (2011). Stochastic unit commitment using multi-cut decomposition algorithm with partial aggregation. In IEEE power and energy society general meeting.

  • Xiong, P., Jirutitijaroen, P., & Singh, C. (2017). Level function method for quasiconvex programming. IEEE Transactions on Power Systems, 32(1), 39–49.

    Google Scholar 

  • Yan, H., Luh, P. B., Guan, X., & Rogan, P. M. (1993). Scheduling of hydro-thermal power systems. IEEE Transactions on Power Systems, 8(3), 1358–1365.

    Google Scholar 

  • Yan, H., Luh, P. B., & Zhang, L. (1994). Scheduling of hydrothermal power systems using the augmented Lagrangian decomposition and coordination technique. In American control conference 1994 (Vol. 2, pp. 1558–1562).

  • Yang, H. T., Yang, P. C., & Huang, C. L. (1996). Evolutionary programming based economic dispatch for units with non-smooth fuel cost functions. IEEE Transactions on Power Systems, 11(1), 112–118.

    Google Scholar 

  • Yang, J.-S., & Chen, N. (1989). Short term hydrothermal coordination using multi-pass dynamic programming. IEEE Transactions on Power Systems, 4(3), 1050–1056.

    Google Scholar 

  • Yang, L., Zhang, C., Jian, J., Meng, K., Xu, Y., & Dong, Z. (2017). A novel projected two-binary-variable formulation for unit commitment in power systems. Applied Energy, 187, 732–745.

    Google Scholar 

  • Ye, H., & Li, Z. (2016). Robust security-constrained unit commitment and dispatch with recourse cost requirement. IEEE Transactions on Power Systems, 31(5), 3527–3536.

    Google Scholar 

  • Yu, Y. W., Luh, P. B., Litvinov, E., Zheng, T. X., Zhao, J. Y., & Zhao, F. (2015). Grid integration of distributed wind generation: Hybrid Markovian and interval unit commitment. IEEE Transactions od Smart Grid, 6(6), 3061–3072.

    Google Scholar 

  • Yu, Z., Sparrow, F. T., Bowen, B., & Smardo, F. J. (2000). On convexity issues of short-term hydrothermal scheduling. Electrical Power and Energy Systems, 20, 451–457.

    Google Scholar 

  • Zaourar, S. (2014). Optimisation convexe non-différentiable et méthodes de décomposition en recherche opérationnelle. Ph.D. thesis, University of Grenoble.

  • Zaourar, S., & Malick, J. (2013). Prices stabilization for inexact unit-commitment problems. Mathematical Methods of Operations Research, 78(3), 341–359.

    Google Scholar 

  • Zaourar, S., & Malick, J. (2014). Quadratic stabilization of benders decomposition (pp. 1–22). Draft submitted, Privately communicated.

  • Zareipour, H. (2012). Short-term electricity market prices: A review of characteristics and forecasting methods. In A. Sorokin, S. Rebennack, P. M. Pardalos, N. A. Iliadis, & M. V. F. Pereira (Eds.), Handbook of networks in power systems I (pp. 89–121). Heidelberg: Springer.

    Google Scholar 

  • Zhai, Q., Li, X., Lei, X., & Guan, X. (2017). Transmission constrained UC with wind power: An all-scenario-feasible MILP formulation with strong nonanticipativity. IEEE Transactions on Power Systems, 32(3), 1805–1817.

    Google Scholar 

  • Zhang, C., & Wang, J. (2014). Optimal transmission switching considering probabilistic reliability. IEEE Transactions on Power Systems.

  • Zhang, D., Luh, P. B., & Zhang, Y. (1999). A bundle method for hydrothermal scheduling. IEEE Transactions on Power Systems, 14, 1355–1361.

    Google Scholar 

  • Zhang, X., Shahidehpour, M., Alabdulwahab, A., & Abusorrah, A. (2016). Hourly electricity demand response in the stochastic day-ahead scheduling of coordinated electricity and natural gas networks. IEEE Transactions on Power Systems, 31(1), 592–601.

    Google Scholar 

  • Zhang, Y., Wang, J., Zeng, B., & Hu, Z. (2017). Chance-constrained two-stage unit commitment under uncertain load and wind power output using bilinear benders decomposition. IEEE Transactions on Power Systems, 32(5), 3637–3647.

    Google Scholar 

  • Zhao, B., Conejo, A. J., & Sioshansi, R. (2017). Unit commitment under gas-supply uncertainty and gas-price variability. IEEE Transactions on Power Systems, 32(3), 2394–2405.

    Google Scholar 

  • Zhao, C., & Guan, Y. (2013). Unified stochastic and robust unit commitment. IEEE Transactions on Power Systems, 28(3), 3353–3361.

    Google Scholar 

  • Zhao, C., & Guan, Y. (2016). Data-driven stochastic unit commitment for integrating wind generation. IEEE Transactions on Power Systems, 31(4), 2587–2595.

    Google Scholar 

  • Zhao, C., Wang, J., Watson, J.-P., & Guan, Y. (2013). Multi-stage robust unit commitment considering wind and demand response uncertainties. IEEE Transactions on Power Systems, 28(3), 2708–2717.

    Google Scholar 

  • Zhao, C., Wang, Q., Wang, J., & Guan, Y. (2014). Expected value and chance constrained stochastic unit commitment ensuring wind power utilization. IEEE Transactions on Power Systems, 29(6), 2696–2705.

    Google Scholar 

  • Zhao, L., & Zeng, B. (2012). Robust unit commitment problem with demand response and wind energy. In Proceedings of IEEE power and energy society general meeting.

  • Zheng, Q., Wang, J., Pardalos, P., & Guan, Y. (2013). A decomposition approach to the two-stage stochastic unit commitment problem. Annals of Operations Research, 210(1), 387–410.

    Google Scholar 

  • Zheng, Q. P., Wang, J., & Liu, A. L. (2015). Stochastic optimization for unit commitment-a review. IEEE Transactions on Power Systems, 30(4), 1913–1924.

    Google Scholar 

  • Zhou, M., Xia, S., Li, G., & Han, X. (2014). Interval optimization combined with point estimate method for stochastic security-constrained unit commitment. Electrical Power and Energy Systems, 63(1), 276–284.

    Google Scholar 

  • Zhu, J. (2009). Optimization of power system operation. Series on power engineering. Hoboken: Wiley-IEEE Press.

    Google Scholar 

  • Zhuang, F., & Galiana, F. D. (1988). Towards a more rigorous and practical unit commitment by Lagrangian relaxation. IEEE Transactions on Power Systems, 3(2), 763–773.

    Google Scholar 

  • Zhuang, F., & Galiana, F. D. (1990). Unit commitment by simulated annealing. IEEE Transactions on Power Systems, 5(1), 311–318.

    Google Scholar 

  • Zorgati, R., & van Ackooij, W. (2011). Optimizing financial and physical assets with chance-constrained programming in the electrical industry. Optimization and Engineering, 12(1), 237–255.

    Google Scholar 

  • Zymler, S., Kuhn, D., & Rustem, B. (2013). Distributionally robust joint chance constraints with second-order moment information. Mathematical Programming, 137(1–2), 167–198.

    Google Scholar 

Download references

Acknowledgements

The first author cites: “1 Corinthians 3:8 : He who plants and he who waters are one, and each will receive his wages according to his labour.” The fourth author is grateful to Andrea Galliani and Massimo Ricci for useful discussions about the future of the balancing markets; the views and the opinions he expressed in this article do not necessary reflect the official policy and position of the ARERA for which he works. The first and third author acknowledge the financial support by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 773897 “plan4res”, and also financial support of PGMO for the original version of this work. The second author acknowledges the support received from Electricite de France and University College Dublin.

Author information

Authors and Affiliations

  1. OSIRIS, EDF R&D, 7 Boulevard Gaspard Monge, 91120, Palaiseau Cedex, France

    W. van Ackooij

  2. EFESE, EDF R&D, 7 Boulevard Gaspard Monge, 91120, Palaiseau Cedex, France

    I. Danti Lopez

  3. University College Dublin, Dublin, Ireland

    I. Danti Lopez

  4. Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, 56127, Pisa, Italia

    A. Frangioni

  5. The Italian Regulatory Authority for Energy, Networks and Environment (ARERA), Piazza Cavour 5, 20121, Milano, Italia

    F. Lacalandra

  6. Supply Chain Consultants, Paris, France

    M. Tahanan

Authors
  1. W. van Ackooij
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Danti Lopez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Frangioni
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. F. Lacalandra
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. M. Tahanan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to W. van Ackooij or A. Frangioni.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

van Ackooij, W., Danti Lopez, I., Frangioni, A. et al. Large-scale unit commitment under uncertainty: an updated literature survey. Ann Oper Res 271, 11–85 (2018). https://doi.org/10.1007/s10479-018-3003-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-018-3003-z

Keywords

Search

Navigation