×

Robust optimization for electricity generation. (English) Zbl 07362319

Summary: We consider a robust optimization problem in an electric power system under uncertain demand and availability of renewable energy resources. Solving the deterministic alternating current (AC) optimal power flow (ACOPF) problem has been considered challenging since the 1960s due to its nonconvexity. Linear approximation of the AC power flow system sees pervasive use, but does not guarantee a physically feasible system configuration. In recent years, various convex relaxation schemes for the ACOPF problem have been investigated, and under some assumptions, a physically feasible solution can be recovered. Based on these convex relaxations, we construct a robust convex optimization problem with recourse to solve for optimal controllable injections (fossil fuel, nuclear, etc.) in electric power systems under uncertainty (renewable energy generation, demand fluctuation, etc.). We propose a cutting-plane method to solve this robust optimization problem, and we establish convergence and other desirable properties. Experimental results indicate that our robust convex relaxation of the ACOPF problem can provide a tight lower bound.
The online supplements are available at https://doi.org/10.1287/ijoc.2020.0956.

MSC:

90Cxx Mathematical programming

Software:

Gurobi; NESTA; JuMP; Ipopt
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Attarha A, Amjady N, Conejo AJ (2018) Adaptive robust AC optimal power flow considering load and wind power uncertainties. Internat. J. Electr. Power Energy Systems 96:132-142.Crossref, Google Scholar · doi:10.1016/j.ijepes.2017.09.037
[2] Bai X, Wei H (2011) A semidefinite programming method with graph partitioning technique for optimal power flow problems. Internat. J. Electr. Power Energy Systems 33(7):1309-1314.Crossref, Google Scholar · doi:10.1016/j.ijepes.2011.06.003
[3] Bai X, Wei H, Fujisawa K, Wang Y (2008) Semidefinite programming for optimal power flow problems. Internat. J. Electr. Power Energy Systems 30(6):383-392.Crossref, Google Scholar · doi:10.1016/j.ijepes.2007.12.003
[4] Bernstein A, Bienstock D, Hay D, Uzunoglu M, Zussman G (2014) Power grid vulnerability to geographically correlated failures—analysis and control implications. IEEE INFOCOM 2014-IEEE Conf. Comput. Comm. (IEEE, Piscataway, NJ), 2634-2642.Google Scholar
[5] Bertsekas DP (2009) Convex Optimization Theory (Athena Scientific, Belmont, MA).Google Scholar · Zbl 1242.90001
[6] Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math. Programming 98(1):49-71.Crossref, Google Scholar · Zbl 1082.90067 · doi:10.1007/s10107-003-0396-4
[7] Bertsimas D, Sim M (2004) The price of robustness. Oper. Res. 52(1):35-53.Link, Google Scholar · Zbl 1165.90565
[8] Bienstock D (2015) Electrical Transmission System Cascades and Vulnerability: An Operations Research Viewpoint (SIAM, Philadelphia).Crossref, Google Scholar · Zbl 1344.90001 · doi:10.1137/1.9781611974164
[9] Bienstock D, Chertkov M, Harnett S (2014) Chance-constrained optimal power flow: Risk-aware network control under uncertainty. SIAM Rev. 56(3):461-495.Crossref, Google Scholar · Zbl 1301.93095 · doi:10.1137/130910312
[10] Boyd S, Vandenberghe L (2004) Convex Optimization (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar · Zbl 1058.90049 · doi:10.1017/CBO9780511804441
[11] Carpentier J (1962) Contribution á l’étude du dispatching économique. Bull. Française Electriciens 3:431-447.Google Scholar
[12] Coffrin C, Gordon D, Scott P (2014) NESTA, the NICTA energy system test case archive. Preprint, submitted November 3, https://arxiv.org/abs/1411.0359.Google Scholar
[13] Coffrin C, Hijazi HL, Van Hentenryck P (2016a) Strengthening the SDP relaxation of AC power flows with convex envelopes, bound tightening, and valid inequalities. IEEE Trans. Power Systems 32(5):3549-3558.Google Scholar
[14] Coffrin C, Hijazi HL, Van Hentenryck P (2016b) The QC relaxation: A theoretical and computational study on optimal power flow. IEEE Trans. Power Systems 31(4):3008-3018.Crossref, Google Scholar · doi:10.1109/TPWRS.2015.2463111
[15] Crainic TG, Hewitt M, Maggioni F, Rei W (2016) Partial Benders decomposition strategies for two-stage stochastic integer programs. Technical report, CIRRELT, Montreal.Google Scholar
[16] Dunning I, Huchette J, Lubin M (2017) JuMP: A modeling language for mathematical optimization. SIAM Rev. 59(2):295-320.Google Scholar · Zbl 1368.90002
[17] Fang X, Hodge B, Du E, Zhang N, Li F (2018) Modelling wind power spatial-temporal correlation in multi-interval optimal power flow: A sparse correlation matrix approach. Appl. Energy 230:531-539.Crossref, Google Scholar · doi:10.1016/j.apenergy.2018.08.123
[18] Gabrel V, Lacroix M, Murat C, Remli N (2014) Robust location transportation problems under uncertain demands. Discrete Appl. Math. 164:100-111.Crossref, Google Scholar · Zbl 1331.90096 · doi:10.1016/j.dam.2011.09.015
[19] Geoffrion AM (1972) Generalized Benders decomposition. J. Optim. Theory Appl. 10(4):237-260.Crossref, Google Scholar · Zbl 0229.90024 · doi:10.1007/BF00934810
[20] Gurobi Optimization (2016) Gurobi Optimizer Reference Manual. Accessed February 19, 2020, https://www.gurobi.com/documentation/9.0/refman/index.html.Google Scholar
[21] Gurvich I, Luedtke J, Tezcan T (2010) Staffing call centers with uncertain demand forecasts: A chance-constrained optimization approach. Management Sci. 56(7):1093-1115.Link, Google Scholar · Zbl 1232.90278
[22] Jabr RA (2006) Radial distribution load flow using conic programming. IEEE Trans. Power Systems 21(3):1458-1459.Crossref, Google Scholar · doi:10.1109/TPWRS.2006.879234
[23] Jabr RA (2013) Adjustable robust OPF with renewable energy sources. IEEE Trans. Power Systems 28(4):4742-4751.Crossref, Google Scholar · doi:10.1109/TPWRS.2013.2275013
[24] Jiang R, Wang J, Guan Y (2012) Robust unit commitment with wind power and pumped storage hydro. IEEE Trans. Power Systems 27(2):800-810.Crossref, Google Scholar · doi:10.1109/TPWRS.2011.2169817
[25] Jiang R, Zhang M, Li G, Guan Y (2014) Two-stage network constrained robust unit commitment problem. Eur. J. Oper. Res. 234(3):751-762.Crossref, Google Scholar · Zbl 1304.90056 · doi:10.1016/j.ejor.2013.09.028
[26] Khodaei A (2014) Resiliency-oriented microgrid optimal scheduling. IEEE Trans. Smart Grid 5(4):1584-1591.Crossref, Google Scholar · doi:10.1109/TSG.2014.2311465
[27] Klima K, Apt J (2015) Geographic smoothing of solar PV: Results from Gujarat. Environ. Res. Lett. 10(10):104001.Crossref, Google Scholar · doi:10.1088/1748-9326/10/10/104001
[28] Kocuk B, Dey SS, Sun XA (2016) Strong SOCP relaxations for the optimal power flow problem. Oper. Res. 64(6):1177-1196.Link, Google Scholar · Zbl 1354.90154
[29] Lavaei J, Low SH (2012) Zero duality gap in optimal power flow problem. IEEE Trans. Power Systems 27(1):92-107.Crossref, Google Scholar · doi:10.1109/TPWRS.2011.2160974
[30] Liu Y, Ferris MC (2015) Security constrained economic dispatch using semidefinite programming. 2015 IEEE Power and Energy Society General Meeting (IEEE, Piscataway, NJ), 1-5.Google Scholar
[31] Lohmann GM, Monahan AH, Heinemann D (2016) Local short-term variability in solar irradiance. Atmospheric Chemistry Phys. 16(10):6365-6379.Crossref, Google Scholar · doi:10.5194/acp-16-6365-2016
[32] Lorca A, Sun XA (2018) The adaptive robust multi-period alternating current optimal power flow problem. IEEE Trans. Power Systems 33(2):1993-2003.Crossref, Google Scholar · doi:10.1109/TPWRS.2017.2743348
[33] Louca R, Bitar E (2017) Robust AC optimal power flow. IEEE Trans. Power Systems 34(3):1669-1681.Google Scholar
[34] Low SH (2014a) Convex relaxation of optimal power flow—part I: Formulations and equivalence. IEEE Trans. Control Network Systems 1(1):15-27.Crossref, Google Scholar · Zbl 1370.90043 · doi:10.1109/TCNS.2014.2309732
[35] Low SH (2014b) Convex relaxation of optimal power flow—part II: Exactness. IEEE Trans. Control Network Systems 1(2):177-189.Crossref, Google Scholar · Zbl 1370.90044 · doi:10.1109/TCNS.2014.2323634
[36] Lubin M, Dvorkin Y, Backhaus S (2016) A robust approach to chance constrained optimal power flow with renewable generation. IEEE Trans. Power Systems 31(5):3840-3849.Crossref, Google Scholar · doi:10.1109/TPWRS.2015.2499753
[37] Maisano J, Radchik A, Ling T (2016) A lognormal model for demand forecasting in the national electricity market. ANZIAM J. 57(3):369-383.Crossref, Google Scholar · doi:10.1017/S1446181115000322
[38] Malvaldi A, Weiss S, Infield D, Browell J, Leahy P, Foley AM (2017) A spatial and temporal correlation analysis of aggregate wind power in an ideally interconnected Europe. Wind Energy 20(8):1315-1329.Crossref, Google Scholar · doi:10.1002/we.2095
[39] Momoh JA, El-Hawary ME, Adapa R (1999) A review of selected optimal power flow literature to 1993. Part II. Newton, linear programming and interior point methods. IEEE Trans. Power Systems 14(1):105-111.Crossref, Google Scholar · doi:10.1109/59.744495
[40] Monticelli A, Pereira MVF, Granville S (1987) Security-constrained optimal power flow with post-contingency corrective rescheduling. IEEE Trans. Power Systems 2(1):175-180.Crossref, Google Scholar · doi:10.1109/TPWRS.1987.4335095
[41] Nemirovsky AS, Yudin DB (1983) Problem Complexity and Method Efficiency in Optimization (Wiley, New York).Google Scholar · Zbl 0501.90062
[42] Nguyen HD, Dvijotham K, Turitsyn K (2019) Constructing convex inner approximations of steady-state security regions. IEEE Trans. Power Systems 34(1):257-267.Crossref, Google Scholar · doi:10.1109/TPWRS.2018.2868752
[43] Phan D, Ghosh S (2014) Two-stage stochastic optimization for optimal power flow under renewable generation uncertainty. ACM Trans. Model. Comput. Simul. 24(1):1-22.Crossref, Google Scholar · Zbl 1322.90055 · doi:10.1145/2553084
[44] Singh R, Pal BC, Jabr RA (2010) Statistical representation of distribution system loads using Gaussian mixture model. IEEE Trans. Power Systems 25(1):29-37.Crossref, Google Scholar · doi:10.1109/TPWRS.2009.2030271
[45] Stott B, Jardim J, Alsaç O (2009) DC power flow revisited. IEEE Trans. Power Systems 24(3):1290-1300.Crossref, Google Scholar · doi:10.1109/TPWRS.2009.2021235
[46] Sundar K, Nagarajan H, Misra S, Lu M, Coffrin C, Bent R (2018) Optimization-based bound tightening using a strengthened QC-relaxation of the optimal power flow problem. Preprint, submitted September 12, https://arxiv.org/abs/1809.04565.Google Scholar
[47] Terry TL (2009) Robust linear optimization with recourse: Solution methods and other properties. Unpublished doctoral dissertation, University of Michigan, Ann Arbor.Google Scholar
[48] Thiele A, Terry T, Epelman M (2009) Robust linear optimization with recourse. Optimization Online 2009(March):2263.Google Scholar
[49] Verma A (2010) Power grid security analysis: An optimization approach. Unpublished doctoral dissertation, Columbia University, New York.Google Scholar
[50] Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Programming 106(1):25-57.Crossref, Google Scholar · Zbl 1134.90542 · doi:10.1007/s10107-004-0559-y
[51] Xie W, Ahmed S (2018) Distributionally robust chance constrained optimal power flow with renewables: A conic reformulation. IEEE Trans. Power Systems 33(2):1860-1867.Crossref, Google Scholar · doi:10.1109/TPWRS.2017.2725581
[52] Zan J, Hasenbein JJ, Morton DP (2014) Asymptotically optimal staffing of service systems with joint QoS constraints. Queueing Systems 78(4):359-386.Crossref, Google Scholar · Zbl 1309.60092 · doi:10.1007/s11134-014-9406-x
[53] Zugno M, Conejo AJ (2015) A robust optimization approach to energy and reserve dispatch in electricity markets. Eur. J. Oper. Res. 247(2):659-671.Crossref, Google Scholar · Zbl 1346.91171 · doi:10.1016/j.ejor.2015.05.081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.