Skip to main content

Advertisement

SpringerLink
Log in

Integrating unimodality into distributionally robust optimal power flow

  • Original Paper
  • Published:
TOP Aims and scope Submit manuscript

Abstract

To manage renewable generation and load consumption uncertainty, chance-constrained optimal power flow (OPF) formulations have been proposed. However, conventional solution approaches often rely on accurate estimates of uncertainty distributions, which are rarely available in reality. When the distributions are not known but can be limited to a set of plausible candidates, termed an ambiguity set, distributionally robust (DR) optimization can reduce out-of-sample violation of chance constraints. Nevertheless, a DR model may yield conservative solutions if the ambiguity set is too large. In view that most practical uncertainty distributions for renewable generation are unimodal, in this paper, we integrate unimodality into a moment-based ambiguity set to reduce the conservatism of a DR-OPF model. We review exact reformulations, approximations, and an online algorithm for solving this model. We extend these results to derive a new, offline solution algorithm. Specifically, this algorithm uses a parameter selection approach that searches for an optimal approximation of the DR-OPF model before solving it. This significantly improves the computational efficiency and solution quality. We evaluate the performance of the offline algorithm against existing solution approaches for DR-OPF using modified IEEE 118-bus and 300-bus systems with high penetrations of renewable generation. Results show that including unimodality reduces solution conservatism and cost without degrading reliability significantly.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. To handle uncertain loads, we denote \(P_L^f\) and \({\tilde{d}}\) as the load forecast and the uncertain load forecast error, respectively. We substitute the vector of loads \(P_L\) with \(P_L^f+{\tilde{d}}\) and the system uncertainty \({\tilde{w}}\) with \({\tilde{w}} + {\tilde{d}}\) in formulation (12). Then, we can use the same reformulation method and solution algorithm as described in Section 4 to solve this extended model.

  2. Note that each intermediate solution comes from a relaxed approximation and so the objective cost is lower than the true objective cost. Here we use a negative optimality gap to illustrate this relation.

References

  • Ahmed S, Shapiro A (2008) Solving chance-constrained stochastic programs via sampling and integer programming. In: State-of-the-art decision-making tools in the information-intensive age. INFORMS TutORials Oper Res:261–269

  • Arrigo A, Kazempour J, De Grève Z, Toubeau J.-F, Vallée F (2021) Embedding dependencies within distributionally robust optimization of modern power systems. arXiv preprint arXiv:2104.08101

  • Arrigo A, Ordoudis C, Kazempour J, De Grève Z, Toubeau J-F, Vallée F (2022) Wasserstein distributionally robust chance-constrained optimization for energy and reserve dispatch: an exact and physically-bounded formulation. Eur J Oper Res 296(1):304–322

    Article  Google Scholar 

  • Baker K, Bernstein A (2019) Joint chance constraints in ac optimal power flow: improving bounds through learning. IEEE Trans Smart Grid 10(6):6376–6385

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Campi M, Calafiore G, Prandini M (2009) The scenario approach for systems and control design. Annu Rev Control 33(2):149–157

    Article  Google Scholar 

  • Charnes A, Cooper W, Symonds G (1958) Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manage Sci 4(3):235–263

    Article  Google Scholar 

  • Coffrin C, Gordon D, Scott P (2016) NESTA, the NICTA energy system test case archive. arXiv preprint arXiv:1411.0359v5

  • Cox MG (1971) An algorithm for approximating convex functions by means by first degree splines. Comput J 14(3):272–275

    Article  Google Scholar 

  • Delage E, Ye Y (2010) Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper Res 58(3):595–612

    Article  Google Scholar 

  • Dharmadhikari SW, Joag-Dev K (1988) Unimodality, convexity, and applications. Academic Press, Cambridge

    Google Scholar 

  • Duan C, Fang W, Jiang L, Yao L, Liu J (2018) Distributionally robust chance-constrained approximate AC-OPF with Wasserstein metric. IEEE Trans Power Syst 33(5):4924–4936

    Article  Google Scholar 

  • Esteban-Pérez A, Morales J.M (2021) Distributionally robust optimal power flow with contextual information. arXiv preprint arXiv:2109.07896

  • Gavrilović MM (1975) Optimal approximation of convex curves by functions which are piecewise linear. J Math Anal Appl 52(2):260–282

    Article  Google Scholar 

  • Ghaoui LE, Oks M, Oustry F (2003) Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper Res 51(4):543–556

    Article  Google Scholar 

  • Grant M, Boyd S (2008) Graph implementations for nonsmooth convex programs. In: Recent Advances in Learning and Control. Lecture Notes in Control and Information Sciences

  • Grant M, Boyd S (2014) CVX: matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx

  • Guo Y, Baker K, Dall’Anese E, Hu Z, Summers TH (2018) Data-based distributionally robust stochastic optimal power flow - Part II: case studies. IEEE Trans Power Syst 34(2):1493–1503

    Article  Google Scholar 

  • Guo Y, Baker K, Dall’Anese E, Hu Z, Summers T (2018) Stochastic optimal power flow based on data-driven distributionally robust optimization. In: IEEE Annual American Control Conference, Milwaukee, WI

  • Guo Y, Baker K, Dall’Anese E, Hu Z, Summers TH (2019) Data-based distributionally robust stochastic optimal power flow - Part I: methodologies. IEEE Trans Power Syst 34(2):1483–1492

    Article  Google Scholar 

  • Hou A.M, Roald L.A (2020) Chance constraint tuning for optimal power flow. In: 2020 International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), pp. 1–6. IEEE

  • Huang W, Zheng W, Hill DJ (2021) Distributionally robust optimal power flow in multi-microgrids with decomposition and guaranteed convergence. IEEE Trans Smart Grid 12(1):43–55

    Article  Google Scholar 

  • Imamoto A, Tang B (2008) A recursive descent algorithm for finding the optimal minimax piecewise linear approximation of convex functions. In: World Congress on Engineering and Computer Science, San Francisco, CA

  • Jabr RA (2013) Adjustable robust OPF with renewable energy sources. IEEE Trans Power Syst 28(4):4742–4751

    Article  Google Scholar 

  • Jabr RA (2020) Distributionally robust cvar constraints for power flow optimization. IEEE Trans Power Syst 35(5):3764–3773

    Article  Google Scholar 

  • Jensen TV, Pinson P (2017) Re-Europe, a large-scale dataset for modeling a highly renewable European electricity system. Sci data 4:170–175

    Article  Google Scholar 

  • Jiang R, Guan Y (2016) Data-driven chance constrained stochastic program. Math Program 158(1):291–327

    Article  Google Scholar 

  • Li B (2019) Distributionally robust optimal power flow with strengthened ambiguity sets. PhD thesis, University of Michigan Ann Arbor

  • Li B, Jiang R, Mathieu J.L (2016) Distributionally robust risk-constrained optimal power flow using moment and unimodality information. In: IEEE Conference on Decision and Control, Las Vegas, NV

  • Li B, Mathieu J.L, Jiang R (2018) Distributionally robust chance constrained optimal power flow assuming log-concave distributions. In: Power Systems Computation Conference, Dublin, Ireland

  • Li B, Vrakopoulou M, Mathieu JL (2019a) Chance constrained reserve scheduling using uncertain controllable loads Part II: analytical reformulation. IEEE Trans Smart Grid 10(2):1618–1625

    Article  Google Scholar 

  • Li B, Jiang R, Mathieu JL (2019b) Ambiguous risk constraints with moment and unimodality information. Math Program 173(1–2):151–192

    Article  Google Scholar 

  • Li B, Jiang R, Mathieu JL (2019c) Distributionally robust chance-constrained optimal power flow assuming unimodal distributions with misspecified modes. IEEE Trans Control Netw Syst 6(3):1223–1234

    Article  Google Scholar 

  • Lu X, Chan KW, Xia S, Zhou B, Luo X (2019) Security-constrained multi-period economic dispatch with renewable energy utilizing distributionally robust optimization. IEEE Trans Sustain Energy 10(2):768–779

    Article  Google Scholar 

  • Lubin M, Dvorkin Y, Backhaus S (2016) A robust approach to chance constrained optimal power flow with renewable generation. IEEE Trans Power Syst 31(5):3840–3849

    Article  Google Scholar 

  • Margellos K, Goulart P, Lygeros J (2014) On the road between robust optimization and the scenario approach for chance constrained optimization problems. IEEE Trans Autom Control 59(8):2258–2263

    Article  Google Scholar 

  • Mieth R, Dvorkin Y (2018) Data-driven distributionally robust optimal power flow for distribution systems. IEEE Control Syst Lett 2(3):363–368

    Article  Google Scholar 

  • Miller B, Wagner H (1965) Chance constrained programming with joint constraints. Oper Res 13(6):930–945

    Article  Google Scholar 

  • Ordoudis C, Nguyen VA, Kuhn D, Pinson P (2021) Energy and reserve dispatch with distributionally robust joint chance constraints. Oper Res Lett 49(3):291–299

    Article  Google Scholar 

  • Pagnoncelli BK, Ahmed S, Shapiro A (2009) Sample average approximation method for chance constrained programming: theory and applications. J Optim Theory Appl 142(2):399–416

    Article  Google Scholar 

  • Papaefthymiou G, Klockl B (2008) MCMC for wind power simulation. IEEE Trans Energy Conv 23(1):234–240

    Article  Google Scholar 

  • Pourahmadi F, Kazempour J (2021) Distributionally robust generation expansion planning with unimodality and risk constraints. IEEE Trans Power Syst 36(5):4281–4295

    Article  Google Scholar 

  • Roald L, Oldewurtel F, Krause T, Andersson G (2013) Analytical reformulation of security constrained optimal power flow with probabilistic constraints. In: IEEE PowerTech Conference, Grenoble, France

  • Roald L, Oldewurtel F, Parys B.V, Andersson G (2015) Security constrained optimal power flow with distributionally robust chance constraints. arXiv preprint arXiv:1508.06061

  • Stellato B (2014) Data-driven Chance Constrained Optimization. Master thesis, ETH Zurich

  • Summers T, Warrington J, Morari M, Lygeros J (2015) Stochastic optimal power flow based on conditional value at risk and distributional robustness. Int J Elect Power Energy Syst 72:116–125

    Article  Google Scholar 

  • Tong X, Sun H, Luo X, Zheng Q (2018) Distributionally robust chance constrained optimization for economic dispatch in renewable energy integrated systems. J Global Optim 70(1):131–158

    Article  Google Scholar 

  • Vandewalle J (1975) On the calculation of the piecewise linear approximation to a discrete function. IEEE Trans Comput 24:843–846

    Article  Google Scholar 

  • Vrakopoulou M, Margellos K, Lygeros J, Andersson G (2013) A probabilistic framework for reserve scheduling and N-1 security assessment of systems with high wind power penetration. IEEE Trans Power Syst 28(4):3885–3896

    Article  Google Scholar 

  • Vrakopoulou M, Li B, Mathieu JL (2019) Chance constrained reserve scheduling using uncertain controllable loads Part I: formulation and scenario-based analysis. IEEE Trans Smart Grid 10(2):1608–1617

    Article  Google Scholar 

  • Wagner M (2008) Stochastic 0–1 linear programming under limited distributional information. Oper Res Lett 36(2):150–156

    Article  Google Scholar 

  • Wang C, Gao R, Qiu F, Wang J, Xin L (2018) Risk-based distributionally robust optimal power flow with dynamic line rating. IEEE Trans Power Syst 33(6):6074-6086

    Article  Google Scholar 

  • Xie W, Ahmed S (2018) Distributionally robust chance constrained optimal power flow with renewables: a conic reformulation. IEEE Trans Power Syst 33(2):1860–1867

    Article  Google Scholar 

  • Xie W, Ahmed S, Jiang R (2019) Optimized bonferroni approximations of distributionally robust joint chance constraints. Math. Program: 1–34

  • Zhang H, Li P (2011) Chance constrained programming for optimal power flow under uncertainty. IEEE Trans Power Syst 26(4):2417–2424

    Article  Google Scholar 

  • Zhang Y, Shen S, Mathieu JL (2017) Distributionally robust chance-constrained optimal power flow with uncertain renewables and uncertain reserves provided by loads. IEEE Trans Power Syst 32(2):1378–1388

    Google Scholar 

Download references

Acknowledgements

This research was supported by the U.S. National Science Foundation Awards CCF-1442495 and CMMI-1662774.

Author information

Authors and Affiliations

  1. Argonne National Laboratory, Lemont, IL, USA

    Bowen Li

  2. Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI, USA

    Ruiwei Jiang

  3. Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA

    Johanna L. Mathieu

Authors
  1. Bowen Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruiwei Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Johanna L. Mathieu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johanna L. Mathieu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, B., Jiang, R. & Mathieu, J.L. Integrating unimodality into distributionally robust optimal power flow. TOP 30, 594–617 (2022). https://doi.org/10.1007/s11750-022-00634-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11750-022-00634-4

Keywords

Mathematics Subject Classification

Search

Navigation