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A more accurate piecewise linear approximation method for quadratic cost curves of thermal generators and its application in unit commitment

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Abstract

In the studies of unit commitment or optimal power flow, to formulate a mixed-integer linear programming model that can be efficiently solved with commercial solvers, it is necessary to approximate the quadratic cost curves of thermal units as piecewise linear (PWL) functions. The conventional approach involves evenly spaced piecewise linear (ES-PWL) interpolation, which often results in relatively large approximation errors. In order to reduce the error, this paper proposes a more accurate PWL method for approximating the quadratic cost functions of thermal units. The method employs a linear least-squares fit instead of linear interpolation within each subinterval and introduces a one-terminal-constraint approach to ensure the continuity of the piecewise function. Subsequently, a straightforward equation is derived, applicable to the widely used ES-PWL interpolation, with the potential to enhance the accuracy of the approximation. Mathematical verification attests that the proposed method substantially diminishes the squared 2-norm error, less than 37.5% of the error associated with ES-PWL interpolation. Subsequent numerical investigations are carried out on a 10-unit system, the IEEE RTS-79, and a real industrial system. The findings validate that all the approximation errors of the proposed method are within 37.5% of the errors associated with the ES-PWL interpolation, meaning that a unit commitment solution that closely approximates the outcome of the quadratic function is obtained. The computational time is also acceptable.

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Availability of data and materials

The data used in the paper have been uploaded to GitHub (https://github.com/PowerSystemGroup/UCDataAndResults/tree/data_and_results_of_paper_A_more_accurate_piecewise_linear_approximation_method_for_quadratic_cost_curves_of_thermal_generators_and_its_application_in_unit_commitment).

Abbreviations

\({\mathbb{G}}\) :

Set of thermal units, indexed by g

\({\mathbb{G}}_{1}\) :

Subset of \({\mathbb{G}}\), including the thermal units with minimum up time equal to 1

\({\mathbb{G}}_{b}\) :

Subset of \({\mathbb{G}}\), including the thermal units connecting to bus b

\({\mathbb{B}}\) :

Set of buses, indexed by b

\({\mathbb{B}}_{b}\) :

Subset of \({\mathbb{B}}\), including the buses connecting to bus b

\({\mathbb{T}}\) :

Set of time periods, indexed by t, from 1 to T

\(A_{g} ,\;B_{g} ,\;C_{g}\) :

Coefficients of production cost function of unit g

\(C_{b,\;d}\) :

Capacity limit of transmission line connecting bus b and bus d

\(C_{g}^{{{\text{su,}}\;{\text{c}}}}\) :

Cold startup cost of unit g

\(C_{g}^{{{\text{su,}}\;{\text{h}}}}\) :

Hot startup cost of unit g

\(D_{b, t}\) :

Load demand of bus b at hour t

\(P_{g}^{\max }\) :

Maximum generation of unit g

\(P_{g}^{\min }\) :

Minimum generation of unit g

\(P_{g}^{{{\text{rp, dn,}}\;\max }}\) :

Maximum ramp-down rate of unit g

\(P_{g}^{{{\text{rp, up,}}\;\max }}\) :

Maximum ramp-up rate of unit g

\(P_{g}^{{{\text{sd,}}\;\max }}\) :

Maximum shutdown capacity of unit g

\(P_{g}^{{{\text{su,}}\;\max }}\) :

Maximum startup capacity of unit g

\(R_{g}^{{{\text{dn,}}\;\max }}\) :

Maximum down spinning reserve contribution of unit g

\(R_{g}^{{{\text{up,}}\;\max }}\) :

Maximum up spinning reserve contribution of unit g

\(R_{t}^{{\text{dn, req}}}\) :

System down spinning reserve requirement at hour t

\(R_{t}^{{\text{up, req}}}\) :

System up spinning reserve requirement at hour t

\(T_{g}^{c}\) :

Cold start hours of unit g

\(T_{g}^{{{\text{dn}},\;\min }}\) :

Minimum down time of unit g

\(T_{g}^{{{\text{up}},\;\min }}\) :

Minimum up time of unit g

\(X_{b,\;d}\) :

Reactance of transmission line connecting bus b and bus d

\(c_{g,\;t}^{{{\text{pd}}}}\) :

Production cost of unit g at hour t

\(c_{g,\;t}^{{{\text{su}}}}\) :

Startup cost of unit g at hour t

\(t_{g,\;t}^{{{\text{dn}}}}\) :

Hours that unit g has been in down state until hour t

\(p_{g,\;t}\) :

Power output of unit g at hour t (non-negative continuous)

\(p_{g,\;t}^{{{\text{ab}}}}\) :

Power output above the minimum generation of unit g at hour t (non-negative continuous)

\(r_{g,\;t}^{{{\text{dn}}}}\) :

Down spinning reserve of unit g at hour t (non-negative continuous)

\(r_{g,\;t}^{{{\text{up}}}}\) :

Up spinning reserve of unit g at hour t (non-negative continuous)

\(u_{g,\;t}\) :

Committed state of unit g at hour t (binary)

\(u_{g,\;t}^{{{\text{sd}}}}\) :

Shutdown state of unit g at hour t (binary)

\(u_{g,\;t}^{{{\text{su}}}}\) :

Startup state of unit g at hour t (binary)

\(\theta_{b,\;t}\) :

Voltage angle of bus b at hour t (continuous)

References

  1. M Carrión JM Arroyo 2006 A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem IEEE Trans Power Syst 21 3 1371 1378

    Article  ADS  Google Scholar 

  2. L Wu 2011 A tighter piecewise linear approximation of quadratic cost curves for unit commitment problems IEEE Trans Power Syst 26 4 2581 2583

    Article  ADS  Google Scholar 

  3. A Viana JP Pedroso 2013 A new MILP-based approach for unit commitment in power production planning Int J Electr Power Energy Syst 44 1 997 1005

    Article  Google Scholar 

  4. SA Kazarlis AG Bakirtzis V Petridis 1996 A genetic algorithm solution to the unit commitment problem IEEE Trans Power Syst 11 1 83 92

    Article  ADS  Google Scholar 

  5. JA Muckstadt SA Koenig 1977 An application of Lagrangian relaxation to scheduling in power-generation systems Oper Res 25 3 387 403

    Article  Google Scholar 

  6. Ott AL (2010) Evolution of computing requirements in the PJM market: Past and future. In: Proceedings IEEE Power & Energy Society General Meeting, Minneapolis, MN, USA, 2010, pp 1–4

  7. Y Fu Z Li L Wu 2013 Modeling and solution of the large-scale security-constrained unit commitment IEEE Trans Power Syst 28 4 3524 3533

    Article  ADS  Google Scholar 

  8. K Doubleday JD Lara B Hodge 2022 Investigation of stochastic unit commitment to enable advanced flexibility measures for high shares of solar PV Appl Energy 321 119337

    Article  Google Scholar 

  9. Y Chen F Liu W Wei S Mei N Chang 2016 Robust unit commitment for large-scale wind generation and run-off-river hydropower CSEE J Power Energy Syst 2 4 66 75

    Article  Google Scholar 

  10. J Ostrowski MF Anjos A Vannelli 2012 Tight mixed integer linear programming formulations for the unit commitment problem IEEE Trans Power Syst 27 1 39 46

    Article  ADS  Google Scholar 

  11. G Morales-España JM Latorre A Ramos 2013 Tight and compact MILP formulation for the thermal unit commitment problem IEEE Trans Power Syst 28 4 4897 4908

    Article  ADS  Google Scholar 

  12. S Atakan G Lulli S Sen 2018 A state transition MIP formulation for the unit commitment problem IEEE Trans Power Syst 33 1 736 748

    Article  ADS  Google Scholar 

  13. L Yang C Zhang J Jian K Meng Y Xu Z Dong 2017 A novel projected two-binary-variable formulation for unit commitment in power systems Appl Energy 187 732 745

    Article  ADS  Google Scholar 

  14. PL Cavalcante 2016 Centralized self-healing scheme for electrical distribution systems IEEE Trans Smart Grid 7 1 145 155

    Article  Google Scholar 

  15. N Alguacil AL Motto AJ Conejo 2003 Transmission expansion planning: a mixed-integer LP approach IEEE Trans Power Syst 18 3 1070 1077

    Article  ADS  Google Scholar 

  16. A Frangioni C Gentile F Lacalandra 2009 Tighter approximated MILP formulations for unit commitment problems IEEE Trans Power Syst 24 1 105 113

    Article  ADS  Google Scholar 

  17. Ahmadi H, Martí JR, Moshref A (2013) Piecewise linear approximation of generators cost functions using max-affine functions. In: Proceedings IEEE Power & Energy Society General Meeting, Vancouver, BC, Canada, 2013, pp 1–5

  18. W Cheney D Kincaid 2012 Numerical Mathematics and Computing 7 Brooks/Cole Boston, MA, USA 178 187

    Google Scholar 

  19. K Eriksson C Johnson D Estep 2004 Applied Mathematics: Body and Soul New York Springer NY, USA 747 749

    Book  Google Scholar 

  20. D Berjón G Gallego C Cuevas F Morán N García 2016 Optimal piecewise linear function approximation for GPU-based applications IEEE Trans Cybern 46 11 2584 2595

    Article  PubMed  Google Scholar 

  21. DA Tejada-Arango S Lumbreras P Sánchez-Martín A Ramos 2020 Which unit-commitment formulation is best? A comparison framework IEEE Trans Power Syst 35 4 2926 2936

    Article  ADS  Google Scholar 

  22. C Chen 2008 Optimal wind-thermal generating unit commitment IEEE Trans Energy Convers 23 1 273 280

    Article  ADS  Google Scholar 

  23. Z Li W Wu J Wang B Zhang T Zheng 2016 Transmission-constrained unit commitment considering combined electricity and district heating networks IEEE Trans Sustain Energy 7 2 480 492

    Article  ADS  Google Scholar 

  24. Y Sun J Dong L Ding 2017 “Optimal day-ahead wind-thermal unit commitment considering statistical and predicted features of wind speeds”, Energy Convers Manage 142 347 356

    Google Scholar 

  25. Z Yang 2019 A binary symmetric based hybrid meta-heuristic method for solving mixed integer unit commitment problem integrating with significant plug-in electric vehicles Energy 170 889 905

    Article  Google Scholar 

  26. K Kwon D Kim 2020 Enhanced method for considering energy storage systems as ancillary service resources in stochastic unit commitment Energy 213 118675

    Article  Google Scholar 

  27. T Jiang 2021 Exploiting flexibility of combined-cycle gas turbines in power system unit commitment with natural gas transmission constraints and reserve scheduling Int J Electr Power Energy Syst 125 106460

    Article  Google Scholar 

  28. Gurobi Optimization, LLC. Gurobi optimizer. 2020. [Online]. Available: https://www.gurobi.com

  29. W Ongsakul N Petcharaks 2004 Unit commitment by enhanced adaptive Lagrangian relaxation IEEE Trans Power Syst 19 1 620 628

    Article  ADS  Google Scholar 

  30. J. Dong “Detailed results of LS paper.”, Dec. 11, 2021, [Online]. Available: https://github.com/PowerSystemGroup/UCDataAndResults/tree/data_and_results_of_paper_A_more_accurate_piecewise_linear_approximation_method_for_quadratic_cost_curves_of_thermal_generators_and_its_application_in_unit_commitment

  31. A Castillo C Laird CA Silva-Monroy JP Watson RO Neill 2016 The unit commitment problem with AC optimal power flow constraints IEEE Trans Power Syst 31 6 1 14

    Article  Google Scholar 

  32. Y Chen Q Guo H Sun Z Li W Wu Z Li 2018 A distributionally robust optimization model for unit commitment based on Kullback-Leibler divergence IEEE Trans Power Syst 33 5 5147 5160

    Article  ADS  Google Scholar 

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Funding

This work was supported by the Jilin Province Science and Technology Development Program under Grant 20220203162SF.

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Authors and Affiliations

  1. State Grid Jilin Electric Power Co., Ltd, Changchun, 130000, China

    Yong Sun

  2. School of Electrical and Electronic Engineering, Changchun University of Technology, Changchun, 130000, China

    Jizhe Dong, Ruiheng Zhang & Danchen Zheng

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  1. Yong Sun
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  2. Jizhe Dong
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  3. Ruiheng Zhang
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  4. Danchen Zheng
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Contributions

YS wrote the program and the manuscript. JD supervised the research and revised the manuscript. RZ revised some figures. DZ prepared the data.

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Correspondence to Jizhe Dong.

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Sun, Y., Dong, J., Zhang, R. et al. A more accurate piecewise linear approximation method for quadratic cost curves of thermal generators and its application in unit commitment. Electr Eng (2024). https://doi.org/10.1007/s00202-024-02254-6

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