A partial outer convexification approach to control transmission lines

Abstract

In this paper we derive an efficient optimization approach to calculate optimal controls of electric transmission lines. These controls consist of time-dependent inflows and switches that temporarily disable single arcs or whole subgrids to reallocate the flow inside the system. The aim is then to find the best energy input in terms of boundary controls in combination with the optimal configuration of switches, where the dynamics is driven by a coupled system of hyperbolic differential equations. We use a well-known three-step optimization approach based on the idea of partial outer convexification, for which we establish that the analytical requirements for its application hold for each fixed spatial discretization of the underlying partial differential equation, provided that combinatorial constraints are only pointwise in time. A comparison with a direct solver yields very promising results, also for problems with from an application viewpoint important switch up-time and down-time constraints, which are not pointwise in time and thus not fully covered by theory.

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    http://www.stromnetz-berlin.de/de/stromversorger.html.

References

  1. 1.

    Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha {\rm BB}\): a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7, 337–363 (1995). (State of the art in global optimization: computational methods and applications (Princeton, NJ, 1995))

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Bahiense, L., Oliveira, G., Pereira, M., Granville, S.: A mixed integer disjunctive model for transmission network expansion. IEEE Trans. Power Syst. 16, 560–565 (2001)

    Article  Google Scholar 

  3. 3.

    Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24, 597–634 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Bergen, J., Vittal, V.: Power Systems Analysis, 2nd edn. Prentice Hall, Upper Saddle River (2000)

    Google Scholar 

  6. 6.

    Berthold, T., Gleixner, A.M.: Undercover: a primal MINLP heuristic exploring a largest sub-MIP. Math. Program. 144, 315–346 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

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

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Bonami, P., Lee, J., Leyffer, S., Wächter, A.: More Branch-and-bound Experiments in Convex Nonlinear Integer Programming. Argonne National Laboratory, Mathematics and Computer Science Division (2011). Preprint ANL/MCS-P1949-0911

  9. 9.

    Bonami, P.: Lift-and-project cuts for mixed integer convex programs. Integer Program. Comb. Optim. 6655, 52–64 (2011)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim. 5, 186–204 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Bonami, P., Cornuéjols, G., Lodi, A., Margot, F.: A feasibility pump for mixed integer nonlinear programs. Math. Program. 119, 331–352 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Bonnans, J.: Mathematical study of very high voltage power networks I. The optimal DC power flow problem. SIAM J. Optim. 7, 979–990 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Colombo, R.M., Guerra, G., Herty, M., Schleper, V.: Optimal control in networks of pipes and canals. SIAM J. Control Optim. 48, 2032–2050 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    da Silva, E., Gil, H., Areiza, J.: Transmission network expansion planning under an improved genetic algorithm. IEEE Trans. Power Syst. 15, 1168–1175 (2000)

    Article  Google Scholar 

  15. 15.

    Dakin, R.J.: A tree-search algorithm for mixed integer programming problems. Comput. J. 8, 250–255 (1965)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Donde, V., Lopez, V., Lesieutre, B., Pinar, A., Yang, C., Meza, J.: Identification of severe multiple contingencies in electric power networks. In: Proceedings 37th North American Power Symposium, LBNL-57994 (2005)

  17. 17.

    Duran, M., Grossmann, I.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Fisher, E., O’Neil, R., Ferris, M.: Optimal transmission switching. IEEE Trans. Power Syst. 23, 1346–1355 (2008)

    Article  Google Scholar 

  19. 19.

    Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Freeman, R., Karbowiak, A.: An investigation of nonlinear transmission lines and shock waves. J. Phys. D 10, 633 (1977)

    Article  Google Scholar 

  21. 21.

    Geißler, B., Martin, A., Morsi, A., Schewe, L.: Using piecewise linear functions for solving MINLPs. In: Mixed Integer Nonlinear Programming, Vol. 154 of IMA Volumes in Mathematics and its Applications, pp. 287–314. Springer, New York (2012)

  22. 22.

    Geoffrion, A.M.: Generalized Benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Göttlich, S., Teuber, C.: Space mapping techniques for the optimal inflow control of transmission lines. Optim. Methods Softw. 33, 120–139 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Göttlich, S., Ziegler, U.: Traffic light control: a case study. Discrete Contin. Dyn. Syst. Ser. S 7, 483–501 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Göttlich, S., Herty, M., Schillen, P.: Electric transmission lines: control and numerical discretization. Optim. Control Appl. Methods 37, 980–995 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Göttlich, S., Potschka, A., Ziegler, U.: Partial outer convexification for traffic light optimization in road networks. SIAM J. Sci. Comput. 39, B53–B75 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Goux, J.-P., Leyffer, S.: Solving large MINLPs on computational grids. Optim. Eng. 3, 327–346 (2002). (Special issue on mixed-integer programming and its applications to engineering)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Gross, G., Galiana, F.D.: Short-term load forecasting. Proc. IEEE 75, 1558–1573 (1987)

    Article  Google Scholar 

  29. 29.

    Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002). (Special issue on mixed-integer programming and its applications to engineering)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Hante, F.M.: Relaxation methods for hyperbolic PDE mixed-integer optimal control problems. Optim. Control Appl. Methods 38, 1103–1110 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Hante, F.M., Sager, S.: Relaxation methods for mixed-integer optimal control of partial differential equations. Comput. Optim. Appl. 55, 197–225 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Jiang, Y.-L.: Mathematical modelling on RLCG transmission lines. Nonlinear Anal. Model. Control 10, 137–149 (2005)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Jung, M.: Relaxations and Approximations for Mixed-Integer Optimal Control. Ph.D. Thesis, Heidelberg University (2013)

  34. 34.

    Jung, M.N., Reinelt, G., Sager, S.: The Lagrangian relaxation for the combinatorial integral approximation problem. Optim. Methods Softw. 30, 54–80 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Kallrath, J.: Gemischt-ganzzahlige Optimierung: Modellierung in der Praxis: Mit Fallstudien aus Chemie, Energiewirtschaft, Metallgewerbe, Produktion und Logistik. Springer, Cham (2013)

    Google Scholar 

  36. 36.

    Kirches, C., Sager, S., Bock, H.G., Schlöder, J.P.: Time-optimal control of automobile test drives with gear shifts. Optim. Control Appl. Methods 31, 137–153 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Kundur, P.: Power System Stability and Control. McGraw-Hill, New York (1994)

    Google Scholar 

  38. 38.

    Lasseter, R.: MicroGrids. IEEE Power Eng. Soc. Winter Meet. 1, 305–308 (2002)

    Article  Google Scholar 

  39. 39.

    Leyffer, S., Linderoth, J., Luedtke, J., Miller, A., Munson, T.: Applications and algorithms for mixed integer nonlinear programming. J. Phys. Conf. Ser. 180, 012014 (2009)

    Article  Google Scholar 

  40. 40.

    Linderoth, J., Savelsbergh, M.: A computational study of search strategies in mixed integer programming. INFORMS J. Comput. 11, 173–187 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Liu, C., Wang, J., Ostrowski, J.: Static switching security in multi-period transmission switching. IEEE Trans. Power Syst. 27, 1850–1858 (2012)

    Article  Google Scholar 

  42. 42.

    Misener, R., Floudas, C.A.: GloMIQO: global mixed-integer quadratic optimizer. J. Glob. Optim. 57, 3–50 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Nannicini, G., Belotti, P.: Rounding-based heuristics for nonconvex MINLPs. Math. Program. Comput. 4, 1–31 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Ostrowski, J., Wang, J.: Network reduction in the transmission-constrained unit commitment problem. Comput. Ind. Eng. 63, 702–707 (2012)

    Article  Google Scholar 

  45. 45.

    Papalexopoulos, A., Hao, S., Peng, T.: An implementation of a neural network based load forecasting model for the EMS. IEEE Trans. Power Syst. 9, 1956–1962 (1994)

    Article  Google Scholar 

  46. 46.

    Pecas Lopes, J., Moreira, C., Madureira, A.: Defining control strategies for microgrids islanded operation. IEEE Trans. Power Syst. 21, 916–924 (2006)

    Article  Google Scholar 

  47. 47.

    Romero, R., Monticelli, A.: A hierarchical decomposition approach for transmission network expansion planning. IEEE Trans. Power Syst. 9, 373–380 (1994)

    Article  Google Scholar 

  48. 48.

    Sager, S.: Numerical Methods for Mixed-integer Optimal Control Problems. Der andere Verlag Tönning, Lübeck (2005)

    Google Scholar 

  49. 49.

    Sager, S.: Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control. J. Process Control 19, 1238–1247 (2009)

    Article  Google Scholar 

  50. 50.

    Sager, S., Bock, H.G., Reinelt, G.: Direct methods with maximal lower bound for mixed-integer optimal control problems. Math. Program. 118, 109–149 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Sager, S., Jung, M., Kirches, C.: Combinatorial integral approximation. Math. Methods Oper. Res. 73, 363–380 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Sager, S., Bock, H.G., Diehl, M.: The integer approximation error in mixed-integer optimal control. Math. Program. 133, 1–23 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Savelsbergh, M.W.P.: Preprocessing and probing techniques for mixed integer programming problems. ORSA J. Comput. 6, 445–454 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Smith, E.M.B., Pantelides, C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 23, 457–478 (1999)

    Article  Google Scholar 

  55. 55.

    Soares, A., Gomes, A., Henggeler-Antunes, C., Cardoso, H.: Domestic Load Scheduling Using Genetic Algorithms. Lecture Notes in Computer Science, vol. 7835, pp. 142–151. Springer, Berlin (2013)

    Google Scholar 

  56. 56.

    Stubbs, R., Mehrotra, S.: A branch-and-cut method for 0–1 mixed convex programming. Math. Program. 86, 515–532 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Van Roy, T.J.: Cross decomposition for mixed integer programming. Math. Program. 25, 46–63 (1983)

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Vielma, J.P., Nemhauser, G.L.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Program. 128, 49–72 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Vielma, J.P., Ahmed, S., Nemhauser, G.: Mixed-integer models for nonseparable piecewise-linear optimization: unifying framework and extensions. Oper. Res. 58, 303–315 (2010)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

SG is supported by the BMBF Project ENets (05M18VMA). AP gratefully acknowledges support by the European Research Council within the ERC Advanced Grant MOBOCON (291 458) and by the German Federal Ministry for Education and Research under Grants 05M2016-MOPhaPro and 05M2018-MOReNet.

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Correspondence to S. Göttlich.

A Details of the Proof of Lemma 3.3

A Details of the Proof of Lemma 3.3

To ensure a clearer presentation of the proof of Lemma 3.3, we provide a detailed derivation for (27) in the following:

$$\begin{aligned}&\Vert [\Phi (t_e,\xi ) - \Phi (t_e,\eta )]\hat{\alpha }\Vert _X\\&=\sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \left( \frac{\lambda _r^+}{\Delta x} (\Theta ^+_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^+(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^+_r(x_d,t_e) + \eta ^+_r(x_d,t_e))\right. \right. \\&\left. \left. \qquad -\,b^r_{11} \xi ^+_r(x_d,t_e) + b^r_{11} \eta _r^+(x_d,t_e) - b^r_{12} \xi ^-_r(x_d,t_e) + b^r_{12} \eta ^-_r(x_d,t_e)\right) \hat{\alpha }_c \right| \Delta x\\&\qquad +\,\sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \left( \frac{\lambda _r^+}{\Delta x} (\Theta ^-_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^-(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^-_r(x_d,t_e) + \eta ^-_r(x_d,t_e))\right. \right. \\&\left. \left. \qquad -\,b^r_{12} \xi ^+_r(x_d,t_e) + b^r_{12} \eta _r^+(x_d,t_e) - b^r_{11} \xi ^-_r(x_d,t_e) + b^r_{11} \eta ^-_r(x_d,t_e)\right) \hat{\alpha }_c \right| \Delta x \end{aligned}$$

Due to \(\hat{\alpha }_c \le 1\), we get

$$\begin{aligned}&\le \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \Big (\Theta ^+_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^+(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^+_r(x_d,t_e) + \eta ^+_r(x_d,t_e)\Big ) \right| \Delta x\\&\qquad +\, \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \Theta ^-_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^-(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^-_r(x_d,t_e) + \eta ^-_r(x_d,t_e)\right) \right| \Delta x\\&\qquad +\, \left( \frac{\lambda ^+}{\Delta x}+ b_{11} + b_{12}\right) n_{oc} \Vert \eta - \xi \Vert _X \\&\le \left( 2 \frac{\lambda ^+}{\Delta x} + b_{11} + b_{12}\right) n_{oc} \Vert \eta -\xi \Vert _X =: L_{oc} \Vert \eta -\xi \Vert _X, \end{aligned}$$

where we exploit that

$$\begin{aligned}&\sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \Big (\Theta ^+_c(\xi _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \Theta _c^+(\eta _r(\cdot ,t_e),x_d,\varvec{u}(t_e)) - \xi ^+_r(x_d,t_e) + \eta ^+_r(x_d,t_e)\Big ) \right| \Delta x\\&= \sum _{r \in A} \sum _{d=1}^{\tilde{l}_r} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \Big ( \xi ^+_r(x_{d-1},t_e) - \eta ^+_r(x_{d-1},t_e) \Big ) \right| \Delta x\\&\quad +\, \sum _{r \in A \setminus A_Q} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \sum _{k \in \delta ^-_{\alpha (r)}} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \xi ^+_k\left( x_{\tilde{l}_k},t_e\right) - \sum _{k \in \delta ^-_{\alpha (r)}} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \eta ^+_k\left( x_{\tilde{l}_k},t_e\right) \right) \right| \Delta x\\&\quad +\, \sum _{r \in A_Q} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \underbrace{u_r(t_e) - u_r(t_e)}_{=0} \right) \right| \Delta x\\&= \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r-1} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right) \right| \Delta x\\&\quad +\, \sum _{r \in A } \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \sum _{k \in A} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \xi ^+_k(x_{\tilde{l}_k},t_e) - \sum _{k \in A} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \eta ^+_k(x_{\tilde{l}_k},t_e) \right) \right| \Delta x\\&\le \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r-1} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right) \right| \Delta x\\&\quad +\,\sum _{r \in A } \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \sum _{k \in A} ~^c\!d^{+}_{rk} \frac{\lambda _k^+}{\lambda _r^+} \left| \xi ^+_k(x_{\tilde{l}_k},t_e) - \eta ^+_k(x_{\tilde{l}_k},t_e) \right| \right) \Delta x\\&= \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r-1} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right) \right| \Delta x\\&\quad +\, \sum _{k \in A } \sum _{c \in \Omega } \underbrace{\sum _{r \in A} ~^c\!d^{+}_{rk}}_{=1} \frac{\lambda _r^+}{\Delta x} \frac{\lambda _k^+}{\lambda _r^+} \left| \xi ^+_k(x_{\tilde{l}_k},t_e) - \eta ^+_k\left( x_{\tilde{l}_k},t_e\right) \right| \Delta x\\&= \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r-1} \left| \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left( \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right) \right| \Delta x\\&\quad +\, \sum _{k \in A } \sum _{c \in \Omega } \frac{\lambda _k^+}{\Delta x} \left| \xi ^+_k\left( x_{\tilde{l}_k},t_e\right) - \eta ^+_k\left( x_{\tilde{l}_k},t_e\right) \right| \Delta x\\&\le \sum _{r \in A} \sum _{d=0}^{\tilde{l}_r} \sum _{c \in \Omega } \frac{\lambda _r^+}{\Delta x} \left| \xi ^+_r(x_d,t_e) - \eta ^+_r(x_d,t_e) \right| \Delta x \end{aligned}$$

The same computations can be done for the remaining summand.

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Göttlich, S., Potschka, A. & Teuber, C. A partial outer convexification approach to control transmission lines. Comput Optim Appl 72, 431–456 (2019). https://doi.org/10.1007/s10589-018-0047-6

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Keywords

  • Transmission lines
  • Optimization
  • Outer convexification

Mathematics Subject Classification

  • 35L65
  • 49J20
  • 90C11
  • 90C35