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Global Optimisation for Energy Systems Global Optimisation for Energy Systems Ksenia Bestuzheva October 2018 A thesis submitted for the degree of Doctor of Philosophy of The Australian National University © Copyright by Ksenia Bestuzheva 2018 All Rights Reserved Except where otherwise stated, this thesis is my own original work. Ksenia Bestuzheva October 2018 1 Abstract The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complex- ity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is com- plicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom. However, nonconvex problems often do not possess any provable convenient properties, and specialised methods are necessary for providing global optimality guarantees. A widely used technique is solving convex relaxations in order to find a bound on the optimal solution. Semidefinite Programming relaxations can provide good quality bounds, but they suffer from a lack of scalability. We tackle this issue by proposing an algorithm that combines decomposition and linearisation approaches. In addition to continuous non-convexities, many problems in Energy Systems model dis- crete decisions and are expressed as mixed-integer nonlinear programs (MINLPs). The for- mulation of a MINLP is of significant importance since it affects the quality of dual bounds. In this thesis we investigate algebraic characterisations of on/off constraints and develop a strengthened version of the Quadratic Convex relaxation of the Optimal Transmission Switching problem. All presented methods were implemented in mathematical modelling and optimisation frameworks PowerTools and Gravity. Contents 1 Introduction6 2 Background 12 2.1 Optimisation Basics................................ 12 2.2 Global Optimisation Methods........................... 14 2.2.1 Upper bounding methods......................... 15 2.2.2 Convex relaxations............................. 15 2.2.3 Spatial branch and bound......................... 16 2.2.4 Mixed-integer programming........................ 16 2.3 The Optimal Power Flow Problem........................ 16 2.3.1 Solution methods.............................. 17 2.3.2 Formulation................................. 17 2.4 Relaxations of OPF................................ 20 2.4.1 Semidefinite programming relaxations.................. 20 2.4.2 Quadratic Convex relaxation....................... 22 2.4.3 Linear relaxations............................. 26 2.5 Optimal Transmission Switching......................... 26 2.6 Relaxations of OTS................................. 27 2.6.1 Quadratic Convex relaxation....................... 28 2.6.2 MISOCP relaxation............................ 29 2.7 Benchmarks..................................... 30 3 Conditions for KT-Invexity 31 3.1 Introduction..................................... 31 3.2 Generalised Convexity............................... 32 3.2.1 Early generalisations............................ 32 3.2.2 Invexity................................... 33 3.3 New Conditions for Kuhn-Tucker Invexity.................... 35 3.3.1 Weak boundary-invexity.......................... 35 3.3.2 Necessary condition for KT-invexity................... 36 3.3.3 Connection between boundary and interior optimality......... 37 3.3.4 Problems with two degrees of freedom.................. 37 1 3.3.5 Local optimality of KKT points...................... 40 3.4 Pseudo-Scalar Product............................... 41 3.4.1 Reformulation of the KKT conditions.................. 42 3.5 Parametrisation of the Boundary of F ...................... 43 3.6 Splitting the Space in Two............................. 46 3.6.1 Behaviour of a concave function on a line................ 46 3.6.2 Boundary optimality on a half-plane................... 47 3.7 Kuhn-Tucker Invexity of Boundary-Invex Problems............... 52 3.7.1 Sequence of crossing points........................ 52 3.7.2 The main theorem............................. 57 3.8 Application: KT-Invexity of AC-OPF...................... 58 3.8.1 Boundary-invex AC-OPF......................... 58 3.8.2 Invexity proof for 1-line AC-OPF..................... 60 3.9 Conclusion..................................... 66 4 Semidefinite Programming Cuts 67 4.1 Introduction..................................... 67 4.2 Background..................................... 68 4.2.1 Graph-theoretic background........................ 68 4.2.2 Semidefinite programming methods.................... 68 4.2.3 Linear cut generation........................... 72 4.3 Deepest Valid Cut................................. 75 4.4 Improved Cuts for SDP Problems......................... 77 4.4.1 Tree decomposition heuristic....................... 78 4.4.2 Dynamic linear cut generation...................... 78 4.4.3 Strengthened polynomial formulation.................. 80 4.5 Application to Optimal Power Flow....................... 81 4.5.1 Bag matrix completion........................... 82 4.5.2 Bag completion propagation........................ 85 4.6 Computational Results............................... 86 4.7 Conclusion..................................... 88 5 Convex Hulls for Quadratic On/Off Constraints 89 5.1 Introduction..................................... 89 5.2 Mixed-Integer Nonlinear Programming...................... 90 5.2.1 Branch and bound............................. 91 5.3 On/Off Constraints................................. 93 5.3.1 Perspective functions............................ 94 5.3.2 Formulating the convex hull........................ 95 5.4 Convex Hull of a Nonmonotone Quadratic Constraint............. 97 5.5 Quadratic Outer Approximations of Trigonometric Functions......... 101 5.6 Application: Optimal Transmission Switching.................. 104 2 5.6.1 Asymmetrical bounds........................... 105 5.6.2 Tightening the big-M constants...................... 106 5.6.3 Relaxations of trigonometric on/off constraints............. 106 5.6.4 On/off lifted nonlinear cuts........................ 107 5.6.5 The complete strengthened QC-OTS model............... 109 5.6.6 Bound propagation............................. 111 5.7 Computational Results............................... 111 5.7.1 Bound propagation strength and performance.............. 112 5.7.2 Results on the QC-OTS models...................... 113 5.7.3 Comparison with MISOCP........................ 114 5.8 Conclusion..................................... 115 6 Conclusion 116 6.1 Main Results.................................... 116 6.2 Future Work.................................... 117 A Improved On/Off Relaxations of Nonlinear Terms 121 A.1 Sine Constraint................................... 121 A.2 Cosine Constraint................................. 123 3 Acknowledgements First of all, I would like to thank my supervisory panel. I am thankful to Dr. Hassan Hijazi for undertaking this journey with me, showing me interesting problems to work on and being an invaluable source of help on all aspects of the research process. Starting from the day when you welcomed me into the world of optimisation by placing a pile of papers in front of me (as I was sitting in the office wide-eyed, still confused by moving to another continent), continuing with the exciting discussions of the research lying ahead and sharing the hardships of model debugging and proof verifying, you have been a great mentor. You have been an example of scientific rigor and curiosity - the qualities that I will continue striving for in my further research career. I am grateful to Prof. Sylvie Thiebaux for watching over the progress of my thesis and giving me valuable strategic advice. I greatly appreciate your guidance and support; despite you being a busy person, I have always felt that I could rely on you if I found myself in any difficult situation. As a passionate scientist and energetic and wise leader, you are an inspiration for me. Many thanks also to Prof. Markus Hegland for being the mathematical sciences expert for this project, ready to help with any difficult proofs, and for inviting me to the group seminars where I could further extend my knowledge of mathematics. I enjoyed the atmosphere of genuine curiosity that you have created and encouraged at those seminars and will do my best to carry this attitude with me. I gratefully acknowledge the funding provided by Data61, CSIRO for this work. I would like to thank Dr. Carleton Coffrin for his contribution to the work on the Quadratic Convex relaxation of Optimal Transmission Switching, as well as others who were part of the optimisation team at
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