Convex Relaxations in Mixed-Integer Optimization – Methods and Control Applications Robin Vujanic Institut f¨ ur Automatik (IfA) Department of Electrical Engineering Swiss Federal Institute of Technology (ETHZ) August 20, 2014 1 / 19
Outline 1 Introduction 2 Optimization of Large Scale Systems 3 Example: Electric Vehicles Charging Coordination 2 / 19
What are Mixed–Integer Optimization Problems / Why do we look into them • many practical and industrial systems entail continuous quantities ◮ physical measurements of voltages ◮ concentrations ◮ and positions in space as well as discrete components ◮ on/off decisions ◮ switches ◮ and logic reasoning (if, or, . . . ) • when the associated control/operation tasks are tackled with optimization, M ixed- I nteger Optimization P roblems (MIPs) arise • but the additional flexibility has a price ... 3 / 19
Computational Issues – a Practical Perspective • experience with instances from supply chain problem • modest size, yet memory blew up # cust. # prod. CPLEX time (sec) 100 25 Min: 10.3 Avg: 63.5 Max: 202.5 300 75 out of memory • computational requirements depend not only on structure and size of the problem, but also the data ◮ bad in a control context The thesis focuses on particular model structures that are of practical interest . For these, we derived computationally attractive approximation schemes , equipped with guarantees . 4 / 19
Thesis Content Part I – Optimization of Large Scale Systems. • Lagrangian Duality • [EEM-13] • Applications: • [MED-14] ◮ Power Systems • [CDC-14] ◮ Supply Chain Optimization • [Submitted, Math. Prog.-13] Part II – Robust Optimization of Uncertain Systems. • [ACC-12] • Linear Programming Relaxations • Applications: • [ECC-13] ◮ Scheduling under Uncertainty • [CDC-13] ◮ PWM Systems 5 / 19
Thesis Content Part I – Optimization of Large Scale Systems. • Lagrangian Duality • [EEM-13] • Applications: • [MED-14] ◮ Power Systems • [CDC-14] ◮ Supply Chain Optimization • [Submitted, Math. Prog.-13] Part II – Robust Optimization of Uncertain Systems. • [ACC-12] • Linear Programming Relaxations • Applications: • [ECC-13] ◮ Scheduling under Uncertainty • [CDC-13] ◮ PWM Systems 5 / 19
Outline 1 Introduction 2 Optimization of Large Scale Systems 3 Example: Electric Vehicles Charging Coordination 6 / 19
Large Scale Optimization – Problem Structure Considered • we consider the problem Rest of the Network aggregated flow � min i c i x i � Substation P : s.t. i H i x i ≤ b critical . . . branch flow x i ∈ X i i = 1 , . . . , I where . . . . . . . . . . . . X i = { x i ∈ R r i × Z z i | A i x i ≤ d i } . . . • large collection of subsystems ◮ subsystem model is X i (mixed–integer) ◮ coupled by shared resources → coupling constraints � i H i x i ≤ b • # of subsystems I ≫ # of coupling constraints m 7 / 19
Problem’s Decomposition Obtain decomposition using duality: � min x i ∈ I c i x i � i ∈ I c i x i + λ ′ ( � min x i ∈ I H i x i − b ) � s.t. i ∈ I H i x i ≤ b ⇒ s.t. x i ∈ X i x i ∈ X i � � � c i x i + λ ′ ( H i x i ) min − λ ′ b ⇒ x i ∈ X i i ∈ I � �� � . = d ( λ ) • Lagrangian dual (or outer) problem: � max d ( λ ) D : s.t. λ ≥ 0 8 / 19
Solutions to the inner problem � � � c i x i + λ ′ ( H i x i ) − λ ′ b d ( λ ) = min x i ∈ X i i ∈ I Consider � � x i ( λ ⋆ ) ∈ arg min c i x i + λ ⋆ ( H i x i ) x i ∈ X i as candidate solution to P Properties of x i ( λ ⋆ ): • satisfy X i constraints • obtained “for free” as by-product of methods that solve D • distributed computations • generally infeasible in the MIP case ! ◮ violate coupling constraints 9 / 19
Primal Recovery Scheme • we show that in x ( λ ⋆ ) only m subsystems may be “problematic” ◮ technique based on Shapley–Folkman–Starr theorem [CDC ’14] ◮ or simplex tableaux argument [MED ’13] ◮ these arguments are used to show bounded duality gap [Ekeland ’76, Bertsekas ’83] • so we propose to consider instead � min x i ∈ I c i x i � P : s.t. i ∈ I H i x i ≤ b − ρ x i ∈ X i ∀ i ∈ I , where � � ρ = m · max x i ∈ X i H i x i − min max x i ∈ X i H i x i i ∈ I Theorem Then x (¯ λ ⋆ ) is feasible for P . [under some uniqueness assumptions] 10 / 19
Performance of the Recovered Solutions • under some technical assumption, ... Theorem The recovered solution x (¯ λ ⋆ ) is feasible and satisfies � � J P ( x (¯ λ ⋆ )) − J ⋆ P ≤ ( m + || ρ || ∞ /ζ ) · x i ∈ X i c i x i − min max x i ∈ X i c i x i • if J ⋆ P grows linearly with | I | , and X i uniformly bounded J ( x (¯ λ ⋆ )) − J ⋆ P → 0 as | I | → ∞ J ⋆ P 11 / 19
Improving Approximations – Conservatism Reduction ρ scales with m but not with I – want to keep it as small as possible • when couplings are determined by certain network topologies Subsystems − − − − − 1 6 7 13 14 19 20 23 24 28 A . . . . . . . . . H 1 H 2 H I A A 1 D B 27 B 26 A 2 28 2 1 C 25 . . . 23 C 3 24 . . . D 22 6 7 13 E 4 21 . . . k -th row E 8 5 12 14 F 20 15 A I F 9 10 11 19 16 17 18 [ H i ] i ∈ I k • can safely use rank([ H i ] i ∈ I k ) instead of m • generally possible to use rank( H ) instead of m 12 / 19
Outline 1 Introduction 2 Optimization of Large Scale Systems 3 Example: Electric Vehicles Charging Coordination 13 / 19
Electric Vehicle (EV) Charging Coordination [CDC ’14] • expected increase of EV presence • substantial additional stress on network & equipment ⇒ need charging coordination • network administrator (DSO) can’t tackle each single unit ⇒ EV aggregator 14 / 19
Aggregator’s Role State of Charge (kWh) 20 SoC desired final SoC 10 • aggregator’s control task is to assign to each EV the time slots 0 0:00 1:30 3:00 4:30 6:00 7:30 when charging can occur Charge Control 1 • compatibly with... 0.5 0 • local requirements 0:00 1:30 3:00 4:30 6:00 7:30 ◮ required final State of Charge Time of the Day ◮ fixed charge rates ◮ battery capacity limits • global objectives ◮ network congestion avoidance (limits set by DSO) ◮ “valley fill”, cost min., . . . from [Lopes ’11] 15 / 19
Computational Experiments • cast as large optimization problem • solve using proposed method: duality+contraction ◮ support extensions (e.g., vehicle–to–grid “V2G”) • population up to 10’000 EVs • computation times ≤ 10 sec (charge only) ◮ greedy subproblem structure 16 / 19
Solutions – Charge and V2G 70 10 base load reference Total Power Flow (MW) 60 base + EVs load iteration #80 8 Power Pro fi les (MW) final iteration 50 6 40 4 30 2 20 10 0 14:00 17:00 20:00 23:00 2:00 5:00 8:00 11:00 0:00 1:30 3:00 4:30 6:00 7:30 Time of the Day Time of the Day (a) reference tracking (b) resulting “valley fill” State of Charge (kWh) 20 SoC 3 desired final SoC line capacity 10 Branch Power Flow (MW) contracted line cap. 2.5 flow iteration #80 final iteration 2 0 0:00 1:30 3:00 4:30 6:00 7:30 1.5 1 Charge Control 1 0.5 0 0 0:00 1:30 3:00 4:30 6:00 7:30 −1 Time of the Day 0:00 1:30 3:00 4:30 6:00 7:30 Time of the Day (c) network limits (d) local requirements 17 / 19
Other Examples or Applications • supply chains optimization – partial shipments [MED ’14] • power systems operation ◮ control of TCLs ◮ large fleet of generators • portfolio optimization for small investors • . . . 18 / 19
Questions? • Prof. Manfred Morari • Prof. Debasish Chatterjee • Dr. Paul Goulart • Prof. Federico Ramponi • Dr. S´ ebastien Mari´ ethoz • Dr. Peter Hokayem • Dr. Peyman Mohajerin • Dr. Apostolos Fertis Esfahani • Dr. Utz-Uwe Haus • Dr. Joe Warrington • Dr. Alexander Fuchs • Dr. Stefan Richter • Dr. Martin Herceg • Dr. Sumit Mitra • Robert Nguyen • Gregory Ledva • Isik Ilber Sirmatel • Marius Schmitt 19 / 19
References • [Ekeland ’76]J. P. Aubin and I. Ekeland, Estimates of the duality gap in nonconvex optimization , Mathematics of Operations Research 1 (1976), no. 3, 225-–245. • [Bertsekas ’83] Dimitri P. Bertsekas, G. Lauer, N. Sandell, and T. Posbergh, Optimal short-term scheduling of large-scale power systems , IEEE Transactions on Automatic Control 28 (1983), no. 1, 1– 11. • [Dawande ’06] Milind Dawande, Srinagesh Gavirneni, and Sridhar Tayur, Effective heuristics for multiproduct partial shipment models , Operations Research 54 (2006), no. 2, 337–352 (en). • [Vujanic ’13] R. Vujanic, P. Mohajerin Esfahani, P. Goulart, S. Mariethoz and M. Morari, Vanishing Duality Gap in Large Scale Mixed-Integer Optimization: a Solution Method with Power System Applications , submitted to Mathematical Programming (2013). • [Lopes ’11] J. Lopes, F. Soares, and P. Almeida, Integration of Electric Vehicles in the Electric Power System , Proceedings of the IEEE, 2011, p.168-183. 19 / 19
BACKUP SLIDES 19 / 19
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