Electric Vehicles Aggregator Optimization A Fast and Solver-Free - PowerPoint PPT Presentation

Electric Vehicles Aggregator Optimization – A Fast and Solver-Free Solution Method Robin Vujanic, Peyman Mohajerin Esfahani, Paul Goulart, Sebastien Mariethoz, Manfred Morari Institut f¨ ur Automatik (IfA) Department of Electrical Engineering Swiss Federal Institute of Technology (ETHZ) May 22, 2015 1 / 16

1 Mixed–Integer Models Considered and Proposed Solution Algorithm 2 Electric Vehicles Charging Coordination 2 / 16

Outline 1 Mixed–Integer Models Considered and Proposed Solution Algorithm 2 Electric Vehicles Charging Coordination 3 / 16

Large Scale Optimization – Problem Structure Considered • we consider the problem  � min i c i x i  � P : s.t. i H i x i ≤ b  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 4 / 16

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 5 / 16

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 • generally infeasible in the MIP case ! ◮ violate coupling constraints 6 / 16

Primal Recovery Scheme • we show that in x ( λ ⋆ ) at most m subproblems may be “problematic” ◮ Shapley–Folkman Theorem (Shapley won Nobel Prize in Eco) ◮ related to bound on duality gap [Ekeland ’76, Bertsekas ’83] • so we propose to consider  � 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 [Vujanic ’14] Then x ( λ ⋆ ) is feasible for P . [under some uniqueness assumption] 7 / 16

Performance of the Recovered Solutions • under some technical assumption, ... Theorem [Vujanic ’14] The recovered solution x (¯ λ ⋆ ) is feasible and satisfies � � J P ( x (¯ λ ⋆ )) − J ⋆ P ≤ ( m + || ρ || ∞ /ζ ) · max x i ∈ X i c i x i − min max x i ∈ X i c i x i i ∈ I • if J ⋆ P grows linearly with | I | , m is fixed and X i uniformly bounded J ( x (¯ λ ⋆ )) − J ⋆ P → 0 as | I | → ∞ J ⋆ P 8 / 16

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 9 / 16

Outline 1 Mixed–Integer Models Considered and Proposed Solution Algorithm 2 Electric Vehicles Charging Coordination 10 / 16

Electric Vehicle (EV) Charging Coordination • expected increase of EV presence • substantial additional stress on network & equipment ⇒ need charging coordination • network administrator (DSO) can’t manage each unit individually ⇒ EV aggregator 11 / 16

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] 12 / 16

Control Strategy • cast as large optimization problem � �  N − 1 � � � � P i u i [ k ] − P ref [ k ]  minimize � �   � � u i  k =0 i ∈ I   � i ∈ I cr P i u i ≤ P max  subject to    �  min i c i x i    e i [0] = E init � ↔ P : s.t. i H i x i ≤ b i e i [ k + 1] = e i [ k ] + ( P i ∆ T ζ i ) u i [ k ]   x i ∈ X i    e i [ N i ] ≥ E ref   i   e i [ k ] ≤ E max   i  u i ∈ { 0 , 1 } N ,  • with ◮ u i [ k ] ∈ { 0 , 1 } : charge decision at step k for EV i ∈ I ◮ e i [ k ]: state of charge of battery i ∈ I ◮ P i : charge rate ◮ E init , E ref , E max : initial, final required and maxium state of charge i i i ◮ ζ i : conversion losses 13 / 16

Computational Experiments • 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 ◮ no external solver needed 14 / 16

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 15 / 16

Other Example Applications • supply chains optimization – partial shipments [Vujanic ’14b] • power systems operation ◮ control of TCLs ◮ large fleet of generators • portfolio optimization for small investors • . . . 16 / 16

Questions? All technical details/proofs on arXiv http://arxiv.org/abs/1411.1973 [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 ’14] R. Vujanic, P. Mohajerin Esfahani, P. Goulart, S. Mariethoz and M. Morari, A Decomposition Method for • Large Scale MILPs, with Performance Guarantees and a Power System Application , submitted to Automatica (2014). [Vujanic ’14b] R. Vujanic, P. Goulart, M. Morari, Large Scale Mixed-Integer Optimization: a Solution Method with • Supply Chain Applications , Mediterranean Conference on Control & Automation, 2014 [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. 16 / 16