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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

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

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Outline

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

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Large Scale Optimization – Problem Structure Considered

• we consider the problem

P :    min

• i cixi

s.t.

• i Hixi ≤ b

xi ∈ Xi i = 1, . . . , I where Xi = {xi ∈ Rri × Zzi | Aixi ≤ di}

• large collection of subsystems

◮ subsystem model is Xi (mixed–integer) ◮ coupled by shared resources → coupling constraints

i Hixi ≤ b

• # of subsystems |I| ≫ # of coupling constraints m

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Problem’s Decomposition

Obtain decomposition using duality: minx

• i∈I cixi

s.t.

• i∈I Hixi ≤ b

xi ∈ Xi ⇒ minx

• i∈I cixi + λ′(

i∈I Hixi − b)

s.t. xi ∈ Xi ⇒

• i∈I

min

xi∈Xi

• cixi + λ′(Hixi)
• − λ′b
• .

=d(λ)

• Lagrangian dual (or outer) problem:

D : max d(λ) s.t. λ ≥ 0

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Solutions to the inner problem

d(λ) =

• i∈I

min

xi∈Xi

• cixi + λ′(Hixi)
• − λ′b

Consider xi(λ⋆) ∈ arg min

xi∈Xi

• cixi + λ⋆(Hixi)
• 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

P :    minx

• i∈I cixi

s.t.

• i∈I Hixi ≤ b − ρ

xi ∈ Xi ∀i ∈ I, where ρ = m · max

i∈I

• max

xi∈XiHixi − min xi∈XiHixi

• Theorem [Vujanic ’14]

Then x(λ⋆) is feasible for P. [under some uniqueness assumption]

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Performance of the Recovered Solutions

• under some technical assumption, ...

Theorem [Vujanic ’14]

The recovered solution x(¯ λ⋆) is feasible and satisfies JP(x(¯ λ⋆)) − J⋆

P ≤ (m + ||ρ||∞/ζ) · max i∈I

• max

xi∈Xi cixi − min xi∈Xi cixi

• if J⋆

P grows linearly with |I|, m is fixed and Xi uniformly bounded

J(x(¯ λ⋆)) − J⋆

P

J⋆

P

→ 0 as |I| → ∞

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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

A B C D E F 1 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 14 20 21 22 23 25 26 27 24 28 Subsystems A B C D E F 7 13 14 19 23 20 24 28 − − − − k-th row [Hi]i∈Ik 1 6 − A1 H1 H2 HI A2 AI . . . . . . . . . . . . . . . . . .

• can safely use rank([Hi]i∈Ik) instead of m
• generally possible to use rank(H) instead of m

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Outline

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

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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

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Aggregator’s Role

• aggregator’s control task is to

assign to each EV the time slots when charging can occur

• compatibly with...
• local requirements

◮ required final State of Charge ◮ fixed charge rates ◮ battery capacity limits

• global objectives

◮ network congestion avoidance

(limits set by DSO)

◮ “valley fill”, cost min., . . . 12 / 16

from [Lopes ’11] 0:00 1:30 3:00 4:30 6:00 7:30 10 20 State of Charge (kWh) SoC desired final SoC 0:00 1:30 3:00 4:30 6:00 7:30 0.5 1 Charge Control Time of the Day

Control Strategy

• cast as large optimization problem

                         minimize

ui N−1

• k=0
• i∈I

Piui[k] − Pref[k]

• subject to
• i∈I cr Piui ≤ Pmax

ei[0] = E init

i

ei[k + 1] = ei[k] + (Pi∆Tζi)ui[k] ei[Ni] ≥ E ref

i

ei[k] ≤ E max

i

ui ∈ {0, 1}N , ↔ P :    min

• i cixi

s.t.

• i Hixi ≤ b

xi ∈ Xi

• with

◮ ui[k] ∈ {0, 1}: charge decision at step k for EV i ∈ I ◮ ei[k]: state of charge of battery i ∈ I ◮ Pi: charge rate ◮ E init

i

, E ref

i

, E max

i

: initial, final required and maxium state of charge

◮ ζi: 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

0:00 1:30 3:00 4:30 6:00 7:30 2 4 6 8 10 Time of the Day Power Profiles (MW) reference iteration #80 final iteration

(a) reference tracking

14:00 17:00 20:00 23:00 2:00 5:00 8:00 11:00 10 20 30 40 50 60 70 Time of the Day Total Power Flow (MW) base load base + EVs load

(b) resulting “valley fill”

0:00 1:30 3:00 4:30 6:00 7:30 0.5 1 1.5 2 2.5 3 Time of the Day Branch Power Flow (MW) line capacity contracted line cap. flow iteration #80 final iteration

(c) network limits

0:00 1:30 3:00 4:30 6:00 7:30 10 20 State of Charge (kWh) SoC desired final SoC 0:00 1:30 3:00 4:30 6:00 7:30 −1 1 Charge Control Time of the Day

(d) local requirements

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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
• . . .

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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