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ENE 2XX: Renewable Energy Systems and Control
LEC 04 : Distributed Optimization of DERs
Professor Scott MouraUniversity of California, Berkeley
Summer 2018
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 1
Distributed vs. Decentralized: What are they?
Distributed
Decentralized
Decentralized
FullyDecentralized
Community
Optimization/Control:
PowerSystems:
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 2
Distributed vs. Decentralized: What are they?
Distributed
Decentralized
Decentralized
FullyDecentralized
Community
Optimization/Control:
PowerSystems:
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 2
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 3
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 3
Source: C. Vlahoplus, G. Litra, P. Quinlan, C. Becker, “Revising the California Duck Curve: An
Exploration of Its Existence, Impact, and Migration Potential,” Scott Madden, Inc., Oct 2016.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 3
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 3
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 4
PEV Energy Storage: How much, when, and where?
A. Langton and N. Crisostomo, “Vehicle-grid integration: A vision for zero-emission transportation interconnected throughout Californias electricity
system,” California Public Utilities Commission, Tech. Rep. R. 13-11-XXX, 2013.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 5
Problem Statement
Goal: Schedule DERs (e.g. PEVs, ESS, TCLs) to flatten California duck curveChallenge: N = 103,106, or 109 DERs to schedule every time slot!!!
minimizeP∈RTh×N
Th∑t=1
(Dt +
N∑n=1
Ptn
)2
subject to Ptn ≤ utn ≤ Ptn, ∀n, ∀t
A Quadratic Program (QP)
Th × N optimization variables
2T × N linear inequality constraints
Enabling Innovation: Use duality theory!!!
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 6
Problem Statement
Goal: Schedule DERs (e.g. PEVs, ESS, TCLs) to flatten California duck curveChallenge: N = 103,106, or 109 DERs to schedule every time slot!!!
minimizeP∈RTh×N
Th∑t=1
(Dt +
N∑n=1
Ptn
)2
subject to Ptn ≤ utn ≤ Ptn, ∀n, ∀t
A Quadratic Program (QP) 100K EVs*, 24 hrs
Th × N optimization variables 2.4M
2Th × N linear inequality constraints 4.8M
*cumulative PEVs sold in CA by mid-2014
Enabling Innovation: Use duality theory!!!
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 6
Problem Statement
Goal: Schedule DERs (e.g. PEVs, ESS, TCLs) to flatten California duck curveChallenge: N = 103,106, or 109 DERs to schedule every time slot!!!
minimizeP∈RTh×N
Th∑t=1
(Dt +
N∑n=1
Ptn
)2
subject to Ptn ≤ utn ≤ Ptn, ∀n, ∀t
A Quadratic Program (QP) 1.5M EVs*, 24 hrs
T × N optimization variables 32M
2T × N linear inequality constraints 64M
*California Gov. Brown 2025 ZEV Goal
Enabling Innovation: Use duality theory!!!
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 6
Problem Statement
Goal: Schedule DERs (e.g. PEVs, ESS, TCLs) to flatten California duck curveChallenge: N = 103,106, or 109 DERs to schedule every time slot!!!
minimizeP∈RTh×N
Th∑t=1
(Dt +
N∑n=1
Ptn
)2
subject to Ptn ≤ utn ≤ Ptn, ∀n, ∀t
A Quadratic Program (QP) 5M EVs*, 24 hrs
T × N optimization variables 120M
2T × N linear inequality constraints 240M
*China’s 2025 EV Goal
Enabling Innovation: Use duality theory!!!
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 6
Problem Statement
Goal: Schedule DERs (e.g. PEVs, ESS, TCLs) to flatten California duck curveChallenge: N = 103,106, or 109 DERs to schedule every time slot!!!
minimizeP∈RTh×N
Th∑t=1
(Dt +
N∑n=1
Ptn
)2
subject to Ptn ≤ utn ≤ Ptn, ∀n, ∀t
A Quadratic Program (QP) 5M EVs*, 24 hrs
T × N optimization variables 120M
2T × N linear inequality constraints 240M
*China’s 2025 EV Goal
Enabling Innovation: Use duality theory!!!
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 6
Optimal PEV Aggregation
minimizeP∈RTh×N
Th∑t=1
(Dt +
N∑n=1
Ptn
)2
+σN∑
n=1
‖Pn‖2
subject to: Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
Plug (zt)? = 12ρ
t into dual problem
maxρ∈RTh
minP∈RTh×N
Th∑t=1
1
4
(ρt)2 − ρt
[1
2ρt − Dt −
N∑n
Ptn
]+ σ
N∑n=1
‖Pn‖2
subject to: Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
The Pn terms decouple along n, yielding N parallel subproblems:
maxρ∈RTh
−1
4‖ρ‖2 + DTρ +
N∑n=1
[min
P∈RTh×NρTPn + σ‖Pn‖2
]subject to: Ptn ≤ Ptn ≤ P
tn, ∀n, ∀t
Each PEV optimizes her own schedule, given ρt from aggregatorParallelized N = 1.5M problemsConstraints remain private
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7
Optimal PEV Aggregation
Define “consensus variable”: zt = Dt +∑N
n Ptn
minimizeP∈RTh×N,z∈RTh
Th∑t=1
(zt)2
+ σ
N∑n=1
‖Pn‖2
subject to: zt = Dt +N∑n
Ptn, ∀t
Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
Plug (zt)? = 12ρ
t into dual problem
maxρ∈RTh
minP∈RTh×N
Th∑t=1
1
4
(ρt)2 − ρt
[1
2ρt − Dt −
N∑n
Ptn
]+ σ
N∑n=1
‖Pn‖2
subject to: Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
The Pn terms decouple along n, yielding N parallel subproblems:
maxρ∈RTh
−1
4‖ρ‖2 + DTρ +
N∑n=1
[min
P∈RTh×NρTPn + σ‖Pn‖2
]subject to: Ptn ≤ Ptn ≤ P
tn, ∀n, ∀t
Each PEV optimizes her own schedule, given ρt from aggregatorParallelized N = 1.5M problemsConstraints remain private
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7
Optimal PEV Aggregation
Strong duality holds. Define dual problem:
maxλ∈RTh
minP∈RTh×N,z∈RTh
Th∑t=1
(zt)2
+λt
[zt − Dt −
N∑n
Ptn
]+ σ
N∑n=1
‖Pn‖2
subject to: Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
Plug (zt)? = 12ρ
t into dual problem
maxρ∈RTh
minP∈RTh×N
Th∑t=1
1
4
(ρt)2 − ρt
[1
2ρt − Dt −
N∑n
Ptn
]+ σ
N∑n=1
‖Pn‖2
subject to: Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
The Pn terms decouple along n, yielding N parallel subproblems:
maxρ∈RTh
−1
4‖ρ‖2 + DTρ +
N∑n=1
[min
P∈RTh×NρTPn + σ‖Pn‖2
]subject to: Ptn ≤ Ptn ≤ P
tn, ∀n, ∀t
Each PEV optimizes her own schedule, given ρt from aggregatorParallelized N = 1.5M problemsConstraints remain private
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7
Optimal PEV Aggregation
Strong duality holds. Define dual problem:
maxλ∈RTh
minP∈RTh×N,z∈RTh
Th∑t=1
(zt)2
+λt
[zt − Dt −
N∑n
Ptn
]+ σ
N∑n=1
‖Pn‖2
subject to: Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
minimize w.r.t. z
f t(zt) = (zt)2 + λtzt,
df t
dzt= 2zt + λt = 0, ⇒ (zt)? = −1
2λt
For convenience, define ρt = −λt. Plug (zt)? = 12ρ
t into dual problem
Plug (zt)? = 12ρ
t into dual problem
maxρ∈RTh
minP∈RTh×N
Th∑t=1
1
4
(ρt)2 − ρt
[1
2ρt − Dt −
N∑n
Ptn
]+ σ
N∑n=1
‖Pn‖2
subject to: Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
The Pn terms decouple along n, yielding N parallel subproblems:
maxρ∈RTh
−1
4‖ρ‖2 + DTρ +
N∑n=1
[min
P∈RTh×NρTPn + σ‖Pn‖2
]subject to: Ptn ≤ Ptn ≤ P
tn, ∀n, ∀t
Each PEV optimizes her own schedule, given ρt from aggregatorParallelized N = 1.5M problemsConstraints remain private
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7
Optimal PEV Aggregation
Plug (zt)? = 12ρ
t into dual problem
maxρ∈RTh
minP∈RTh×N
Th∑t=1
1
4
(ρt)2 − ρt
[1
2ρt − Dt −
N∑n
Ptn
]+ σ
N∑n=1
‖Pn‖2
subject to: Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
The Pn terms decouple along n, yielding N parallel subproblems:
maxρ∈RTh
−1
4‖ρ‖2 + DTρ +
N∑n=1
[min
P∈RTh×NρTPn + σ‖Pn‖2
]subject to: Ptn ≤ Ptn ≤ P
tn, ∀n, ∀t
Each PEV optimizes her own schedule, given ρt from aggregator
Parallelized N = 1.5M problems
Constraints remain private
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7
Optimal PEV Aggregation
Plug (zt)? = 12ρ
t into dual problem
maxρ∈RTh
minP∈RTh×N
Th∑t=1
1
4
(ρt)2 − ρt
[1
2ρt − Dt −
N∑n
Ptn
]+ σ
N∑n=1
‖Pn‖2
subject to: Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
The Pn terms decouple along n, yielding N parallel subproblems:
maxρ∈RTh
−1
4‖ρ‖2 + DTρ +
N∑n=1
[min
P∈RTh×NρTPn + σ‖Pn‖2
]subject to: Ptn ≤ Ptn ≤ P
tn, ∀n, ∀t
Each PEV optimizes her own schedule, given ρt from aggregator
Parallelized N = 1.5M problems
Constraints remain private
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7
Provable Convergence w/ Bounds
Define g(ρ) = −1
4‖ρ‖2 + DTρ +
N∑n=1
[min
P∈RTh×NρTPn + σ‖Pn‖2
]s. to Ptn ≤ Ptn ≤ P
tn, ∀n, ∀t
Theorem: Linear Convergence RateThe dual problem has a unique solution ρ?, and the gradient ascentalgorithm with step-size α = −2σ/(N + σ) converges linearly according to
g(ρ?)− g(ρk) ≤[
N
N + σ
]k(g(ρ?)− g(ρ0))
Similar theorems for
Incremental stochastic gradient method (constant step-size)
Incremental stochastic gradient method (decreasing step-size)
Incorporate uncertainty in Dt and PEV availability (SOCP)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 8
Optimal DER Aggregation
maxρ∈RTh
−1
4‖ρ‖2 + DTρ +
N∑n
minP∈RTh×N
ρTPn + σ
N∑n=1
‖Pn‖2
s. to Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
ρt is time-varying price incentive uniformly provided to each DER.Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 9
Distributed Algorithm
maxρ∈RTh
−1
4‖ρ‖2 + DTρ +
N∑n
minP∈RTh×N
ρTPn + σ
N∑n=1
‖Pn‖2
s. to Ptn ≤ Ptn ≤ Ptn, ∀n, ∀t
Algorithm 1 Gradient Ascent (constant step size)
Initialize ρ = ρ0; Choose α = −2σ/(N + σ)for k = 1, · · · , kmax
(1) Inner Optimization: Optimize charge schedule for each PEV nfor n = 0,1, · · · ,N...Solve, Pkn = arg minPtn≤Ptn≤P
tn
(ρk)TPn + σ∑N
n=1 ‖Pn‖2
end for(2) Outer Optimization: Update dual variable ρ...ρk+1 = ρk + α · ∇g(ρk)
...ρk+1 = ρk + α[−1
2ρk + D +
∑Nn=1 P
kn
]end for
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 10
CENTRALIZED
Central Controller
PEV PEV
Personal Charge
Schedule
Mobility Data Vehicle Data EVSE Data
Social Coordinator
PEV PEV
Uniform Incen=ve
Charge Schedule
Self-‐Op=mize
+ Global optimality+ Complete controllability
- Communication infrastructure- Privacy concerns- Scalability- Modularity
DISTRIBUTED
Central Controller
PEV PEV
Personal Charge
Schedule
Mobility Data Vehicle Data EVSE Data
Social Coordinator
PEV PEV
Uniform Incen=ve
Charge Schedule
Self-‐Op=mize
+ Communication light+ Privacy preserving+ Modular+ Scalable
- Lacks global optimality- Analysis more difficult
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 11
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 12
C. Le Floch, F. Belletti, S. J. Moura, “Optimal Charging of Electric Vehicles for Load Shaping: a Dual SplittingFramework with Explicit Convergence Bounds,” IEEE Transactions on Vehicle Electrification, vol. 2, no. 2, pp.190-199, June 2016. DOI: 10.1109/TTE.2016.2531025.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 12
Optimal PEV Aggregation
Goal: Aggregate N PEVs into “Virtual Power Plant,” sell in Day Ahead Market
Challenges: (i) N = 106 EVs; (ii) assets stochastic; (iii) privacy
Local System i = 1,2, · · ·N
Gi : Power imported from grid[kW]
Si : Power gen. from solar [kW]
EVi : Power to charge EV [kW]
Li : Power of loads [kW]
Day Ahead Market
Clearing price p ∈ R24 is stochastic.
p = E(p)
Σ = Cov(p) ∈ R24×24
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 13
Optimal PEV Aggregation
Goal: Aggregate N PEVs into “Virtual Power Plant,” sell in Day Ahead Market
Challenges: (i) N = 106 EVs; (ii) assets stochastic; (iii) privacy
Local System i = 1,2, · · ·N
Gi : Power imported from grid[kW]
Si : Power gen. from solar [kW]
EVi : Power to charge EV [kW]
Li : Power of loads [kW]
Day Ahead Market
Clearing price p ∈ R24 is stochastic.
p = E(p)
Σ = Cov(p) ∈ R24×24
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 13
Objective Function
J = pTGΣ︸ ︷︷ ︸expected cost
+α · GTΣΣGΣ︸ ︷︷ ︸
variance
+δ
2
N∑i=1
(‖EVi‖22 + ‖Gi‖22
)︸ ︷︷ ︸
battery degradation & transformer strain
(1)
where GΣ =N∑i=1
Gi, EVΣ =N∑i=1
EVi (2)
Case Study: CAISO Day Ahead Market
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 14
Objective Function
J = pTGΣ︸ ︷︷ ︸expected cost
+α · GTΣΣGΣ︸ ︷︷ ︸
variance
+δ
2
N∑i=1
(‖EVi‖22 + ‖Gi‖22
)︸ ︷︷ ︸
battery degradation & transformer strain
(1)
where GΣ =N∑i=1
Gi, EVΣ =N∑i=1
EVi (2)
Case Study: CAISO Day Ahead Market
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 14
Local Constraints
Local System i = 1,2, · · ·N Supply = Demand
Lti + EVti = Sti + Gt
i , ∀ t (3)
Grid import/export limits
Gti ≤ Gt
i ≤ Gti , ∀ t (4)
EV battery energy & power limits
evti ≤t∑
τ=1
EVτi ∆t ≤ evti , ∀ t (5)
EVti ≤ EVt
i ≤ EVti , ∀ t (6)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 15
Aggregated EV Energy & Power Limits
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 16
Aggregated EV Energy & Power Limits
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 16
Aggregated EV Energy & Power Limits
AGGREGATE individual EV energy &power limits
evtΣ ≤ A · EVΣ ≤ evtΣ, ∀ t (7)
EVtΣ ≤ EVt
Σ ≤ EVtΣ, ∀ t (8)
evtΣ is r.v., e.g. ∼ N (evtΣ, (σtev)2)
evtΣ is r.v.
EVtΣ is r.v.
EVtΣ is r.v.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 16
Aggregated EV Energy & Power Limits
AGGREGATE individual EV energy &power limits
evtΣ ≤ A · EVΣ ≤ evtΣ, ∀ t (7)
EVtΣ ≤ EVt
Σ ≤ EVtΣ, ∀ t (8)
evtΣ is r.v., e.g. ∼ N (evtΣ, (σtev)2)
evtΣ is r.v.
EVtΣ is r.v.
EVtΣ is r.v.
Relax inequalities into chance con-straints
Pr(evtΣ ≤ A · EVΣ
)≥ η (9)
Pr(A · EVΣ ≤ evtΣ
)≥ η (10)
Pr(EVt
Σ ≤ EVtΣ
)≥ η (11)
Pr(EVt
Σ ≤ EVtΣ
)≥ η (12)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 16
Problem Summary
minimize J = pTGΣ + α · GTΣΣGΣ +
δ
2
N∑i=1
(‖EVi‖22 + ‖Gi‖22
)(13)
subject to GΣ =N∑i=1
Gi, EVΣ =N∑i=1
EVi (14)
evtΣ ≤ A · EVΣ ≤ evtΣ, ∀ t (15)
EVtΣ ≤ EVt
Σ ≤ EVtΣ, ∀ t (16)
∀ i = 1,2, · · · ,N; locali : Lti + EVti = Sti + Gt
i , ∀ t (17)
Gti ≤ Gt
i ≤ Gti , ∀ t (18)
evti ≤t∑
τ=1
EVτi ∆t ≤ evti , ∀ t (19)
EVti ≤ EVt
i ≤ EVti , ∀ t (20)
A Quadratic Program (QP) 48 · N vars , 144 · N constraints
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 17
Problem Summary
minimize J = pTGΣ + α · GTΣΣGΣ +
δ
2
N∑i=1
(‖EVi‖22 + ‖Gi‖22
)(13)
subject to GΣ =N∑i=1
Gi, EVΣ =N∑i=1
EVi (14)
evtΣ ≤ A · EVΣ ≤ evtΣ, ∀ t (15)
EVtΣ ≤ EVt
Σ ≤ EVtΣ, ∀ t (16)
∀ i = 1,2, · · · ,N; locali : Lti + EVti = Sti + Gt
i , ∀ t (17)
Gti ≤ Gt
i ≤ Gti , ∀ t (18)
evti ≤t∑
τ=1
EVτi ∆t ≤ evti , ∀ t (19)
EVti ≤ EVt
i ≤ EVti , ∀ t (20)
A Quadratic Program (QP) 100K EVs* 4.8M vars , 14.4M constraints
*cumulative PEVs sold in CA by mid-2014Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 17
Problem Summary
minimize J = pTGΣ + α · GTΣΣGΣ +
δ
2
N∑i=1
(‖EVi‖22 + ‖Gi‖22
)(13)
subject to GΣ =N∑i=1
Gi, EVΣ =N∑i=1
EVi (14)
evtΣ ≤ A · EVΣ ≤ evtΣ, ∀ t (15)
EVtΣ ≤ EVt
Σ ≤ EVtΣ, ∀ t (16)
∀ i = 1,2, · · · ,N; locali : Lti + EVti = Sti + Gt
i , ∀ t (17)
Gti ≤ Gt
i ≤ Gti , ∀ t (18)
evti ≤t∑
τ=1
EVτi ∆t ≤ evti , ∀ t (19)
EVti ≤ EVt
i ≤ EVti , ∀ t (20)
A Quadratic Program (QP) 5M EVs* 240M vars , 720M constraints
*China’s 2025 GoalProf. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 17
Distributed Algorithm
ν∗ can be regarded as grid power export price
Bµ∗ can be regarded as EV charging price
Convergence TheoremThe distributed algorithm solves the original problem, and it convergessub-linearly w.r.t. the number of iterations between the aggregator and“prosumers”.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 18
Distributed Algorithm
ν∗ can be regarded as grid power export price
Bµ∗ can be regarded as EV charging price
Convergence TheoremThe distributed algorithm solves the original problem, and it convergessub-linearly w.r.t. the number of iterations between the aggregator and“prosumers”.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 18
Simulations
Nesterov’s acceleration method uses the concept of momentum:
maxρ
g(ρ) (21)
ρk+1 = ρk +k − 1
k + 2
(ρk − ρk−1
)+ α · ∇g(ρk) (22)
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 19
Resources
C. Le Floch, F. Belletti, SJM, “Optimal Charging of Electric Vehicles for Load Shaping: a Dual SplittingFramework with Explicit Convergence Bounds,” IEEE Transactions on Vehicle Electrification, vol. 2, no.2, pp. 190-199, June 2016. DOI: 10.1109/TTE.2016.2531025.
B. Travacca, S. Bae, J. Wu, SJM, “Stochastic Day Ahead Load Scheduling for Aggregated DistributedEnergy Resources,” IEEE Conference on Control Technology and Applications, Kohala Coast, HI, 2017.DOI: 10.1109/CCTA.2017.8062774.
Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 20