ENE 2XX: Renewable Energy Systems and Control

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:


Distributed vs. Decentralized: What are they?

Distributed

Decentralized

Decentralized

FullyDecentralized

Community

Optimization/Control:

PowerSystems:




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.




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.


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!!!


Problem Statement


minimizeP∈RTh×N

Th∑t=1

(Dt +

N∑n=1

Ptn

)2


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



Problem Statement


minimizeP∈RTh×N

Th∑t=1

(Dt +

N∑n=1

Ptn

)2


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



Problem Statement


minimizeP∈RTh×N

Th∑t=1

(Dt +

N∑n=1

Ptn

)2


A Quadratic Program (QP) 5M EVs*, 24 hrs



*China’s 2025 EV Goal



Problem Statement


minimizeP∈RTh×N

Th∑t=1

(Dt +

N∑n=1

Ptn

)2


A Quadratic Program (QP) 5M EVs*, 24 hrs



*China’s 2025 EV Goal



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


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



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



maxρ∈RTh

−1

4‖ρ‖2 + DTρ +

N∑n=1

[min



tn, ∀n, ∀t




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


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



maxρ∈RTh

−1

4‖ρ‖2 + DTρ +

N∑n=1

[min



tn, ∀n, ∀t




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


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



maxρ∈RTh

−1

4‖ρ‖2 + DTρ +

N∑n=1

[min



tn, ∀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



maxρ∈RTh

−1

4‖ρ‖2 + DTρ +

N∑n=1

[min



tn, ∀n, ∀t

Each PEV optimizes her own schedule, given ρt from aggregator

Parallelized N = 1.5M problems

Constraints remain private



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



maxρ∈RTh

−1

4‖ρ‖2 + DTρ +

N∑n=1

[min



tn, ∀n, ∀t

Each PEV optimizes her own schedule, given ρt from aggregator

Parallelized N = 1.5M problems

Constraints remain private


Provable Convergence w/ Bounds

Define g(ρ) = −1

4‖ρ‖2 + DTρ +

N∑n=1

[min


]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)


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


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



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.


https://ecal.berkeley.edu/pubs/TTE_DualSplittingEVsDuckCurve.pdf


http://dx.doi.org/10.1109/TTE.2016.2531025


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



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


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


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


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)


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.



AGGREGATE individual EV energy &power limits


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)


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 ≤ 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


EVti ≤ EVt

i ≤ EVti , ∀ t (20)

A Quadratic Program (QP) 48 · N vars , 144 · N constraints


Problem Summary


δ

2

N∑i=1

(‖EVi‖22 + ‖Gi‖22

)(13)


Gi, EVΣ =N∑i=1

EVi (14)


EVtΣ ≤ EVt

Σ ≤ EVtΣ, ∀ t (16)


i , ∀ t (17)

Gti ≤ Gt

i ≤ Gti , ∀ t (18)

evti ≤t∑

τ=1


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


δ

2

N∑i=1

(‖EVi‖22 + ‖Gi‖22

)(13)


Gi, EVΣ =N∑i=1

EVi (14)


EVtΣ ≤ EVt

Σ ≤ EVtΣ, ∀ t (16)


i , ∀ t (17)

Gti ≤ Gt

i ≤ Gti , ∀ t (18)

evti ≤t∑

τ=1


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


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”.



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”.


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)


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.




http://dx.doi.org/10.1109/TTE.2016.2531025

https://ecal.berkeley.edu/pubs/CCTA17_Stochastic_DER.pdf

https://ecal.berkeley.edu/pubs/CCTA17_Stochastic_DER.pdf

https://doi.org/10.1109/CCTA.2017.8062774

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