ENE 2XX: Renewable Energy Systems and Control LEC 04 : Distributed - PowerPoint PPT Presentation

ENE 2XX: Renewable Energy Systems and Control LEC 04 : Distributed Optimization of DERs Professor Scott Moura University 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? Community Optimization/ Distributed Decentralized Control: Power Systems: Decentralized Fully Decentralized Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 2

Distributed vs. Decentralized: What are they? Community Optimization/ Distributed Decentralized Control: Power Systems: Decentralized Fully Decentralized 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 curve Challenge: N = 10 3 , 10 6 , or 10 9 DERs to schedule every time slot!!! � � 2 T h N � � D t + P t minimize P ∈ R Th × N n t = 1 n = 1 t P t n ≤ u t n ≤ P n , ∀ n , ∀ t subject to A Quadratic Program (QP) T h × N optimization variables 2 T × N linear inequality constraints 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 curve Challenge: N = 10 3 , 10 6 , or 10 9 DERs to schedule every time slot!!! � � 2 T h N � � D t + P t minimize P ∈ R Th × N n t = 1 n = 1 t P t n ≤ u t n ≤ P ∀ n , ∀ t subject to n , A Quadratic Program (QP) 100K EVs*, 24 hrs T h × N optimization variables 2.4M 2 T h × N linear inequality constraints 4.8M *cumulative PEVs sold in CA by mid-2014 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 curve Challenge: N = 10 3 , 10 6 , or 10 9 DERs to schedule every time slot!!! � � 2 T h N � � D t + P t minimize P ∈ R Th × N n t = 1 n = 1 t P t n ≤ u t n ≤ P ∀ n , ∀ t subject to n , A Quadratic Program (QP) 1.5M EVs*, 24 hrs T × N optimization variables 32M 2 T × N linear inequality constraints 64M *California Gov. Brown 2025 ZEV Goal 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 curve Challenge: N = 10 3 , 10 6 , or 10 9 DERs to schedule every time slot!!! � � 2 T h N � � D t + P t minimize P ∈ R Th × N n t = 1 n = 1 t P t n ≤ u t n ≤ P ∀ n , ∀ t subject to n , A Quadratic Program (QP) 5M EVs*, 24 hrs T × N optimization variables 120M 2 T × N linear inequality constraints 240M *China’s 2025 EV Goal 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 curve Challenge: N = 10 3 , 10 6 , or 10 9 DERs to schedule every time slot!!! � � 2 T h N � � D t + P t minimize P ∈ R Th × N n t = 1 n = 1 t P t n ≤ u t n ≤ P ∀ n , ∀ t subject to n , A Quadratic Program (QP) 5M EVs*, 24 hrs T × N optimization variables 120M 2 T × 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 � � 2 T h N N � � � D t + P t � P n � 2 + σ minimize P ∈ R Th × N n t = 1 n = 1 n = 1 t P t n ≤ P t n ≤ P ∀ n , ∀ t subject to: n , Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7

Optimal PEV Aggregation Define “consensus variable”: z t = D t + � N n P t n T h N z t � 2 + σ � � � � P n � 2 minimize P ∈ R Th × N , z ∈ R Th t = 1 n = 1 N � z t = D t + P t n , ∀ t subject to: n t P t n ≤ P t n ≤ P n , ∀ n , ∀ t 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: � � T h N N z t � 2 + λ t � � � � z t − D t − P t � P n � 2 + σ max min n λ ∈ R Th P ∈ R Th × N , z ∈ R Th n t = 1 n = 1 t P t n ≤ P t n ≤ P n , ∀ n , ∀ t subject to: 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: � � T h N N z t � 2 + λ t � � � � z t − D t − P t � P n � 2 + σ max min n λ ∈ R Th P ∈ R Th × N , z ∈ R Th n t = 1 n = 1 t P t n ≤ P t n ≤ P n , ∀ n , ∀ t subject to: minimize w.r.t. z f t ( z t ) = ( z t ) 2 + λ t z t , df t ⇒ ( z t ) ⋆ = − 1 dz t = 2 z t + λ t = 0 , 2 λ t For convenience, define ρ t = − λ t . Plug ( z t ) ⋆ = 1 2 ρ t into dual problem Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7

Optimal PEV Aggregation Plug ( z t ) ⋆ = 1 2 ρ t into dual problem � � T h N N ρ t � 2 − ρ t � 1 1 � � 2 ρ t − D t − � P t � P n � 2 max min + σ n 4 ρ ∈ R Th P ∈ R Th × N t = 1 n n = 1 t P t n ≤ P t subject to: n ≤ P n , ∀ n , ∀ t Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7

Optimal PEV Aggregation Plug ( z t ) ⋆ = 1 2 ρ t into dual problem � � T h N N ρ t � 2 − ρ t � 1 1 � � 2 ρ t − D t − � P t � P n � 2 max min + σ n 4 ρ ∈ R Th P ∈ R Th × N t = 1 n n = 1 t P t n ≤ P t subject to: n ≤ P n , ∀ n , ∀ t The P n terms decouple along n , yielding N parallel subproblems: � � N ρ ∈ R Th − 1 � 4 � ρ � 2 + D T ρ P ∈ R Th × N ρ T P n + σ � P n � 2 + max min n = 1 t subject to: P t n ≤ P t n ≤ P n , ∀ n , ∀ t Each PEV optimizes her own schedule, given ρ t from aggregator Parallelized N = 1 . 5 M problems Constraints remain private Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 7

Provable Convergence w/ Bounds N � � g ( ρ ) = − 1 � 4 � ρ � 2 + D T ρ P ∈ R Th × N ρ T P n + σ � P n � 2 + Define min n = 1 t s. to P t n ≤ P t n ≤ P n , ∀ n , ∀ t Theorem: Linear Convergence Rate The dual problem has a unique solution ρ ⋆ , and the gradient ascent algorithm with step-size α = − 2 σ/ ( N + σ ) converges linearly according to � � k N g ( ρ ⋆ ) − g ( ρ k ) ≤ ( g ( ρ ⋆ ) − g ( ρ 0 )) N + σ Similar theorems for Incremental stochastic gradient method (constant step-size) Incremental stochastic gradient method (decreasing step-size) Incorporate uncertainty in D t and PEV availability (SOCP) Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 8

Optimal DER Aggregation N N ρ ∈ R Th − 1 � � 4 � ρ � 2 + D T ρ P ∈ R Th × N ρ T P n + σ � P n � 2 max + min n n = 1 t s. to P t n ≤ P t n ≤ P ∀ n , ∀ t n , ρ t 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 N N ρ ∈ R Th − 1 � � 4 � ρ � 2 + D T ρ P ∈ R Th × N ρ T P n + σ � P n � 2 + max min n n = 1 t s. to P t n ≤ P t n ≤ P ∀ n , ∀ t n , Algorithm 1 Gradient Ascent (constant step size) Initialize ρ = ρ 0 ; Choose α = − 2 σ/ ( N + σ ) for k = 1 , · · · , k max (1) Inner Optimization: Optimize charge schedule for each PEV n for n = 0 , 1 , · · · , N ( ρ k ) T P n + σ � N ... Solve, P k n = 1 � P n � 2 n = arg min P t t n ≤ P t n ≤ P n end for (2) Outer Optimization: Update dual variable ρ ... ρ k + 1 = ρ k + α · ∇ g ( ρ k ) � � ... ρ k + 1 = ρ k + α 2 ρ k + D + � N − 1 n = 1 P k n end for Prof. Moura | Tsinghua-Berkeley Shenzhen Institute ENE 2XX | LEC 04 - Distributed Opt of DERs Slide 10