Distributed algorithms for network optimization under non-sparse constraints Ashish Cherukuri and Jorge Cort´ es Mechanical and Aerospace Engineering University of California, San Diego Allerton Conference on Communication, Contol, and Computing Monticello, Sep 28-30, 2016
Need for network optimization is pervasive Optimizing agent operation given limited network resources power networks: generation, transmission, distribution, consumption wireless communication networks: throughput, routing, topology sensor&robotic networks: data gathering, fusion, estimation, life Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 2 / 16
Grid of the future: from vertical to flat Integration of renewables and distributed energy resources (DERs) From small number of large generators to large number of smaller generators advent of renewables, distributed energy generation large-scale grid optimization problems, highly dynamic traditional top-down approaches impractical, inefficient Rethinking of operational&infrastructure design for efficiency and emission targets Optimized coordination for allowing&dispatching power flows originating from any point, handle dynamic loads, robust against failures, privacy, plug-and-play September 22, 2014 January 21, 2015 Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 3 / 16
Network optimization with non-sparse constraints Network of n agents communicating over connected undirected graph convex cost function: f i : R → R , ∀ i local constraint: x m i ≤ x i ≤ x M i , ∀ i global constraint: Ax = b , with b ∈ R m and non-sparse A ∈ R m × n Network optimization problem � n minimize i =1 f i ( x i ) subject to Ax = b x m ≤ x ≤ x M Objective: distributed algorithmic solution under local exchanges: only neighbors communicate with each other information: i knows f i , x m i , x M and ([ A ] k , b k ) for k such that [ A ] k,i � = 0 i Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 4 / 16
Sample scenario: I Economic dispatch Group of n power generators aim to meet power demand while minimizing total cost of generation and respecting individual generator constraints Economic dispatch problem � n minimize i =1 f i ( P i ) � n subject to i =1 P i = L P m ≤ P ≤ P M load constraint is global and generator constraints are local m = 1, A = [1 , . . . , 1], and b = L Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 5 / 16
Sample scenario: II Sensitivity analysis-based optimal power flow 1 Given operating point, group of n power generators seek to determine cost-effective change in generation to meet change in demand while accounting for flow constraints Linearized optimal power flow � N g i =1 f i (∆ P g minimize i ) � N g i =1 ∆ P g i = � N l i =1 ∆ P d j + Λ ⊤ ∆ P g subject to � ∆ P g � P f ≤ Ψ f ≤ P ∆ P d P g ≤ ∆ P g ≤ P g change in losses and flows represented using shift factors power balance and flow constraints are global as Λ and Ψ are non-sparse 1 K. E. Van Horn, A. D. Dom´ ınguez-Garc´ ıa, and P. W. Sauer. “Measurement-based real-time security-constrained economic dispatch,” IEEE Transaction on Power Systems , vol. 31, no. 5, pp. 3548-3560, 2016. Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 6 / 16
Outline 1 Introduction Motivation Problem statement 2 Exact reformulations Using consensus Using auxiliary variables 3 Perturbation analysis General constraints Affine constraints Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 7 / 16
Exact reformulation using consensus x i ∈ R n Decision variable for agent i is copy of network state ˆ x n ) ∈ ( R n ) n x 1 ; ˆ x 2 ; . . . ; ˆ Collective decision variable ˆ x = (ˆ ( ˜ A i , ˜ b i ) are submatrices formed by rows k of A and b where [ A ] k,i � = 0 Original problem Exact reformulation � n � n x i min i =1 f i ( x i ) min i =1 f i (ˆ i ) x i = ˜ ˜ s.t. Ax = b s.t. A i ˆ b i , ∀ i x m ≤ x ≤ x M x m x i i ≤ x M i ≤ ˆ i , ∀ i ( L ⊗ I n )ˆ x = 0 n 2 L is graph Laplacian All constraints are local (computable using information exchange with neighbors) in the reformulated problem! Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 8 / 16
Exact reformulation using consensus x i ∈ R n Decision variable for agent i is copy of network state ˆ x n ) ∈ ( R n ) n x 1 ; ˆ x 2 ; . . . ; ˆ Collective decision variable ˆ x = (ˆ ( ˜ A i , ˜ b i ) are submatrices formed by rows k of A and b where [ A ] k,i � = 0 Original problem Exact reformulation � n � n x i min i =1 f i ( x i ) min i =1 f i (ˆ i ) x i = ˜ ˜ s.t. Ax = b s.t. A i ˆ b i , ∀ i x m ≤ x ≤ x M x m x i i ≤ x M i ≤ ˆ i , ∀ i ( L ⊗ I n )ˆ x = 0 n 2 L is graph Laplacian Proposition Original problem and consensus-based formulation have the same optimizers Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 8 / 16
Exact reformulation using consensus x i ∈ R n Decision variable for agent i is copy of network state ˆ x 1 ; ˆ x 2 ; . . . ; ˆ x n ) ∈ ( R n ) n Collective decision variable ˆ x = (ˆ ( ˜ A i , ˜ b i ) are submatrices formed by rows k of A and b where [ A ] k,i � = 0 Original problem Exact reformulation � n � n x i min i =1 f i ( x i ) min i =1 f i (ˆ i ) x i = ˜ ˜ s.t. Ax = b s.t. A i ˆ b i , ∀ i x m ≤ x ≤ x M x m x i i ≤ x M i ≤ ˆ i , ∀ i ( L ⊗ I n )ˆ x = 0 n 2 L is graph Laplacian Distributed implementation: size of the interchanged messages is order n either communication complexity or time complexity suffers Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 8 / 16
Exact reformulation using auxiliary variables for k ∈ { 1 , . . . , m } , let y k ∈ R n be auxiliary variable for k -th constraint decision variable for agent i is ( x i , { y k i } m k =1 ) Original problem Exact reformulation � n � n min i =1 f i ( x i ) min i =1 f i ( x i ) s.t. Ax = b b k diag([ A ] k ) x + L y k = n e k e k , ∀ k s.t. x m ≤ x ≤ x M 1 ⊤ x m ≤ x ≤ x M � 1 , if [ A ] k,i � = 0 ek ∈ R n is defined by ek i = 0 , otherwise All constraints are local in the reformulated problem! Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 9 / 16
Exact reformulation using auxiliary variables for k ∈ { 1 , . . . , m } , let y k ∈ R n be auxiliary variable for k -th constraint decision variable for agent i is ( x i , { y k i } m k =1 ) Original problem Exact reformulation � n � n min i =1 f i ( x i ) min i =1 f i ( x i ) s.t. Ax = b b k � ( y k i − y k n e k e k s.t. [ A ] k,i x i + j ) = i , ∀ k, i x m ≤ x ≤ x M 1 ⊤ j ∈N i x m ≤ x ≤ x M � 1 , if [ A ] k,i � = 0 ek ∈ R n is defined by ek i = 0 , otherwise All constraints are local in the reformulated problem! Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 9 / 16
Exact reformulation using auxiliary variables for k ∈ { 1 , . . . , m } , let y k ∈ R n be auxiliary variable for k -th constraint decision variable for agent i is ( x i , { y k i } m k =1 ) Original problem Exact reformulation � n � n min i =1 f i ( x i ) min i =1 f i ( x i ) s.t. Ax = b b k diag([ A ] k ) x + L y k = n e k e k , ∀ k s.t. x m ≤ x ≤ x M 1 ⊤ x m ≤ x ≤ x M � 1 , if [ A ] k,i � = 0 ek ∈ R n is defined by ek i = 0 , otherwise Proposition Original problem and reformulation have same optimizers diag([ A ] k ) x + L y k = Key fact: 1 ⊤ n e k e k � b k � yields [ A ] k x = b k n 1 ⊤ Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 9 / 16
Exact reformulation using auxiliary variables for k ∈ { 1 , . . . , m } , let y k ∈ R n be auxiliary variable for k -th constraint decision variable for agent i is ( x i , { y k i } m k =1 ) Original problem Exact reformulation � n � n min i =1 f i ( x i ) min i =1 f i ( x i ) s.t. Ax = b b k diag([ A ] k ) x + L y k = n e k e k , ∀ k s.t. x m ≤ x ≤ x M 1 ⊤ x m ≤ x ≤ x M � 1 , if [ A ] k,i � = 0 ek ∈ R n is defined by ek i = 0 , otherwise Distributed implementation: size of the interchanged messages is of order m + 1 scalable implementation when m and n independent Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 9 / 16
Comparison Economic dispatch problem �� n � � n i =1 c i P 2 � min i =1 P i = L } i four cases, number of generators ( n ): 5, 15, 25, 35 same primal-dual dynamics for both formulations No. of steps to convergence for differ- Volume of communication at each it- ent network sizes eration for different network sizes Cherukuri & Cort´ es (UCSD) Distributed network optimization September 28, 2016 10 / 16
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