Incremental Cost Consensus Algorithm in a Smart Grid Environment Y3.F.C1 Distributed Control of FREEDM System Ziang Zhang PI: Dr. Mo-Yuen Chow Department of Electrical and Computer Engineering North Carolina State University Raleigh, North Carolina Future Renewable Electric Energy Delivery and Management Systems Center
Outline • Background • Motivations & Goal • Technical Approach • Incremental Cost Consensus Algorithm – Problem Formulation – Convergence Rate Analysis • Future Plan 2 Future Renewable Electric Energy Delivery and Management Systems Center
Background System Demonstration: - Intelligent Energy Management Enabling Technology: - Distributed Grid Intelligence (DGI) Fundamental Technology: - System Theory Modeling and Control (SMC) 3 Future Renewable Electric Energy Delivery and Management Systems Center
Outline • Background • Motivations & Goal • Technical Approach • Incremental Cost Consensus Algorithm – Problem Formulation – Convergence Rate Analysis • Future Plan 4 Future Renewable Electric Energy Delivery and Management Systems Center
Motivation & Goal Challenges for the Current Power Grid • Lack of support for Distributed Generation and Renewable Energy • Lack of flexibility and adaptability • Vulnerability to Cyber attack, natural disasters and human errors • $100 Billion annual loss due to power quality problems • Aging Components Solution: Take advantages from the new technologies -- make the grid smarter Project Goal Design and implement high performance distributed controls to achieve real- time intelligent power allocation in FREEDM system. 5 Future Renewable Electric Energy Delivery and Management Systems Center
Outline • Background • Motivations & Goal • Technical Approach • Incremental Cost Consensus Algorithm – Problem Formulation – Convergence Rate Analysis • Future Plan 6 Future Renewable Electric Energy Delivery and Management Systems Center
Central Control vs. Distributed Control Puppet vs. School of fish Distributed Control [1] Central Control System Puppets and Puppeteer School of fish Controller Puppeteer (Single) Fish (Multiple) Information Puppeteer know the position Each fish only know the position available to the of every part of puppet of neighbors (Local) controller (Global) Control Goal Keep certain pattern of style Keep certain pattern of shape and moving around and moving around • Iain D. Couzin , Jens Krause, Nigel R. Franks and Simon A. Levin, “Effective leadership and decision -making in animal groups on the move”, Nature 433, 513-516 (3 February 2005) • … 7 Future Renewable Electric Energy Delivery and Management Systems Center
Central Control vs. Distributed Control Central Controlled System Distributed Controlled System • Control algorithm is relatively • Relieved the computational burden for a Pros simple single controller • … • Ease of heavy data exchange demand • Single point of failure will not necessarily affect the others • Controllers do not need the entire system state information • … • Computational limitation of • Only part of the system states are Cons central controller available to each distributed controller • Communication limitation of • Normally need complex algorithms and central controller designs • Single point of failure will affect • … the entire system • … Usages Normally more appropriate for Normally more appropriate for large-scale systems with simple control systems need sophisticated control 8 Future Renewable Electric Energy Delivery and Management Systems Center
What is consensus? Consensus [1] Consensus A school of fish Chorus Goal: swimming towards one Goal: Synchronize the melody same direction [1]. Larissa Conradt and Timothy J. Roper, “Consensus decision making in animals”, Trends in Ecology & Evolution, Volume 20, Issue 8, August 2005, Pages 449-456. 9 Future Renewable Electric Energy Delivery and Management Systems Center
How can consensus be reached? Consensus Network Independent Physical Systems (Generators) Each of them follow their own dynamic A sufficient condition for reach consensus: If there is a directed spanning tree* exists in the communication network, then consensus can be reached. ** * Spanning tree: a minimal set of edges that connect all nodes ** Wei Ren Randal W. Beard Ella M. Atkins , “ A Survey of Consensus Problems in Multi- agent Coordination”, 2005 American Control Conference June, 2005. Portland, OR, USA 10 Future Renewable Electric Energy Delivery and Management Systems Center
Networked Control System Picture from EPRI Future Renewable Electric Energy Delivery and Management Systems Center
Cyber Physical System Networked Control System Cyber Layer Physical Layer Picture from EPRI Future Renewable Electric Energy Delivery and Management Systems Center
Agent-based Distributed Control Network Cyber Layer Power Grid Physical Layer Future Renewable Electric Energy Delivery and Management Systems Center
Graph Theory Modeling Adjacency matrix of a finite graph G on n vertices is the n × n matrix where the entry a ij is the number of edges from vertex i to vertex j , a ij =0 represent that agent i cannot receive information from agent j 1 1 1 1 0 1 1 2 4 4 A 1 0 0 1 1 D 0 1 0 0 2 3 2 2 Adjacency matrix 1 1 Example Network 0 2 2 diag (2,1,1) A L ; Row-stochastic matrix 2 1 1 L 1 1 0 1 0 1 Laplacian matrix Future Renewable Electric Energy Delivery and Management Systems Center
First-order Consensus Algorithm Consensus problem modeling Local information state • i First-order system i i , 1,..., n • i • Consensus algorithm: Scalar From Matrix Form Continuous – time n a ( ), i 1,..., n L i ij i j n j 1 n Discrete-time [ k 1] d [ ], k i 1,..., n [ k 1] D [ ] k i ij j n j 1 Where L n is the Laplacian matrix associated with A, and D n is Row-stochastic matrix associated with A. k 0 1 2 3… 1/ 2 1/ 4 1/ 4 1 ξ 1 -2 -2*1/2+1/4+3/4 = 0 0 0… D 1/ 2 1/ 2 0 1/ 2 0 1/ 2 ξ 2 1 -2*1/2+1*1/2 = -0.5 -0.25 - 0.125… 2 3 ξ 3 3 -2*1/2+3*1/2 = 0.5 0.25 0.125… Future Renewable Electric Energy Delivery and Management Systems Center 15
Outline • Background • Motivations & Goal • Technical Approach • Incremental Cost Consensus Algorithm – Problem Formulation – Convergence Rate analysis • Future Plan 16 Future Renewable Electric Energy Delivery and Management Systems Center
Decentralized Economic Dispatch Assumptions: • All the signals are “good” • No security issue • No generation limitation (in this presentation) • The cost functions are quadratic • The power grid topology is fixed 17 Future Renewable Electric Energy Delivery and Management Systems Center
Decentralized Economic Dispatch Economic Dispatch Problem -- A constrained optimization problem 2 ) 2 )+(310+7.85P 2 +0.94P 2 Min: Cost = (561+7.92P 1 +0.562P 1 s.t. : P 1 + P 2 = 500 3D view Contour Graph 18 Future Renewable Electric Energy Delivery and Management Systems Center
Decentralized Economic Dispatch Economic Dispatch Problem -- A constrained optimization problem Min: Cost = f(P 1, P 2 ) s.t. : g(P 1, P 2 )= P 1 + P 2 -500 At the optimal point: ∇ f(x, y)= λ ∇ g(x, y) 19 Future Renewable Electric Energy Delivery and Management Systems Center
Incremental Cost Consensus Decentralize the Economic Dispatch Problem Using Consensus Network: When using Lagrange multiplier method solving Economic Dispatch Problem, each generator will have the same Incremental Cost at the minimum cost point Conventional Central Controlled Communication Distribute Controlled Incremental Topology for a 3-bus system Cost Consensus Network Future Renewable Electric Energy Delivery and Management Systems Center
Incremental Cost Consensus Mathematical Formulation : 2 , i =1,2,…m C i (P Gi )= α i + β i P Gi + γ i P Gi Assume the fuel-cost curve of each generating unit is known and expressed where C i (P Gi ) is the cost of generation for unit i. in terms of the output power: P Gi is the output power of unit i The objective is to minimize total cost of operation: C T = Σ C i (P Gi ). Subject to constrains: Σ P Di - Σ P Gi =0; IC i = ∂ C i (P Gi ) / ∂ P Gi = λ i From the conventional economic dispatch we know: λ i [ k+1 ] = Σ d ij λ j [ k ] , Pick λ as the information state, use the first order discrete consensus algorithm : where d ij is the ( i,j ) entry of row-stochastic matrix D n . λ i [ k+1 ] = Σ d ij λ j [ k ] + ε Δ P , The consensus algorithm for the leader where ε is a scalar which controls the convergence speed. (mediator/ coordinator) generator becomes: Δ P = Σ P Di - Σ P Gi . 21 Future Renewable Electric Energy Delivery and Management Systems Center
Incremental Cost Consensus Flow Chart: 22 Future Renewable Electric Energy Delivery and Management Systems Center
Recommend
More recommend
