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Optimal Secondary frequency control connecting AGC with economic dispatch Changhong Zhao Na Li Lijun Chen Steven Low LIDS, MIT Telecommunication CMS CU-Boulder Caltech CMU 02/04/2014 Frequency control economic dispatch unit commitment secondary freq control primary freq control sec min dynamic model e.g. swing eqtn 5 min 60 min day power flow model e.g. DC/AC power flow year Frequency control economic dispatch secondary freq control Optimally schedule gens/loads, and the inter-area power flows Maintain the nominal freq and the inter-area power flows ∆ Total Gen = ∆Total Load sec min dynamic model e.g. swing eqtn 5 min power flow model e.g. DC/AC power flow Area 1 Area 2 Area 3 Inter-area power flows Frequency control To Improve secondary freq control sec 1. Optimally schedule gens/loads within a area economic dispatch min dynamic model e.g. swing eqtn Maintain the nominal freq and the inter-area power flows 2. Optimally (re)schedule gens/loads among it is permitted) ∆ Total areas Gen =(if ∆Total Load 5 min power flow model e.g. DC/AC power flow Area 1 Area 2 Area 3 Inter-area power flows Frequency control To Improve 1. Optimally schedule gens/loads within a area economic dispatch secondary freq control Maintain the nominal freq and the inter-area power flows 2. Optimally (re)schedule gens/loads among areas (if it is permitted) sec min dynamic model e.g. swing eqtn 5 min power flow model e.g. DC/AC power flow Motivation: Power system efficiency Frequency control To Improve 1. Optimally schedule gens/loads within a area economic dispatch secondary freq control Maintain the nominal freq and the inter-area power flows 2. Optimally (re)schedule gens/loads among areas (if it is permitted) sec min dynamic model e.g. swing eqtn 5 min power flow model e.g. DC/AC power flow Challenge: Physical dynamics + existing control mechanisms (AGC) Physical dynamics: Swing Dynamics - Swing + ωi : Mechanical power : Load ωi : Frequency Pij : Power Flow Mi, Di: Constant parameters Bus i: control area/ balance authority + - Swing ωj Variables: the deviations from reference (steady state) values Power Flow dynamics - Swing + - Pij Flow ωi + - -1 + - Swing ωj : Mechanical power : Load ωi : Frequency Pij : Power Flow Tij: Constant parameter Assumptions: Lossless (resistance=0) Fixed voltage magnitudes Small deviation of angles Turbine-Governor Control + 1/R1 - Governor - Swing + - Pij Flow ωi + - -1 + 1/R2 Governor + - Swing ωj : Mechanical power : Load ωi : Frequency Pij : Power Flow : Power command input Ti, Ri: Constant parameters (Area Control Error) ACE-based AGC B1 + + 1/R1 + ACE Governor - Swing + - Pij -1 + + B2 Flow ωi + - -1 ACE + - 1/R2 Governor + - Swing ωj : Mechanical power : Load ωi : Frequency Pij : Power Flow : Power command input Ki, Bi: Constant parameters Recap: System Dynamics with AGC i Pi m Swing dynamics Pij Pi L Power Flow Dynamics Turbine-Governor Control ACE-based AGC Suppose the system is in steady state (all variables = 0) Disturbance, e.g. Pi L Drive the system to a new steady state (i) Economically Dispatch (ii) Our objective to minimize generation cost j Recap: System Dynamics with AGC i Pi m Swing dynamics Pij Pi L Power Flow Dynamics Turbine-Governor Control ACE-based AGC Suppose the system is in steady state (all variables = 0) Disturbance, e.g. Pi L Drive the system to a new steady state (i) (ii) Generation cost Power flow balance Economic AGC j Recap: System Dynamics with AGC i Pi m Swing dynamics Pij Pi L Power Flow Dynamics Turbine-Governor Control ACE-based AGC Question: How to modify AGC to be economic AGC? Drive the system to a new steady state (i) (ii) Economic AGC j Results: Economic AGC i Pi m Swing dynamics Pij Pi L Power Flow Dynamics Drive the system to a new steady state (i) (ii) Economic AGC j Results: Economic AGC i Pi m Swing dynamics Power Flow Dynamics Theorem(Li, Chen, Zhao, Low 2013): Any trajectory P M (t ), (t ), P(t ) converges to P M * , * , P* • P M * is optimal to economic dispatch * • 0 • P* is a feasible power flow Pij Pi L j Results: Economic AGC i Pi m Swing dynamics Pij Pi L Power Flow Dynamics Local computation Local communication — Additional local variable — Frequency, Power flow, Power Command: local measurable signals — A decentralized algorithm; Local information and communications — Not just local… j Not Just Local… i Pi m Swing dynamics Pij Pi L Power Flow Dynamics Only use information/signals that are easy to measure or calculate j Extension to load freq control i Pi m Swing dynamics Power Flow Dynamics Drive the system to a new steady state (i) (ii) Load disutility Pij Pi L Similar mechanisms apply to load control Optimal gens/loads control j Tool: reverse/forward engineering System Dynamics & Existing Control Economic Generation Control (i) (ii) Analogy ? solve Optimization Problem Reverse Tool: reverse/forward engineering System Dynamics & Existing Control Economic Generation Control (i) (ii) Modified Optimization Problem Equivalent Forward Tool: reverse/forward engineering System Dynamics & Modified Control Economic Generation Control (i) (ii) solve Modified Optimization Problem Equivalent Forward Case Study 1 2 4 control areas: a 39-bus New England Transmission Network Source: Ilic, et.al. IEEE TPS, vol. 8, no. 1, 1993 4 • • • • 3 At time t = 10s, load increase 0.2 pu at area 4 (Nonlinear) power flow model; Nonzero resistance Sample rate =15s (e.g., ACE is reset for every 15s) Each area has a same cost function (optimal value should be that each area increase generation by 0.05 pu ) Mechanical Power, PM (pu) Mechanical Power Generation 0.3 0.2 Area 1 Area 2 Area 3 Area 4 0.1 0 -0.1 0 0.3 Mechanical Power, PM(pu) ACE-based AGC 100 200 300 400 600 Area 1 Area 2 Area 3 Area 4 Economic AGC 0.2 500 0.1 0 -0.1 0 100 200 300 Time (s) 400 500 600 Total generation cost 1.4 Generation cost 1.2 1 ACE-based AGC 0.8 0.6 Economic AGC 0.4 0.2 Optimal value of economic dispatch 0 0 100 200 300 Time (s) 400 500 600 Frequency Frequency (Hz) 60.01 60.005 ACE-based AGC 60 Area 1 Area 2 Area 3 Area 4 59.995 59.99 59.985 Frequency (Hz) 0 60.01 100 200 300 60.005 400 500 600 Economic AGC 60 Area 1 Area 2 Area 3 Area 4 59.995 59.99 59.985 0 100 200 300 Time (s) 400 500 600 Conclusion economic dispatch secondary freq control primary freq control 1. Optimally schedule gens/loads within a area Maintain the nominal frequency and the inter-area power flows 2. Optimally (re)schedule gens/loads among areas (if it is permitted) Minor Modification (additional local computation and communication based on local measurable signals) Back Up System dynamics Economic Energy Control c (P ) m min i i i over Pi m , s. t. Pi m = Pi L i 0 Partial primal-dual gradient algorithm min Reverse Engineering over s. t. 1 2 ( Pi m )2 D2i i 2 i Pi M , Pi M = Pi L Dii Pi M = Pi L P P j:i j ij k :k i ki P P j:i j ij k :k i ki Economic AGC Economic Energy Control c (P ) m min i i i over Pi m , s. t. Pi m = Pi L P P j:i j ij k :k i i 0 Forward Engineering Partial primal-dual gradient algorithm min 1 2 ( Pi m )2 D2i i 2 Pi M , s. t. Pi M = Pi L Dii Pi M = Pi L i i over ci ( Pi m ) D2i i 2 P P j:i j ij k :k i Pi M Pi L ki j:i j ij k :k i ki ki