Discrete Optimal Control & Analysis of a PEM Fuel Cell to Grid (V2G) System

Transcription

1 Discrete Optimal Control & Analysis of a PEM Fuel Cell to Grid (V2G) System Scott Moura Siddartha Shankar ME Winter 2007 Professor Huei Peng April 24, 2007 ME 561 Design of Digital Control System, Winter 2007 Slide 1 of 15

2 Outline Introduction Modeling Fuel Cell System & Battery Power Grid Demand Cycle Modeling Control Problem Formulation Presentation of Results Linearization Discrete System Analysis & LQR Design Luenberger Estimator Design Discussion of Results Open Loop System Analysis LQR Weight Selection Observer Pole Placement & Estimation Analysis Conclusions & Recommendations Acknowledgements & References ME 561 Design of Digital Control System, Winter 2007 Slide 2 of 15

3 Introduction Fuel Cell technology Abundant energy source H 2 High efficiency (50-70%) Clean energy source (zero emissions) Hybrid Technology Hybrid concept is developing in many engineering fields, esp. the auto industry Fuel Cell/Battery leverages advantages of each energy source V2G Concept Enables the use of renewable energy sources Adds energy storage capacity element to grid Distributed generation (DG) decentralizes grid 5% of California s vehicle fleet can provide 10% peak power for entire state [1] Consumer may sell power back to the grid More expensive FCV becomes a more profitable investment [1] W. Kempton, J. Tomic, S. Letendre, A. N. Brooks and T. Lipman, "Vehicle-to-grid power: Battery, hybrid, and fuel cell vehicles as resources for distributed electric power in California," California Air Resources Board, Tech. Rep. UCD-ITS-RR-01-03, June 2001, ME 561 Design of Digital Control System, Winter 2007 Slide 3 of 15

4 Fuel Cell System & Battery Voltage COMPRESSOR MOTOR SUPPLY MANIFOLD COOLER & HUMIDIFIER BATTERY Battery Power POWER BUS Stack Stack Current Voltage Power Output H2 List of components FC stack Humidifier Supply Manifold Compressor Air Supply COMPRESSOR CATHODE SIDE FUEL CELL STACK ANODE SIDE H2 STORAGE TANK Fuel Cell stack model adapted from [Lin, Kim, Peng, Grizzle] Battery model from ADVISOR Resistive Equivalent Circuit Model Main functions of battery model V oc = f ( soc, T ), Rint = f ( soc, T ) P batt = f ( soc, Voc, Rint ) Isothermal Operation Assumption I batt V = oc I SOC = Q batt max V 2 oc 2R 4P int batt R Q SOC = int max Q I max batt dt ME 561 Design of Digital Control System, Winter 2007 Slide 4 of 15

5 Power Grid Demand Cycle Modeling Representative grid power demand cycle Adapted from California Independent System Operator (CAISO) daily demand forecast 24 hour cycle (6am to 6am) Power Demand (kw) Grid Power Demand Cycle 6 6am 9am 12am 3am 6pm 9pm 12pm 3pm 6am Time (hour) Scaled for medium size office or apartment complex Augmented with white Gaussian noise to simulate stochastic nature of power demand ME 561 Design of Digital Control System, Winter 2007 Slide 5 of 15

6 Control Problem Formulation Regulate battery state of charge (SOC) Maximize fuel cell system (FCS) efficiency Reject disturbances (grid power demand) Accurately estimate SOC Power Grid Demand Cycle Control Inputs: u I st = e cm u Plant Fuel Cell System - w + Battery Multiplexer y Process Disturbance: w = P grid Dynamic States: Measurements: x ωcp p = sm Q used Voc y = η fc u Full-State Feedback (-K) Observer Based Compensator ˆx Luenberger Observer u ME 561 Design of Digital Control System, Winter 2007 Slide 6 of 15

7 Linearization 0 30 Operating Points u A = 42.6V 0 w = 8500W 0 Linear System Model δ = δ + δ + δ x rad 2646 sec = 111,570Pa 1.8C x A x B y G w δ y = Cδx+ Dδy + Hδw V y = In terms of perturbation variables δ ( ) = ( ) ( ) 0 represents the perturbation about the nominal operating point Verification V oc Response of Original vs. Linearized Models Original Linearized Response of original vs. linearized model for sinusoidal control inputs and power demand cycle disturbance. η fcs Time (sec) 10 ME 561 Design of Digital Control System, Winter 2007 Slide 7 of 15

8 Discrete System Analysis & LQR Design Continuous to Discrete Transformation Sampling Time T = 0.05s 4 times faster than fastest plant dynamics Linear Quadratic Regulator Performance Index Continuous Discrete ME 561 Design of Digital Control System, Winter 2007 Slide 8 of 15

9 Luenberger Estimator Design Observer Pole Placement Closed loop full state feedback dynamics Rule of thumb: Select observer dynamics to be 3x faster than closed loop system p = 3 eig ( A-BK ) p ( ) 3 observer = eig A -BK observer Why cube the eigenvalues for discrete time? Proof: Z-transform definition 3x faster in continuous time Substitute into Z-transform definition Re-apply Z-transform definition State Estimator Dynamics: Complete Closed Loop Dynamics: Separation Principle Poles of FB Control & Poles of Observer are independent! ME 561 Design of Digital Control System, Winter 2007 Slide 9 of 15

10 Open Loop System Analysis Zero-Pole Cancellation Analyze physical significance Open Circuit Voltage (Integrator) FCS Efficiency (Gain) A Closer Look: Open Circuit Voltage (Integrator) FCS Efficiency (Gain) Battery modeled as capacitive element Zero bandwidth Constant 90 phase lag Proportional to I st Infinite Bandwidth Constant 180 phase lag P K I fcs, net η fcs = LHV m H2 st ME 561 Design of Digital Control System, Winter 2007 Slide 10 of 15

11 LQR Weight Selection by Parameter Sweep Select Q and R to meet desired specifications: V oc acheives steady state in less than 1 min FCS operates at optimal efficiency for as long as possible Stack current cannot exceed 310A (saturated control action) Recall Performance Index Select diagonal matrices Set q 1 = q 2 = r 2 = 0 Initially, no desire to regulate I st, V oc, or e cm Parameter Sweep Methodology 1. Fix one parameter and vary the other 2. Observe performance Fast Response Huge Control Action Tradeoff between response speed and control actuation Slow Response Small Control Action ME 561 Design of Digital Control System, Winter 2007 Slide 11 of 15

12 LQR Weight Selection by Multiobjective Optimization Multiobjective Optimization: Minimize SOC rise time Maximize average FCS efficiency Small q 3 Large r 1 Tradeoff observed between each objective Large q 3 Small r 1 Pareto Front Analysis: Plot each objective on an axis Lower region exhibits typical Pareto Front behavior Red points indicate the design exceeds the I st < 310A limit INFEASIBLE Pareto Front visualization method is very intuitive Effective method for LQR weight selection ME 561 Design of Digital Control System, Winter 2007 Slide 12 of 15

13 Observer Pole Placement & Estimation Analysis Observer Pole Placement Fast poles Advantage: Estimation error decays rapidly Disadvantage: Assumes perfect sensors and/or noise-free environment Slow poles Advantage: Less sensitive to process disturbances and measurement noise Disadvantage: Estimation error decays slowly Estimation Analysis No state experiences greater than 5% error Estimation error imitates power grid cycle Estimator picks up on process disturbance Error reduces dramatically for 10x faster poles NOTE: Perfect sensors assumed ME 561 Design of Digital Control System, Winter 2007 Slide 13 of 15

14 Conclusions & Recommendations Conclusions Pole-zero cancellation analysis elucidates physical significance of transfer function models LQR weight selection is a multi-objective optimization problem Fast observer poles track process disturbance Linearization assumes inputs, as well as states, do not deviate far from the linearization points A tradeoff exists between fast SOC response and low stack current control input Recommendations Gain scheduling to accommodate nonlinearity of power disturbance cycle Kalman estimator design to reject process disturbance and sensor noise ME 561 Design of Digital Control System, Winter 2007 Slide 14 of 15

15 Acknowledgements & References Thank you for your support & advice throughout the course! Professor Huei Peng Dongsuk Kum Dr. Hosam Fathy Professor Jessy Grizzle Key References Dongsuk Kum, S. Moura, Plant/Control Optimization of a PEM Hybrid Fuel Cell V2G System, University of Michigan, ME 555, Chan-Chiao Lin, Min-Joong Kim, H. Peng and J. W. Grizzle, "System-level model and stochastic optimal control for a PEM fuel cell hybrid vehicle," Transactions of the ASME. Journal of Dynamic Systems, Measurement and Control, vol. 128, pp , J. T. Pukrushpan, A. G. Stefanopoulou and H. Peng, Control of Fuel Cell Power Systems: Principles, Modeling, Analysis and Feedback Design., vol. XVII, Springer, 2004, pp H. Peng and George T.C Chiu, Chapter 9 Linear Quadratic Optimal Control, in Design of Digital Control Systems coursepack, ME 561 Design of Digital Control System, Winter 2007 Slide 15 of 15