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PhD Thesis

PhD Thesis

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  • 1. 1 Control relevant modeling and nonlinear state estimation applied to SOFC-GT power systems Rambabu Kandepu 04-12-2007
  • 2. 2 Contents • Motivation • Modeling and control of SOFC-GT power system • Nonlinear state estimation • Conclusions
  • 3. 3 Motivation • Increase in energy demand – Population growth – Industrialization • Dependency on oil and gas • Global warming
  • 4. 4 Motivation • Solution to energy demand increase – Efficient of energy conversion – Technology with low emissions – Using renewable energy sources • Distributed generation – Avoid transmission and distribution losses – Wind turbines, biomass, small scale hydro, fuel cells etc
  • 5. 5 Fuel cells • Electrochemical device • Advantages – High efficiency – Low emissions – No moving parts • Different types – Electrolyte – Temperature • SOFC – Solid components – High operating temperature – More fuel flexibility – Internal reforming
  • 6. 6 SOFC-GT system Fuel Fuel cell stack Load Gas Air turbine • Tight integration between SOFC and GT • Low complexity models – Relevant dynamics
  • 7. 7 SOFC-GT system
  • 8. 8 Modeling - SOFC • Assumptions – All variables are uniform – Thermal inertia of gases is neglected – Pressure losses are neglected for energy balance – Ideal gas behavior • Reactions CH 4 + H 2O ⇔ CO + 3H 2 1 O + 2e − → O 2 − 2 2 CO + H 2O ⇔ CO2 + H 2 2− − H2 + O → H 2 O + 2e CH 4 + 2 H 2O ⇔ CO2 + 4 H 2
  • 9. 9 Modeling - SOFC • Mass balance (anode and cathode) dN i • • M = N i ,in − N i ,out + ∑ aij rj Anode Electrolyte dt j =1 Cathode • Energy balance (one volume) N N M dTs ms C s P = − P + ∑ Fan ,i (han ,i − hi ) + ∑ Fca ,i (hca ,i − hi ) − ∑ ΔH j rj dt i =1 i =1 j =1
  • 10. 10 Modeling - SOFC • Voltage RT ⎛ pH 2 pO22 ⎞ 1 E = E0 + ln ⎜ ⎟ V = E − Vloss 2 F ⎜ pH 2O ⎝ ⎟ ⎠ • Fuel Utilization (FU) = fuel utilized / fuel supplied • Distributed nature of SOFC • All models are developed in gPROMS Fuel Anode inlet Anode outlet Anode inlet Anode outlet Volume − I Volume − II Air Cathode inlet Cathode outlet Cathode inlet Cathode outlet
  • 11. 11 SOFC model evaluation • Evaluated against a detailed model 1200 Detailed model Simple model with one volume 1150 Simple model with two volumes Temperature (K) 1100 1050 1000 950 0 100 200 300 400 500 600 700 Time (min)
  • 12. 12 Control structure design • Dynamic load operation is necessary • Manipulated variable (1) – Fuel flow rate • Controlled variables (2) – Fuel utilization (FU) – SOFC temperature • Load as a disturbance • Need for a process redesign
  • 13. 13 Control structure design • Three possible options – Air blow-off – Extra fuel source – Air by-pass • Control structure Load disturbance FU ref Fuel FU Controller 1 flow Tref Hybrid system T Controller 2 Air blow-off -
  • 14. 14 SOFC-GT control
  • 15. 15 SOFC-GT control P m fuel FU FUr PI FU Controller 2 Hybrid System TSOFC TSOFCr ωr PI PI Ig ω Controller 3 Controller 1 TSOFC ω
  • 16. 16 SOFC-GT control – double shaft Controlled variables 8 fuel flow rate (g/s) 6 4 2 0 5 10 15 20 25 30 time (sec) Manipulated variables air blow-off rate (kg/s) 0.1 0.05 0 0 5 10 15 20 25 30 time (sec)
  • 17. 17 SOFC-GT control • Model Predictive Control (MPC) to include constraints – FU – Steam to carbon ratio – SOFC temperature change • Not all states are measurable • State estimation is necessary
  • 18. 18 State estimation • Need for state estimation • Nonlinear state estimation – Extended Kalman Filter (EKF) – Unscented Kalman Filter (UKF) – Comparison – Constraint handling – Results • Conclusions
  • 19. 19 State estimation • Important for process control and performance monitoring • Uncertainties; Model, measurement and noise sources • Represent the model state by an probability distribution function (pdf) • State estimation propagates the pdf over time in some optimal way • Gaussian pdf
  • 20. 20 Nonlinear state estimation • Extended Kalman Filter (EKF) – Most common way to apply KF to a nonlinear system • High order EKFs – Computationally not feasible • Ensemble Kalman Filter (EnKF) – Mostly for large scale systems (reservoir models) • Unscented Kalman Filter (UKF) – Simple and effective • Moving Horizon Estimation (MHE) – Computationally demanding
  • 21. 21 EKF principle y = g ( x); x ∈ n a random vector g: n → m , nonlinear function ( How to compute the pdf of y, given the Gaussian pdf x, Px of x ?) EKF y = g ( x) PyEKF = ( ∇g ) Px ( ∇g ) T where ( ∇g ) is the Jacobian of g ( x) at x
  • 22. 22 UKF principle • UKF principle y = g ( x); x ∈ n a random vector g: n → m , nonlinear function ( ) How to compute the pdf of y, given the Gaussian pdf x, Px of x ? UKF approximates the pdf. It uses true nonlinear process and observation models.
  • 23. 23 UKF principle • UKF principle
  • 24. 24 Comparison • Example = 58.26 = 2686
  • 25. 25 EKF Comparison UKF 110 110 Xmean 100 EKF Ymean 100 Xmean true 90 Ymean 90 ukf Ymean linearization true 80 Px=16 80 Ymean sigma points true transformed sigma points 70 Py =2686 70 Px=16 EKF 60 Py =2304 60 y=g(x)=x2 y=g(x)=x2 true Py =2686 50 50 UKF Py =2816 40 40 30 30 20 20 10 10 0 0 0 5 10 0 5 10 x x 58.26
  • 26. 26 Algorithms: EKF and UKF Nonlinear system
  • 27. 27 Algorithms: EKF and UKF EKF UKF Prediction step: Calculate Jacobians / sigma points transformation Prediction step: Calculate mean and covariance Correction step: Calculate Jacobians/ sigma points transformation Correction step: Kalman update equations
  • 28. 28 State constraint handling • No general way in KF theory – Projecting unconstrained state estimate onto boundary • Systematic approach in MHE – Solving a nonlinear problem at each time step • A simple method is introduced in UKF
  • 29. 29 State constraint handling - EKF xk−1 covariance xkEKF, C xkEKF
  • 30. 30 State constraint handling - UKF UKF, t=k xk−1 Transformed sigma points covariance x-kUKF
  • 31. 31 Constraint handling
  • 32. 32 Constraint handling UKF • Constraint handling method – Projections at different steps • Sigma points • Transformed sigma points • Transformed sigma points through measurement function – Inequality constraints
  • 33. 33 Constraint handling- example • Gas phase reversible reaction 3 true UKF 2 EKF A 1 C 0 -1 0 1 2 3 4 5 6 7 8 9 10 time (sec) true 4 UKF EKF 3 B C 2 1 0 1 2 3 4 5 6 7 8 9 10 time (sec)
  • 34. 34 Comparison (EKF and UKF) • Nonlinear systems – Induction motor and Van der Pol Oscillator – Faster convergence with UKF • Robustness to model errors – Van der Pol oscillator • Better performance with UKF • Higher order nonlinear system – SOFC-GT hybrid system (18 states)
  • 35. 35 Comparison (EKF and UKF) Comparison of estimated states of an induction motor: components of stator flux 1 true UKF EKF 0.5 1 x 0 0 5 10 15 20 25 30 35 40 45 50 time (sec) 0 true UKF EKF -0.5 2 x -1 -1.5 0 5 10 15 20 25 30 35 40 45 50 time (sec)
  • 36. 36 Comparison (EKF and UKF) • SOFC-GT system – Higher order nonlinear system (18 states) – Turbine shaft speed plot
  • 37. 37 Conclusions – state estimation • The UKF is a promising option – Simple and easy to implement – No need for Jacobians – Computational load is comparable to EKF – Improved performance • Faster convergence • Robustness to model errors and initial choices • Simple constraint handling method works
  • 38. 38 Thank you for your attention ☺