A Switched A Switched Extremum Extremum Seeking Approach Seeking Approach to Maximum Power Point Tracking to Maximum Power Point Tracking in Photovoltaic Systems in Photovoltaic Systems Scott J. Scott J. Moura Moura Grid Integration of Alternative Energy Sources Grid Integration of Alternative Energy Sources – Professor Ian Professor Ian Hiskens Hiskens New Contributions New Contributions • Examine extremum seeking (ES) methods for MPPT • Apply a switching scheme that enables asymptotic convergence Motivation Motivation • Increase energy conversion efficiency • Mathematically guarantee asymptotic convergence, eliminate limit cycles • Use control theoretic techniques and models developed in class Photovoltaic System Model Photovoltaic System Model 3 200 W/m 2 3 0 ° C -2 10 -1 10 0 10 1 unov Fcn, V Lyapunov Fcn Switch Threshold 0 5 10 15 20 0 1 2 3 Current [A] Characteristic Curve ES Trajectory Initial Operating Point Converged Solution Maximum Power Point 40 Impact of Varying Environmental Conditions Impact of Varying Environmental Conditions Simulate switched extremum seeking controller for 1000 W/m 2 to 500 W/m 2 step change in incident solar irradiation at 200 ms Lyapunov Lyapunov-Based Switching Scheme Based Switching Scheme • Use stability proof by Krstic and Wang, Automatica, 2000. • Compute the average system: x a = 1/T ∫ T x(τ)dτ • Linearize the averaged system about the extremum point: A = [df a /dx a ] eq • Show the Jacobian A is Hurwitz: Re[eig(A)] < 0 • Develop a Lyapunov function to track proximity to the extremum point • Use the Jacobian A to solve the following Lyapunov equation: PA + A T P = -Q, Q>0 • Pose a quadratic Lyapunov function: V(x a ) = 1/2·x a T Px a • Switched duty ratio controller for DC/DC converter • Perturbation amplitude decays exponentially Switched Switched Extremum Extremum Seeking Seeking 0 0.1 0.2 0.3 0.4 0.5 1 1.5 2 2.5 Current [A] 0 0.1 0.2 0.3 0.4 0.8 0.85 0.9 0.95 1 Duty Raio Time [sec] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 5 10 15 20 Voltage [V] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 10 20 30 40 Power [W] Time [sec] 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 Current [A] 400 W/m 2 600 W/m 2 800 W/m 2 1000 W/m 2 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 Current [A] 20 ° C 40 ° C 60 ° C 0 5 10 15 20 25 0 10 20 30 40 50 Power [W] Voltage [V] 0 5 10 15 20 25 0 10 20 30 40 50 Power [W] Voltage [V] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 10 -4 10 -3 10 -2 Lyapu Time [sec] 0 5 10 15 20 0 10 20 30 Power [W] Voltage [V] Future Work Future Work • Investigate impact of shading effects • Prove stability of switched system • Investigate alternative perturbation signals (e.g. square, triangle, stochastic) • Implement and analyze proposed algorithm on an experimental setup Conclusions Conclusions • Extremum seeking provably converges to the maximum power point • Switched Lyapunov scheme allows asymptotic convergence • Uses averaging and Lyapunov stability theory • Independent of specific PV array • Does not require periodic tuning • Requires only the existing voltage and current sensors