A Switched A Switched ExtremumExtremum Seeking Approach Seeking Approach to Maximum Power Point Tracking to Maximum Power Point Tracking in Photovoltaic Systemsin Photovoltaic SystemsScott J. Scott J. MouraMouraGrid Integration of Alternative Energy Sources Grid Integration of Alternative Energy Sources –– Professor Ian Professor Ian HiskensHiskens

New ContributionsNew Contributions• Examine extremum seeking (ES) methods for MPPT

• Apply a switching scheme that enables asymptotic convergence

MotivationMotivation• Increase energy conversion efficiency

• Mathematically guarantee asymptotic convergence, eliminate limit cycles

• Use control theoretic techniques and models developed in class

Photovoltaic System ModelPhotovoltaic System Model

3 200 W/m2

3 0°C

-2

10-1

100

101

Lyap

unov

Fcn

, V

Lyapunov FcnSwitch Threshold

0 5 10 15 200

1

2

3

Cur

rent

[A]

Characteristic CurveES TrajectoryInitial Operating PointConverged SolutionMaximum Power Point

40

Impact of Varying Environmental ConditionsImpact of Varying Environmental ConditionsSimulate switched extremum seeking controller for 1000 W/m2 to 500 W/m2 step change

in incident solar irradiation at 200 ms

LyapunovLyapunov--Based Switching SchemeBased Switching Scheme• Use stability proof by Krstic and Wang, Automatica, 2000.

• Compute the average system: xa = 1/T ∫T x(τ)dτ• Linearize the averaged system about the extremum point: A = [dfa/dxa]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 + ATP = -Q, Q>0• Pose a quadratic Lyapunov function: V(xa) = 1/2·xa

TPxa

• Switched duty ratio controller

for DC/DC converter

• Perturbation amplitude decays exponentially

Switched Switched ExtremumExtremum SeekingSeeking

0 0.1 0.2 0.3 0.40.5

1

1.5

2

2.5

Cur

rent

[A]

0 0.1 0.2 0.3 0.40.8

0.85

0.9

0.95

1

Dut

y R

aio

Time [sec]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.45

10

15

20

Vol

tage

[V]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410

20

30

40

Po

we

r [W

]

Time [sec]

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

Cur

ren

t [A

]

200 W/m

400 W/m2

600 W/m2

800 W/m2

1000 W/m2

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

Cur

ren

t [A

]

20°C

40°C

60°C

0 5 10 15 20 250

10

20

30

40

50

Pow

er [W

]

Voltage [V]0 5 10 15 20 25

0

10

20

30

40

50

Pow

er [W

]

Voltage [V]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410

-4

10-3

10-2

Lyap

unov

Fcn

, VTime [sec]

0 5 10 15 200

10

20

30

Pow

er [W

]

Voltage [V]

Future WorkFuture 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

ConclusionsConclusions• 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