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EE152 Green Electronics Photovoltaics 10/10/13 Prof. William Dally Computer Systems Laboratory Stanford University
EE152 Green Electronics
Photovoltaics 10/10/13
Prof. William Dally Computer Systems Laboratory
Stanford University
Course Logistics • HW3 Due Tuesday • Lab2 Signed off this week • Lab3 Out • Quiz1 – Next Thursday 10/17
10−1 100 101 102 10310−4
10−2
100
102
104
|H(s
)|
10−1 100 101 102 103−200
−150
−100
−50
0�
H(s
) (de
gree
s)
t (rad/s)
t1 = 20.5116
e1 = −103.699
Summary of Feedback Control • Plant is described by ODEs (possibly non-linear) • Controller drives plant input(s) to achieve goal • Feedback control, input is function of “error” • Stable if – ζ >= 1 – H(s) has phase margin at unity gain
• PD and PI controllers – Derivative feedback stabilizes 2nd order system – Integral feedback cancels residual error (but avoid wind up)
• Bode Plots – Show phase margin at unity gain – Zeros ramp up and advance phase 90 degrees – Poles ramp down and retard phase 90 degrees
• Motor control with current limit
G(s) = C(s)P(s)1+C(s)P(s)
=H (s)1+H (s)
C(s) P(s)X(s) E(s) A(s) Y(s)+
_
Example – Control of Boost Converter
Want transfer function from Dh (duty factor of high switch) to Vd
�I = (Vin �DhVd)tcyL
i(s) = �dh(s)hVdisL
� vd(s)hDhisL
�Vd =
✓DhI �
Vd
R
◆tcyC
vd(s) =hDhii(s)
sC� vd(s)
sRC
vd(s)
✓1 +
1
sRC
◆=
hDhii(s)sC
vd(s) (sRC + 1) = hDhiRi(s)
vd(s) =hDhiRi(s)
sRC + 1=
hDhii(s)sC + 1
R
vd(s) = �hDhihVdidh(s) + hDhi2vd(s)s2LC + sL
R
vd(s)
✓s2LC + s
L
R+ hDhi2
◆= �hDhihVdidh(s)
vd(s)
dh(s)= � hDhihVdi
s2LC + sLR + hDhi2
vd(s)
dh(s)= �
hDhihVdiLC
s2 + s 1RC + hDhi2
LC
1
�I = (Vin �DhVd)tcyL
i(s) = �dh(s)hVdisL
� vd(s)hDhisL
�Vd =
✓DhI �
Vd
R
◆tcyC
vd(s) =hDhii(s)
sC� vd(s)
sRC
vd(s)
✓1 +
1
sRC
◆=
hDhii(s)sC
vd(s) (sRC + 1) = hDhiRi(s)
vd(s) =hDhiRi(s)
sRC + 1=
hDhii(s)sC + 1
R
vd(s) = �hDhihVdidh(s) + hDhi2vd(s)s2LC + sL
R
vd(s)
✓s2LC + s
L
R+ hDhi2
◆= �hDhihVdidh(s)
vd(s)
dh(s)= � hDhihVdi
s2LC + sLR + hDhi2
vd(s)
dh(s)= �
hDhihVdiLC
s2 + s 1RC + hDhi2
LC
1
Derive small signal model
�I = (Vin �DhVd)tcyL
i(s) = �dh(s)hVdisL
� vd(s)hDhisL
�Vd =
✓DhI �
Vd
R
◆tcyC
vd(s) =hDhii(s)
sC� vd(s)
sRC
vd(s)
✓1 +
1
sRC
◆=
hDhii(s)sC
vd(s) (sRC + 1) = hDhiRi(s)
vd(s) =hDhiRi(s)
sRC + 1=
hDhii(s)sC + 1
R
vd(s) = �hDhihVdidh(s) + hDhi2vd(s)s2LC + sL
R
vd(s)
✓s2LC + s
L
R+ hDhi2
◆= �hDhihVdidh(s)
vd(s)
dh(s)= � hDhihVdi
s2LC + sLR + hDhi2
vd(s)
dh(s)= �
hDhihVdiLC
s2 + s 1RC + hDhi2
LC
1
−20
0
20
40
60
Mag
nitu
de (d
B)
103 104 105 106−180
−135
−90
−45
0
Phas
e (d
eg)
Bode DiagramGm = Inf dB (at Inf rad/s) , Pm = 1.74 deg (at 3.32e+05 rad/s)
Frequency (rad/s)
Plant Step Response
0 0.2 0.4 0.6 0.8 1 1.2x 10−3
0
20
40
60
80
100
120
140
160
Step Response
Time (seconds)
Ampl
itude
tcy ~ 180us
Validate to SPICE
0µs 50µs 100µs 150µs 200µs 250µs 300µs 350µs 400µs 450µs 500µs-3V
0V
3V
6V
9V
12V
15V
18V
21V
24V
27V
30VV(vd)
To bring phase back, add a zero at ω=1000 Add a pole to give infinite DC gain Lower gain to bring ω1 to 1e4 P=.0001 Q=1E-7 R=1000
−80
−60
−40
−20
0
20
Mag
nitu
de (d
B)
103 104 105 106 107 108−90
−45
0
45
90
Phas
e (d
eg)
Bode DiagramGm = Inf , Pm = −90 deg (at 1e+07 rad/s)
Frequency (rad/s)
Forward Transfer of Controller x Plant
−40
−30
−20
−10
0
10
20
Mag
nitu
de (d
B)
103 104 105 106−135
−90
−45
0
Phas
e (d
eg)
Bode DiagramGm = Inf , Pm = 85.4 deg (at 9.45e+03 rad/s)
Frequency (rad/s)
Step Response
0 0.5 1 1.5 2 2.5 3 3.5 4x 10−3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (seconds)
Ampl
itude
Boost Control • Develop small-signal model for plant dynamics
– Linearize around nominal values for Vd and Dh
– Note that these dynamics depend on load (R) – Change significantly for DCM
• Plot forward transfer function • Derive controller to fix phase margin
– Zero to advance phase 90 degrees – Pole to give infinite DC gain – Lower proportional gain to place crossover well below ωsw
– Also important to have enough derivative gain to avoid large-signal effects on startup.
• Check combined response
PV and MPPT
Energy Conversion
Photovoltaic System
Solar Panel
Solar Panel
Solar Panel
Solar Panel
Solar Panel
Solar Panel
Photovoltaic Array
PV Controller and Inverter
Batteries
400V DC 240V AC60 Hz
48V DC
To Grid
16
17
M
215 Installation and Operation
C
opyright � 2012 Enphase Energy
141-00012 Rev 04
29
Sam
ple Wiring D
iagram – M
215, 240 VA
C
Electrons absorb energy from photons
Equivalent Circuit
RSHISC
RS
D1 VC
+
_
D2
IV-Curve
Typical Panel CS6P 60 cells in series ~0.5V per cell 3 strings of 20 with bypass diode on each string
IV Curve from SPICE Model
Peak-Power Tracking • Find point on IV curve where power is maximized.
Start at any point (v(0),i(0)) “Dither” v, v(i+1) = v(i) + Δv Check result: if(p(i+1) < p(i)) v(i+1) = v(i) Try both directions: Δv = -Δv
MPP Tracking – The Movie
Start at (35 V, 5.5A) P=192.5
Dither by DV = 0.5V to V = 35.5V (35.5V, 4.7A) P=166.9 < 192.5
(35.5V, 4.7A) P=166.9 < 192.5 Bad Move – Go Back to (35, 5.5)
Dither by -0.5V to 34.5V (34.5, 6.2) P=213.9 > 192.5
(34.5, 6.2) P=213.9 > 192.5 Keep move and keep going
Move to 34.0 (34.0, 6.7) P=227.8 > 213.9
(34.0, 6.7) P=227.8 > 213.9 Keep move and keep going
(33.5, 7.0) P=234.5> 227.8 Keep move and keep going
(33.0, 7.3) P=240.9 > 234.5 Keep move and keep going
(32.5, 7.5) P=243.75 > 240.9 Keep move and keep going
(32.0, 7.6) P=243.2 < 243.75 Abandon Move and Go Back!
Operate at (32.5, 7.5) P=243.8 With occasional forays to 32.0 and 33.0
“Hillclimbing” On the Power Curve
Compound Power Curve
Compound Power Curve (2 Panels)
Not convex How do you find maximum power point?
Three Panels
Typical String of 10 PV Panels
Search Strategies for Non-Convex MPPT • Exhaustion
– Try every operating point • Random
– Randomly pick new points – keep if better • Hierarchical
– Try every point – with coarse spacing – Try every point near best point with finer spacing – Repeat
• Acquire and Track – One of the above to acquire MPPT (e.g., hierarchical) – Then gradient search to track – Periodically revisit (devote some fraction of string time to this)
• Optimal method depends on – Shape of curve – How fast the curve changes – How the curve changes
Good Optimization Depends on Understanding The Problem
• Collect lots of data – Time series of IV curves from typical strings
• Understand the data • What causes “dips”
– Bad panels • Static offset in current
– Fixed shading – trees, buildings, etc… • Periodic offset – same time each day
– Variable shading – clouds, etc… • Unpredictable shading – but shifts across panels in one direction
• Develop algorithms • Test on data
An Example of Optimization • Trade-off parameters against one another to maximize
a figure of merit.
• In this case, parameters are panel voltage and current.
• Figure of merit is power.
• Optimization is done real-time because temperature and irradiance change. – Sometimes optimization is done at design time, or calibration
time.
MPPT Power Path (Boost Converter with Energy Meter)
Ci
VPV
PV Panel
RS
M1G
CO
L1
Load
M2G
VL
IPV
Cycle Waveforms
350 355 360 365 370 375 3802
4
6
8il(
A)
350 355 360 365 370 375 38034.5
35
35.5
v in (V
)
350 355 360 365 370 375 38043
43.5
44
44.5
v out (V
)
t (µs)
Size input cap Ci for acceptable ripple
Size output cap Co for acceptable ripple
Size inductor L to set ripple
SPICE
Longer Simulation
0 2 4 6 8 10 12 14 1610
20
30
40
v in(V
)
0 2 4 6 8 10 12 14 160
5
10i pv
(A)
0 2 4 6 8 10 12 14 1620
40
60
v out(V
)
0 2 4 6 8 10 12 14 160
0.2
0.4
D
0 2 4 6 8 10 12 14 1650
100150200250
P (W
)
t (ms)
Summary of PV • PV cells/strings are voltage-dependent current
sources (Diode in parallel with current source) • PV controllers regulate their input voltage/current to
maximize power – Maximum power-point tracking
• Can apply almost any converter topology – Boost used for illustration – Regulate input rather than output
• Gradient search for convex optimization • More sophisticated search needed for multi cell/panel
string
In Future Lectures • Midterm Review • Magnetics • Transformers and bridge converters • Grounding • Inverters • Soft Switching • Batteries
