Transcript Real Sources - The University of Texas at Austin

Photovoltaic modules
n-type substrate
Bias voltage
p-type substrate
Id
• Vd is the diode voltage
• I0 is the reverse saturation current caused by
thermally generated carriers
• At 25 C:
Vd
 0.026

Id  I0  e
 1


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Ideal diode
Real diode
I0
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 qVkTd

I d  I 0  e  1


Photovoltaic modules
The current source
shifts the reversed
diode curve upwards
ISC
VOC
Same curve
The bias source
(voltage source)
is replaced by a
current source
powered by the
photons
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ISC
p-n junction is
equivalent to
a diode
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Reverse v-i
curve for the
diode
Photovoltaic modules
• From Kirchoff’s current law:
I PV  I SC  I d  I SC
 qVkTd

 I 0  e  1


For a more efficient MPP
tracking it is desirable that the
output current of the PV cells is
constant
• The open circuit voltage is
VOC  V ( I PV

kT  I SC
 0) 
ln 
 1
q  I0

Maximum power point
Power
P  I PVVPV
Pmax  0.7 • Voc • Isc
Current
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Photovoltaic modules
• Dependence on temperature and insolation:
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Photovoltaic modules
• More on the dependence on temperature and irradiance (Power / unit of area):
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Photovoltaic modules
• For a more realistic representation we can consider the following (equivalent
to a diode’s model):
• 1) Effect current leakage
slope 
ISC
Rp
I PV  ( I SC  I d ) 
V
Rp
• 2) Effect of internal ohmic resistance
+
Vd
ISC
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RS
 qVkTd

 I 0  e  1


+
I PV  I SC
V
where
Vd = V+IRS
This is a transcendental
equation
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V  IRS
1
Rp
Photovoltaic modules
• Both effects can be combined to obtain the more realistic (and complex)
steady state model:
+
ISC
Rp
RS
Vd
-
I PV  I SC
+
V
-
 qVkTd
 Vd
 I 0  e  1 

 Rp
where
Vd = V+IRS
This is a transcendental
equation
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Photovoltaic modules
Capacitive effect
• As with any diode, there is an associated capacitance. However, this
capacitance is relatively small, so the effects on the output can often be
neglected. Therefore, PV modules can follow a rapidly changing load very well.
•One undesirable effect of the capacitance is that it makes PV cells more
susceptible to indirect atmospheric discharges.
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Photovoltaic modules
• PV cells are combined to form modules (panels). Modules may be combined
to form arrays.
More modules (or cells)
in series
More modules (or cells)
in parallel
• When modules are connected in
parallel, the array voltage is that of the
module with the lowest voltage.
•When several modules are connected
in series to achieve a higher array
voltage, the array’s current equals that of
the module delivering the lowest current.
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Photovoltaic modules
(Rp+Rs)(n-1)Imodule
• A shadowed module
degrades the performance of
the entire array
+
+
One module with 50%
shadow
One module with 100%
shadow
(n-1)Vmodule
Two modules with 100%
shadow
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Photovoltaic modules
• Bypass diodes can mitigate the effects of shadows but they don’t solve the
issue completely.
• A better solution will be presented when discussing power electronics
interfaces.
No shade
Shaded without
bypass diode
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Shaded with
bypass diode
Photovoltaic modules
•Of course, one issue with solar power is its variability, both
• Deterministic (day vs. night).
• Stochastic (clouds).
Measured solar radiation components
Predicted solar radiation on PV module
Solar Measurem ents, File UTAUSTIN_NREL.dat
GH
DN
DH
Licor_PA
Pred. Using UTAUSTIN_NREL.dat, Lat. 30.29, Long. Shift -7.74
Licor_GH
Meth. 1 = 5.00 kWH/m2,
1200
Licor_PA = 5.72 kWH/m2.
1200
1000
1000
800
800
W/m2
W/m2
Meth. 2 = 5.58 kWH/m2.
600
600
400
400
200
200
0
0
5
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7
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Year 2009, Day 32, Feb. 01
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Year 2009, Day 32, Feb. 01, Tilt 30.29, Azim uth 180
Ultracapacitors
compensation
Batteries or large
ultracapacitors arrangement
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Lead-acid batteries
• Positive electrode: Lead dioxide (PbO2)
• Negative electrode: Lead (Pb)
• Electrolyte: Solution of sulfuric acid (H2SO4) and water (H2O)
H 2O
PbO2
Pb
H 2O
H 2O
H 2O
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H 2O
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Lead-acid batteries
• Lead-acid batteries are the most inexpensive type of batteries.
• Lead-acid batteries are not suitable for applications with often and deep
discharges.
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Lead-acid batteries
• Lead-acid batteries are very sensitive to temperature effects. It can be
expected that battery temperature exceeding 77°F (25°C) will decrease
expected life by approximately 50% for each 18°F (10°C) increase in average
temperature. [Tyco Electronics IR125 Product Manual]. Internal resistance
changes with temperature
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Lead-acid batteries
“A New Battery Model for use with Battery Energy Storage
Systems and Electric Vehicles Power Systems”
H.L. Chan, D. Sutanto
“A New Dynamic Model for Lead-Acid Batteries”
N. Jantharamin, L. Zhangt
• All models imply one issue when connecting batteries of different capacity in
parallel: since the internal resistances depend on the capacity, the battery with
the lower capacity may act as a load for the battery with the higher capacity.
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Lead-acid batteries
• Battery capacity is often measured in Ah (Amperes-hour) at a given discharge
rate (often 8 or 10 hours).
• Due to varying internal resistance the capacity is less if the battery is
discharged faster (Peukert effect)
• Lead-acid batteries capacity ranges from a few Ah to a few thousand Ah.
http://polarpowerinc.com/info/operation20/operation25.htm
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Lead-acid batteries
• The output voltage changes during the discharge due to the change in internal
voltage and resistances with the state of charge.
Coup de Fouet
Patent 6924622
Battery capacity measurement
Anbuky and Pascoe
Tyco Electronics 12IR125 Product Manual
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Lead-acid batteries
• Methods:
• Constant voltage
• Constant current
• Constant current / constant voltage
• Cell equalization problem: as the number of cells in series increases, the
voltage among the cells is more uneven. Some cells will be overcharged and
some cells will be undercharged. This issue leads to premature cell failure
• As the state of charge increases, the internal resistance tends to decrease.
Hence, the current increases leading to further increase of the state of charge
accompanied by an increase in temperature. Both effects contribute to further
decreasing the internal resistances, which further increases the current and the
temperature….. This positive feedback process is called thermal runaway.
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Lead-acid batteries
• Most calculations are based on some specific rate of discharge and then a linear
discharge is assumed.
•The linear assumption is usually not true. The nonlinearity is more evident for faster
discharge rates. For example, in the battery below it takes about 2 hours to dischage the
battery at 44 A but it takes 4 hours to discharge the battery at 26 A. Of course, 26x2 is
not 44.
• A better solution is to consider the manufacturer discharge curves and only use a linear
approximation to interpolate the appropriate discharge curve.
• In the example below, the battery can deliver 10 A continuously for about 12 hours.
Since during the discharge the voltage is around 12 V, the power is 120 W and the
energy is about 14.5 kWh
10 A continuous
discharge curve
approximation
Discharge
limit
Nominal curve
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Wind generators
• The output in all types of generators have an ac component.
• The frequency of the ac component depends on the angular speed of the wind
turbine, which does not necessarily matches the required speed to obtain an
output electric frequency equal to that of the grid (unless you tradeoff efficiency)
• For this reason, the output of the generator is always rectified.
• The rectification stage can also be used to regulate the output voltage.
• If ac power at a given frequency is needed, an inverter must be also added.
• There are 2 dynamic effects in the model: the generator dynamics and the
wind dynamics.
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Wind generators
• Consider a mass m of air moving at a speed v. The kinetic energy is
1
WK  mv 2
2
• Then power is
P
dWK 1 dm 2

v
dt
2 dt
The last expression assumes an static wind behavior (i.e. v is constant)
•The mass flow rate dm/dt is
• Thus,
dm
  Av
dt
P
1
 Av3
2
• But, power from the wind is different from the generator mechanical power
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Wind generators
SW Windpower
Whisper 200
1 kW
Rotor diameter: 2.7 m
SW Windpower
Whisper 500
3 kW
Rotor diameter: 4.5 m
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Wind generators
• Wind speed probability (then generated power, too) is an stochastic function.
• Wind speed probability can be represented using a Rayleigh distribution,
which is a special case of a Weibull distribution.
• The Rayleigh distribution appears when a 2-dimentional vector has
characteristics that:
• are normally distributed
• are uncorrelated
• have equal variance.
• A typical probability density distribution
for wind speed is shown next. Rayleigh
distributions approximates these curves.
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Microturbines
• Microturbines are essentially low-power versions of traditional gas turbines
used in large power plants.
• Typical power outputs of microturbines range from a few tens of kW to a few
hundred of kW.
• Natural gas is the most common fuel, but other hydrocarbons, such as
kerosene, or bio-fuels can be used, too.
Exhaust
Recuperator
Natural Gas
Air
Combustion
Chamber
Generator
Compressor
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Turbine
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Microturbines
Capstone
30 kW and 60 kW units
Ingersoll
70 kW Induction microturbine
250 kW synchronous microturbine
Wilson TurboPower
300 kW
Mariah Energy
30 kW and 60 kW units
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Microturbines and Internal Combustion Engines
• Microturbines:
•High-frequency output is rectified (and inverted again in ac microgrids).
Generator output frequency is in the order of a few kHz (e.g. 1600 Hz for
Capstone’s 30 kW microturbine).
• Power shaft rotates at high speeds, usually on the order of 50,000 to
120,000 rpm
• Very reliable technology (Essentially microturbines are aircraft’s APU’s).
Critical parts: bearings and generator.
• Generator technologies: Synchronous and permanent magnet
• Moderately fast dynamic response
• Internal combustion engines
• Generator technologies: Synchronous and permanent magnet
• Model similar to that of microturbines without the rectifiers. The output can
be made to have a fixed frequency
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Fuel Cells
• Fuel cells convert chemical energy directly into electrical energy.
• Difference with batteries: fuel cells require a fuel to flow in order to produce
electricity.
• Heat is produced from chemical reaction and not from combustion.
• Types of fuel cells:
• Proton exchange membrane (PEMFC)
• Direct Methanol fuel cell (DMFC)
• Alkaline fuel cell (AFC)
• Phosphoric acid fuel cell (PAFC) (*)
• Molten-carbonate fuel cell (MCFC) (*)
• Solid-oxide fuel cell (SOFC) (*)
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Fuel Cells
• Example: PEMFC
• The hydrogen atom’s electron and proton are separated at the anode.
• Only the protons can go through the membrane (thus, the name
proton exchange membrane fuel cell).
dc current
Heat
Oxygen
Hydrogen
Water
Catalyst (Pt)
Anode (-)
Membrane
(Nafion)
Catalyst (Pt)
Cathode (+)
H 2  2 H   2e 
1/ 2O2  2 H   2e   1H 2O
O2  2H 2  2H 2O
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(Er  1.23 V )
Fuel Cells
• The Tafel equation yields the cell’s output voltage Ec considering additional
loosing mechanisms:
Ec  Er  b log(i / i0 )  ir
• The first term is the reversible cell voltage (1.23V in PEMFCs)
• The last term represents the ohmic losses, where i is the cell’s current density,
and r is the area specific ohmic resistance.
• The second term represent the losses associated with the chemical kinetic
performance of the anode reaction (activation losses). This term is obtained
from the Butler-Volmer equation and its derivation is out of the scope of this
course.
• In the second term, i0 is the exchange current density for oxygen reaction and
b is the Tafel slope:
RT
b
n log(e)
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Fuel Cells
• In the last equation R is the universal gas constant (8.314 Jmol-1K-1), F is the
Faraday constant, T is the temperature in Kelvins, n is the number of electrons
per mole (2 for PEMFC), and β is the transfer coefficient (usually around 0.5).
Hence, b is usually between 40 mV and 80 mV.
• The Tafel equation assumes that the reversible voltage at the cathode is 0 V,
which is only true when using pure hydrogen and no additional limitations, such
as poisoning, occur.
• The Tafel equation do not include additional loosing mechanisms that are
more evident when the current density increases. These additional mechanisms
are:
• Fuel crossover: fuel passing through the electrolyte without reacting
• Mass transport: hydrogen and oxygen molecules have troubles reaching
the electrodes.
• Tafel equation also assumes that the reaction occurs at a continuous rate.
That is, implicit in the analysis is the notion that fuel cells output current should
be constant or nearly constant.
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Fuel Cells
Er = 1.23 V
Maximum power
operating point
Er =1.23V
b=60mV,
i0=10-6.7Acm-2
r=0.2Ωcm2
Activation loss
region
Ohmic loss region
(linear voltage to current
relationship)
Actual PEMFCs efficiency vary between 35% and 60%
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Mass transport loss region
Fuel Cells
• This past curve represent the steady state output of a fuel cell.
• The steady state output depends on the fuel flow:
Amrhein and Krein “Dynamic Simulation for Analysis of Hybrid Electric Vehicle
System and Subsystem Interactions, Including Power Electronics”
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Fuel Cells
• A very good dynamic model of a PEMFC is discussed in: Wang, Nehrir, and
Shaw, “Dynamic Models and Model Validation for PEM Fuel Cells Using
Electrical Circuits.” IEEE Transactions on Energy Conversion, vol 20, no. 2,
June 2005.
• Some highlight for this model:
Basic circuit
• Rohm: represents ohmic losses
• Ract: represents the activation losses (related with 2nd term in Tafel equation)
• Rconc: losses related with mass transport.
• C: capacitance related with the fact that there are opposing charges buildup
between the cathode and the membrane.
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Fuel Cells
• Model for the internal fuel cell voltage E
Equal to Er


N cell RT
where, f1 ( I , T )  
ln pH* 2 pO* 2  N cell k E (T  298)
2F
f 2 ( I )  N cell Ed ,cell
• Comments:
• The voltage drop related with fuel and oxidant delay is represented by
Ed,cell.
•The fuel cell output voltage depends on hydrogen’s and oxygen’s pressure
• The fuel cell output voltage also depends on the temperature.
• The time constants for these chemical, mechanical, and thermodynamic
effects are much larger than electrical time constants.
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Fuel Cells
• Ed,cell can be calculated from the following dynamic equation:
E d ,cell ( s)  e I( s)
 es
 es  1


dEd ,cell (t )
dt

1
e
Ed ,cell (t )  e
di(t )
dt
where τe is the ovreall flow delay.
• In steady state, both derivatives are zero, so Ed,cell = 0. But when the load
changes, di(t)/dt is not zero, so Ed,cell will be a non-trivial function of time that will
affect the fuel cell internal output voltage.
•When considering fuel cells dynamic behavior, they all tend to have a slow
response caused by the capacitance effect in slide 19, the flow delays, the
mechanical characteristics of the pumps, and the thermodynamic
characteristics.
• Thermodynamic characteristics were introduced in the model through an
analogous electric circuit, as shown in the next slide.
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Fuel Cells
• Simulation model and equivalent electric circuit for the thermodynamic block:
• Fuel cells have a slow dynamic response, as shown in the next figure that
evaluates the response of a fuel cell to multiple fast step load changes:
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