Photovoltaic modules

1 © Alexis Kwasinski, 2011

Photovoltaic modules

p-type substrate

n-type substrate

Bias voltage

Ideal diodeReal diode

0 1dqV

kTdI I e

Id

• Vd is the diode voltage• I0 is the reverse saturation current caused by thermally generated carriers• At 25 C:

0.0260 1

dV

dI I e

I0


ISC

Reverse v-i curve for the

diode

ISC

VOC

p-n junction is equivalent to a diode

Same curve

The bias source (voltage source) is replaced by a current source powered by the photons

The current source shifts the reversed diode curve upwards



• From Kirchoff’s current law:

• The open circuit voltage is

0 1dqV

kTPV SC d SCI I I I I e

0

( 0) ln 1SCOC PV

IkTV V I

q I

PV PVP I V

Current

Power

Maximum power point

Pmax 0.7 • Voc • Isc


For a more efficient MPP tracking it is desirable that the

output current of the PV cells is constant


• Dependence on temperature and insolation:



• More on the dependence on temperature and irradiance (Power / unit of area):



• For a more realistic representation we can consider the following (equivalent to a diode’s model):

• 1) Effect current leakage

• 2) Effect of internal ohmic resistance

ISC

ISC

( )PV SC dp

VI I I

R

1

p

slopeR

SV IR

Rp

RS

Vd

+

V

-

+

-

0 1dqV

kTPV SCI I I e

where Vd = V+IRS

This is a transcendental equation



• Both effects can be combined to obtain the more realistic (and complex) steady state model:

ISC Rp

RS

Vd V

--

0 1dqV

dkTPV SC

p

VI I I e

R

where Vd = V+IRS

This is a transcendental equation

++



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.



• 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.



(n-1)Vmodule

-

++

-

(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

Two modules with 100% shadow



• 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 with bypass diode

Shaded without bypass diode



Solar Measurements, File UTAUSTIN_NREL.dat

0

200

400

600

800

1000

1200

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Year 2009, Day 32, Feb. 01

W/m

2

GH DN DH Licor_PA Licor_GH

Pred. Using UTAUSTIN_NREL.dat, Lat. 30.29, Long. Shift -7.74

0

200

400

600

800

1000

1200

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Year 2009, Day 32, Feb. 01, Tilt 30.29, Azimuth 180

W/m

2

Meth. 1 = 5.00 kWH/m2, Meth. 2 = 5.58 kWH/m2. Licor_PA = 5.72 kWH/m2.

•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

Ultracapacitors compensation

Batteries or large ultracapacitors arrangement



Lead-acid batteries• Positive electrode: Lead dioxide (PbO2)

• Negative electrode: Lead (Pb)

• Electrolyte: Solution of sulfuric acid (H2SO4) and water (H2O)

PbPbO2

H2O

H2O

H2O

H2O H2O


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.


• 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

Lead-acid batteries


• 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.

“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

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

Lead-acid batteries


• The output voltage changes during the discharge due to the change in internal voltage and resistances with the state of charge.

Tyco Electronics 12IR125 Product Manual

Coup de Fouet

Patent 6924622 Battery capacity measurement Anbuky and Pascoe

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.

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

Discharge limit

Nominal curve

10 A continuous discharge curve approximation

Lead-acid batteries


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.


• Consider a mass m of air moving at a speed v. The kinetic energy is

• Then power is

The last expression assumes an static wind behavior (i.e. v is constant) •The mass flow rate dm/dt is

• Thus,

• But, power from the wind is different from the generator mechanical power

21

2KW mv

21

2KdW dm

P vdt dt

31

2P Av

dmAv

dt

Wind generators


SW WindpowerWhisper 200

1 kWRotor diameter: 2.7 m

SW WindpowerWhisper 500

3 kWRotor diameter: 4.5 m

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. Rayleighdistributions approximates these curves.

Wind generators


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.

Natural Gas

Air

GeneratorCompressor

Recuperator

CombustionChamber

Turbine

Exhaust


Microturbines

Capstone30 kW and 60 kW units

Ingersoll70 kW Induction microturbine

250 kW synchronous microturbine

Wilson TurboPower300 kW

Mariah Energy30 kW and 60 kW units


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


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) (*)


• 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).

Hydrogen Oxygen

Water

2 2 2H H e

Heat

2 21/ 2 2 2 1O H e H O

Membrane(Nafion)Catalyst (Pt)

Anode (-)Catalyst (Pt)Cathode (+)

dc current

2 2 22 2 ( 1.23 )rO H H O E V

Fuel Cells


• The Tafel equation yields the cell’s output voltage Ec considering additional loosing mechanisms:

• 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:

log( )

RTb

n e

0log( / )c rE E b i i ir

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.

Fuel Cells


Maximum power operating point

Er = 1.23 V

Activation loss region Ohmic loss region

(linear voltage to current relationship)

Mass transport loss region

Er =1.23Vb=60mV,i0=10-6.7Acm-2

r=0.2Ωcm2

Actual PEMFCs efficiency vary between 35% and 60%

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”

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:

• 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.

Basic circuit

Fuel Cells


• Model for the internal fuel cell voltage E

where,

• 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.

Equal to Er

2 2

* *1( , ) ln ( 298)

2cell

H O cell E

N RTf I T p p N k T

F

2 ,( ) cell d cellf I N E

Fuel Cells


• Ed,cell can be calculated from the following dynamic equation:

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.

,, ,

( ) 1 ( )E ( ) I( ) ( )

1d celle

d cell e d cell ee e

dE ts di ts s E t

s dt dt

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:

Fuel Cells

Photovoltaic modules

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