In this course, you will learn the purpose of each component in an equivalent-circuit model of a lithium-ion battery cell, how to determine their parameter values from lab-test data, and how to use them to simulate cell behaviors under different load profiles. By the end of the course, you will be able to: - State the purpose for each component in an equivalent-circuit model - Compute approximate parameter values for a circuit model using data from a simple lab test - Determine coulombic efficiency of a cell from lab-test data - Use provided Octave/MATLAB script to compute open-circuit-voltage relationship for a cell from lab-test data - Use provided Octave/MATLAB script to compute optimized values for dynamic parameters in model - Simulate an electric vehicle to yield estimates of range and to specify drivetrain components - Simulate battery packs to understand and predict behaviors when there is cell-to-cell variation in parameter values
2.4.1: How do I use the ECM to simulate constant voltage?
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From the lesson
Simulating battery packs in different configurations
In this module, you will learn how to generalize the capability of simulating the voltage response of a single battery cell to a profile of input current versus time to being able to simulate constant-voltage and constant-power control of a battery cell, as well as different configurations of cells built into battery packs.
Taught By
Gregory Plett
Professor
Welcome to week four of this course.
This is where things start to get
really interesting I think because you're
going to see how to use some of the techniques
that you've learned and apply them to real problems.
So so far in this course,
you've learned how to make equations that
describe the operation of a lithium ion battery cell.
You've also seen how to conduct tests in
the laboratory and how to use
the data that you collected from those tests,
in order to determine
the static and dynamic model parameter values.
Then you learned how to use
these mathematical models combined
with the parameter values from
the lab tests in order to be able to
simulate the performance of a single battery cell.
In particular, you've learned how to predict
a battery cell's output voltage response
to a change in its input current.
So this week, we're going to look at
some other model applications.
In this lesson, you will learn how to simulate
a constant voltage battery response and later,
you will learn how to simulate
a constant power battery response.
Then in later lessons this week,
you will learn how to simulate battery packs comprising
many cells connected together in series and in parallel.
When we are charging a battery cell,
it's common first to apply a constant current charge to
the cell until the battery cell
achieves some maximum terminal voltage.
Then, once we have reached that maximum terminal voltage,
instead of supplying a constant current to this cell,
we clamp the voltage at the terminal's constant and we
allow current to vary
until the batteries cell is fully charged.
So up until now you have seen how to simulate how
a cell's voltage responds when we
supply a profile of current versus time,
but you have not yet learned how to
compute the profile of current versus time that
will keep a battery cell's voltage
constant and that's what we're going to do now.
To see how to do this,
it is important to examine
the enhanced self-correcting cell model once again,
and I have reproduced
the equations of this model on the slide.
The state equation states or says,
that the state of the model right now,
is a function of the values of
this state at the previous time,
combined with values of
the input current from the previous time.
It will be really important for all of
our applications this week to notice that
the present state is not a function of the present input,
but is a function of past inputs only.
Next, let's look at the output equation
or the voltage equation of the model.
The voltage is computed as the open-circuit voltage plus
the hysteresis voltage minus diffusion voltages
and minus an ohmic voltage drop.
The open-circuit voltage contribution
is a function of the present state,
but remember that the present state
is not a function of the present input.
It is a function only of previous inputs.
Similarly, the hysteresis value
right now is a function of
the present state and therefore is a function of
previous input current values only.
Also, the diffusion voltages
are a function of the present state,
and so therefore, are a function only
of previous values of the input current.
So when we look at the model and
we look at the part of the voltage,
that is a function of the input current right now,
the only part of the equation that talks about
that is the ohmic voltage drop across resistor R_0.
This observation is critical for being able
to simulate a constant voltage situation.
So to recap, cell voltage has two portions,
one portion I will call a fixed part.
Now, it's not fixed over all time,
but at this point in time,
this fixed part is determined by all of
the previous input current values and cannot
vary at this point in
time due to the input current right now.
So it's fixed in that sense.
Every time step it changes, but right now,
the value of this part is
fixed by what happened in the past.
So we have a fixed part,
we also have a variable part and the variable part
does depend on the present input current.
We're going to make use of this observation to develop
a way to simulate a constant voltage scenario.
To condense the mathematical notation,
make it a little bit easier to talk about,
we define the fixed part to be
v_f at time step k. So this is
the open circuit voltage plus
the hysteresis voltage and with
the combination of the diffusion voltages as well.
Then to simulate a specific constant voltage,
we can calculate the input current that
produces the desired output voltage in a few easy steps.
So we remember that the voltage is equal to
the fixed part minus R_0 multiplying the input current.
Then, we rearrange the equation such that
the input current is isolated on
the left-hand side just using standard algebra.
We find that the input current,
in order to create a desired output voltage,
is equal to the fixed voltage,
minus the desired, voltage all divided by R_0.
So at any point in time,
when we require fixing the output voltage at some level,
we can use this equation to compute
the input current at that point in
time that is guaranteed to do so.
Then we simply simulate
the enhanced self-correcting cell model with
this value of input current for this point in time,
just like you learned how to do in earlier lessons.
So remember one application for this is
a common type of battery cell
charging protocol especially done in laboratories,
where we charge this cell using constant current
until we achieve a certain voltage level,
and then we hold the voltage constant.
So let's have a look at this in some octave code.
Here we look at some octave code
that can be used to simulate
a constant current constant voltage charge
of a battery cell.
The script here begins with some comments,
and then clears the work space and closes all figures,
if any happen to be open.
It clears the command window also.
So we're starting fresh.
Then it loads and example cell model
from storage into the work space.
Here I am loading a model of an E1 cell but
this could be done with any
enhanced self-correcting cell model.
The simulation then does some initializations.
It sets the maximum duration of
the simulation to be 3,001 time
samples which in this case corresponds to 3,001 seconds.
Then it sets the temperature to 25 degrees Celsius.
After that, it loads
some enhanced self-correcting cell model parameter values
from the model data structure into a local variable,
so that we can simulate the cell more easily.
Finally, it sets
the maximum charge voltage to 4.15 volts,
which is the point we will use for
switching from the constant current portion of
our charging into the constant voltage segment
of our charging profile.
Next, the octave code
preallocates some storage into which
we can save values of our variables so that we can
have a look at them later on using plotting functions.
For example, the storez variable
will be used to store state of
charge at every point in the simulation,
the storev variable will store a voltage,
the storei variable will store current,
and the storep variable will store power.
Then we initialize the simulation with
an initial cell state of charge of z equals 50 percent,
the resistor-capacitor current state
is initialized to zero,
the history usage state is initialized to minus one
corresponding to this cell having recently been charged,
and the sign of the input current is
initialized to minus one in a similar way.
We also define the value of constant current that we will
use in this simulation and
the constant current portion to be nine Amperes.
All of these things can be
changed and this code will be provided to
you on the Coursera website for you
to be able to experiment with as well.
The code on this slide implements
the simulation of the constant
current, constant voltage charge.
We begin by entering a loop,
a for loop that iterates over all
of the time allocated for charging.
The first step inside of this loop is to
compute the fixed part of voltage that applies right now,
by computing the open-circuit voltage, the histories,
its contribution, and the diffusion voltage contribution.
Then we compute the input current
necessary or that would be necessary to
force the output voltage to the
maximum 4.15 volts when charging.
Now, at early points in charging,
the current required to take
a cell from low state of charge to
a terminal voltage of 4.15
volts would be an enormous amount of current,
that would greatly exceed
the constant current threshold
of nine amperes that we have set.
So if this value that we
compute in this equation happens to be
greater in magnitude than this threshold of nine amperes,
then we use nine amperes as the input current.
Otherwise, we use this particular computed value.
So what we're doing, is we are
guaranteeing that the cell voltage never exceeds 4.15
volts and at the same time guaranteeing that
the input current never exceeds
the constant current charging rate of nine amperes.
That's how we implement the constant
current, constant voltage charging.
So those were really the tricky lines in
this simulation to come up with that logic for how would
we know when to
do a constant current versus a constant voltage step.
Now I think you can see that it's really
quite simple after you learn the trick.
Everything else in this code
should look very familiar to you after
having studied the SimCell.m function last week.
For example, the next step updates
the state of charge equation and
then we update the
resistor-capacitor, resistor currents equation.
Then we compute the hysteresis rate factor and
we update the hysteresis state and finally,
we update the sign of the input current.
At the end of all of this, we store
the present values of state of charge and
cell voltage and input
current and power for later examination.
So the simulation is now complete and the code on
this slide shows you how you can
plot some results from the simulation.
We begin by creating a time vector having correct length
against which we can plot the voltage and
current and power and state of charge.
Then in figure one,
we plot the state of charge of
the cell in percent versus time,
in figure two we plot the terminal voltage versus time,
in figure three we plot the input current versus time,
and finally, in figure four we plot
the cell power versus time.
The figures on this slide show you
some example results from the simulator.
In the top left plot,
you see state of charge versus time,
on the top right you see terminal voltage versus time,
in the bottom left,
you see cell input current versus time,
and finally, on the bottom right,
you can see cell power versus time.
If we look first at the top right figure,
you can easily see that
the voltage is increasing as the cell is
being charged under
the constant current part of the profile.
Then when the voltage achieves
its maximum value of 4.15 volts,
you can see that the voltage is held constant
and steady until the end of the simulation.
This is exactly what we would hope to see.
In the bottom left figure,
you can see that the input current
is indeed held constant at
the constant current value until
the maximum voltage is achieved
around time 1,500 seconds,
after that the magnitude of current decreases
towards zero as the terminal voltage is held constant.
By the way, this is exactly what we see
when we're charging real battery cells also,
this is not an artifact of simulation.
While the simulation uses a model that
is not completely precise,
the trends that you see and the basic idea that you see
here is exactly what we see when we're
charging a battery cell in the laboratory.
In the top left figure, you can see state of
charge and how it increases
linearly over time during
the constant current portion of the test.
Then you see that when we switch to
the constant voltage portion of the test,
the state of charge
still increases but not as rapidly and so
the slope becomes non-constant as
state of charge slowly
approaches the desired final values.
By the way, this is a demonstration
of one reason why it's possible
for many systems to charge quickly up
to about 80 or 90 percent state of charge.
But then why it takes so much longer
to charge up to the final state of charge,
the full state of charge of a battery pack.
The reason is if we charged
quickly all the way up to a 100 percent state of charge,
we would exceed the
manufacturer voltage ratings of the cell.
But we can get maybe 80 percent of
the way quite quickly without
exceeding those voltage limits.
Finally, the bottom right figure shows power versus time.
For the first interval,
we know that current remains constant but voltage is
increasing and so the magnitude of
power is actually increasing.
Then for the second interval,
the magnitude of current is
decreasing while voltage is held constant
and therefore the magnitude of power decreases over time.
Notice I've been careful talking about
magnitude of current and magnitude of power.
When we're charging both of
these quantities are negative,
so I am talking about the size decaying,
perhaps, towards zero where you can see that
the signed value is actually
increasing but the magnitude is decreasing.
That brings us to the end of this lesson and in summary,
you've learned that our discrete-time, enhanced,
self-correcting battery model of cell voltage has
a fixed portion and
an instantaneous portion at every time step.
This observation is really important,
it turns out to be
critical for everything that we're going to look
at this week regarding different ways of
simulating battery cells and battery packs.
The specific example that we looked at in this lesson,
is having an ability to
simulate a constant voltage scenario.
We applied this to
learning how to simulate a constant current,
constant voltage charging profile.
In the next lesson, you will learn how to
exploit the same feature of the cell model having
a fixed portion and
an instantaneous portion to be
able to simulate a constant power,
constant voltage charging profile as well.
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