Remember that in the first course of this specialization,

you learned that the power limit calculations are

predictive and I will review this idea with you here.

At every point in time,

we must compute a value of power that we transmit to the load.

That the load is permitted to use for the next future time horizon of

Delta T seconds and we are guaranteeing that it is safe for the load to do that.

So, at this point in time,

I might predict that 10 kilowatts of

discharged power is acceptable for the next 10 seconds.

So, if the load discharges the battery pack at

a constant level of 10 kilowatts for 10 seconds,

then I'm guaranteeing that the battery will be completely

safe over that interval and no voltage limits or

current limits or power limits will be violated if the load does this.

In particular, we're guaranteeing right now that no cell voltage can go below

some minimum design voltage threshold and also

perhaps that no state of charge will go below some design threshold and so forth.

So, we compute the power limit,

we transmit it to the load and we say that this power limit

is guaranteed for the next Delta T seconds,

but then we keep on recomputing the power limits and we transmit them

to the load much more frequently than once every Delta T seconds.

This allows us to have some error in our power limits calculations and

then have those errors be corrected before the load has caused damage.

If at some point we decide that the power that we first

computed was too much and it would cause some damage,

some voltage is to be too high or too low.

So, the name that I give this idea and perhaps others do

also is a moving horizon power limit calculation.

We're looking over some future time horizon and giving a power limit estimate,

but as we move in time,

we keep on looking at that time horizon as it's progressing into the future.

The remainder of this particular lesson is devoted to

formalizing the problem that we desire to solve.

Later lessons, we'll look at how to compute power limits based on this formal definition.

So, specifically, our objective is to

address one of the following three different problems.

The first has to do with computing the level of permitted discharged power.

We base this computation on the present battery packs state.

We estimate from that starting point

the maximum discharge power that can be maintained constant

over a future time horizon of Delta T seconds without

violating some preset design limits on cell terminal voltage,

cell's state of charge,

maximum design power or maximum design current.

The second problem that we might address is actually exactly the same,

but it has to do with charged power instead of discharge power.

So, we begin with the present battery packs state,

and we compute a maximum absolute charged power that can be maintained constant over

a Delta T second future time horizon without violating

preset design limits on voltage or state of charge or power or current.

The third problem is simply a combination of the previous two.

Many applications require that we compute limits

both on discharged power and on charge power,

and they might have different Delta T second

moving time horizons for the two different computations.

So, our calculation algorithm that we come up

with must be general enough to allow for all of these possibilities.

When we approach these problems to solve them,

we are going to use some specific notation that I introduced on this slide.

We will use the letter N to denote the total number of cells in a battery pack.

We will use the notation v sub n of t to

indicate cell voltage for cell number N in the battery pack at

time t. This terminal voltage must always remain inside of

design limits that span from V min up to V max,

the minimum to the maximum voltage.

We use the symbol Z subscript N as a function of

time to indicate state of charge of cell N in the battery pack

at time t. Every state of charge must remain within

the design limit spanning from Z min up to Z max.

We use the notation P subscript N of t

to denote the cell power of cell number N in the battery pack at

time t. Each one of these powers must remain

within limits spanning from P min up to P max.

Finally, we denote cell current as I subscript N of t where we enforce

design limits on every cell's current that must exist between i min and i max.

When we solve the power limit problem including all of these limits,

it might seem like we are solving a very specific problem.

But, we actually can see this as a very general problem.

There are many scenarios that can be really accommodated and by

this problem if we allow some of the limits to go to infinity for example.

So, if there are no design limits on minimum and maximum current,

then we let i min go to negative infinity and i

max go to positive infinity and everything that we look at will still work.

Or if there is no design limits on power,

we set the minimum power to negative infinity and

the positive power to maximum infinity and so forth.

So, the method that you will learn about is actually very general and it

encompasses pretty much any scenario that I

could imagine that you need to compute power limits for.

The limits that we talked about on the previous slide do not need to be static or fixed.

They can be functions of temperature,

they could be functions of any other particular factor

that depends on the battery pack operating condition.

We could even apply different limits to different cells in

the battery pack if we had some reason for doing so.

In our equations, we will assume that discharge current and discharge power

both have a positive sign and that

charge current and charge power both have negative sign.

This is the standard convention that we've been

using throughout the entire specialization.

But if you choose to use the opposite sign convention, of course,

it's quite simple to do so simply by

multiplying by negative one in the appropriate places.

As a final node on notation,

the battery pack under consideration is assumed to comprise N

as cell modules wired in series and then within each cell module,

Np cells wired together in parallel.

Either, one of these counts,

Ns or Np may be equal to one or greater than one,

and then of course the total number of cells and

the battery pack is equal to Ns multiplied by Np.

To summarize this lesson,

we've started studying the final major topic in this specialization which

has to do with estimating cell power and battery pack power limits.

The question that we're trying to answer is,

how quickly may we add or remove energy from

the battery pack without violating some preset design limits?

As a review, I reminded you that we estimate

these power limits over a moving future time horizon of Delta T seconds.

The overall problem is to estimate either

discharge power or charge power or both over that horizon.

Then we update these power limit estimates at

a rate that's faster than once every Delta T seconds,

more than once per future time horizon interval.

Then after this review of moving time horizon estimation,

I introduced some notational conventions that we will use.

So, we're now ready to study some methods to estimate power.