Frequency Regulation of a Power Grid
I have a battery that is used to regulate the frequency of a power grid. That is, as the grid frequency varies about it’s ideal value, $f_{nom}$, the battery imports and exports power from and to the grid to keep the frequency within its operational limits. The battery charges (imports energy) from the grid when the frequency goes above from, and conversely when the frequency goes below $f_{nom}$, the battery discharges (exports energy).
If we consider the grid frequency as a random variable, F - lets say it is sampled every 1 second for a year - think of it as a time series. F is distributed about a mean, $f_{nom}$. Let’s assume it’s normally distributed.
Q1) I would like to know the number of seconds in a year (for a given set of values for f) that f is within the range $f_{nom}$ to $+\infty$. That is, the amount of time the battery is importing energy from the power grid.
I have a function $s(k)$ that models the state of charge (the amount of energy within the battery, the unit of measure is kWh). This is given by:
$s(k+1) = s(k) + \eta P(f).dt , when P(f) is \geq0$
$s(k+1) = s(k) + \frac{1}{\eta}P(f).dt , when P(f) is <0$
$P(f)$ is some function that represents the amount of instantaneous power in/out of the battery, as the frequency varies as defined above. $s(k)$ goes up and down as the battery charges and discharges according to the varying grid frequency.
Q2) For a given time series represented by f, what is the maximum value s reaches. This represents the minimum size the battery energy needs to be to regulate a given frequency series represented by f.