A method was required to account for the degradation of a battery's capacity as part of the NSW Farmers' PV + Battery Financial Analysis Calculator. Some preliminary desktop research showed that, in general, batteries have a 'signature' of degradation involving two stages. The first stage is characterised by a slow rate of capacity decline; it is followed by the second stage, characterised by an accelerated rate of decline, which is indicative of the end of the battery's cycle life. This is shown, as analysed for five individual lithium-ion batteries, in the figure below:
These degradation lines were indexed to represent movements against 100% of original capacity and 100% of their cycle life (end of cycle life was assumed to be indicated by the last Ah value and cycle recorded for each battery in Figure 1).
The two steps involved in this process are shown in the following figures:
Using this information, an average cubic formula was derived to estimate capacity degradation for any particular battery, given the portion of cycles (out of its total lifetime of cycles) that it has experienced:
So, for our current template function, the value of capacity retention at end of life (i.e. cycle life = 100% or x=1) is given by:
In other words, 76.88% of the capacity of the battery will remain by the completion of its cycle life.
This is shown plotted in the following figure.
*Note: The above equation is implemented piecewise, as negative values of cycle life (i.e. negative x values) and capacity retention below 0 (i.e. negative y values) are ignored.++
The next step of the process involves obtaining a single average factor to apply to de-rate the capacity of the battery over the analysis time frame. To do this, we calculate the average capacity retained in the battery over the course of the calculator's analysis period. This is merely the integral of the function up to the cycles used, divided by the cycles used, or:
Since x represents the portion of cycle life used (cycles used/ total cycle life), this eventually simplifies to:
For example, let’s assume that our battery has a cycle-life of 3,650 cycles and that for the length of our analysis, 3,376 cycles are used up. Thus:
So the battery has exhausted about 92% of its cycle life. The average capacity retention over the entire period of the analysis is therefore:
So over the period of the analysis, spanning 3,376 cycles used by the battery, the average available capacity in the battery will be 94.36% of its original capacity. We therefore use this factor to de-rate the capacity of the battery over the whole analysis period.
However, other batteries may experience similar degradation profiles but retain different capacities at the end of their cycle lives. So a final process involves modifying the function so that it may represent different batteries. This requires us to adjust the mapping of x to reflect the needed values. We use lookup tables (or a calculated inverse function) to find the appropriate adjustment factor by which to multiply our x values to give us the signature profile we require.
Our final x value (cycle life used during analysis) is then multiplied by our adjustment factor and entered into our ‘Average capacity over analysis’ function. The output from that function serves as a the de-rating factor over the capacity of the battery.
In the back-end of the calculator tool, the derived functions shown here are not used plainly, as this would introduce errors when the functions' results drop below zero. Instead, the calculator uses a prepopulated lookup table to obtain the 'adjusted x' value and calculates the derating factor by averaging the values up to the 'adjusted x' of a table listing the output of the 'capacity retained function' that has negative values truncated to zero.
Although this overall method is unequivocally primitve, it is adequate for the level of analysis that is required for the Solar PV + Battery Financial Analysis Calculator. More importantly, it avoids the need to do an iterative 15-minute interval analysis for the 10+ years of the model that, while more accurate, would slow the calculator down significantly and result in a decidedly unpleasant online experience.
In short, the calculator will de-rate a battery in several steps.
For instance, assuming that a given battery has a rated capacity of 7kWh...
- only 6.4kWh is useful, due to depth-of-discharge limitations;
- degradation de-rating means an average of only 5.99kWh of available capacity over the analysis period of 10 years;
- additional discharge and charging efficiencies are then considered in the iterative analysis;
- by the end of the analysis, the battery has 287 cycles left but only 79% of its capacity retention.
If you would like to see these calculations in action, check out the NSW Farmers' Solar PV + Battery Financial Analysis Calculator.