What’s a Battery? 1. Definition and Formulation (the Two Period Case)

               First principles: a battery is a machine with limited capacity to store energy, whose primary purpose is to store energy for later use.  This definition captures the need for a battery to be a machine (excluding living creatures whose metabolic functions might be described in this way); whose capacity is limited (i.e., the boundaries of the description of the machine must include its limits, in contrast to hypothetical machines with internal capabilities to increase their capacities, such as systems storing compressed gasses or liquids in underground caverns of unknown size); with a primary purpose of storing energy (as opposed to machines for which storage is an incidental function or part of a different productive process); and whose storage is for use later in time (i.e., it is unable to borrow energy from a later point in time for use now, although a current use could certainly be accompanied by a plan to add back to available energy at a later time).

From this definition a simple formal math representation may be derived.  A battery’s limited capacity may be represented by C.  At least two time periods should be designated, a period in which energy is stored and a later period in which energy is used.  If the entire capacity of a battery is considered usable, then efficiency losses during use (sometimes called “discharge”) are internalized in the representation of available energy e; that is, energy up to the amount e is considered usable, although in fact the energy required to create e “units of usable energy,” whatever that means, might be larger than e.  If the process (sometimes called “charging”) of storing e units of usable energy required e units of energy, that process would be perfectly efficient with respect to this representation.  In general, ec units of energy are required to store e units of energy, where ec > e.  If e/ec does not vary depending on current storage or the amount stored in a given period, e/ec may be considered a constant efficiency factor denoted eff.  The amount of energy required to charge the battery fully may be represented by EC, where EC > C, and EC*eff=C for constant eff.

The implications of this simplest battery model for representing decisions about whether to charge or discharge are:

1.      A battery may begin the current period in a state of energy storage C0 between 0 and C.

2.      The battery may charge up to (C – C0) units of energy in the current period.  Charging will require an amount of energy greater than (C – C0) and less than EC.

3.      The battery may discharge up to C0 units of energy in the current period.

4.      The battery operator may decide to retain or increase energy stored until the end of the current period in order to have the option of using it in a future period.

5.      The optionality of future use does not equate to future use.

The economic value of this simple battery is maximized when the charge/discharge decisions made in the current period 0 and the later period 1 are such that they maximize the objective function “Maximize (Value of discharge in period 0 – Cost of charge in period 0) + (Value of optionality of discharge in period 1)”, which may be written

     Max DValue(d0) – Cost(ec0) + OValue(e1)

    where 1. d0 <= C0

               2. ec0 * eff = c0 < (C – C0)

               3. e1 = C0 + ec0 * eff – d0

               3. OValue(e1) >= DValue(e1)

Constraint 3 is a statement that the value of either optionality or energy beyond period 1 is zero, so that if the battery is used not for energy but some other purpose covered by “optionality,” that optionality must have a value higher than the value of discharged energy in that period.  The optionality value may be considered to include the value of energy beyond the horizon of the problem to be solved: i.e., it includes a liquidation value.