A battery whose value is realized over two periods, starting from no initial storage (i.e., C0 = 0 or Cmin if the battery is represented as having a nonzero minimum energy state, but Cmin is not usable during period 0 or period 1) cannot discharge in period 0, assuming each period allows only one directional decision on energy usage. If a single period can include both charge and discharge, then there may be multiple values associated with a single net charge or discharge, and the maximum net value may be taken as the value associated with an equivalent single decision to charge or discharge.
Period 0 requires a net charge greater than or equal to zero, and only the net energy resulting from the Period 0 charge is available at the beginning of Period 1. The option value OValue(e1) is associated with the energy above Cmin available at the beginning of Period 1.
In the pure two-period model, energy remaining at the end of Period 1 has no value, so that the optionality value must exceed the energy value for the battery to be used for its optionality rather than its energy in Period 1; with no loss of generality, the option value may be taken to include energy value DValue(e1) if the energy value exceeds the option value.
The battery in this case must receive, or expect to receive, revenues in Period 1 that exceed the costs of storing energy in Period 0. If prices are known with certainty, the battery can be scheduled to charge enough to cover maximum discharge in Period 1 if the arbitrage is profitable. In the case that prices are not known, a schedule may not yield profits with certainty, though it may in expectation. Of course, I want to apply this model to the case of participation in an energy market, that is to say a case of repeated market runs, but I’ll note the model at this point isn’t quite ready to be used for that purpose: for one thing, when a battery charges because of expected revenues from discharge, but doesn’t discharge, that alters the next market run’s initial storage from the “no initial storage” case.
The issue of end-of-horizon storage can be assumed away by requiring the battery be returned to its “no initial storage” state at the end of Period 1. We can imagine real world cases that approximate such a constraint, such as batteries with operational constraints that require zero storage at the end of the scheduling horizon (perhaps because the use for charging in each repeated market’s “Period 0” is contractually required, as it is for solar+storage hybrid resources), or system conditions that very reliably make use of a battery’s entire storage capacity by the end of the scheduling horizon. In such cases, every day is a new day unless the battery operator is employing game-theoretic strategies that yield differential outcomes against other strategic players (i.e., partial or no combined charge-discharge schedules on some days in exchange for other more lucrative days).
Aside from making a self-scheduling discharge to charge and then discharge based on forecast values (more typically, electricity prices), another application of the “no initial storage” case that doesn’t require modeling of repeated market runs is to use the concept of a contingent two-part bid, often referred to as an “arbitrage bid” by those who would like to see something like it implemented in wholesale markets. The contingent two-part bid consists of a paired charge bid and discharge bid, such as that one of the two bids is accepted only if the other is also accepted. The beauty of reducing the battery model to two periods with no initial storage is that it reveals the difficulties, not to say impracticality, of accepting contingent two-part bids from more than one battery: although the difference between “discharge price” and “charge price,” also known as the arbitrage, can be nicely ordered among batteries, there is not necessarily a monotonic ordering of the bid pairs behind these arbitrages based on the market objective function (usually the minimization of total electricity costs required to match total supply and total demand). In fact, contingent bids for multiple two-period batteries would require a mixed integer linear program formulation which by itself could be an arbitrarily hard combinatoric problem: potentially harder, in other words, than the traditional electric system unit commitment problem that serves as the backbone of all modern electricity markets.
This very simple model allows consideration of a contentious point in the calculation of “default energy bids” that may in principle be imposed on batteries as a version of cost-based bidding by system operators or regulators, in cases where market power is or might be exercised to the disadvantage of the market as a whole. The California ISO, in its ESDER 4 market enhancements initiative, proposed a default energy bid calculation as
DEB = max( charging cost/efficiency + a variable cost term, opportunity cost of discharge)
In the two period case, it’s clear that the variable cost term can be adjusted for any single market run to equal (opportunity cost of discharge – charging cost/efficiency), so that the first term in the maximum equals the second. Discussion of the DEB calculation has focused on the question of whether the opportunity cost term is needed or legitimate in the day ahead market, when the CAISO as system operator can at least in theory determine the optimal combination of charging and discharge that maximizes the value of the battery.
A proposal not to include opportunity cost in the DEB calculation must be based on an assumption that a variable cost component, which as a “master file” parameter does not vary from one day to the next, is equivalent to an opportunity cost component based on the anticipated benefit of using the battery in the current market versus reserving it for future use. But the terms are clearly not equivalent under any price uncertainty: in particular, the combination of a charging cost term that varies daily (per the DEB proposal) with an invariant variable cost term will clearly not be equivalent to an opportunity cost term that varies daily.
It is clear that a battery operator’s objective (and it might be supposed, the CAISO’s long-term objective) should be to see the battery discharged in markets with the highest values over the lifetime of the battery, and that the variable cost term by itself would not lead to such highest value discharges; instead, it would lead to the battery being discharged in every market for which discharge was economic (i.e., had non-negative net revenue) until the battery’s degradation made such discharge impossible.
