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formation of water has a negligible effect on the capacity. Here, we referred to such electrolyte maintenance as battery servicing. ππ@ +πΆ π» π βππE@ +2πΆπ +π» π (II-4) EEEJ EE Thus, we have an overall linear fade rate (Rfade, in units of % capacity loss per cycle), which encompasses fade from electrolyte decay (rED, also in units of % capacity loss per cycle) and fade from crossover (Rfade-rED), both of which are also assumed linear. The product of fcap, the fraction of original capacity accessed at any given time, and bs, the nominal battery size, determines the capacity accessed in a given cycle. The fraction of original capacity accessed is a function of Rfade, as calculated in Equation II-5: f =cap (t=R)-R !nR (II-5) cap max fade cyc where πK is the number of cycles passed since the last rebalancing event (i.e., t = R), and capmax !/! is the maximum accessible capacity. The capmax decreases as the average oxidation state drifts with each cycle as a function of rED. The capmax can be defined by Equation II-6: cap (t)=100% - r ! nS (t) (II-6) max ED cyc where πL is the number of cycles passed since the last servicing event (at which point capmax = !/! 100%, by definition). These cycle counters are defined in Equation set II-7: 1. At the beginning of the battery life, the capacity is 100%. For simplicity, the nominal capacity is defined as the product of the power rating and duration of the battery. In actuality, the accessible capacity of a battery is dependent on many factors including current density [77], the depth of discharge [78], and average oxidation state [79], so impacts of this assumption should be explored in future work. 2. Capacity fades linearly at a specified rate (Rfade) according to Equation II-5. It is assumed that the battery undergoes one charge/discharge cycle per day during the expected lifetime. 3. Rebalancing occurs when the capacity fades to a predefined lower limit (caplim). The charging cost is scaled by the number of electrons required per vanadium ion to charge the battery to a nR/S (t) = iΜ0, t = 0 or upon rebalancing (R) or servicing (S) cyc iΜnR/S (t -1) +1, t > 0 and not rebalancing (R) or servicing (S) (II-7) iΜ cyc The following assumptions have been used within the modeling framework: 26PDF Image | Bringing Redox Flow Batteries to the Grid
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