Battery Model

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Summary

The battery model tracks the internal state of the energy storage system and derives the resulting power flows at each timestep. It runs in sequence: degradation updates the battery’s capacity and efficiency parameters, then the state of charge is updated based on available charge/discharge power, and finally DC and AC power are derived from the change in stored energy. Round-trip DC efficiency losses are applied during charging, and inverter efficiency losses are applied to both charge and discharge.

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Inputs

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Outputs

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Detailed Description

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Initial Conditions

At the first timestep ($t = 1$), the battery is initialized to full charge with year-1 parameters: $$SOC_1 = E_{max,1}, \qquad \eta_{RTE,1} = \eta_{RTE,year1}, \qquad P_{DC} = 0$$

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Degradation

For all subsequent timesteps ($t \geq 2$), degradation is calculated before the SOC update so that the battery operates with its current degraded parameters.

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System Age

$$\text{SystemAge} = \frac{t}{8760 \times 60 / \Delta t}$$ where 8760 is the number of hours per year. The denominator converts from timesteps to years.

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Cycle Degradation

Cycle degradation accumulates based on the previous timestep’s discharge energy, normalized by the previous maximum capacity. This means each discharge cycle degrades the battery proportionally to its depth. Cumulative capacity cycle degradation: $$D_{E,cycle} = \frac{P_{AC,d,prev} / \eta_{inv}}{E_{max,prev}} \times d_{E,cycle} + D_{E,cycle,prev}$$ Cumulative RTE cycle degradation: $$D_{\eta,cycle} = \frac{P_{AC,d,prev} / \eta_{inv}}{E_{max,prev}} \times d_{\eta,cycle} + D_{\eta,cycle,prev}$$ The term $P_{AC,d,prev} / \eta_{inv}$ converts the previous AC discharge back to DC to normalize against the DC-rated capacity.

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Calendar Degradation

Calendar degradation is proportional to system age, independent of usage: Cumulative capacity calendar degradation: $$D_{E,cal} = d_{E,cal} \times \text{SystemAge}$$ Cumulative RTE calendar degradation: $$D_{\eta,cal} = d_{\eta,cal} \times \text{SystemAge}$$

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Degraded Parameters

Cycle and calendar degradation effects are additive: $$E_{max} = E_{max,1} \times \left(1 - (D_{E,cycle} + D_{E,cal})\right)$$ $$\eta_{RTE} = \eta_{RTE,1} \times \left(1 - (D_{\eta,cycle} + D_{\eta,cal})\right)$$

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State of Charge

The SOC update uses the degraded $E_{max}$ and $\eta_{RTE}$ from above. When both charge and discharge flags are set (possible with the LGIA Excess algorithm), the charge flag takes priority. Charging (Version 12): $$SOC = \min\!\left( SOC_{prev} + P_{charge} \times \frac{\Delta t}{60} \times \eta_{RTE},\; E_{max} \right)$$ Charging (Versions 3–11): $$SOC = \min\!\left( SOC_{prev} + P_{charge} \times \eta_{RTE},\; E_{max} \right)$$ Discharging (Version 12): $$SOC = \max\!\left( SOC_{prev} - P_{discharge} \times \frac{\Delta t}{60},\; 0 \right)$$ Discharging (Versions 3–11): $$SOC = \max\!\left( SOC_{prev} - P_{discharge},\; 0 \right)$$ Idle ($F_c = \text{FALSE}$ and $F_d = \text{FALSE}$): $$SOC = SOC_{prev}$$ The $\Delta t / 60$ factor converts the power (W) over the timestep interval (minutes) to energy (Wh). Round-trip DC efficiency ($\eta_{RTE}$) is applied only during charging — the full charge power enters the battery, but only a fraction is stored. During discharge, no RTE loss is applied; the stored energy is released directly.

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Battery DC Power

DC power is derived from the change in SOC, reversing the efficiency adjustments applied during the SOC update. Charging (SOC increased, Version 12): $$P_{DC} = \frac{(SOC - SOC_{prev}) / \eta_{RTE}}{\Delta t / 60}$$ Charging (Versions 3–11): $$P_{DC} = \frac{SOC - SOC_{prev}}{\eta_{RTE}}$$ Discharging (SOC decreased, Version 12): $$P_{DC} = \frac{SOC - SOC_{prev}}{\Delta t / 60}$$ Discharging (Versions 3–11): $$P_{DC} = SOC - SOC_{prev}$$ $P_{DC}$ is positive during charging and negative during discharging.

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Battery AC Power

The inverter converts between DC and AC. Efficiency losses apply in opposite directions: AC Discharge (when $P_{DC} < 0$): $$P_{AC,d} = -P_{DC} \times \eta_{inv}$$ AC Charge (when $P_{DC} > 0$): $$P_{AC,c} = \frac{P_{DC}}{\eta_{inv}}$$ During discharge, the inverter converts DC to AC with efficiency losses. During charge, the inverter converts AC to DC, requiring more AC input than DC output.

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Loss Reporting

Two losses are calculated for reporting purposes. They do not appear as separate steps in the power flow — they are embedded in the SOC and AC power calculations above. DC Round-Trip Efficiency Loss (charging only, when $P_{DC} > 0$): $$L_{RTE} = (1 - \eta_{RTE}) \times |P_{DC}|$$ Inverter Efficiency Loss (charge and discharge): $$L_{inv} = (1 - \eta_{inv}) \times |P_{DC}|$$

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