# Storage characterization

**Introduction**

KAIROS supports accurate modeling of electricity storage systems, incorporating key operational constraints and temporal dynamics. These constraints are essential for representing short- and long-duration storage assets in unit commitment, dispatch optimization, and energy planning models. The formulation includes energy balance equations, operational limits, and power flow restrictions.

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**1. Energy Balance Equation**

The energy level of a storage unit evolves based on charging and discharging activity:

$$
SL_{s,t} - SL_{s,t-1} = ( s_{cha,s,t} · S_{CEf,s,t} - s_{dis,s,t} / S_{DEf,s,t} ) · Dur_{t} 
$$ 

Where:
* {math}`SL_{s,t}`: energy stored at time (in MWh),
* {math}`s_{cha,s,t}`: charging power (MW),
* {math}`s_{dis,s,t}`: discharging power (MW),
* {math}`S_{CEf,s,t}`: charging efficiency (unitless),
* {math}`S_{DEf,s,t}`: discharging efficiency (unitless),
* {math}`Dur_{t}`: duration of each time step (in hours).

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**2. Maximum Storage Level**

The energy stored in the system must remain within physical bounds:

$$
SL_{s,t} \le S_{lmax,s,t} 
$$

Where:
* {math}`S_{lmax,s,t}`: is the maximum storage capacity (in MWh),

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**3. Maximum Charging Rate**

Charging is limited by converter and infrastructure capacity:

{math}`s_{cha,s,t} \le S_{cmax,s,t} `

Where:
* {math}`S_{cmax,s,t}`: is the maximum allowable charging power (in MW).

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**4. Maximum Discharging Rate**

Similarly, the discharging power is constrained by technical ratings:

{math}`s_{dis,s,t} \le S_{dmax,s,t} `

Where:
* {math}`S_{dmax,s,t}``: is the maximum discharging power (in MW),

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**Conclusion**

These constraints provide a comprehensive and flexible representation of electricity storage within KAIROS. They enable detailed analysis of energy shifting, arbitrage, grid support, and the role of storage in high-renewable energy systems.