# Title

**Introduction**

KAIROS models short-term flexibility sources on the demand side (DSM) to simulate operational responses over sub-hourly to daily horizons. These mechanisms allow modeling of load shifting and flexible demand behavior. The following constraints ensure that these technologies are correctly represented in the optimization problem.

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**1. Power consumption incorporating DSM**

The consumption by each demand category includes the effect of DSM, which can shift load positively or negatively:

$$
d_{d,t} = Dem_{d,t} + dsm_{d,t} 
$$

Where:

* {math}`d_{d,t}`: actual consumption,
* {math}`Dem_{d,t}`: baseline demand (MW),
* {math}`dsm_{d,t}`: load shifted via demand-side management (MW). Positive values increase consumption, negative values reduce it.

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**2. DSM constraints (Upper and Lower Bounds)**

DSM is modeled as a virtual storage that must return to neutrality over time:

**Upper Bound (DSM discharge limit):**
$$
AD_{d} · LSP_{d} \ge sum_{tt=1}^{t-1} dsm_{d,t} · Dur_{t} 
$$

**Lower Bound (DSM charge limit):**
$$
-AD_{d} · LSP_{d} \le sum_{tt=1}^{t-1} dsm_{d,t} · Dur_{t} 
$$

Where:

* {math}`Dur_{t}`: duration of time block in hours,
* {math}`LSP_{d}`: load shifting potential (in hours),
* {math}`AD_{d}`: average demand.

These bounds enforce that DSM does not inject or absorb more energy than its defined storage-like capability.

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**3. Unserved Energy (ENS) Constraint**

KAIROS limits unserved energy as a fraction of average demand to maintain reliability:
$$
ens_{d,t} \le AD_{d}
$$

Where:

* {math}`ens_{d,t}`: unserved energy at time ,

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**4. Node-Level Energy Balance Constraint**

Each node in the network maintains an energy balance that includes all sources and sinks:
$$
\sum_{g(n)} g_{g,t} - \sum_{g(n)} pump_{g,t} + \sum_{s(n)} ( s_{dis,s,t} - s_{cha,s,t} ) + \sum_{d(n)} ens_{d,t} - \sum_{l(n)} flow_{l,t} - \sum_{l(n)} 0.5 · loss_{l,t} =  
sum_{d(n)} d_{d,t} - \sum_{l(n)} 0.5 · loss_{l,t} 
$$

Where:

* {math}`g_{g,t}`: generation,
* {math}`pump_{g,t}`: pumping load (if any),
* {math}`s_{dis,s,t}`: storage discharging,
* {math}`s_{cha,s,t}`: storage charging,
* {math}`ens_{d,t}`: unserved energy,
* {math}`d_{d,t}`: net consumption,
* {math}`flow_{l,t}`: flow from the current node,
* {math}`loss_{l,t}`: power losses.

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**Conclusion**

These constraints support the realistic and effective modeling of short-term storage and DSM flexibility in KAIROS, enabling more resilient and cost-efficient energy system operation under variable renewable conditions and demand fluctuations.