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.
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:
\(d_{d,t}\): actual consumption,
\(Dem_{d,t}\): baseline demand (MW),
\(dsm_{d,t}\): load shifted via demand-side management (MW). Positive values increase consumption, negative values reduce it.
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:
\(Dur_{t}\): duration of time block in hours,
\(LSP_{d}\): load shifting potential (in hours),
\(AD_{d}\): average demand.
These bounds enforce that DSM does not inject or absorb more energy than its defined storage-like capability.
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:
\(ens_{d,t}\): unserved energy at time ,
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:
\(g_{g,t}\): generation,
\(pump_{g,t}\): pumping load (if any),
\(s_{dis,s,t}\): storage discharging,
\(s_{cha,s,t}\): storage charging,
\(ens_{d,t}\): unserved energy,
\(d_{d,t}\): net consumption,
\(flow_{l,t}\): flow from the current node,
\(loss_{l,t}\): power losses.
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.