Thermal and renewable generation
Introduction
KAIROS models thermal and renewable power generation with high fidelity, capturing operational and physical constraints that govern real-world generator behavior. These constraints play a critical role in producing realistic and implementable unit commitment and dispatch schedules, particularly in systems with high renewable penetration, tight reserve margins, or strong technical requirements.
1. Generation Capacity Limits
KAIROS enforces upper and lower bounds on generator output at every time step, based on physical constraints and availability profiles.
Maximum Generation Constraint:
\(P_{g,t} \le P_{max,g} · u_{g,t} · A_{g,t}\)
where:
\(P_{max,g}\) is the generator’s installed capacity,
\(u_{g,t}\) is the binary online status variable (1 if on, 0 if off),
\(A_{g,t}\) is a continuous availability factor (e.g., derating, resource availability, outages, fuel limits).
Minimum Generation Constraint:
\(P_{g,t} \ge P_{min,g} · u_{g,t}\)
This ensures that generators operate above their technical minimum output when they are online.
2. Startup Costs and Temperature States
Thermal units transition between hot, warm, and cold start modes. Their startup cost and startup time depend on the temperature state , determined by downtime duration.
Startup Trigger Constraint:
This ensures that a startup (i.e., transition from off to on) activates exactly one of the startup state variables.
\(\sum_{temp} σ_{g,t,temp} \ge u_{g,t} - u_{g,t-1} \)
Shutdown Trigger Constraint:
This defines the shutdown variable as the difference in online status from the previous period, capturing a transition from on to off.
\(w_{g,t} \ge u_{g,t-1} - u_{g,t} \)
Warm and Cold Startup Enforcement Constraints: Let \(T^{warm}_{g}\) and \(T^{cold}_{g}\), be the thresholds for warm and cold starts respectively:
If the unit has been off for at least \(T^{warm}_{g}\): \(σ_{g,t,warm} +σ_{g,t,cold} \ge u_{g,t} - \sum_{k=t-Twarm}^{t-1} u_{g,k} \)
If the unit has been off for at least \(T^{cold}_{g}\): \(σ_{g,t,cold} \ge u_{g,t} - \sum_{k=t-Tcold}^{t-1} u_{g,k} \)
These constraints force the selection of warm or cold startup categories when the generator has remained offline long enough.
3. Minimum Up/Down Time Constraints
To prevent unrealistic cycling, KAIROS enforces logical consistency in generator operation with respect to their minimum required online and offline durations.
Minimum Up Time Constraint: If a unit has started within the last periods, it must remain online:
\(u_{g,t} \ge \sum_{k=t-Ton}^{t-1} \sum_{temp} σ_{g,k,temp} \)
This ensures that once a generator is started, it stays online for at least its minimum up time.
Minimum Down Time Constraint: If a unit has been shut down within the last periods, it cannot be brought back online:
\(u_{g,t} \le 1 - \sum_{k=t-Toff}^{t-1} w_{g,k} \)
This ensures that a generator remains offline for at least its minimum down time after being shut down.
These constraints support a realistic representation of thermal generator behavior over time.
4. Ramp Constraints
Capture gradual changes in output: \(P_{g,t} - P_{g,t-1} \le RUp_g + P_{max,g} · \sum_{k=t-Toff}^{t-1} \sum_{temp}σ_{g,k,temp} \) \(P_{g,t} - P_{g,t-1} \ge RDn_g - P_{max,g} · \sum_{k=t-Toff}^{t-1} w_{g,k} \)
Where , ensure output jumps allowed during transitions.
5. Heat Rate Modeling (Non-Convex Efficiency)
Thermal efficiency is modeled via segmented heat rate curves as follows:
Fuel consumption: \(f_{g,t} = \sum_{temp} σ_{g,t,temp} · SF_{temp} + \sum_{seg} gseg_{g,t,seg} · HR_{g,seg} \)
Generation per segment: \(g_{g,t} = \sum_{seg} gseg_{g,t,seg} \)
Ensures the segments are activated in order: \(gseg_{g,t,seg} \ge gseg_{max,g,t,seg} · gact_{g,t,seg} \) \(gseg_{g,t,seg+1} \le gseg_{max,g,t,seg+1} · gact_{g,t,seg} \)
Conclusion
KAIROS’s thermal unit formulations integrate startup states, ramping behavior, non-convex heat rates, emissions, and minimum on/off times. These constraints form a technically rigorous, operationally realistic foundation for unit commitment, dispatch, market pricing, and resilience analysis in modern energy systems.