# 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.

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**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:**

{math}`P_{g,t} \le P_{max,g} · u_{g,t} · A_{g,t}`

where:

* {math}`P_{max,g}` is the generator’s installed capacity,
* {math}`u_{g,t}` is the binary online status variable (1 if on, 0 if off),
* {math}`A_{g,t}` is a continuous availability factor (e.g., derating, resource availability, outages, fuel limits).

**Minimum Generation Constraint:**

{math}`P_{g,t} \ge P_{min,g} · u_{g,t}`

This ensures that generators operate above their technical minimum output when they are online.

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**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.

{math}`\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.

{math}`w_{g,t} \ge u_{g,t-1} - u_{g,t} `

**Warm and Cold Startup Enforcement Constraints:** Let {math}`T^{warm}_{g}` and {math}`T^{cold}_{g}`, be the thresholds for warm and cold starts respectively:

If the unit has been off for at least {math}`T^{warm}_{g}`:
{math}`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 {math}`T^{cold}_{g}`:
{math}`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.

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**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:

{math}`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:

{math}`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.

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**4. Ramp Constraints**

Capture gradual changes in output:
{math}`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} `
{math}`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.

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**5. Heat Rate Modeling (Non-Convex Efficiency)**

Thermal efficiency is modeled via **segmented heat rate curves** as follows:

*Fuel consumption:*
{math}`f_{g,t} = \sum_{temp} σ_{g,t,temp} · SF_{temp} + \sum_{seg} gseg_{g,t,seg} · HR_{g,seg} `

*Generation per segment:*
{math}`g_{g,t} = \sum_{seg} gseg_{g,t,seg} `

*Ensures the segments are activated in order:*
{math}`gseg_{g,t,seg} \ge gseg_{max,g,t,seg} · gact_{g,t,seg} `
{math}`gseg_{g,t,seg+1} \le gseg_{max,g,t,seg+1} · gact_{g,t,seg} `

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**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.