# Hydropower

**Introduction**

Hydropower modeling in KAIROS captures the essential operational and environmental aspects of reservoir and run-of-river systems. This includes accurate representation of power-water conversions, pumping operations, reservoir balances, and physical limitations of both storage and waterways. These formulations ensure realistic dispatch, planning, and reliability analysis involving hydroelectric resources.

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**1. Power-Water Coupling for Turbining and Pumping**

Power generation and pumping are directly linked to water flows via efficiency factors that translate MWh into m³:

**Turbination Constraint:**
{math}`p_{g,t} = a^{tur}_{g} · q^{tur}_{g,t}`

**Pumping Constraint:**
{math}`pp_{g,t} = a^{pump}_{g} · q^{pump}_{g,t}`

Where:

* {math}`p_{g,t}`, is the power generation (in MW),
* {math}`pp_{g,t}`, is the power consumption for pumping (in MW),
* {math}`q^{tur}_{g,t}`, is the turbination water use (in m³/h),
* {math}`q^{pump}_{g,t}`, is the pumping water use (in m³/h),
* {math}`a^{tur}_{g}`, are the water-to-energy conversion factors for turbining in m³/MWh,
* {math}`a^{pump}_{g}`, are the water-to-energy conversion factor for pumping in m³/MWh.

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**2. Maximum Pumping Capacity Constraint**

Pumping operations are limited by the mechanical capacity of the pumps:
{math}`p^{pump}_{g,t} \le p^{pump}_{max,g,t}`

Where is the installed pumping capacity (MW).

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**3. Reservoir Energy Balance**

The energy balance in the reservoir integrates storage level changes and water flows:
{math}`L_{h,t} - L_{h,t-1} = Dur_t · ( \sum_{g(h)} q^{pump}_{g,t} - q^{tur}_{g,t} - q^{spill}_{g,t} ) + \sum_{w(h)} w^{in}_{w,t} + rw^{in}_{h,t} - rw^{out}_{h,t} )`

Where:

* {math}`L_{h,t}`: storage level at time (in m³),
* {math}`Dur_{t}`: duration of time period (in hours),
* {math}`w^{in}_{w,t}`: incoming controlled water flows (e.g., upstream discharges),
* {math}`rw^{in}_{h,t}`: natural inflows (e.g., rainfall, snowmelt),
* {math}`rw^{out}_{h,t}`: agricultural and environmental outflows,
* {math}`q^{tur}_{g,t}`: water turbinated for power,
* {math}`q^{pump}_{g,t}`: water added via pumping,
* {math}`q^{spill}_{g,t}`: spillage (overflow).

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**4. Water Flow Balance**

Total water flow in the system must respect conservation and operational logic:
{math}`
w^{flow}_{w,t} = w^{in}_{w,t} - w^{out}_{w,t} + \sum_{g(w)} (q^{tur}_{g,t} + q^{spill}_{g,t} - q^{pump}_{g,t}) ) 
`

This balance ensures that all input and output flows are accounted for in the hydraulic cycle.

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**5. Reservoir and Waterway Limits**

Hydropower operations are constrained by physical bounds where reservoirs are constrained by their minimum and maximum reservoir capacities (in m³) and waterways are constrained by the operational flow limits of the waterway.

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**Conclusion**

These hydropower constraints allow KAIROS to model multipurpose reservoir systems with high realism, balancing energy, environmental, and operational needs across complex time horizons and geographies.