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.
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: \(p_{g,t} = a^{tur}_{g} · q^{tur}_{g,t}\)
Pumping Constraint: \(pp_{g,t} = a^{pump}_{g} · q^{pump}_{g,t}\)
Where:
\(p_{g,t}\), is the power generation (in MW),
\(pp_{g,t}\), is the power consumption for pumping (in MW),
\(q^{tur}_{g,t}\), is the turbination water use (in m³/h),
\(q^{pump}_{g,t}\), is the pumping water use (in m³/h),
\(a^{tur}_{g}\), are the water-to-energy conversion factors for turbining in m³/MWh,
\(a^{pump}_{g}\), are the water-to-energy conversion factor for pumping in m³/MWh.
2. Maximum Pumping Capacity Constraint
Pumping operations are limited by the mechanical capacity of the pumps: \(p^{pump}_{g,t} \le p^{pump}_{max,g,t}\)
Where is the installed pumping capacity (MW).
3. Reservoir Energy Balance
The energy balance in the reservoir integrates storage level changes and water flows: \(L_{h,t} - L_{h,t-1} = Dur_t · ( \sum_{g(h)} q^{pump}_{g,t} - q^{tur}_{g,t} - q^{spill}_{g,t} ) + \sum_{w(h)} w^{in}_{w,t} + rw^{in}_{h,t} - rw^{out}_{h,t} )\)
Where:
\(L_{h,t}\): storage level at time (in m³),
\(Dur_{t}\): duration of time period (in hours),
\(w^{in}_{w,t}\): incoming controlled water flows (e.g., upstream discharges),
\(rw^{in}_{h,t}\): natural inflows (e.g., rainfall, snowmelt),
\(rw^{out}_{h,t}\): agricultural and environmental outflows,
\(q^{tur}_{g,t}\): water turbinated for power,
\(q^{pump}_{g,t}\): water added via pumping,
\(q^{spill}_{g,t}\): spillage (overflow).
4. Water Flow Balance
Total water flow in the system must respect conservation and operational logic: \( w^{flow}_{w,t} = w^{in}_{w,t} - w^{out}_{w,t} + \sum_{g(w)} (q^{tur}_{g,t} + q^{spill}_{g,t} - q^{pump}_{g,t}) ) \)
This balance ensures that all input and output flows are accounted for in the hydraulic cycle.
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.
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.