Coverage Variables and Constraints

A new LP variable is created for each demand (demand site, instream flow requirement or hydropower demand), which will equal its "coverage" (percent of demand satisfied).  The coverage (for demand D) cannot exceed 100%

CoverageD <= 100%

The inflow to a demand site (DS) from all its transmission links (Src) will equal its supply requirement times its coverage.

  InflowSrc,DS = SupplyRequirementDS x CoverageDS

which can be rewritten as

  InflowSrc,DS - SupplyRequirementDS x CoverageDS = 0

Whereas demand sites cannot receive more water than their supply requirement, instream flow requirements (IFR) and hydropower requirements (H) can be exceeded by actual flows.  Therefore, the equation is written as an inequality.

FlowIFR >= SupplyRequirementIFR x CoverageIFR

VolumeThroughTurbineH >= SupplyRequirementH x CoverageH

which can be rewritten as

FlowIFR - SupplyRequirementIFR x CoverageIFR >= 0

VolumeThroughTurbineH - SupplyRequirementH x CoverageH >= 0

The LP constraint would be a row in the LP matrix, with coefficients of 1 for the flow variables (inflow to demand sites, streamflow at the instream flow requirement node, or turbine flow), and -1 times the supply requirement as the coefficient for the coverage variable. The entire row would be set equal to 0 (demand sites) or greater than or equal to 0 (instream flow and hydropower requirements).

For example, if Demand Site A had a requirement of 100 units, but in solving the LP was allocated only 30, the coverage would be 30%:

30 =100 x 30%

If a system hydropower priority and energy demand has been set, WEAP will create LP variables and constraints for each reservoir or run of river hydropower contributing to the system hydropower demand, representing the hydropower energy generated at each hydropower plant (H).

EnergyGeneratedH = VolumeThroughTurbineH x HydroGenerationFactorH

For the coverage constraint, the system hydropower energy generated by all contributing hydropower plants (H) will equal or exceed the system hydropower energy demand times the system hydropower energy coverage.

EnergyGeneratedH >= SystemHydropowerEnergyRequirementD x CoverageD

which can be rewritten as

EnergyGeneratedH - SystemHydropowerEnergyRequirementD x CoverageD >= 0

Because WEAP tries to satisfy all demands with the same priority equally (in terms of percentage of demand), additional constraints are added to the LP. Each coverage variable is set equal to a new variable that represents the final coverage (CoverageFinal). In this way, all the coverages being solved for must be equal.

CoverageFinal =CoverageD1

CoverageFinal =CoverageD2