Supplement A - Mathematical Formulation

3.1 Modelling Approach

scope

3.2 Mathematical Formulation

Decision variables, parameter..

3.3 Objective function

Objective Function

The objective is to minimize the total discounted system cost, which includes generation, storage, and transmission costs, while subtracting the discounted salvage values at the end of the planning horizon:

$\min \; Z = \sum_{y \in \text{year}} \left[ \sum_{r \in \text{regions}} \left( \sum_{t \in \text{technologies}} \frac{\text{TotalCost}_{y,r,t}}{(1+\text{DiscountRate})^{(y - \min(\text{year}))}} + \sum_{s \in \text{storages}} \frac{\text{TotalStorageCost}_{y,r,s}}{(1+\text{DiscountRate})^{(y - \min(\text{year}))}} \right) + \sum_{l \in \text{links}} \frac{\text{TotalTransmissionCost}_{y,l}}{(1+\text{DiscountRate})^{(y - \min(\text{year}))}} \right] - \sum_{y \in \text{year}} \left[ \sum_{r \in \text{regions}} \left( \sum_{t \in \text{technologies}} \frac{\text{SalvageValue}_{y,r,t}}{(1+\text{DiscountRate})^{(y - \min(\text{year}))}} + \sum_{s \in \text{storages}} \frac{\text{StorageSalvageValue}_{y,r,s}}{(1+\text{DiscountRate})^{(y - \min(\text{year}))}} \right) + \sum_{l \in \text{links}} \frac{\text{TransmissionSalvageValue}_{y,l}}{(1+\text{DiscountRate})^{(y - \min(\text{year}))}} \right]$

where:

$y$ = planning year
$r$ = region
$t$ = generation technology
$s$ = storage technology
$l$ = transmission link
$\text{DiscountRate}$ = annual discount rate

This objective function ensures that all future costs are discounted to their present value, and that salvage values from remaining asset life are properly subtracted to obtain the net present system cost.

Model implementation in Julia:

@objective(ESM, Min,
      sum(TotalCost[y,r,t]/ (1+DiscountRate)^(y - minimum(year)) for t in technologies for r in regions, y in year)
    + sum(TotalStorageCost[y,r,s]/ (1+DiscountRate)^(y - minimum(year)) for s in storages for r in regions, y in year)
    + sum(TotalTransmissionCost[y,l] / (1+DiscountRate)^(y - minimum(year)) for l in links, y in year)
    - sum(SalvageValue[y, r, t] / (1+DiscountRate)^(y - minimum(year)) for t in technologies for r in regions, y in year)
    - sum(StorageSalvageValue[y, r, s] / (1+DiscountRate)^(y - minimum(year)) for s in storages for r in regions, y in yea

3.4 Constraints

Generation must meet demand

$\sum_{t \in T} FuelProductionByTechnology_{y,r,h,t,f} + \sum_{s \in S} StorageDischarge_{y,r,s,h,f} + \sum_{rr \in R} Import_{y,r,rr,h,f} = Demand_{y,r,f} \cdot DemandProfile_{r,h,f} + \sum_{t \in T} FuelUseByTechnology_{y,r,h,t,f} + Curtailment_{y,r,h,f} + \sum_{s \in S} StorageCharge_{y,r,s,h,f} + \sum_{rr \in R} Export_{y,r,rr,h,f}$

Calculate the total cost

$\left( \sum_{f \in F} \sum_{h \in H} FuelProductionByTechnology_{y,r,h,t,f} \right) \cdot VariableCost_{y,t} \cdot YearlyDifferenceMultiplier_{y} + NewCapacity_{y,r,t} \cdot InvestmentCost_{y,t} = TotalCost_{y,r,t}$

Salvage value (zero and positive cases)

$SalvageValue_{y,r,t} = \begin{cases} 0, & \text{si } y + OperationalLife_t - 1 \leq \max(Y) \\[6pt] NewCapacity_{y,r,t} \cdot InvestmentCost_{y,t} \cdot \left(1 - \dfrac{\max(Y) - y + 1}{OperationalLife_t}\right), & \text{si } y + OperationalLife_t - 1 > \max(Y) \end{cases}$

Dispatchable production limited by installed capacity

$FuelProductionByTechnology_{y,r,h,t,f} \leq OutputRatio_{t,f} \cdot AccumulatedCapacity_{y,r,t} \cdot CapacityFactor_{r,h,t}$

Variable renewables (must equal maximum output)

$FuelProductionByTechnology_{y,r,h,t,f} = OutputRatio_{t,f} \cdot AccumulatedCapacity_{y,r,t} \cdot CapacityFactor_{r,h,t}$

Use function

$FuelUseByTechnology_{y,r,h,t,f} = InputRatio_{t,f} \cdot \sum_{ff \in F} FuelProductionByTechnology_{y,r,h,t,ff}$

Technology emissions

$\sum_{f \in F} \sum_{h \in H} FuelProductionByTechnology_{y,r,h,t,f} \cdot EmissionRatio_{t} = AnnualTechnologyEmissions_{y,r,t}$

Annual emissions limit

$\sum_{t \in T} \sum_{r \in R} AnnualTechnologyEmissions_{y,r,t} \leq AnnualEmissionLimit_{y}$

Max installed capacity

$AccumulatedCapacity_{y,r,t} \leq MaxCapacity_{y,r,t}$

Capacity accounting

$\sum_{\substack{yy \in Y \\ yy \leq y, \; yy + OperationalLife_t > y}} NewCapacity_{yy,r,t} + ResidualCapacity_{y,r,t} = AccumulatedCapacity_{y,r,t}$

Storage constraints

Charge limit

$StorageCharge_{y,r,s,h,f} \leq \frac{AccumulatedStorageEnergyCapacity_{y,r,s,f}}{E2PRatio_s}$

Discharge limit

$StorageDischarge_{y,r,s,h,f} \leq \frac{AccumulatedStorageEnergyCapacity_{y,r,s,f}}{E2PRatio_s}$

Storage balance (h>1)

$StorageLevel_{y,r,s,h,f} = StorageLevel_{y,r,s,h-1,f} \cdot StorageLosses_{s,f} + StorageCharge_{y,r,s,h,f} \cdot StorageChargeEfficiency_{s,f} - \frac{StorageDischarge_{y,r,s,h,f}}{StorageDischargeEfficiency_{s,f}}$

Storage start (h=1)

$StorageLevel_{y,r,s,1,f} = 0.5 \cdot AccumulatedStorageEnergyCapacity_{y,r,s,f} \cdot StorageLosses_{s,f} + StorageCharge_{y,r,s,1,f} \cdot StorageChargeEfficiency_{s,f} - \frac{StorageDischarge_{y,r,s,1,f}}{StorageDischargeEfficiency_{s,f}}$

Max storage level

$StorageLevel_{y,r,s,h,f} \leq AccumulatedStorageEnergyCapacity_{y,r,s,f}$

Annual balance

$StorageLevel_{y,r,s,n\_hour,f} = 0.5 \cdot AccumulatedStorageEnergyCapacity_{y,r,s,f}$

Storage cost

$TotalStorageCost_{y,r,s} = \sum_{f \in F} NewStorageEnergyCapacity_{y,r,s,f} \cdot InvestmentCostStorage_{y,s}$

Max storage capacity limit

$\sum_{f \in F} AccumulatedStorageEnergyCapacity_{y,r,s,f} \leq MaxStorageCapacity_{y,r,s}$

Storage capacity accounting

$\sum_{\substack{yy \in Y \\ yy \leq y}} NewStorageEnergyCapacity_{yy,r,s,f} = AccumulatedStorageEnergyCapacity_{y,r,s,f}$

Storage salvage value

$StorageSalvageValue_{y,r,s} = \begin{cases} 0, & \text{si } y + StorageOperationalLife_s - 1 \leq \max(Y) \\[6pt] InvestmentCostStorage_{y,s} \cdot \left(1 - \dfrac{\max(Y) - y + 1}{StorageOperationalLife_s}\right), & \text{si } y + StorageOperationalLife_s - 1 > \max(Y) \end{cases}$

Trade constraints

Import-export balance

$Import_{y,r,rr,h,f} = Export_{y,rr,r,h,f} \cdot (1 - TradeLossFactor_f \cdot TradeDistance_{r,rr})$

Max import capacity

$Import_{y,r,rr,h,f} \leq MaxTradeCapacity_{y,r,rr,f}$

Total emission limit (model horizon)

$\sum_{y \in Y} \sum_{r \in R} \sum_{t \in T} \sum_{f \in F} \sum_{h \in H} FuelProductionByTechnology_{y,r,h,t,f} \cdot EmissionRatio_t \cdot YearlyDifferenceMultiplier_y \leq ModelPeriodEmissionLimit$