Mathematical Formulation
Objective function
The objective function equation of the planning mode is the sum of all the regional and cross-border transmission link costs discounted to the reference year. While, in the operational mode, the objective function is just the sum of the fixed and variable costs with their related taxes within the modeled year.
Planning mode
Total Objective Function
Regional Objective Function
Trades Objective Function
Operation mode
Total Objective Function
Regional Objective Function
Trades Objective Function
Equations
costs
calculating the components of the objective function including the investment, fixed and variable operation and maintenance and decommissioning costs followed by the related taxes considered for each unit of investment or fixed cost of the technologies. Carbon taxes are also included to be applied for the carbon-intensive technologies. Alongside the related costs of technologies, some revenues are considered in the objective function with a negative sign. These revenues are including subsidies that are applied to some technologies based on the national policies and the salvage values. The Hypatia model considers the economic life time of the technologies in the investment cost calculation. Therefore, each required investment in a specific year “y” is divided into a stream of annuities during several years (from “y+1” to “y+ELIFE”) which is determined by the technology-specific economic lifetime, depreciation rate and time value of money.
Note
In Hypatia, the inter-regional links are modeled as technologies. Therefore all the below equations for calculating the objective function cost components and intermediate variables except all the taxes and subsidies have been correspondingly written for the links.
Investment Cost
Investment Salvage Value
Fixed Cost
Taxes & Subsidies
Decommissioning Cost
Variable Cost
Carbon Tax
Capacity
Accumulated New Installed Capacity
Total Installed Capacity
Decomissioned Capacity
Calculates the annual decommissioning capacities based on the previously installed new capacities in the vintage years of the horizon.
Emission
Constraints
Energy balance
Guarantees the balance between the supply and demand sides of the energy system.
Note
All the technologies within Hypatia have one input carrier or one output carrier except for the conversion plus techs where the production and use of each carrier must be calculated from the following eaquation:
Trade balance
Ensures that the amounts of imports and exports among any pair of regions are completely balanced.
Resource & Technology Availability
Ensures that the production of each technology does not exceed its available activity based technology capacity factor and resource capacity factor.
Capacity
Maximum & Minimum Regional Total Capacity
Maximum and minimum allowed annual total installed capacity for each technology in each region.
Maximum & Minimum Regional New Capacity
Maximum and minimum allowed annual new installed capacity for each technology in each region.
Maximum & Minimum Overall Total Capacity
Maximum and minimum allowed annual aggregated total installed capacity for each technology over all the regions.
Maximum & Minimum Overall New Capacity
Maximum and minimum allowed annual aggregated new installed capacity for each technology over all the regions.
Activity
Maximum & Minimum Regional Production
Maximum and minimum allowed production of each technology in each region.
Maximum & Minimum Overall Production
Maximum and minimum aggregated production of each technology over all the regions.
Output to Input Activity Ratio
CO2 Equivalent Emissions
Regional Emission cap
Ensures regional maximum allowed annual carbon emissions.
Overall Emission cap
Ensures global maximum allowed annual carbon emissions over all the regions.